We improve the Bend-and-Break result of Miyaoka and Mori by establishing the optimal degree bound. Our result also yields optimal bounds on lengths of extremal rays of log canonical pairs.

We develop a unified framework for nonlinear subdivision schemes on complete metric spaces (CMS). We begin with CMS preliminaries and formalize refinement in CMS, retaining key structural properties, such as locality. We prove a convergence theorem under contractivity and demonstrate its applicability. To address schemes where contractivity is unknown, we introduce two notions of proximity. Our proximity methods relate a nonlinear scheme to another nonlinear scheme with known contractivity, rather than to a linear scheme, as in much of the literature. Specifically, the first type proximity compares the two schemes after a single refinement step and, as in the classical theory, yields convergence from sufficiently dense initial data. The proximity of the second type monitors alignment across all refinement levels and provides strong convergence without density assumptions. We formulate and prove the corresponding theorems, and illustrate them with various examples, such as schemes over metric spaces of compact sets in $\R^n$ and schemes over the Wasserstein space, as well as a geometric Hermite metric space. These results extend subdivision theory beyond Euclidean and manifold-valued data for data in metric spaces.

In this study, we present a tensor--train framework for nonintrusive operator inference aimed at learning discrete operators and using them to predict solutions of physical governing equations. Our framework comprises three approaches: full--order tensor--train operator inference, full--order quantized tensor--train operator inference, and reduced--order tensor--train operator inference. In each case, snapshot data is represented in tensor--train format--either through compression or cross interpolation--enabling the efficient handling of extremely large datasets with significantly reduced computational effort compared to standard methods. The effectiveness of each approach is demonstrated through numerical experiments related to Computational Fluid Dynamics and benchmarked against the standard reduced--order operator inference method, highlighting the advantages of the tensor--train representations in both accuracy and scalability.

We study the relativistic and non-relativistic Vlasov equation driven by short-range interaction potentials and identify the large time dynamics of solutions. In particular, we construct global-in-time solutions launched from small initial data and prove that they scatter along the forward free flow to well-behaved limits as $t \to \infty$. Moreover, we prove the existence of wave operators for such a regime and, upon constructing the aforementioned time asymptotic limits, use the wave operator formulation to prove for the first time that the relativistic scattering states converge to their non-relativistic counterparts as $c \to \infty$.

We introduce and analyze a discrete soft-decision channel called the linear reliability channel (LRC) in which the soft information is the rank ordering of the received symbol reliabilities. We prove that the LRC is an appropriate approximation to a general class of discrete modulation, continuous noise channels when the noise variance is high. The central feature of the LRC is that its combinatorial nature allows for an extensive mathematical analysis of the channel and its corresponding hard- and soft-decision maximum likelihood (ML) decoders. In particular, we establish explicit error exponents for ML decoding in the LRC when using random codes under both hard- and soft-decision decoding. This analysis allows for a direct, quantitative evaluation of the relative advantage of soft-decision decoding. The discrete geometry of the LRC is distinct from that of the BSC, which is characterized by the Hamming weight, offering a new perspective on code construction for soft-decision settings.

The generalized diamond group is the semi-direct product $G$ of the abelian group ${\mathbb R}^m$ by the $(2n+1)$-dimensional Heisenberg group $H_n$. We construct the generic representations of $G$ on the Fock space by extending those of $H_n$. Then we study the Berezin correspondence and the complex Weyl correspondence in connection with a generic representation $\pi$ of $G$, proving in particular that these correspondences are covariant with respect to $\pi$. We give also some explicit formulas for the Berezin symbols and the complex Weyl symbols of the representation operators $\pi(g)$ for $g\in G$. These results are applied to recover various formulas involving the Moyal product. Moreover, we relate $\pi$ to a coadjoint orbit of $G$ in the spirit of the Kirillov-Kostant method of orbits. This allows us to establish that the complex Weyl correspondence is a Stratonovich-Weyl correspondence for $\pi$.

We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which tracks them along the homotopy. Doing so produces a representation of the target solutions, bypassing the need to hold all solutions in memory. We discuss several applications of this datatype ranging from solution counting to data compression.

We consider a loosely coupled, non-iterative Robin-Robin coupling method proposed and analyzed in [Numer. Algorithms, 99:921-948, 2025] for a parabolic-parabolic interface problem. We modify the first step of the scheme so that several error difference quantities remain higher order convergence without requiring additional assumptions. Numerical results are presented to support our findings.

For domains that are easily represented by structured meshes, robust geometric multigrid solvers can quickly provide the numerical solution to many discretized elliptic PDEs. However, for complicated domains with unstructured meshes, constructing suitable hierarchies of meshes becomes challenging. We propose a framework for mapping computations from such complex domains to regular computational domains via diffeomorphisms, enabling the use of robust geometric-style multigrid. This mapping facilitates regular memory accesses during solves, improving efficiency and scalability, especially on massively parallel processors such as GPUs. Moreover, we show that the diffeomorphic mapping itself may be approximately learned using an invertible neural network, facilitating automated application to geometries where no analytic mapping is readily available.

We characterize all minimizers of the vector-valued Allen-Cahn equation in $\mathbb{R}^2$ under the assumption that the potential $W$ has three wells and that the associated degenerate metric does not satisfy the usual strict triangle inequality. These minimizers depend on one variable only in a suitable coordinate system. In particular, we show that no minimizing solutions to $ \Delta u=\nabla W(u)$ on $\mathbb{R}^2$ can approach the three distinct values of the potential wells.

We consider a spatial stochastic model for a pathogen population growing inside a host that attempts to eliminate the pathogens through its immune system. The pathogen population is divided into different types. A pathogen can either reproduce by generating a pathogen of its own type or produce a pathogen of a new type that does not yet exist in the population. Pathogens with living ancestral types are protected against the host's immune system as long as their progenitors are still alive. Each pathogen type without living ancestral types is eliminated by the immune system after a random period, independently of the other types. When a pathogen type is eliminated from the system, all pathogens of this type die simultaneously. In this paper, we determine the conditions on the set of model parameters that dictate the survival or extinction of the pathogen population when the dynamics unfold on graphs with an infinite tree structure.

Integrated sensing and communication (ISAC) is envisioned as a key technology in 6G networks, owing to its potential for high spectral and cost efficiency. As a promising solution for extremely large-scale arrays, reconfigurable holographic surfaces (RHS) can be integrated with ISAC to form the holographic ISAC paradigm, where enlarged radiation apertures of RHS can achieve significant beamforming gains, thereby improving both communication and sensing performance. In this paper, we investigate holographic beamforming designs for ISAC systems, which, unlike existing holographic beamforming schemes developed for RHS-aided communications, requires explicit consideration of mutual coupling effects within RHS. This is because, different from prior works only considering communication performance, ISAC systems incorporate sensing functionality, which is sensitive to sidelobe levels. Ignoring mutual coupling in holographic beamforming can lead to notable undesired sidelobes, thus degrading sensing performance. The consideration of mutual coupling introduces new challenges, i.e., it induces non-linearity in beamforming problems, rendering them inherently non-convex. To address this issue, we propose a tractable electromagnetic-compliant holographic ISAC model that characterizes mutual coupling in a closed form using coupled dipole approximations. We then develop an efficient mutual coupling aware holographic beamforming algorithm to suppress sidelobes and enhance ISAC performance. Numerical results validate effectiveness of the proposed algorithm.

Motivated by the $L$-space conjecture, we prove left-orderability of certain Dehn fillings on integral homology solid tori with techniques first appearing in the work of Culler-Dunfield. First, we use the author's previous results to construct arcs of representations originating at ideal points detecting Seifert surfaces inside certain 3-manifolds. This, combined with the holonomy extension locus techniques of Gao, proves that Dehn fillings near 0 of such 3-manifolds are left-orderable. We then explicitly verify the hypotheses of the main theorem for an infinite collection of odd pretzel knots, establishing previously unknown intervals of orderable Dehn fillings. This verifies the $L$-space conjecture for a new infinite family of closed 3-manifolds.

In recent years, the success of large language models (LLMs) has inspired growing interest in exploring their potential applications in wireless communications, especially for channel prediction tasks. However, directly applying LLMs to channel prediction faces a domain mismatch issue stemming from their text-based pre-training. To mitigate this, the ``adapter + LLM" paradigm has emerged, where an adapter is designed to bridge the domain gap between the channel state information (CSI) data and LLMs. While showing initial success, existing adapters may not fully exploit the potential of this paradigm. To address this limitation, this work provides a key insight that learning representations from the spectral components of CSI features can more effectively help bridge the domain gap. Accordingly, we propose a spectral-attentive framework, named SCA-LLM, for channel prediction in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. Specifically, its novel adapter can capture finer spectral details and better adapt the LLM for channel prediction than previous methods. Extensive simulations show that SCA-LLM achieves state-of-the-art prediction performance and strong generalization, yielding up to $-2.4~\text{dB}$ normalized mean squared error (NMSE) advantage over the previous LLM based method. Ablation studies further confirm the superiority of SCA-LLM in mitigating domain mismatch.

We study the category $F$-$\textbf{Mat}_\bullet$ of matroids over an idyll $F$. We show that $F$-$\textbf{Mat}_\bullet$ is a proto-exact category, a non-additive generalization of an exact category by Dyckerhoff and Kapranov. We further show that $F$-$\textbf{Mat}_\bullet$ is proto-abelian in the sense of André. As an application, we establish that the category $\textbf{TRS}_\bullet^\Sigma$ of tropical toric reflexive sheaves associated to a fan $\Sigma$, introduced by Khan and Maclagan, is also proto-exact and proto-abelian. We then investigate the stability of modular tropical toric reflexive sheaves within the framework of proto-abelian categories and reformulate Harder-Narasimhan filtrations in this setting.

A manifold is said to be $n$-plectic if it is equipped with a closed, nondegenerate $(n+1)$-form. This thesis develops the theory of \emph{relative $n$-plectic structures}, where the classical condition is replaced by a closed, nondegenerate \emph{relative} $(n+1)$-form defined with respect to a smooth map. Analogous to how $n$-plectic manifolds give rise to $L_\infty$-algebras of observables, we show that relative $n$-plectic structures naturally induce corresponding $L_\infty$-algebras. These structures provide a conceptual bridge between the frameworks of quasi-Hamiltonian $G$-spaces and $2$-plectic geometry. As an application, we examine the relative $2$-plectic structure canonically associated to quasi-Hamiltonian $G$-spaces. We show that every quasi-Hamiltonian $G$-space defines a closed, nondegenerate relative $3$-form, and that the group action induces a Hamiltonian infinitesimal action compatible with this structure. We then construct explicit homotopy moment maps as $L_\infty$-morphisms from the Lie algebra $\mathfrak{g}$ into the Lie $2$-algebra of relative observables, extending the moment map formalism to the higher and relative geometric setting.

A ubiquitous feature of data of our era is their extra-large sizes and dimensions. Analyzing such high-dimensional data poses significant challenges, since the feature dimension is often much larger than the sample size. This thesis introduces robust and computationally efficient methods to address several common challenges associated with high-dimensional data. In my first manuscript, I propose a coherent approach to variable screening that accommodates nonlinear associations. I develop a novel variable screening method that transcends traditional linear assumptions by leveraging mutual information, with an intended application in neuroimaging data. This approach allows for accurate identification of important variables by capturing nonlinear as well as linear relationships between the outcome and covariates. Building on this foundation, I develop new optimization methods for sparse estimation using nonconvex penalties in my second manuscript. These methods address notable challenges in current statistical computing practices, facilitating computationally efficient and robust analyses of complex datasets. The proposed method can be applied to a general class of optimization problems. In my third manuscript, I contribute to robust modeling of high-dimensional correlated observations by developing a mixed-effects model based on Tsallis power-law entropy maximization and discussed the theoretical properties of such distribution. This model surpasses the constraints of conventional Gaussian models by accommodating a broader class of distributions with enhanced robustness to outliers. Additionally, I develop a proximal nonlinear conjugate gradient algorithm that accelerates convergence while maintaining numerical stability, along with rigorous statistical properties for the proposed framework.

We introduce the Closest Point Heat Method (CPHM), a novel approach for solving the surface Eikonal equation on general smooth surfaces. Building on the strengths of the classical heat method, such as simplicity of implementation and computational efficiency, CPHM integrates closest point techniques to reduce dependence on surface meshes. This embedding framework naturally extends the heat method to implicit surfaces while preserving both its efficiency and intrinsic geometric properties. Numerical experiments on benchmark geometries confirm the accuracy and convergence of the proposed method and demonstrate its effectiveness on complex shapes.

In this work, we propose the first fully first-order method to compute an epsilon stationary Stackelberg equilibrium with convergence guarantees. To achieve this, we first reframe the leader follower interaction as single level constrained optimization. Second, we define the Lagrangian and show that it can approximate the leaders gradient in response to the equilibrium reached by followers with only first-order gradient evaluations. These findings suggest a fully first order algorithm that alternates between (i) approximating followers best responses through gradient descent and (ii) updating the leaders strategy via approximating the gradient using Lagrangian.

For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.

This paper presents a novel, mathematically rigorous framework for autoencoder-type deep neural networks that combines optimal control theory and low-rank tensor methods to yield memory-efficient training and automated architecture discovery. The learning task is formulated as an optimization problem constrained by differential equations representing the encoder and decoder components of the network and the corresponding optimality conditions are derived via a Lagrangian approach. Efficient memory compression is enabled by approximating differential equation solutions on low-rank tensor manifolds using an adaptive explicit integration scheme. These concepts are combined to form OCTANE (Optimal Control for Tensor-based Autoencoder Network Emergence) -- a unified training framework that yields compact autoencoder architectures, reduces memory usage, and enables effective learning, even with limited training data. The framework's utility is illustrated with application to image denoising and deblurring tasks and recommendations regarding governing hyperparameters are provided.

A shape possesses Rupert's property if a hole can be cut through it such that a second identical copy of the shape can cleanly pass straight through the interior of the first. Such a passage proving cubes are Rupert was first shown more than 300 years ago. It remains open whether every polyhedron in three dimensions is Rupert. We propose a customized subgradient method providing high-accuracy local numerical optimization of the quality of a passage for a given polyhedron. From extensive numerical searches, we improve these best-known passages for more than half of the Platonic, Archimedean, and Catalan solids and for numerous Johnson solids. Our high accuracy solves support a new conjecture of a simple form for the Tetrahedron's optimal passage. Despite our computational search, three Archimedean and two Catalan solids remain open, providing further negative evidence against the conjecture that all polyhedrons are Rupert.

In this paper, we conduct a systematic numerical analysis of the spectral properties of the collocation and mass matrices in the isogeometric least-squares collocation method (IGA-L), for the approximation of the Poisson problem with homogeneous Dirichlet boundary conditions. This study primarily focuses on the spectral properties of the IGA-L collocation and mass matrices in relation to the isogeometric discretization parameters, such as the mesh size, degree, regularity, spatial dimension, and the number and distribution of the collocation points. Through a comprehensive numerical investigation, we provide estimations for the condition number, as well as the maximum and minimum singular values, in relation to the mesh size, degree and regularity. Moreover, in this paper we also study the effect of the number and distribution of the collocation points on the spectral properties of the collocation matrix, providing insights into the optimization of the collocation points for achieving better-conditioned linear systems.

We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In particular, we construct a Godeaux surface whose singular locus consists of two $\mathsf A_1$ and two $\mathsf A_3$ singularities. We show that this surface admits a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover which is a smooth minimal surface of general type with invariants $K^2=8$ and $p_g=0$, i.e. a fake quadric. Together with the result in the Appendix, this provides the first explicit construction of a fake quadric that does not arise as a quotient of a product of curves.

The main purpose of this paper is to give an explicit description of the moduli space of semistable sheaves of rank two on a stable curve C obtained by gluing two smooth curves at a point. We prove that the moduli space is irreducible and birational to a projective bundle over the moduli space of stable vector bundles on each component curve, independently of the choice of polarization. As an application, we compute the virtual Poincare polynomial of the moduli space.

Two-dimensional steady supersonic ramp flows are important and well-studied flow patterns in aerodynamics. Vimercati, Kluwick and Guardone [J. Fluid Mech., 885 (2018) 445--468] constructed various self-similar composite wave solutions to the supersonic flow of Bethe-Zel'dovich-Thompson (BZT) fluids past compressible and rarefactive ramps. We study the stabilities of the self-similar fan-shock-fan and shock-fan-shock composite waves constructed by Vimercati et al. in that paper. %In order to study the stabilities of the composite waves, we solve some classes of shock free boundary problems. In contrast to ideal gases, the flow downstream (or upstream) of a shock of a BZT fluid may possibly be sonic in the sense of the flow velocity relative to the shock front. In order to study the stabilities of the composite waves, we establish some a priori estimates about the type of the shocks and solve some classes of sonic shock free boundary problems. We find that the sonic shocks are envelopes of one out of the two families of wave characteristics, and not characteristics. This results in a fact that the flow downstream (or upstream) a sonic shock is not $C^1$ smooth up to the shock boundary. We use a characteristic decomposition method and a hodograph transformation method to overcome the difficulty cased by the singularity on sonic shocks, and derive several groups of structural conditions to establish the existence of curved sonic shocks.

In 2025, Mosić defined the weak CMP inverse utilizing a minimal rank weak Drazin inverse instead of the Drazin inverse. The weak CMP inverse is a new wider class of generalized inverses, of which the CMP and MPCEP inverse are particular cases. In this paper, we provide several expressions, along with a number of new characterizations and properties for the weak CMP inverse. Moreover, we investigate the relationships between the weak CMP inverse and some well-known generalized inverses, such as the Moore-Penrose inverse and weak MPD inverse. Finally, we show that the weak CMP inverse, weak MPD inverse and weak DMP inverse are all strong Bott-Duffin $(e,f)$-inverses.

The TODIM method (an acronym in Portuguese for interactive and multiple criteria decision-making) with entropy weights is influenced by rank reversal, a phenomenon where the order of two alternatives changes following the addition of another alternative. Research on rank reversal has predominantly focused on single decision-making methods. To the best of our knowledge, the reduction of rank reversal probability in hybrid methods, such as the entropy-weight TODIM method, remains an unresolved challenge. To address this, this paper introduces the expanded evaluation matrix, which incorporates virtual alternatives, to reduce the probability of rank reversal in the entropy-weight TODIM method. A simulation study is conducted to assess the effectiveness of the expanded evaluation matrix in mitigating rank reversal. The results demonstrate that the expanded evaluation matrix significantly reduces the rank reversal probability. A case study on selecting energy storage technology showcases the potential real-world applications of the expanded evaluation matrix. The reliability of the expanded evaluation matrix is further validated through sensitivity and comparative analyses. Given the simplicity and ease of implementation of the expanded evaluation matrix, it can be readily adapted to other decision-making methods and holds substantial potential for broad application.

Gaussian mixture models (GMMs) are fundamental statistical tools for modeling heterogeneous data. Due to the nonconcavity of the likelihood function, the Expectation-Maximization (EM) algorithm is widely used for parameter estimation of each Gaussian component. Existing analyses of the EM algorithm's convergence to the true parameter focus primarily on either the two-component case or multi-component settings with both known mixing probabilities and known, isotropic covariance matrices. In this work, we establish the minimax optimal rate of convergence of the EM algorithm for multi-component GMMs in full generality. The required separation condition between Gaussian components for EM to converge is the weakest known to date. We develop two distinct analytical approaches, each tailored to a different regime of separation, reflecting two complementary perspectives on the use of EM: parameter estimation and clustering. As a byproduct of our analysis, we show that the EM algorithm, when used for community detection, also achieves the minimax optimal rate of misclustering error under milder separation conditions than spectral clustering and Lloyd's algorithm, an interesting result in its own right. Our analysis allows the number of components, the minimal mixing probabilities, the separation between Gaussian components as well as the dimension to grow with the sample size. Simulation studies corroborate the theoretical findings.

In decision making, the cognitive fuzzy set (CFS) is a useful tool in expressing experts' complex assessments of alternatives. The distance of CFS, which plays an important role in decision analyses, is necessary when the CFS is applied in solving practical issues. However, as far as we know, the studies on the distance of CFS are few, and the current Minkowski distance of CFS ignores the hesitancy degree of CFS, which might cause errors. To fill the gap of the studies on the distance of CFS, because of the practicality of the Hausdorff distance, this paper proposes the improved cognitive fuzzy Minkowski (CF-IM) distance and the cognitive fuzzy Hausdorff (CF-H) distance to enrich the studies on the distance of CFS. It is found that the anti-perturbation ability of the CF-H distance is stronger than that of the CF-IM distance, but the information utilization of the CF-IM distance is higher than that of the CF-H distance. To balance the anti-perturbation ability and information utilization of the CF-IM distance and CF-H distance, the cognitive fuzzy combined (CF-C) distance is proposed by establishing the linear combination of the CF-IM distance and CF-H distance. Based on the CF-C distance, a combined-distanced-based score function of CFS is proposed to compare CFSs. The proposed score function is employed in lung cancer pain evaluation issues. The sensitivity and comparison analyses demonstrate the reliability and advantages of the proposed methods.

For a rigid object $M$ in an algebraic triangulated category $\mathcal{T}$, a functor pr$(M)\to\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ is constructed, which essentially takes an object to its `presentation', where pr$(M)$ is the full subcategory of $\mathcal{T}$ of objects finitely presented by $M$, $A$ is the endomorphism algebra of $M$ and $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ is the homotopy category of complexes of finitely projective $A$-modules concentrated in degrees $-1$ and $0$. This functor is shown to be full and dense and its kernel is described. It detects isomorphisms, indecomposability and extriangles. In the Hom-finite case, it induces a bijection from the set of isomorphism classes of basic relative cluster-tilting objects of pr$(M)$ to that of basic silting complexs of $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$, which commutes with mutations. These results are applied to cluster categories of self-injective quivers with potential to recover a theorem of Mizuno on the endomorphism algebras of certain 2-term silting complexes. As an interesting consequence of the main results, if $\mathcal{T}$ is a 2-Calabi--Yau triangulated category and $M$ is a cluster-tilting object such that $A$ is self-injective, then $\mathbb{P}$ is an equivalence, in particular, $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ admits a triangle structure. In the appendix by Iyama it is shown that for a finite-dimensional algebra $A$, if $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ admits a triangle structure, then $A$ is necessarily self-injective.

This paper addresses the problem of stochastic optimization with decision-dependent uncertainty, a class of problems where the probability distribution of the uncertain parameters is influenced by the decision-maker's actions. While recent literature primarily focuses on solving or analyzing these problems by directly imposing hypotheses on the distribution mapping, in this work we explore some of these properties for a specific construction by means of the moving support and a density function. The construction is motivated by the Bayesian approach to bilevel programming, where the response of a follower is modeled as the uncertainty, drawn from the moving set of optimal responses which depends on the leader's decision. Our main contribution is to establish sufficient conditions for the Lipschitz continuity of the expected value function. We show that Lipschitz continuity can be achieved when the moving support is a Lipschitz continuous set-valued map with full-dimensional, convex, compact values, or when it is the solution set of a fully linear parametric problem. We also provide an example showing that the sole Lipschitz assumption on the moving set itself is not sufficient and that additional conditions are necessary.

Solving large-scale optimization problems is a bottleneck and is very important for machine learning and multiple kinds of scientific problems. Subspace-based methods using the local approximation strategy are one of the most important methods. This paper discusses different and novel kinds of advanced subspaces for such methods and presents a new algorithm with such subspaces, called MD-LAMBO. Theoretical analysis including the subspaces' properties, sufficient function value decrease, and global convergence is given for the new algorithm. The related model construction on the subspaces is given under derivative-free settings. In numerical results, performance profiles, and truncated Newton step errors of MD-LAMBO using different model-driven subspaces are provided, which show subspace-dependent numerical differences and advantages of our methods and subspaces.

In this paper, we study a bilinear saddle point problem of the form $\min_{x}\max_{y} F(x) + \langle Ax, y \rangle - G(y)$, where $F$ and $G$ are $\mu_F$- and $\mu_G$-strongly convex functions, respectively. By incorporating Nesterov acceleration for strongly convex optimization, we first propose an optimal first-order discrete primal-dual gradient algorithm. We show that it achieves the optimal convergence rate $\mathcal{O}\left(\left(1 - \min\left\{\sqrt{\frac{\mu_F}{L_F}}, \sqrt{\frac{\mu_G}{L_G}}\right\}\right)^k\right)$ for both the primal-dual gap and the iterative, where $L_F$ and $L_G$ denote the smoothness constants of $F$ and $G$, respectively. We further develop a continuous-time accelerated primal-dual dynamical system with constant damping. Using the Lyapunov analysis method, we establish the existence and uniqueness of a global solution, as well as the linear convergence rate $\mathcal{O}(e^{-\min\{\sqrt{\mu_F},\sqrt{\mu_G}\}t})$. Notably, when $A = 0$, our methods recover the classical Nesterov accelerated methods for strongly convex unconstrained problems in both discrete and continuous-time. Numerical experiments are presented to support the theoretical convergence rates.

In this paper, we study the non-relativistic limit of the two-dimensional cubic nonlinear Klein-Gordon equation with a small parameter $0<\varepsilon \ll 1$ which is inversely proportional to the speed of light. We show the cubic nonlinear Klein-Gordon equation converges to the cubic nonlinear Schrödinger equation with a convergence rate of order $O(\varepsilon^2)$. In particular, for the defocusing case with high regularity initial data, we show error estimates of the form $C(1+t)^N \varepsilon^2$ at time $t$ up to a long time of order $\varepsilon^{-\frac{2}{N+1}}$, while for initial data with limited regularity, we also show error estimates of the form $C(1+t)^M\varepsilon$ at time $t$ up to a long time of order $\varepsilon^{-\frac{1}{M+1}}$. Here $N$ and $M$ are constants depending on initial data. The idea of proof is to reformulate nonrelativistic limit problems to stability problems in geometric optics, then employ the techniques in geometric optics to construct approximate solutions up to an arbitrary order, and finally, together with the decay estimates of the cubic Schrödinger equation, derive the error estimates.

Transposed Poisson algebra was introduced as a dual notion of the Poisson algebra by switching the roles played by the commutative associative operation and Lie operation in the Leibniz rule defining the Poisson algebra. Let $H$ be a Hopf algebra with a bijective antipode and $A$ an $H$-comodule transposed Poisson algebra. Assume that there exists an $H$-colinear map which is also an algebra map from $H$ to the transposed Poisson center of $A$. In this paper we generalize the fundamental theorem of $(A, H)$-Hopf modules to transposed Poisson $(A, H)$-Hopf modules and deduce relative projectivity in the category of transposed Poisson $(A, H)$-Hopf modules.

The purpose of this paper is to present a novel and elegant summation formula for $H_w$, the Kronecker-Hurwitz class number. Specifically, for any prime $p$, we have the formula: $$ \sum_{t^2<p} H_w(t^2-p) = \frac{p-2}{3}. $$

We introduce and study the class of positive weakly (q,r)-dominated multilinear operators between Banach lattices. This notion extends classical domination and summability concepts to the positive multilinear setting and generates a new positive multi-ideal. A Pietsch domination theorem and a polynomial version are established. Finally, we provide a tensorial representation that yields an isometric identification with the dual of an appropriate completed tensor product.

Motivated by applications in background-independent quantum gravity, we discuss the quantization of labeled and unlabeled finite multigraphs with a maximum edge count. We provide a unified way to represent quantum multigraphs with labeled or unlabeled vertices, which enables the study of quantum multigraphs as dynamical microscopic degrees of freedom and not just as representations of relations among quantum states of particles. The quantum multigraphs represent a quantum mechanical treatment of the relations themselves and give rise to Hilbert space realizations of relations. After defining the Hilbert space, we focus on quantum simple graphs and explore the thermodynamics resulting from two simple models, a free Hamiltonian and an Ising-type Hamiltonian (with interactions among nearest-neighbor edges). We show that removing the distinction among vertices by considering unlabeled vertices gives rise to a qualitatively different thermodynamics. We find that the free theory of labeled quantum simple graphs is the Erdős--Rényi--Gilbert $G(N,p)$ model of random graphs. This model has analytic free energy and hence no thermodynamic phase transition. On the other hand, the unlabeled quantum graphs give rise to proper thermodynamic phase transitions in both the free and the ferromagnetic Ising models, characterized by divergence in the specific heat and critical slowing near the critical temperature. The thermodynamic phase transition has an order parameter given as the fraction of vertices in the largest connected component. Although this is similar to the phase transition in the $G(N,p)$ model, in this case it represents the actual thermodynamic phase transition.

In this article, we study Bohr-type inequalities involving a parameter or convex combinations for $K$-quasiconformal, sense-preserving harmonic mappings in $\mathbb{D}$, where the analytic part is subordinate to a convex function. Moreover, we establish similar inequalities when the subordinating function is chosen from the class of concave univalent functions with pole $p$, as well as from the family of concave univalent functions with opening angle $\pi\alpha$. The results generalize several existing results.

We construct a new family of skein exact triangles for link Floer homology. The skein triples are described by a triple of rational tangles $(R_0,R_1,R_{2n+1})$, where $R_0$ is the trivial tangle and $R_k$ is obtained from it by applying $k$ positive half-twists. We also set up an appropriate framework for potential construction of further skein exact triangles corresponding to arbitrary triples of rational tangles.

Topological dynamics constitutes the study of asymptotic properties of orbits under flows or maps on the Hausdorff phase space. Hyperbolic dynamics is the study of differentiable flows or maps that are usually characterized by the presence of expanding and contracting directions for the associated derivative on some manifold. We study some topological dynamics, essentially the property of `proximality', of two prototype examples of hyperbolic dynamical systems - \emph{Arnold's Cat Map} and \emph{Smale's Horseshoe Map} as an attempt to find some analogies in these two directions of study.

An open problem posed by Gaboriau is whether the product of any two infinite countable groups has fixed price one. We provide an affirmative answer if the two groups are finitely generated and their growths satisfy a specific condition. The proof uses the propagation method to construct a Poisson horoball process as a weak factor of i.i.d., where each horoball is equipped with a marking that depends only on the first coordinate, in an i.i.d. manner. Then, a low-cost graphing of this process is constructed using the markings of the horoballs and adding a percolation with small intensity.

Let $(X, d, \mu)$ be a space of homogeneous type and $\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\Omega) \to L^q(\Omega)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\frac{1}{p_0} - \frac{1}{a} = \frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\frac{\alpha}{2}}$ or for Riesz transform type operators $\nabla \Delta^{-\frac{\alpha}{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H^1_L$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a'}(X)$ into $\text{BMO}_L$.

In a series of papers we investigated the following question: given a family $\calF$ of binary forms having nonzero discriminant and integer coefficients, for each $d\geqslant 3$, we estimate the number of integers $m$ with $|m|\leqslant N$ which are represented by an element in $\calF$ of degree $\geqslant d$. Under suitable assumptions, asymptotically as $N\to\infty$, the main term in the estimate is given by the forms in $\calF$ having degree $d$ (if any), while the forms of degree $>d$ contribute only to the error term. The present text is devoted to fewnomials \[ a_0X^{kr}+a_1X^{k(r-1)}Y^k+\cdots +a_{r-1}X^kY^{k(r-1)}+a_rY^{kr} \] with fixed $r\geqslant 1$ and varying $k,a_0,a_1,\dots,a_r$.

This paper extends the study of arc crossing change, a local operation on knot diagrams recently introduced by Cericola, from knot diagrams to link diagrams. We consider two types of arc crossing change on link diagrams and discuss when they are unknotting operations. Furthermore, we show that any two crossing points in an alternating knot diagram are arc crossing change admissible.

We find some bounds for the internal radii of stable and unstable manifolds of points in terms of their Lyapunov exponents under the assumption of the existence of a dominated splitting.

In this paper, by means of divergence-free condition, we establish an anisotropic energy conservation class enabling one component of velocity in the largest space $L^{3} (0,T; B^{1/3}_{3,\infty})$ for the 3D inviscid incompressible fluids, which extends the celebrated result obtained by Cheskidov, Constantin, Friedlander and Shvydkoy in [15, Nonlinearity 21 (2008)]. For viscous flows, we generalize famous Lions's energy conservation criteria to allow the horizontal components and vertical part of velocity to have different integrability.

For a connected subcubic graph $G\neq K_1$ let $V_i(G) = \{v \in V(G) ~|~ d_G(v)=i\}$ for $1 \leq i \leq 3.$ Given $c_1, c_2, c_ 3 \in \mathbb{R}^+$ and $ d \in \mathbb{R}$, we show several results of type $\alpha(G) \geq c_1|V_1(G)| + c_2|V_2(G)| + c_3|V_3(G)| - d.$ We also derive classes of graphs $G$ showing sharpness of these lower bounds on the independence number $\alpha(G)$ of $G$.

We study the two--dimensional magnetic Schrödinger operator with a penetrable circular wall modeled by a $\delta$--interaction. Using the boundary triple approach we classify all self--adjoint extensions and obtain Krein's resolvent formula, showing that the essential spectrum coincides with the Landau levels. The wall breaks their infinite degeneracy and produces a spectral broadening: each Landau level becomes an accumulation point of discrete eigenvalues from one side. In the circular case, rotational symmetry reduces the eigenvalue problem to scalar equations with explicit Weyl coefficients. We prove strict monotonicity, ensuring that each angular momentum channel contributes at most one eigenvalue per gap, and derive asymptotics showing that the boundary coefficients decay faster than any exponential, explaining the strong localization of the broadened spectrum. Numerical simulations are consistent with these results.

This survey was written on the occasion of the course I gave at the Winterbraids XIV workshop in Bordeaux (2025). Its main purpose is to present the techniques that have proven most effective in the study of parabolic subgroups of Artin groups, with particular emphasis on the parabolic subgroups intersection problem. The survey highlights the core ideas and strategies behind them, aiming to give the reader a concise and accessible entry point to the essential methods.

The Laplacian of a general trace polynomial function defined on the special orthogonal group $SO(N)$ is explicitly computed. An invariant flag of spaces generated by trace polynomials is constructed. The matrix of the Laplace-Beltrami operator on $SO(N)$ for this flag of vector spaces takes an upper block triangular form. As a consequence of this construction, the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on $SO(N)$ can be computed in an iterative manner. For the particular cases of the special orthogonal groups $SO(3)$ and $SO(4)$ the complete list of eigenvalues is obtained and the corresponding irreducible characters of the representation for these groups are expressed as trace polynomials.

Let $R$ be a commutative ring with identity and $M$ a unitary $R$-module. The purpose of this paper is to introduce the concept of semi-$n$-submodules as an extension of semi $n$-ideals and $n$-submodules. A proper submodule $N$ of $M$ is called a semi $n$-submodule if whenever $r\in R$, $m\in M$ with $r^{2}m\in N$, $r\notin\sqrt{0}$ and $Ann_{R}(m)=0$, then $rm\in N$. Several properties, characterizations of this class of submodules with many supporting examples are presented. Furthermore, semi $n$-submodules of amalgamated modules are investigated.

A pair of plane curves with the same combinatorics is said to be (a) a Zariski pair if the plane curves have different embedded topology, and (b) a strong Ziegler pair if their Milnor algebra are not isomorphic. We show that some examples of Zariski pairs are also strong Ziegler pairs.

A family of explicit modified Euler methods (MEMs) is constructed for long-time approximations of super-linear SODEs driven by multiplicative noise. The proposed schemes can preserve the same Lyapunov structure as the continuous problems. Under a non-contractive condition, we establish a non-asymptotic error bound between the law of the numerical approximation and the target distribution in Wasserstein-1 ($\mathcal{W}_1$) distance through a time-independent weak convergence rate for the proposed schemes. As a by-product of this weak error estimate, we obtain an $\mathcal{O}(\tau|\ln \tau|)$ convergence rate between the exact and numerical invariant measures.

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this conclusion. Our main result implies that if, for a bounded simply-connected planar domain, the n-th eigenvalue of the magnetic Dirichlet Laplacian with uniform magnetic field is smaller than the corresponding eigenvalue for a disk of the same area, then the Fraenkel asymmetry of that domain tends to zero in the strong magnetic field limit. Comparable results are also derived for the magnetic Dirichlet Laplacian on rectangles, as well as the magnetic Dirac operator with infinite mass boundary conditions on smooth domains. As part of our analysis, we additionally provide a new estimate for the torsion function on rectangles.

To address the order degradation at critical points in the WENO3-Z scheme, some improvements have been proposed , but these approaches generally fail to consider the occurrence of critical points at arbitrary positions within grid intervals, resulting in their inability to maintain third-order accuracy when a first-order critical point (CP1) occurs. Also, most previous improved schemes suffer from a relatively large exponent p of the ratio of global to local smoothness indicators, which adversely affects the numerical resolution. Concerning these limitations, introduced here is an accuracy-optimization lemma demonstrating that the accuracy of nonlinear weights can be enhanced providing that smoothness indicators satisfy specific conditions, thereby establishing a methodology for elevating the accuracy of nonlinear weights. Leveraging this lemma, a local smoothness indicator is constructed with error terms achieving second-order in smooth regions and fourth-order at CP1, alongside a global smoothness indicator yielding fourth-order accuracy in smooth regions and fifth-order at CP1, enabling the derivation of new nonlinear weights that meet accuracy requirements even when employing p=1. Furthermore, a resolution-optimization lemma is proposed to analyze the relationship between parameters in local smoothness indicators and resolution. By integrating theoretical analysis with numerical practices, free parameters in non-normalized weights and local smoothness indicators are determined under the balance of numerical resolution and robustness, which leads to the development of WENO3-ZES4, a new WENO3-Z improvement that preserves the optimal order at CP1 especially with p=1. 1D and 2D validating tests show that the new scheme consistently achieves third-order in the case of CP1 regardless of its position and exhibits good resolution as well as preferable robustness.

In this paper, we prove a Liouville theorem for the $2$-Hessian equation on the Heisenberg group $\mathbb{H}^n$. The result is obtained by choosing a suitable test function and using integration by parts to derive the necessary integral estimates.

This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion. If a string link is the product of some string link diagram $T$ and the parallel n-cable of a framed long knot diagram $D$, then there is a canonical arc $push$ in $M_n$, defined by pushing $T$ through the n-cable of $D$. In this paper we apply the combinatorial 1-cocycles from the HOMFLYPT and Kauffman polynomials in $M_n$ with values in the corresponding skein modules to this canonical arc in $M_n$. Some of the 1-cocycles lead to linear systems of equations in the skein modules, for each couple of diagrams $D$ and $D'$. If the system has no solution in the Laurent polynomials then $D$ and $D'$ represent different knots. We give first examples where we distinguish knots without any knot invariants. In particular, we distinguish the knot $9_{42}$ from its mirror image with equations coming from the HOMFLYPT polynomial. Notice that the knot $9_{42}$ and its mirror image share the same HOMFLYPT polynomial. On the other hand, each solution of the system gives rather fine information about any regular isotopy which connects $D$ with $D'$.

For the discrete-time additive white generalized Gaussian noise channel with a generalized input power constraint, with the respective shape and power parameters >= 1, we derive an upper bound on the optimal block error exponent. Explicit asymptotic upper bounds in the limit of a large block length n are given for three special cases: the Laplace noise channel and the Gaussian noise channel with the average absolute value constraint, and for the Laplace noise channel with the second power constraint. The derivation uses the method of types with finite alphabets of sizes depending on the block length n and with the number of types sub-exponential in n.

Railcars, as part of the rolling stock, perform regular transportation tasks with respect to a service level agreement (SLA) and undergo preventive maintenance at regular intervals based on the recommendations of train manufacturers. When unexpected failures occur, they need to enter corrective maintenance immediately. However, this reactive approach may result in large SLA violations and an excessive number of corrective maintenance actions. In this study, we utilize a predictive maintenance approach based on the reliability of a railcar. In particular, we propose a stochastic programming model, in which railcar failure scenarios are generated from a Weibull distribution, a common assumption in the reliability literature. The model incorporates both SLA and track-capacity considerations and is solved through the Sample Average Approximation (SAA) method. We generate random instances to compare the stochastic model and a deterministic model adopted from the literature with respect to several system parameters. Our results show that the stochastic model achieves lower total costs, fewer SLA violations, and a reduced number of corrective interventions compared with deterministic approaches, while effectively managing track-capacity constraints. Our results underscore the importance of the predictive approach in the context of the railcar maintenance problem.

We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of derivatives of a matrix w.r.t. another matrix. This tensor form makes possible for us to unify ordinary differential equations (ODEs) with partial differential equations (PDEs), and facilitates solution to them in some cases. For our purpose, we also extend the outer product and contractive product of tensors (matrices) to a more general case through any partition of the modes, present some identities in terms of these products, initialize the definition of partial Tucker decompositions (TuckD) of a tensor, and use the partial TuckD to simplify the PDEs. We also present a tensor form for the Lyapunov function. Our results in the products of tensors and matrices help us to establish some important equalities on the derivatives of matrices and tensors. An algorithm based on the partial Tucker decompositions (TuckD) to solve the PDEs is given, and a numerical example is presented to illustrate the efficiency of the algorithm.

We study an interacting particle system of a finite number of labelled particles on the integer lattice, in which particles have intrinsic masses and left/right jump rates. If a particle is the minimal-label particle at its site when it tries to jump left, the jump is executed. If not, `momentum' is transferred to increase the rate of jumping left of the minimal-label particle. Similarly for jumps to the right. The collision rule is `elastic' in the sense that the net rate of flow of mass is independent of the present configuration, in contrast to the exclusion process, for example. We show that the particle masses and jump rates determine explicitly, via a concave majorant of a simple `potential' function associated to the masses and jump rates, a unique partition of the system into maximal stable subsystems. The internal configuration of each stable subsystem remains tight, while the location of each stable subsystem obeys a strong law of large numbers with an explicit speed. We indicate connections to adjacent models, including diffusions with rank-based coefficients.

We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation $ -\Delta u= f(u)$ in the upper half-space $\mathbb{R}^{N}_{+}$. Some Liouville-type theorems are also proven in the case of differential inequalities in $\mathbb{R}^{N}_{+}$, even without imposing any boundary condition. Although subject to dimensional restrictions, our results apply to a broad family of functions $f$. In particular, they apply to all non-negative $f$ that behaves at least linearly at infinity.

We consider the rank-1 inhomogeneous random graph in the Brownian regime in the critical window. Aldous studied the weights of the components, and showed that this ordered sequence converges in the $\ell^2$-topology to the ordered excursions of a Brownian motion with parabolic drift when appropriately rescaled (this http URL), as the number of vertices $n$ tends to infinity. We show that, under the finite third moment condition, the same conclusion holds for the ordered component sizes. This in particular proves a result claimed by Bhamidi, Van der Hofstad and Van Leeuwaarden (this https URL). We also show that, for the large components, the ranking by component weights coincides with the ranking by component sizes with high probability as $n \to \infty$.

This article explores a variant of Kac's famous problem, "Can one hear the shape of a drum?", by addressing a geometric inverse problem in acoustics. Our objective is to reconstruct the shape of a cuboid room using acoustic signals measured by microphones placed within the room. By examining this straightforward configuration, we aim to understand the relationship between the acoustic signals propagating in a room and its geometry. This geometric problem can be reduced to locating a finite set of acoustic point sources, known as image sources. We model this issue as a finite-dimensional optimization problem and propose a solution algorithm inspired by super-resolution techniques. This involves a convex relaxation of the finite-dimensional problem to an infinite-dimensional subspace of Radon measures. We provide analytical insights into this problem and demonstrate the efficiency of the algorithm through multiple numerical examples.

Given $u$ and $v$ in $\mathbb{Z}^d$, say that $u$ is visible from $v$ if the segment from $u$ to $v$ contains exactly two elements, which are $u$ and $v$. Take $X$ "uniformly at random in $\mathbb{Z}^d$" and colour each vertex $u$ of $\mathbb{Z}^d$ in white if $u$ is visible from $X$ and in black otherwise. Previous independent works of Pleasants-Huck and of the third author give a precise meaning to this definition. This paper is dedicated to the study of this random colouring from the point of view of percolation theory: given a reasonable graph structure on $\mathbb{Z}^d$, how many infinite black (resp. white) connected components are there?

In this article, we have analyzed semi-discrete finite element approximation and full discretization of the Stochastic semilinear generalized Benjamin-Bona-Mahony equation in a bounded convex polygonal domain driven by additive Wiener noise. We use the finite element method for spatial discretization and the semi-implicit method for time discretization and derive a strong convergence rate with respect to both parameters (spatial and temporal). Numerical experiments have also been performed to support theoretical bounds.

The main objective of this manuscript is to investigate the global behavior of the solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as a supercritical source term which is a combined power-type nonlinearities. The global existence of the solutions is obtained provided that the energy sink dominates the energy source in an appropriate sense. In more general scenarios, we prove the global existence of the solutions if the initial history value $u_0$ is taken from a subset of a suitable potential well. Based on global existence results, the energy decay rate is derived which depends on the relaxation kernel as well as the growth rate of the damping term. In addition, we study blow-up of solutions when the source is stronger than dissipation.

In this paper, we define a braid invariant, the purified determinant $P(b)$ of a braid $b$, considering the determinant of the crossing matrix, and show that $P(b_1 b_2)=P(b_2 b_1)$ for any pair of $n$-braids $b_1$ and $b_2$.

We introduce an abstract Hilbert space-valued framework of Markovian lifts for stochastic Volterra equations with operator-valued Volterra kernels. Our main results address the existence and characterisation of possibly multiple limit distributions and stationary processes, a law of large numbers including a convergence rate, and the central limit theorem for time averages of the process within the Gaussian domain of attraction. As particular examples, we study Markovian lifts based on Laplace transforms in a weighted Hilbert space of densities and Markovian lifts based on the shift semigroup on the Filipović space. We illustrate our results for the case of fractional stochastic Volterra equations with additive or multiplicative Gaussian noise.

The symmetric inverse monoid $I_X$ on a set $X$ consists of all bijective functions whose domain and range are subsets of $X$ under the usual composition and inversion of partial functions. For an arbitrary infinite set $X$, we classify all maximal subsemigroups and maximal inverse subsemigroups of $I_X$ which contain the symmetric group Sym($X$) or any of the following subgroups of Sym($X$): the pointwise stabiliser of a finite subset of $X$, the stabiliser of an ultrafilter on $X$, or the stabiliser of a partition of $X$ into finitely many parts of equal cardinality.

The main objective of this paper is to study the hierarchical exact controllability for a parabolic equation with Hardy potential by Stackelberg-Nash strategy. In linear case, we employ Lax-Milgram theorem to prove the existence of an associated Nash equilibrium pair corresponding to a bi-objective optimal control problem for each leader, which is responsible for an exact controllability property. Then the observability inequality of a coupled parabolic system is established by using global Carleman inequalities, which results in the existence of a leader that drives the controlled system exactly to any prescribed trajectory. In semilinear case, we first prove the well-posedness of the coupled parabolic system to obtain the existence of Nash quasi-equilibrium pair and show that Nash quasi-equilibrium is equivalent to Nash equilibrium. Based on these results, we establish the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory by Leray-Schauder fixed point theorem.

We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an exponential function, and apply this to the case of standard fields of transseries, such as $\log$-$\exp$ transseries and $\omega$-series.

The numerical flow iteration method has recently been proposed as a memory-slim solution method for the Vlasov-Poisson equation. It stores the temporal evolution of the electric field and reconstructs the solution in each time step by following the characteristics backwards in time and reconstructing the solution from the initial distribution. If the number of time steps gets large, the computational cost of this reconstruction may get prohibitive. Given a representation of the intermediate solution, the time intervals over which the characteristic curves need to be solved backwards in time can be reduced by restarting the numerical flow iteration after certain time intervals. In this paper, we propose an algorithm that reconstructs a low-rank representation of the solution at the restart times using the blackbox approximation. The proposed algorithm reduces the computational complexity compared to the pure numerical flow iteration from quadratic to linear in the number of times step while still keeping its memory complexity. On the other hand, our numerical results demonstrate that the methods preserves the property of the numerical flow iteration of showing much less dissipation of filaments compared to the semi-Lagrangian method.

We classify Riemannian $\text{spin}^c$ manifolds carrying a type I imaginary generalized Killing spinor, by explicitly constructing a parallel spinor on each leaf of the canonical foliation given by the Dirac current. We also provide a class of Riemannian $\text{spin}^c$ manifolds carrying a type II imaginary generalized Killing spinor, by considering spacelike hypersurfaces of Lorentzian $\text{spin}^c$ manifolds. We carry out much of the work in the setting of semi-Riemannian $\text{spin}^c$-manifolds carrying generalized Killing spinors, allowing us to draw conclusions in this setting as well. In this context, the Dirac current is not always a closed vector field. We circumvent this in even dimensions, by considering a modified Dirac current, which is closed in the cases when the original Dirac current is not. On the path to these results, we also study semi-Riemannian manifolds carrying closed and conformal vector fields.

This survey reviews recent advances connecting link homology theories to invariants of smooth 4-manifolds and extended topological quantum field theories. Starting from joint work with Morrison and Walker, I explain how functorial link homologies that satisfy additional invariance conditions become diagram-independent, give rise to braided monoidal 2-categories, extend naturally to links in the 3-sphere, and globalize to skein modules for 4-manifolds. Later developments show that these skein lasagna modules furnish invariants of embedded and immersed surfaces and admit computation via handle decompositions. I then survey structural properties, explicit computations, and applications to exotic phenomena in 4-manifold topology, and place link homology and skein lasagna modules within the framework of extended topological quantum field theories.

We consider the finest grading of the algebra of regular functions of a trinomial variety. An explicit description of locally nilpotent derivations that are homogeneous with respect to this grading is obtained.

In this brief note, we prove a min-min equality for a clean tangled clutter, that the rainbow covering number is equal to the connectivity of its setcore.

A pseudo-Riemannian Einstein manifold with a Killing spinor and Killing constant $\lambda$ induces on its nondegenerate hypersurfaces a pair of spinors $\phi,\psi$ and a symmetric tensor $A$, corresponding to the second fundamental form. Viewed as an intrinsic object, $(\phi,\psi,A,\lambda)$ is known as a harmful structure; this notion generalizes nearly hypo and nearly half-flat structures to arbitrary dimension and signature. We show that when $A$ is a multiple of the identity the harmful structure is determined by a Killing spinor. We characterize left-invariant harmful structures on Lie groups in terms of Clifford multiplication by some special elements induced by the structure constants and metric. This enables us to classify left-invariant harmful structures on unimodular metric Lie groups of definite or Lorentzian signature and dimension $\leq 4$, under the assumption that the symmetric tensor $A$ is diagonalizable over $\mathbb{R}$. These pseudo-Riemannian Lie groups are principal orbits of cohomogeneity one Einstein metrics of Riemannian, Lorentzian or anti-Lorentzian signature with a Killing spinor.

Let $\overline{B}$ be a smooth projective varieity, and $Z \subset \overline{B}$ a simple normal crossing divisor. Assume that $B = \overline{B} - Z$ admits a variation of pure, polarized Hodge structure. The divisor $Z$ is naturally stratified, and Schmid's nilpotent orbit theorem defines a family/variation of nilpotent orbits along each strata. We study the rich geometric structure encoded by this family, its relationship to the induced (quotient) variation of pure Hodge structure on the strata, and establish a relationship between the extension data in the nilpotent orbits and the normal bundles of the smooth irreducible components of $Z$.

We treat shared value problems for rational functions $R(z)$ and their derivative $R'(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and $\partial_{z} R = \lambda w \cdot R'(w)$ on ${\mathbb C} \setminus \{ 0 \}$ and $\widehat{\mathbb C}$, again with rational functions $R$. In ${\mathbb C} \setminus \{ 0 \}$ this is related to shared values of meromorphic functions $f : {\mathbb C} \to \widehat{\mathbb C}$ and $f'$ through $f(z)=R(w)$ with $w=\exp(\lambda z)$, while on $\widehat{\mathbb C}$ this is connected to shared limit values in a similar fashion.

The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes, we investigate the problem of deep holes of a class of twisted Reed-Solomon codes in this paper. Firstly, we provide the necessary and sufficient conditions for $\boldsymbol{a}=(a_{0},a_{1},\cdots,a_{n-k-1})\in\mathbb{F}_{q}^{n-k}$ to be the syndrome of some deep hole of $TRS_{k}(\mathcal{A},l,\eta)$. Next, we consider the problem of determining all deep holes of the twisted Reed-Solomon codes $TRS_{k}(\mathbb{F}_{q}^{*},k-1,\eta)$. Specifically, we prove that there are no other deep holes of $TRS_{k}(\mathbb{F}_{q}^{*},k-1,\eta)$ for $\frac{3q+2\sqrt{q}-8}{4}\leq k\leq q-5$ when q is even, and $\frac{3q+3\sqrt{q}-5}{4}\leq k\leq q-5$ when q is odd. We also completely determine their deep holes for $q-4\leq k\leq q-2$ when $q$ is even.

In this paper, we generalize the spectral correspondence for parabolic Higgs bundles established by Diaconescu--Donagi--Pantev to the relative setting. We show that the relative moduli space of $\vec{\xi}$-parabolic Higgs bundles on a curve can be realized as the relative moduli space of pure dimension one sheaves on a family of holomorphic symplectic surfaces. This leads us to formulate the image of the relative moduli space under the Hitchin map in terms of linear systems on the family of surfaces. Then we explore the relationship between the geometry of these linear systems and the so-called $OK$ condition introduced by Balasubramanian--Distler--Donagi in the context of six-dimensional superconformal field theories. As applications, we obtain (a) the non-emptiness of the moduli spaces and (b) the Deligne--Simpson problem and its higher genus analogue. In particular, we prove a conjecture proposed by Balasubramanian--Distler--Donagi that the $OK$ condition is sufficient for solving the Deligne--Simpson problem.

Suwa investigated the unit group scheme of the group ring associated with a finite flat group scheme and provided a characterization of torsors possessing the normal base property for such schemes. In this paper, we examine the unit group scheme of the group ring for a specific non-commutative finite flat group scheme and characterize torsors with the normal base property in this context. Moreover, in connection with the Noether problem for Hopf algebras proposed by Kassel and Masuoka, we compute the quotient of the unit group scheme under the action of this non-commutative finite flat group scheme.

In this paper, we present a new factor of IID process based on the local algorithm introduced by Díaz, Serna, and Wormald (2007). This new approach allows us to improve the previously known upper bounds on the minimum and maximum bisection width and the maximum cut of random d-regular graphs for d > 4 by introducing a new recoloring phase after the termination of the original algorithm. As an application, we show that random 5-regular graphs asymptotically almost surely admit an internal partition, i.e., a partition of the vertex set into two nonempty classes so that every vertex has at least half of its neighbors in its own class.

We consider a one dimensional affine switched system obtained from a formal limit of a two dimensional linear system. We show this is equivalent to minimising the average digit in beta representations with unrestricted digits. We give a countable set of $\beta$ for which the result is given by the usual (greedy) beta expansion, an interval of values for which it is strictly less, and a conditional lower bound for all $\beta$.

We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove \emph{a posteriori} error bounds in the $L^\infty(L^2)$-norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically.

The focus of this work is on the homogeneous and non-homogeneous Dirichlet problem for the Laplacian in bounded Lipschitz domains (BLD). Although it has been extensively studied by many authors, we would like to return to a number of fundamental questions and known results, such as the traces and the maximal regularity of solutions. First, to treat non-homogeneous boundary conditions, we rigorously define the notion of traces for non regular functions. This approach replaces the non-tangential trace notion that has dominated the literature since the 1980s. We identify a functional space E = \{v\in H^{1/2}(\Omega);\nabla v\in [H^1/2(\Omega)]'\} for which the trace operator is continuous from $E$ into $L^2(\Gamma)$. Second, we address the regularity of solutions to the Laplace equation with homogeneous Dirichlet conditions. Using specific equivalent norms in fractional Sobolev spaces and Grisvard's results for polygons and polyhedral domains, we prove that maximal regularity $H^{3/2}$ holds in any BLD $\Omega$, for all right-hand sides in the dual of $H^{1/2}_{00}(\Omega)$. This conclusion contradicts the prevailing claims in the literature since the 1990s. Third, we describe some criteria which establish new uniqueness results for harmonic functions in Lipschitz domains. In particular, we show that if $u\in H^{1/2}(\Omega)$ or $u\in W^{1, 2N/(N+1)}(\Omega)$, is harmonic in $\Omega$ and vanishes on $\Gamma$, then $u= 0$. These criteria play a central role in deriving regularity properties. Finally, we revisit the classical Area Integral Estimate. Using Grisvard's work and an explicit function given by Necas, we show that this inequality cannot hold in its stated form. Since this estimate has been widely used to argue that $H^{3/2}$-regularity is unattainable for data in the dual of $H^{1/2}_{00}(\Omega)$, our counterexample provides a decisive clarification.

In this paper, we introduce three QUBO (Quadratic Unconstrained Binary Optimization) relaxations for the sparsest $k$-subgraph (SkS) problem: a quadratic penalty relaxation, a Lagrangian relaxation, and an augmented Lagrangian relaxation. The effectiveness of these approaches strongly depends on the choice of penalty parameters. We establish theoretical results characterizing the parameter values for which the QUBO relaxations are exact. For practical implementation, we propose three iterative algorithms, which have in their kernel the QUBO relaxations, that update the penalty parameters at each iteration while approximately solving the internal QUBO problems with simulated annealing and quantum processing units. Extensive numerical experiments validate our theoretical findings on exact relaxations and demonstrate the efficiency of the proposed iterative algorithms.

In optimal transport, quadratic regularization is an alternative to entropic regularization when sparse couplings or small regularization parameters are desired. Here quadratic regularization means that transport couplings are penalized by the squared $L^2$ norm, or equivalently the $\chi^2$ divergence. While a number of computational approaches have been shown to work in practice, quadratic regularization is analytically less tractable than entropic, and we are not aware of a previous theoretical convergence rate analysis. We focus on the gradient descent algorithm for the dual transport problem in continuous and semi-discrete settings. This problem is convex but not strongly convex; its solutions are the potential functions that approximate the Kantorovich potentials of unregularized optimal transport. The gradient descent steps are straightforward to implement, and stable for small regularization parameter -- in contrast to Sinkhorn's algorithm in the entropic setting. Our main result is that gradient descent converges linearly; that is, the $L^2$ distance between the iterates and the limiting potentials decreases exponentially fast. Our analysis centers on the linearization of the gradient descent operator at the optimum and uses functional-analytic arguments to bound its spectrum. These techniques seem to be novel in this area and are substantially different from the approaches familiar in entropic optimal transport.

Evaluating network-wide fairness is challenging because it is not a static property but one highly sensitive to Service Level Agreement (SLA) parameters. This paper introduces a complete analytical framework to transform fairness evaluation from a single-point measurement into a proactive engineering discipline centered on a predictable sensitivity landscape. Our framework is built upon a QoE-Imbalance metric whose form is not an ad-hoc choice, but is uniquely determined by a set of fundamental axioms of fairness, ensuring its theoretical soundness. To navigate the fairness landscape across the full spectrum of service demands, we first derive a closed-form covariance rule. This rule provides an interpretable, local compass, expressing the fairness gradient as the covariance between a path's information-theoretic importance and its parameter sensitivity. We then construct phase diagrams to map the global landscape, revealing critical topological features such as robust "stable belts" and high-risk "dangerous wedges". Finally, an analysis of the landscape's curvature yields actionable, topology-aware design rules, including an optimal "Threshold-First" tuning strategy. Ultimately, our framework provides the tools to map, interpret, and navigate the landscape of system sensitivity, enabling the design of more robust and resilient networks.

Brox's diffusion is a typical one-dimensional singular diffusion, which was introduced by Brox (1986) as a continuous analogue of Sinai's random walk. In this paper, we will establish quenched heat kernel estimates for short time and annealed heat kernel estimates for large time of Brox's diffusion. The proofs are based on Brox's construction via the scale-transformation and the time-change arguments as well as the theory of resistance forms for symmetric strongly recurrent Markov processes. We emphasize that, since the reference measure of Brox's diffusion does not satisfy the so-called volume doubling conditions neither for the small scale nor the large scale, the existing methods for heat kernel estimates of diffusions in ergodic media do not work, and new techniques will be introduced to establish both quenched and annealed heat kernel estimates of Brox's diffusions, which take into account different oscillation properties for one-dimensional Brownian motion in random environments.

Substantial progress has recently been made in the understanding of the cutoff phenomenon for Markov processes, using an information-theoretic statistics known as varentropy [Sal23; Sal24; Sal25a; PS25]. In the present paper, we propose an alternative approach which bypasses the use of varentropy and exploits instead a new W-TV transport inequality, combined with a classical parabolic regularization estimate [BGL01; OV01]. While currently restricted to non-negatively curved processes on smooth spaces, our argument no longer requires the chain rule, nor any approximate version thereof. As applications, we recover the main result of [Sal25a] establishing cutoff for the log-concave Langevin dynamics, and extend the conclusion to a widely-used discrete-time sampling algorithm known as the Proximal Sampler.

Nonsmooth Riemannian optimization has attracted increasing attention, especially in problems with sparse structures. While existing formulations typically involve convex nonsmooth terms, incorporating nonsmooth difference-of-convex (DC) penalties can enhance recovery accuracy. In this paper, we study a class of nonsmooth Riemannian optimization problems whose objective is the sum of a smooth function and a nonsmooth DC term. We establish, for the first time in the manifold setting, the equivalence between such DC formulations (with suitably chosen nonsmooth DC terms) and their $\ell_0$-regularized or $\ell_0$-constrained counterparts. To solve these problems, we propose an inexact Riemannian proximal DC (iRPDC) algorithmic framework, which returns an $\epsilon$-Riemannian critical point within $\mathcal{O}(\epsilon^{-2})$ outer iterations. Within this framework, we develop several practical algorithms based on different subproblem solvers. Among them, one achieves an overall iteration complexity of $\mathcal{O}(\epsilon^{-3})$, which matches the best-known bound in the literature. In contrast, existing algorithms either lack provable overall complexity or require $\mathcal{O}(\epsilon^{-3})$ iterations in both outer and overall complexity. A notable feature of the iRPDC algorithmic framework is a novel inexactness criterion that not only enables efficient subproblem solutions via first-order methods but also facilitates a linesearch procedure that adaptively captures the local curvature. Numerical results on sparse principal component analysis demonstrate the modeling flexibility of the DC formulaton and the competitive performance of the proposed algorithmic framework.

The bilinear form u^\top f(A) v of matrix functions appears in many application problems, where u, v \in R^n\), A \in R^{n * n}\), and f(z) is a given analytic this http URL IDR(s) method effectively reduces computational complexity and storage requirements by introducing dimension reduction techniques, while maintaining the numerical stability of the algorithm. This paper studies the numerical algorithm and posterior error estimation for the matrix function bilinear form u^{\top} f(A) v based on the IDR(s) method. Through the error analysis of the IDR(s) algorithm, the corresponding error expansion is derived, and it is verified that the leading term of the error expansion serves as a reliable posterior error estimate. Based on this, in this paper a corresponding stopping criterion is proposed. This approach is dedicated to improving computational efficiency, especially by showing excellent performance in handling ill-posed and large-scale problems.

Let $\psi: (M,g)\longrightarrow (N,h)$ be a smooth map between Riemannian manifolds. The tension field of $\psi$ can be regarded as a map from $(M,g)$ into the Riemannian vector bundle $\psi^{-1}TN$, equipped with the Sasaki metric $G_{S}$. In this paper, we study certain aspects of two types of maps: those whose tension fields are harmonic maps (called $HM$-tensional maps) and those whose tension fields are harmonic sections (called $HS$-tensional maps).

We consider an arbitrary network of $M/M/\infty$ queues with controlled transitions between queues. We consider optimal control problems where the costs are linear functions of the state and inputs over a finite or infinite horizon. We provide in both cases an explicit characterization of the optimal control policies. We also show that these do not involve state feedback, but they depend on the network topology and system parameters. The results are also illustrated with various examples.

This work introduces a new efficient iterative solver for the reconstruction of real-time cone-beam computed tomography (CBCT), which is based on the Prior Image Constrained Compressed Sensing (PICCS) regularization and leverages the efficiency of Krylov subspace methods. In particular, we focus on the setting where a sequence of under-sampled CT scans are taken on the same object with only local changes (e.g. changes in a tumour size or the introduction of a surgical tool). This is very common, for example, in image-guided surgery, where the amount of measurements is limited to ensure the safety of the patient. In this case, we can also typically assume that a (good) initial reconstruction for the solution exists, coming from a previously over-sampled scan, so we can use this information to aid the subsequent reconstructions. The effectiveness of this method is demonstrated in both a synthetic scan and using real CT data, where it can be observed that the PICCS framework is very effective for the reduction of artifacts, and that the new method is faster than other common alternatives used in the same setting.

We study the distribution of normal subgroups in non-torsion, regular branch multi-EGS groups and show that the congruence completions of such groups have bounded finite central width. In particular, we show that the profinite completion of the Fabrykowski--Gupta group acting on the $p$-adic tree has central width 2 for every odd prime $p$. The methods used also apply to the family of Sunic groups, which closely resemble the Grigorchuk group.

It is known that a surface with parallel mean curvature vector field in a Riemannian product of two surfaces of constant Gaussian curvature carries a holomorphic quadratic differential. In this paper we consider the Riemannian product of a sphere and a hyperbolic plane of opposite Gaussian curvatures and study the parallel mean curvature surfaces for which the differential vanishes. In particular, we classify all parallel mean curvature spheres, for which the differential vanishes for topological reasons.

This paper investigates the spectral properties of the Hilbert-Brunn-Minkowski operator $L_K$ to derive stability estimates for geometric inequalities, including the local Brunn-Minkowski inequality. By analyzing the eigenvalues of $L_K$, we establish the uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem in $\mathbb{R}^{n}$ for $\frac{1-3n^2}{2n}\leq p<-n$ with $\lambda_2(-L_K)\geq \frac{n-1}{2n-1+p}$. Furthermore, we extend this uniqueness result to the range $-2n-1 \leq p<-n$ with $\lambda_2(-L_K)\geq \frac{-p-1}{n-1}$, assuming the origin-centred condition.

The polyconvexity of a strain-energy function is nowadays increasingly presented as the ultimate material stability condition for an idealized elastic response. While the mathematical merits of polyconvexity are clearly understood, its mechanical consequences have received less attention. In this contribution we contrast polyconvexity with the recently rediscovered true-stress-true-strain monotonicity (TSTS-M${}^{++}\!$) condition. By way of explicit examples, we show that neither condition by itself is strong enough to guarantee physically reasonable behavior for ideal isotropic elasticity. In particular, polyconvexity does not imply a monotone trajectory of the Cauchy stress in unconstrained uniaxial extension which TSTS-M${}^{++}\!$ ensures. On the other hand, TSTS-M${}^{++}\!$ does not impose a monotone Cauchy shear stress response in simple shear which is enforced by Legendre-Hadamard ellipticity and in turn polyconvexity. Both scenarios are proven through the construction of appropriate strain-energy functions. Consequently, a combination of polyconvexity, ensuring Legendre-Hadamard ellipticity, and TSTS-M${}^{++}\!$ seems to be a viable solution to Truesdell's Hauptproblem. However, so far no isotropic strain-energy function has been identified that satisfies both constraints globally at the same time. Although we are unable to deliver a valid solution here, we provide several results that could prove helpful in the construction of such an exceptional strain-energy function.

The fluid antenna system (FAS) employs reconfigurable antennas for high spatial gains in compact spaces, enhancing physical layer flexibility. Channel state information (CSI) acquisition is vital for port selection and FAS optimization. Greedy algorithms rely on signal assumptions, and model-free methods face high complexity. A flexible, low-complexity solution is needed for massive connectivity in FAS. Based on expectation maximization-approximate message passing (EM-AMP) framework, efficient matrix computations and adaptive learning without prior model knowledge naturally suit CSI acquisition for FAS. We propose a EM-AMP variant exploiting FAS geographical priors, improving estimation precision, accelerating convergence, and reducing complexity in large-scale deployment. Simulations validate the efficacy of the proposed algorithm.

Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \[ n=a_1x_1+a_2x_2+\cdots+a_kx_k, \quad (x_1, x_2, \cdots, x_k\in\mathbb{Z}_{\ge 0},~\gcd(x_1, x_2, \cdots, x_k)=1). \] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem.

We propose a novel feedback controller for a class of uncertain higher-order nonlinear systems, subject to delays in both state measurement and control input signals. Building on the prescribed performance control framework, a delay-dependent performance correction mechanism is introduced to ensure the boundedness of all signals in the closed-loop and to keep the output tracking error strictly within a dynamically adjusted performance envelope. This mechanism adapts in response to large delays that may cause performance degradation. In the absence of delays, the correction term vanishes, and the controller recovers the nominal (user-defined) performance envelope. The effectiveness of the proposed approach is validated through simulation studies.

We study degree two unirational parameterizations of geometrically rational surfaces over the real field.

The problem of bi-equivariant extension of continuous maps of binary $G$-spaces is considered. The concept of a structural map of distributive binary $G$-spaces is introduced, and a theorem on the bi-equivariant extension of structural maps is proven. A theorem on the bi-equivariant extension of continuous maps defined on the cross sections of a distributive binary $G$-space is also proven.

We establish the optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities along smooth anticanonical divisors on smooth K-unstable del Pezzo surfaces.

For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. Applications will be provided. We prove also a $C_p$-version of Rosenthal's theorem by showing that for an infinite compact $X$ the space $C_p(X)$ contains a closed copy of $c_{0}(\Gamma)$ (with the pointwise topology) for some uncountable set $\Gamma$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$. Illustrating examples, additional supplementing $C_p$-theorems and comments are included.

We introduce DDE-Solver, a Maple package designed for solving Discrete Differential Equations (DDEs). These equations are functional equations relating algebraically a formal power series F(t, u) with polynomial coefficients in a "catalytic" variable u, with specializations of it with respect to the catalytic variable. Such equations appear in enumerative combinatorics, for instance in the enumeration of maps. Bousquet-Melou and Jehanne showed in 2006 that when these equations are of a fixed point type in F, then F is an algebraic series. In the same paper, they proposed a systematic method for computing annihilating polynomials of these series. Bostan, Safey El Din and the author of this paper recently designed new efficient algorithms for computing these witnesses of algebraicity. This paper provides combinatorialists an automated tool in hand that solves DDEs using these algorithms. We also compare the timings of all these algorithms on DDEs from the literature.

On a volume doubling metric measure space endowed with a family of $p$-energies such that the Poincaré inequality and the cutoff Sobolev inequality with $p$-walk dimension $\beta_p$ hold, for $p$ in an open interval $I\subseteq (1,+\infty)$, we prove the following dichotomy: either $\beta_p=p$ for all $p\in I$, or $\beta_p>p$ for all $p\in I$.

We consider parabolic evolution equations with Lipschitz continuous and strongly monotone spatial operators. By introducing an additional variable, we construct an equivalent system where the operator is a Lipschitz continuous mapping from a Hilbert space $Y \times X$ to its dual, with a Lipschitz continuous inverse. Resulting Galerkin discretizations can be solved with an inexact Uzawa type algorithm. Quasi-optimality of the Galerkin approximations is guaranteed under an inf-sup condition on the selected `test' and `trial' subspaces of $Y$ and $X$. To circumvent the restriction imposed by this inf-sup condition, an a posteriori condition for quasi-optimality is developed that is shown to be satisfied whenever the test space is sufficiently large.

In this article, we are interested in convexity properties of the free energy for stationary point processes on $\mathbb R$ w.r.t.\ a new geometry inspired by optimal transport. We will show for a rich class of pairwise interaction energies A) quantified strict convexity of the free energy implying uniqueness of minimizers B) existence of a gradient flow curve of the free energy w.r.t. the new metric converging exponentially fast to the unique minimizer. Examples for energies for which A holds include logarithmic or Riesz interactions with parameter $0<s<1$, examples for which A and B hold are hypersingular Riesz or Yukawa interactions.

We consider the dynamics of a class of weakly interacting, gapless $1d$ fermionic systems, in presence of small external perturbations slowly varying in space and in time. We consider the evolution of the expectation values of the charge density and of the current density, in the thermodynamic limit and for low enough temperatures. We prove the validity and the asymptotic exactness of linear response in the limit of vanishing space-time variation of the perturbation, and we provide the explicit expression of the response of the system. The proof relies on the representation of the real time Duhamel expansion in terms of Euclidean correlation functions, for which we provide sharp estimates using rigorous renormalization group methods. The asymptotic exactness of linear response holds thanks to a cancellation for the scaling limit of the correlations that is reminiscent of bosonization, and which is derived rigorously using emergent chiral Ward identities.

We introduce and construct a novel type of canonical metric: the semi-flat constant scalar curvature Kähler (semi-flat cscK) current, which naturally arises in Calabi-Yau fibrations. For a given elliptic surface $X$ with a holomorphic section, We explicitly construct the desired semi-flat cscK current and analyze its behavior along singular parts. We establish its uniqueness under the condition that $X$ possesses at least one singular fiber other than of type $I_b$ or $I_b^*$. These results contribute to a geometric uniformization program for elliptic surfaces.

We show how to extract alternative solutions for optimization problems solved by Benders Decom- position. In practice, alternative solutions provide useful insights for complex applications; some solvers do support generation of alternative solutions but none appear to support such generation when using Benders Decomposition. We propose a new post-processing method that extracts multiple optimal and near-optimal solutions using the cut-pool generated during Benders Decomposition. Further, we provide a geometric framework for understanding how the adaptive approximation in Benders Decomposition re- lates to alternative solutions. We demonstrate this technique on stochastic programming and interdiction modeling, and we highlight use cases that require the ability to enumerate all optimal solutions.

For the edge ideal $I(\D)$ of a weighted oriented graph $\D$, we prove that its symbolic powers $I(\D)^{(t)}$ are Cohen-Macaulay for all $t\geqslant 1$ if and only if the underlying graph $G$ is composed of a disjoint union of some complete graphs. We also completely characterize the Cohen-Macaulayness of the ordinary powers $I(\D)^t$ for all $t\geqslant 2$. Furthermore, we provide a criterion for determining whether $I(\D)^t=I(\D)^{(t)}$.

For an $r$-graph $F$, define Sidorenko exponent $s(F)$ as $$s(F):= \sup \{s \geq 0: \exists \text{$r$-graph $H$ s.t. } t_F(H) = t_{K^{(r)}_r} (H)^s > 0\},$$ where $t_{H_1}(H_2)$ denotes the homomorphism density of $H_1$ in $H_2$. The celebrated Sidorenko's conjecture states that $s(F) = e(F)$ holds for every bipartite graph $F$. It is known that for all $r \geq 3$, the $r$-uniform version of Sidorenko's conjecture is false, and only a few hypergraphs are known to be Sidorenko. In this paper, we discover a new broad class of Sidorenko hypergraphs and obtain general upper bounds on $s(F)$ for certain hypergraphs related to dominating hypergraphs. This makes progress toward a problem raised by Sprio and Nie. We also discover a new connection between Sidorenko exponents and upper bounds on the extremal numbers of a large class of hypergraphs, which generalizes the hypergraph analogue of Kővári--Sós--Turán theorem proved by Erdős.

For an increasing weighted tree $G_\omega$, we obtain an asymptotic value and a sharp bound on the index stability of the depth function of its edge ideal $I(G_\omega)$. Moreover, if $G_\omega$ is a strictly increasing weighted tree, we provide the minimal free resolution of $I(G_\omega)$ and an exact formula for the regularity of all powers of $I(G_\omega)$.

An oriented $k$-uniform hypergraph, or oriented $k$-graph, is said to satisfy Property O if, for every linear ordering of its vertex set, there is some edge oriented consistently with this order. The minimum number $f(k)$ of edges in a $k$-graph with Property O was first studied by Duffus, Kay, and Rödl, and later improved by Kronenberg, Kusch, Lamaison, Micek, and Tran. In particular, they established the bounds $k! + 1 \le f(k) \le \left(\lfloor\tfrac{k}{2}\rfloor+1 \right) k! - \lfloor\tfrac{k}{2}\rfloor(k-1)!$ for every $k \ge 2$. In this note, we extend the study of Property O to the linear setting. We determine the minimum number $f'(k)$ of edges in a linear $k$-graph up to a $\operatorname{poly}(k)$ multiplicative factor, showing that $\frac{(k!)^2}{2e^2k^4} \le f'(k) \le (1+o(1)) \cdot 4 k^6 \ln^2 k \cdot (k!)^2$. Our approach also yields bounds on the minimum number $n'(k)$ of vertices in an oriented linear $k$-graph with Property O. Additionally, we explore the minimum number of edges and vertices required in a linear $k$-graph satisfying the newly introduced Erdős--Szekeres properties.

Schanuel Conjecture contains all ``reasonable" statements that can be made on the values of the exponential function. In particular it implies the Lindemann-Weierstrass Theorem. In my Ph.D. I showed that Schanuel Conjecture has a geometrical origin: it is equivalent to the Grothendieck-André periods Conjecture applied to a 1-motive without abelian part. In this paper, we state a conjecture in Schanuel style, which will imply conjectures in Lindemann-Weierstrass style, for the semi-elliptic exponential function, that is for the exponential map of an extension G of an elliptic curve E by a multiplicative group. We propose the semi-elliptic Conjecture, which concerns the exponential function, the Weierstrass $\wp,$ $\zeta$ functions and Serre functions. The case of a trivial extension has been treated in \cite{BW}, where we introduced the split semi-elliptic Conjecture. As in Schanuel's case, we expect that the semi-elliptic Conjecture contains all ``reasonable" statements that can be made on the values of the exponential function, of the Weierstrass $\wp$, $\zeta$ functions and of Serre functions. We show that the semi-elliptic Conjecture has a geometrical origin (as Schanuel Conjecture): it is equivalent to the Grothendieck-André periods Conjecture applied to a 1-motive whose underlying abelian part is an elliptic curve. We prove the Grothendieck-André periods Conjecture for 1-motives defined by an elliptic curve with algebraic invariants and complex multiplication and by torsion points. We introduce the $\sigma$-Conjecture which involves the Weierstrass $\wp$, $\zeta$ and $\sigma$ functions and we show that this conjecture is a consequence of the Grothendieck-André periods Conjecture applied to an adequate 1-motive.

We develop the Bernstein-Zelevinsky theory for quasi-split real classical groups and employ this framework to establish an Euler-Poincaré characteristic formula for general linear groups. The key to our approach is establishing the Casselman-Wallach property for the homology of the Jacquet functor, which also provides an affirmative resolution to an open question proposed by A. Aizenbud, D. Gourevitch and S. Sahi. Furthermore, we prove the vanishing of higher extension groups for arbitrary pairs of generic representations, confirming a conjecture of Dipendra Prasad. We also utilize the Bernstein-Zelevinsky theory to establish two additional results: the Leibniz law for the highest derivative and a unitarity criterion for general linear groups. Lastly, we apply the Bernstein-Zelevinsky theory to prove the Hausdorffness and exactness of the twisted homology of split even orthogonal groups.

This paper studies decentralized optimization problem $f(\mathbf{x})=\frac{1}{m}\sum_{i=1}^m f_i(\mathbf{x})$, where each local function has the form of $f_i(\mathbf{x}) = {\mathbb E}\left[F(\mathbf{x};{\xi}_i)\right]$ which is $(L_0,L_1)$-smooth but possibly nonconvex and the random variable ${\xi}_i$ follows distribution ${\mathcal D}_i$. We propose a novel algorithm called decentralized normalized stochastic gradient descent (DNSGD), which can achieve the $\epsilon$-stationary point on each local agent. We present a new framework for analyzing decentralized first-order methods in the relaxed smooth setting, based on the Lyapunov function related to the product of the gradient norm and the consensus error. The analysis shows upper bounds on sample complexity of ${\mathcal O}(m^{-1}(L_f\sigma^2\Delta_f\epsilon^{-4} + \sigma^2\epsilon^{-2} + L_f^{-2}L_1^3\sigma^2\Delta_f\epsilon^{-1} + L_f^{-2}L_1^2\sigma^2))$ per agent and communication complexity of $\tilde{\mathcal O}((L_f\epsilon^{-2} + L_1\epsilon^{-1})\gamma^{-1/2}\Delta_f)$, where $L_f=L_0 +L_1\zeta$, $\sigma^2$ is the variance of the stochastic gradient, $\Delta_f$ is the initial optimal function value gap, $\gamma$ is the spectral gap of the network, and $\zeta$ is the degree of the gradient dissimilarity. In the special case of $L_1=0$, the above results (nearly) match the lower bounds on decentralized nonconvex optimization in the standard smooth setting. We also conduct numerical experiments to show the empirical superiority of our method.

We prove optimal Lipschitz regularity for weak solutions of the measure-valued $p$-Poisson equation $-\Delta_p u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$. Here $p \in (1,2)$, $\Gamma$ is a compact and connected $C^2$-hypersurface without boundary, and $Q$ is a positive $W^{2,\infty}$-density. This equation can be understood as a nonlinear interface transmission problem. Our main result extends previous studies of the linear case and provides further insights on a delicate limit case of (linear and nonlinear) potential theory.

In this paper, we propose the Bregman Douglas-Rachford splitting (BDRS) method and its variant Bregman Peaceman-Rachford splitting method for solving maximal monotone inclusion problem. We show that BDRS is equivalent to a Bregman alternating direction method of multipliers (ADMM) when applied to the dual of the problem. A special case of the Bregman ADMM is an alternating direction version of the exponential multiplier method. To the best of our knowledge, algorithms proposed in this paper are new to the literature. We also discuss how to use our algorithms to solve the discrete optimal transport (OT) problem. We prove the convergence of the algorithms under certain assumptions, though we point out that one assumption does not apply to the OT problem.

The extended-domain method is an appealingly simple strategy for applying spectral methods to complex geometries, but its theoretical stability properties, particularly for non-normal operators, are not fully understood. This paper provides a rigorous stability analysis based on the Lebesgue constant and reveals a fundamental stability dichotomy at the heart of the method. We first prove a surprising result: for the self-adjoint Poisson equation, the method is unstable, with a Lebesgue constant that grows super-polynomially due to the ill-conditioning of spectral differentiation. In stark contrast, we prove that for the non-self-adjoint convection-diffusion equation, the method becomes stable. We show that the first-order convection term regularizes the operator, leading to a provably polynomial bound on the Lebesgue constant. These results, extended here to multiple dimensions and variable coefficients, provide a complete theoretical foundation for this practical method, establishing the precise conditions under which it is stable and highlighting a non-trivial interplay between operator components.

Inverse microstructure design plays a central role in materials discovery, yet remains challenging due to the complexity of structure-property linkages and the scarcity of labeled training data. We propose Design-GenNO, a physics-informed generative neural operator framework that unifies generative modeling with operator learning to address these challenges. In Design-GenNO, microstructures are encoded into a low-dimensional, well-structured latent space, which serves as the generator for both reconstructing microstructures and predicting solution fields of governing PDEs. MultiONet-based decoders enable functional mappings from latent variables to both microstructures and full PDE solution fields, allowing a multitude of design objectives to be addressed without retraining. A normalizing flow prior regularizes the latent space, facilitating efficient sampling and robust gradient-based optimization. A distinctive feature of the framework is its physics-informed training strategy: by embedding PDE residuals directly into the learning objective, Design-GenNO significantly reduces reliance on labeled datasets and can even operate in a self-supervised setting. We validate the method on a suite of inverse design tasks in two-phase materials, including effective property matching, recovery of microstructures from sparse field measurements, and maximization of conductivity ratios. Across all tasks, Design-GenNO achieves high accuracy, generates diverse and physically meaningful designs, and consistently outperforms the state-of-the-art method. Moreover, it demonstrates strong extrapolation by producing microstructures with effective properties beyond the training distribution. These results establish Design-GenNO as a robust and general framework for physics-informed inverse design, offering a promising pathway toward accelerated materials discovery.

In this note, we first study the notion of subexponential decay (SD) for countable groups with respect to a length function, which generalizes the well-known rapid decay (RD) property, first discovered by Haagerup in 1979. Several natural properties and examples are studied, especially including groups that have SD, but not RD. This consideration naturally has applications in $C^*$-algebras. We also consider in this setting a permanence theorem for decay in amalgamated free products (proved also recently by Chatterji--Gautero), and demonstrate that it is in a precise sense optimal.

We formulate an effective variant of the Yau-Tian-Donaldson conjecture, then review effective results on K-stability of spherical varieties, that is, K-stability criterions which can be effectively computed given the combinatorial data associated with the variety. We focus on the standard notion of K-stability as defined by Donaldson for constant scalar curvature Kähler metrics.

In this paper we study the existence of minimizers for interaction energies with the presence of external potentials. We consider a class of subharmonic interaction potentials, which include the Riesz potentials $|{\bf x}|^{-s},\,\max\{0,d-2\}<s<d$ and its anisotropic counterparts. The underlying space is taken as $\mathbb{R}^d$ or a half-space with possibly curved boundary. We give a sufficient and almost necessary condition for the existence of minimizers, as well as the uniqueness of minimizers. The proof is based on the observation that the Euler-Lagrange condition for the energy minimizer is almost the same as that for the maximizer of the height functional, defined as the essential infimum of the generated potential. We also give two complimentary results: a simple sufficient condition for the existence of minimizers for general interaction/external potentials, and a slight improvement to the known result on the existence of minimizers without external potentials.

Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want the paths to be far apart, say at distance at least $c$? One might hope that we can find either $k+1$ paths pairwise far apart, or $k$ sets of bounded radius that separate $S$ and $T$, where the bound on the radius is some $\ell$ that depends only on $k,c$ (the ``coarse Menger conjecture''). We showed in an earlier paper that this is false for all $k\ge 2$ and $c\ge3$. To do so we gave a sequence of finite graphs, counterexamples for larger and larger values of $\ell$ with $k=2$, $c=3$. Our counterexamples contained subdivisions of uniform binary trees with arbitrarily large depth as subgraphs. Here we show that for any binary tree $T$, the coarse Menger conjecture is true for all graphs that contain no subdivision of $T$ as a subgraph, that is, it is true for graphs with bounded path-width (and, further, for graphs with bounded coarse path-width). This is perhaps surprising, since it is false for bounded tree-width.

The porous medium type reaction-diffusion equation and the Hele-Shaw problem are two free boundary problems linked through the incompressible (Hele-Shaw) limit. We investigate and compare the sharp power concavities of the pressures on their respective supports for the two free boundary problems. For the pressure of the porous medium type reaction-diffusion equation, the $\frac{1}{2}$-concavity preserves all the time, while $\alpha$-concavity for $\alpha\in[0,\frac{1}{2})\cup(\frac{1}{2},1]$ does not persist in time. In contrast, in the case of the pressure for the Hele-Shaw problem, $\alpha$-concavity with $\alpha\in[0,\frac{1}{2}]$ is maintained all the while and $\frac{1}{2}$ acts as the largest index. The intuitive explanation for the difference between the two free boundary problems is that, although the Hele-Shaw problem is the incompressible limit of the porous medium-type reaction-diffusion equation, it is no longer a degenerate parabolic equation. Furthermore, for the pressure of the porous medium type reaction-diffusion equation, the non-degenerate estimate is established by means of the derived concave properties, indicating that the spatial Lipschitz regularity in the whole space is sharp.

This article investigates the question of disorder relevance for the continuous-time Random Walk Pinning Model (RWPM) and completes the results of our companion paper. The RWPM considers a continuous time random walk $X=(X_t)_{t\geq 0}$, whose law is modified by a Gibbs weight given by $\exp(\beta \int_0^T \mathbf{1}_{\{X_t=Y_t\}} dt)$, where $Y=(Y_t)_{t\geq 0}$ is a quenched trajectory of a second (independent) random walk and $\beta \geq 0$ is the inverse temperature. The random walk $Y$ has the same distribution as $X$ but a jump rate $\rho \geq 0$, interpreted as the disorder intensity. For fixed $\rho\ge 0$, the RWPM undergoes a localization phase transition as $\beta$ crosses a critical threshold $\beta_c(\rho)$. The question of disorder relevance then consists in determining whether a disorder of arbitrarily small intensity $\rho$ changes the properties of the phase transition. We focus our analysis on the case of transient $\gamma$-stable walks on $\mathbb{Z}$, i.e. random walks in the domain of attraction of a $\gamma$-stable law, with $\gamma\in (0,1)$. In the present paper, we show that disorder is relevant when $\gamma \in (0,\frac23]$, namely that $\beta_c(\rho)>\beta_c(0)$ for every $\rho>0$. We also provide lower bounds on the critical point shift, which are matching the upper bounds obtained in our companion paper. Interestingly, in the marginal case $\gamma = \frac23$, disorder is always relevant, independently of the fine properties of the random walk distribution. When $\gamma \in (\frac23,1)$, our companion paper proves that disorder is irrelevant (in particular $\beta_c(\rho)=\beta_c(0)$ for $\rho$ small enough). We provide here an upper bound on the free energy in the regime $\gamma\in (\frac 2 3,1)$ that highlights the fact that although disorder is irrelevant, it still has a non-trivial effect on the phase transition, at any $\rho>0$.

For fixed genus g and varying finite marking set A, the gluing and forgetful maps give the spaces of holomorphic forms on the moduli space of stable A-marked curves of genus g has the structure of an FA-module, i.e., a functor from the category of finite sets to vector spaces. We prove that the resulting FA-modules of holomorphic k-forms are simple, for k less than or equal to 18, whenever they are nonzero. Conditional upon the conjectured vanishing of holomorphic 19-forms and 20-forms in genus 3, for 15 and 16 marked points, respectively, this extends to k less than or equal to 20.

The Random Walk Pinning Model (RWPM) is a statistical mechanics model in which the trajectory of a continuous time random walk $X=(X_t)_{t\geq 0}$ is rewarded according to the time it spends together with a moving catalyst. More specifically for a system of size $T$, the law of $X$ is tilted by the Gibbs factor $\exp(\beta \int_0^T \mathbf{1}_{\{X_t=Y_t\}} dt)$, where $\beta \geq 0$ is the inverse temperature. The moving catalyst $Y=(Y_t)_{t\ge 0}$ is given by the quenched trajectory of a second continuous-time random walk, with the same distribution as $X$ but a different jump rate $\rho\geq 0$, interpreted as the disorder intensity. For fixed $\rho\ge 0$, the RWPM undergoes a localization phase transition when $\beta$ passes a critical value $\beta_c(\rho)$. We thoroughly investigate the question of disorder relevance to determine whether a disorder of arbitrarily small intensity affects the features of the phase transition. We focus our analysis on the case of transient $\gamma$-stable walks on $\mathbb{Z}$, i.e. random walks in the domain of attraction of a $\gamma$-stable law, with $\gamma\in (0,1)$. In the present paper, we derive lower bounds for the free energy, which results in either a proof of disorder irrelevance or upper bounds on the critical point shift. More precisely, when $\gamma \in(\frac23,1)$, our estimates imply that that $\beta_c(\rho)=\beta_c(0)$ and $\rho$ is small, showing disorder irrelevance. When $\gamma\in (0,\frac23]$ our companion paper shows that $\beta_c(\rho)>\beta_c(0)$ for every $\rho>0$, showing disorder relevance: we derive here upper bounds on the critical point shift, which are matching the lower bounds obtained in our companion paper. For good measure, our analysis also includes the case of the simple random walk of $\mathbb{Z}^d$ (for $d\ge 3$) for which no upper bound on the critical point shift was previously known.

We introduce discontinuous spectral-element methods of arbitrary order that are well balanced, conservative of mass, and conservative or dissipative of total energy (i.e., a mathematical entropy function) for a covariant flux formulation of the rotating shallow water equations with variable bottom topography on curved manifolds such as the sphere. The proposed methods are based on a skew-symmetric splitting of the tensor divergence in covariant form, which we implement and analyze within a general flux-differencing framework using tensor-product summation-by-parts operators. Such schemes are proven to satisfy semi-discrete mass and energy conservation on general unstructured quadrilateral grids in addition to well balancing for arbitrary continuous bottom topographies, with energy dissipation resulting from a suitable choice of numerical interface flux. Furthermore, the proposed covariant formulation permits an analytical representation of the geometry and associated metric terms while satisfying the aforementioned entropy stability, conservation, and well-balancing properties without the need to approximate the metric terms so as to enforce discrete metric identities. Numerical experiments on cubed-sphere grids are presented in order to verify the schemes' structure-preservation properties as well as to assess their accuracy and robustness within the context of several standard test cases characteristic of idealized atmospheric flows. Our theoretical and numerical results support the further development of the proposed methodology towards a full dynamical core for numerical weather prediction and climate modelling, as well as broader applications to other hyperbolic and advection-dominated systems of partial differential equations on curved manifolds.

Every composition of two strictly singular operators is compact on the Baernstein space $B_p$ for $1 < p < \infty$ and on the $p$-convexified Schreier space $S_{p}$ for $1 \leq p < \infty$. Furthermore, every subsymmetric basic sequence in $B_p$ (respectively, $S_p$) is equivalent to the unit vector basis for $\ell_p$ (respectively, $c_0$), and the Banach spaces $B_p$ and $S_p$ contain block basic sequences whose closed span is not complemented.

We study the semi-discrete formulation of one-dimensional partial optimal transport with quadratic cost, where a probability density is partially transported to a finite sum of Dirac masses of smaller total mass. This problem arises naturally in applications such as risk management, the modeling of crowd motion, and sliced partial transport algorithms for point cloud registration. Unlike higher-dimensional settings, the dual functional in the unidimensional case exhibits reduced regularity. To overcome this difficulty, we introduce a regularization procedure based on thickening the density along an auxiliary dimension. We prove that the maximizers of the regularized dual problem converge to those of the original dual problem, with quadratic rate in the introduced thickness. We further provide a numerical scheme that leverages the regularized functional, and we validate our analysis with simulations that confirm the quadratic convergence rate. Finally, we compare the semi-discrete and fully discrete settings, demonstrating that our approach offers both improved stability and computational efficiency for unidimensional partial transport problems.

In this short note, we prove several infinite family of congruences for some restricted partitions introduced by Pushpa and Vasuki (2022) (thereby, also proving a conjecture of Dasappa et. al. (2023)). We also prove some isolated congruences which seem to have been missed by earlier authors. Our proof techniques uses both elementary means as well as the theory of modular forms.

The fluid antenna system (FAS) concept is an emerging paradigm that promotes the utilization of the feature of shape and position reconfigurability in antennas to broaden the design of wireless communication systems. This also means that spatial diversity can be exploited in an unconventional way. However, a rigorous framework for error probability analysis of FAS under realistic spatially correlated channels has been lacking. In this paper, we fill this gap by deriving a tight, closed-form asymptotic expression for the symbol error rate (SER) that establishes the fundamental scaling law linking the system's SER to the channel's spatial correlation structure. A key insight of our analysis is that the achievable diversity gain is governed not by the number of antenna ports, but by the channel's effective rank. To find this critical parameter, we propose a novel dual-pronged approach. First of all, we develop a geometry-based algorithm that extracts distinct performance thresholds from the channel's eigenvalue spectrum. Second, we theoretically prove that the effective rank converges to a fundamental limit dictated solely by the antenna's normalized aperture width. We further establish the equivalence between the threshold identified by the geometric algorithm and the derived theoretical limit, providing rigorous validation for the proposed method. Our effective rank model achieves higher accuracy than existing approaches in the literature. Building on this framework, we offer a complete characterization of diversity and coding gains. The analysis leads to a definitive design insight: FAS performance improvements are fundamentally driven by enlarging the antenna's explorable aperture, which increases the effective channel rank, whereas increasing port density within a fixed aperture yields diminishing returns.

The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the component of interest, such as the location of the optimum. Nuisance parameters are eliminated via profile likelihood, which naturally handles constraints. As a direct instantiation, we develop a MINimalist Thompson Sampling (MINTS) algorithm. Our framework accommodates structured problems, including continuum-armed Lipschitz bandits and dynamic pricing. It also provides a probabilistic lens on classical convex optimization algorithms such as the center of gravity and ellipsoid methods. We further analyze MINTS for multi-armed bandits and establish near-optimal regret guarantees.

The stability of plasmas is a challenging topic. We study a control problem for plasma living in a uniform external magnetic field. Linear analysis for any equilibrium $\mu$ is investigated through the Laplace-Fourier transform approach. Moreover, the Penrose condition, which characterizes the stability of equilibrium, is derived for Bernstein modes and verified numerically. Based on the linear analysis, a general control strategy is introduced, with recovering the free-streaming solution as a specific example. We finally present several examples to control instabilities, including Gaussian equilibria and Dory-Guest-Harris instability.

Out-of-time-order correlation functions (OTOCs) and their higher-order generalizations present important probes of quantum information dynamics and scrambling. We introduce a solvable many-body quantum model, which we term boundary scrambling, for which the full dynamics of higher-order OTOCs is analytically tractable. These dynamics support a decomposition into free cumulants and unify recent extensions of the eigenstate thermalization hypothesis with predictions from random quantum circuit models. We obtain exact expressions for (higher-order) correlations between matrix elements and show these to be stable away from the solvable point. The solvability is enabled by the identification of a higher-order Markovian influence matrix, capturing the effect of the full system on a local subsystem. These results provide insight into the emergence of random-matrix behavior from structured Floquet dynamics and show how techniques from free probability can be applied in the construction of exactly-solvable many-body models.

Federated Learning (FL) is a novel, multidisciplinary Machine Learning paradigm where multiple clients, such as mobile devices, collaborate to solve machine learning problems. Initially introduced in Kone{č}n{ý} et al. (2016a,b); McMahan et al. (2017), FL has gained further attention through its inclusion in the National AI Research and Development Strategic Plan (2023 Update) of the United States (Science and on Artificial Intelligence, 2023). The FL training process is inherently decentralized and often takes place in less controlled settings compared to data centers, posing unique challenges distinct from those in fully controlled environments. In this thesis, we identify five key challenges in Federated Learning and propose novel approaches to address them. These challenges arise from the heterogeneity of data and devices, communication issues, and privacy concerns for clients in FL training. Moreover, even well-established theoretical advances in FL require diverse forms of practical implementation to enhance their real-world applicability. Our contributions advance FL algorithms and systems, bridging theoretical advancements and practical implementations. More broadly, our work serves as a guide for researchers navigating the complexities of translating theoretical methods into efficient real-world implementations and software. Additionally, it offers insights into the reverse process of adapting practical implementation aspects back into theoretical algorithm design. This reverse process is particularly intriguing, as the practical perspective compels us to examine the underlying mechanics and flexibilities of algorithms more deeply, often uncovering new dimensions of the algorithms under study.

Computing the permanent of a non-negative matrix is a computationally challenging, \#P-complete problem with wide-ranging applications. We introduce a novel permanental analogue of Schur's determinant formula, leveraging a newly defined \emph{permanental inverse}. Building on this, we introduce an iterative, deterministic procedure called the \emph{permanent process}, analogous to Gaussian elimination, which yields constructive and algorithmically computable upper bounds on the permanent. Our framework provides particularly strong guarantees for matrices exhibiting approximate diagonal dominance-like properties, thereby offering new theoretical and computational tools for analyzing and bounding permanents.

While modal extensions of decidable fragments of first-order logic are usually undecidable, their monodic counterparts, in which formulas in the scope of modal operators have at most one free variable, are typically decidable. This only holds, however, under the provision that non-rigid constants, definite descriptions and non-trivial counting are not admitted. Indeed, several monodic fragments having at least one of these features are known to be undecidable. We investigate these features systematically and show that fundamental monodic fragments such as the two-variable fragment with counting and the guarded fragment of standard first-order modal logics $\mathbf{K}_{n}$ and $\mathbf{S5}_{n}$ are decidable. Tight complexity bounds are established as well. Under the expanding-domain semantics, we show decidability of the basic modal logic extended with the transitive closure operator on finite acyclic frames; this logic, however, is Ackermann-hard.

The dynamics of gene regulatory networks is governed by the interaction between deterministic biochemical reactions and molecular noise. To understand how gene regulatory networks process information during cell state transitions, we study stochastic dynamics derived from a Boolean network model via its representation on the parameter space of Gaussian distributions, equipped with the Fisher information metric. This reformulation reveals that the trajectories of optimal information transfer are gradient flows of the Kullback-Leibler divergence. We demonstrate that the most efficient dynamics require isotropic decay rates across all nodes and that the noise intensity quantitatively determines the potential differentiation between the initial and final states. Furthermore, we show that paths minimizing biological cost correspond to metric geodesics that require noise suppression, leading to biologically irrelevant deterministic dynamics. Our approach frames noise and decay rates as fundamental control parameters for cellular differentiation, providing a geometric principle for the analysis and design of synthetic networks.

Preconditioning with the quantum Fisher information matrix (QFIM) is a popular approach in quantum variational algorithms. Yet the QFIM is costly to obtain directly, usually requiring more state preparation than its classical counterpart: the classical Fisher information matrix (CFIM). We rigorously prove that averaging the classical Fisher information matrix over Haar-random measurement bases yields $\mathbb{E}_{U\sim\mu_H}[F^U(\boldsymbol{\theta})] = \frac{1}{2}Q(\boldsymbol{\theta})$ for pure states in $\mathbb{C}^N$. Furthermore, we obtain the variance of CFIM ($O(N^{-1})$) and establish non-asymptotic concentration bounds ($\exp(-\Theta(N)t^2)$), demonstrating that using few random measurement bases is sufficient to approximate the QFIM accurately, especially in high-dimensional settings. This work establishes a solid theoretical foundation for efficient quantum natural gradient methods via randomized measurements.

We consider a communication scenario over a discrete memoryless interference channel or multiple access channel without feedback, where transmitters exploit classical, quantum, or no-signaling cooperation. In this scenario, several previous works have shown that the sum capacities of channels involving pseudo-telepathy games can be enhanced by quantum or no-signaling cooperation. However, a full characterization of which channels admit such an improvement remains open. By focusing on the common characteristics of previously studied channels, we propose a broader class of channels for which quantum or no-signaling cooperation increases the sum capacity. Channels in this class are associated with a pseudo-telepathy game, with channel inputs specified as tuples of questions and answers from the game. In addition, when the channel inputs satisfy the winning condition of the game, the channel decomposes into parallel weakly symmetric sub-channels and is less noisy compared to the case when the inputs do not meet the winning condition.

We propose a novel framework for analyzing the geometric structure of horseshoes arising in three- and four-dimensional Hénon-type maps by introducing paperfolding structures as geometric templates. These structures capture the folding and stacking mechanisms characteristic of high-dimensional chaotic dynamics, offering a combinatorial and visual language to describe complex horseshoe formations. By systematically relating iterated paperfolding patterns to the observed geometries in multidimensional maps, our approach provides a concrete method for visualizing and classifying the topological features of chaotic sets in dimensions higher than two. This framework offers new insight into the organization of chaos in higher-dimensional discrete dynamical systems.

In this work, we show that the quantum mechanical notions of density operator, positive operator-valued measure (POVM), and the Born rule, are all simultaneously encoded in the categorical notion of a natural transformation of functors. In particular, we show that given a fixed quantum system, there exists an explicit bijection from the set of density operators on the associated Hilbert space to the set of natural transformations between the canonical measurement and probability functors associated with the system, which formalize the way in which quantum effects (i.e., POVM elements) and their associated probabilities are additive with respect to a coarse-graining of measurements.

Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of communities, impact efficient network navigation, and fundamentally shape connectivity and system behavior. We introduce a novel topological data analysis weighting scheme for graphs, based on chordless cycles, aimed at estimating the dimensionality of networks in a data-driven way. We further show that the resulting descriptors can effectively estimate network dimensionality using a neural network architecture trained in a synthetic graph database constructed for this purpose, which does not need retraining to transfer effectively to real-world networks. Thus, by combining cycle-aware filtrations, algebraic topology, and machine learning, our approach provides a robust and effective method for uncovering the hidden geometry of complex networks and guiding accurate modeling and low-dimensional embedding.

We study a missing-value imputation method, termed kNNSampler, that imputes a given unit's missing response by randomly sampling from the observed responses of the $k$ most similar units to the given unit in terms of the observed covariates. This method can sample unknown missing values from their distributions, quantify the uncertainties of missing values, and be readily used for multiple imputation. Unlike popular kNNImputer, which estimates the conditional mean of a missing response given an observed covariate, kNNSampler is theoretically shown to estimate the conditional distribution of a missing response given an observed covariate. Experiments demonstrate its effectiveness in recovering the distribution of missing values. The code for kNNSampler is made publicly available (this https URL).

This article proposes a paraconsistent framework for evaluating similarity in knowledge bases. Unlike classical approaches, this framework explicitly integrates contradictions, enabling a more robust and interpretable similarity measure. A new measure $ S^* $ is introduced, which penalizes inconsistencies while rewarding shared properties. Paraconsistent super-categories $ \Xi_K^* $ are defined to hierarchically organize knowledge entities. The model also includes a contradiction extractor $ E $ and a repair mechanism, ensuring consistency in the evaluations. Theoretical results guarantee reflexivity, symmetry, and boundedness of $ S^* $. This approach offers a promising solution for managing conflicting knowledge, with perspectives in multi-agent systems.

Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In this paper, we develop an alternative, algorithm-centered framework for understanding and improving the robustness against errors. For a given quantum algorithm and error model, we derive worst-case fidelity bounds which can be explicitly computed to certify the robustness. We consider general error models including coherent and (Markovian) incoherent errors and allowing for set-based error descriptions to address uncertainty or time-dependence in the errors. Our results give rise to guidelines for robust algorithm design and compilation by optimizing our theoretical robustness measure. Numerical results on algorithm analysis and robust optimization demonstrate the practicality of the framework.

We analyze gradient descent with Polyak heavy-ball momentum (HB) whose fixed momentum parameter $\beta \in (0, 1)$ provides exponential decay of memory. Building on Kovachki and Stuart (2021), we prove that on an exponentially attractive invariant manifold the algorithm is exactly plain gradient descent with a modified loss, provided that the step size $h$ is small enough. Although the modified loss does not admit a closed-form expression, we describe it with arbitrary precision and prove global (finite "time" horizon) approximation bounds $O(h^{R})$ for any finite order $R \geq 2$. We then conduct a fine-grained analysis of the combinatorics underlying the memoryless approximations of HB, in particular, finding a rich family of polynomials in $\beta$ hidden inside which contains Eulerian and Narayana polynomials. We derive continuous modified equations of arbitrary approximation order (with rigorous bounds) and the principal flow that approximates the HB dynamics, generalizing Rosca et al. (2023). Approximation theorems cover both full-batch and mini-batch HB. Our theoretical results shed new light on the main features of gradient descent with heavy-ball momentum, and outline a road-map for similar analysis of other optimization algorithms.

We consider an n-qubit quantum system with a Hamiltonian, defined by an expansion in the Pauli basis, and propose a new algorithm for classical computing the exponential of the Hamiltonian. The algorithm is based on the representation of the exponential by the Dunford-Cauchy integral, followed by an efficient computation of the resolvent, and is suitable for Hamiltonians that are sparse in the Pauli basis. The practical efficiency of the algorithm is demonstrated by two illustrative examples.

Using inverse planning tools to create radiotherapy treatment plans is an iterative process, where clinical trade-offs are explored by changing the relative importance of different objectives and rerunning the optimizer until a desirable plan is found. We seek to optimize hundreds of radiosurgery treatment plans, corresponding to different weightings of objectives, fast enough to incorporate interactive Pareto navigation of clinical trade-offs into the clinical workflow. We apply the alternating direction method of multipliers (ADMM) to the linear-program formulation of the optimization problem used in Lightning. We implement both a CPU and a GPU version of ADMM in Matlab and compare them to Matlab's built-in, single-threaded dual-simplex solver. The ADMM implementation is adapted to the optimization procedure used in the clinical software, with a bespoke algorithm for maximizing overlap between low-dose points for different objective weights. The method is evaluated on a test dataset consisting of 20 cases from three different indications, with between one and nine targets and total target volumes ranging from 0.66 to 52 cm3, yielding speedups of 1.6-97 and 54-1500 times on CPU and GPU, respectively, compared to simplex. Plan quality was evaluated by rerunning the ADMM optimization 20 times, each with a different random seed, for each test case and for nine objective weightings per case. The resulting clinical metrics closely mimicked those obtained when rerunning the simplex solver, verifying the validity of the method. In conclusion, we show how ADMM can be adapted for radiosurgery plan optimization, allowing hundreds of high-quality Gamma Knife treatment plans to be created in under two minutes on a single GPU, also for very large cases.

The unadjusted Langevin algorithm is widely used for sampling from complex high-dimensional distributions. It is well known to be biased, with the bias typically scaling linearly with the dimension when measured in squared Wasserstein distance. However, the recent paper of Chen et al. (2024) identifies an intriguing new delocalization effect: For a class of distributions with sparse interactions, the bias between low-dimensional marginals scales only with the lower dimension, not the full dimension. In this work, we strengthen the results of Chen et al. (2024) in the sparse interaction regime by removing a logarithmic factor, measuring distance in relative entropy (a.k.a. KL-divergence), and relaxing the strong log-concavity assumption. In addition, we expand the scope of the delocalization phenomenon by showing that it holds for a class of distributions with weak interactions. Our proofs are based on a hierarchical analysis of the marginal relative entropies, inspired by the authors' recent work on propagation of chaos.

The lack of a unified theoretical framework for characterizing band inversions across different crystal symmetries hinders the rapid development of topological photonic band engineering. To address this issue, we have constructed a framework constrained by symmetry $k \cdot p$ that universally models bands near high-symmetry points for symmetric photonic crystals $C_6$, $C_4$, $C_3$, and $C_2$. This framework enables a coefficient-free quantitative diagnosis of band topology. We have demonstrated the power of this framework by systematically engineering band inversions. In $C_6$ crystals, we induce a reopening of the linear gap at $\Gamma$. In $C_4$ systems, mirror symmetry enforces a characteristic quadratic coupling leading to distinct spectral features. Our analysis further reveals that a lone $E$ doublet prevents inversion at the $\Gamma$ point in $C_3$ symmetry, while $C_2$ symmetry facilitates a unique inversion of $Y$ pointsints with anisotropic gap. This symmetry-first, fit-free approach establishes a direct link between experimental band maps and the extraction of fundamental topological parameters. It offers a universal tool for inversion and coupling-order identification.

We introduce a new Markov Chain called the Cycle Walk for sampling measures of graph partitions where the partition elements have roughly equal size. Such Markov Chains are of current interest in the generation and evaluation of political districts. We present numerical evidence that this chain can efficiently sample target distributions that have been difficult for existing sampling Markov chains.

Integrated Sensing and Communication (ISAC) is one of the pivotal supporting technologies for next-generation wireless communication networks. As an emerging technical means, the new six-dimensional movable antenna (6DMA) system can effectively improve the communication and sensing performance of ISAC systems. However, related research on the architecture based on three-dimensional rotatable antennas (RA) remains relatively limited. Especially under the influence of non-ideal channel characteristics, how to balance system performance and hardware cost control through beamforming technology in non-ideal channels has become a crucial issue to be solved urgently. Given the significant advantages of hybrid beamforming technology in balancing system performance and hardware complexity, this paper focuses on the channel model considering effective aperture loss under the RA architecture and studies the sub-connected hybrid beamforming design for multi-user ISAC systems. Aiming at the non-convex nature with coupled variables in this problem, this paper first transforms the complex fractional objective function using the Fractional Programming (FP) framework, and then proposes an algorithm based on the Alternating Optimization (AO) architecture, which achieves optimization by alternately solving five subproblems. Among them, the closed-form update expression of the array rotation angle is derived through the Karush-Kuhn-Tucker (KKT) conditions, and a two-stage Gradient Ascent (GA) based method is proposed to optimize the antenna rotation angle. Simulation results show that compared with the traditional fixed-position antenna (FPA), the proposed method can significantly improve the overall performance of the system.

In federated learning, multiple parties train models locally and share their parameters with a central server, which aggregates them to update a global model. To address the risk of exposing sensitive data through local models, secure aggregation via secure multiparty computation has been proposed to enhance privacy. At the same time, perfect privacy can only be achieved by a uniform distribution of the masked local models to be aggregated. This raises a problem when working with real valued data, as there is no measure on the reals that is invariant under the masking operation, and hence information leakage is bound to occur. Shifting the data to a finite field circumvents this problem, but as a downside runs into an inherent accuracy complexity tradeoff issue due to fixed point modular arithmetic as opposed to floating point numbers that can simultaneously handle numbers of varying magnitudes. In this paper, a novel secure parameter aggregation method is proposed that employs the torus rather than a finite field. This approach guarantees perfect privacy for each party's data by utilizing the uniform distribution on the torus, while avoiding accuracy losses. Experimental results show that the new protocol performs similarly to the model without secure aggregation while maintaining perfect privacy. Compared to the finite field secure aggregation, the torus-based protocol can in some cases significantly outperform it in terms of model accuracy and cosine similarity, hence making it a safer choice.

A binary word is Sturmian if the occurrences of each letter are balanced, in the sense that in any two factors of the same length, the difference between the number of occurrences of the same letter is at most 1. In digital geometry, Sturmian words correspond to discrete approximations of straight line segments in the Euclidean plane. The Dorst-Smeulders coding, introduced in 1984, is a 4-tuple of integers that uniquely represents a Sturmian word $w$, enabling its reconstruction using $|w|$ modular operations, making it highly efficient in practice. In this paper, we present a linear-time algorithm that, given a binary input word $w$, computes the Dorst-Smeulders coding of its longest Sturmian prefix. This forms the basis for computing the Dorst-Smeulders coding of an arbitrary binary word $w$, which is a minimal decomposition (in terms of the number of factors) of $w$ into Sturmian words, each represented by its Dorst-Smeulders coding. This coding could be leveraged in compression schemes where the input is transformed into a binary word composed of long Sturmian segments. Although the algorithm is conceptually simple and can be implemented in just a few lines of code, it is grounded in a deep analysis of the structural properties of Sturmian words.

Given encoded 3D point cloud geometry available at the decoder, we study the problem of lossy attribute compression in a multi-resolution B-spline projection framework. A target continuous 3D attribute function is first projected onto a sequence of nested subspaces $\mathcal{F}^{(p)}_{l_0} \subseteq \cdots \subseteq \mathcal{F}^{(p)}_{L}$, where $\mathcal{F}^{(p)}_{l}$ is a family of functions spanned by a B-spline basis function of order $p$ at a chosen scale and its integer shifts. The projected low-pass coefficients $F_l^*$ are computed by variable-complexity unrolling of a rate-distortion (RD) optimization algorithm into a feed-forward network, where the rate term is the sparsity-promoting $\ell_1$-norm. Thus, the projection operation is end-to-end differentiable. For a chosen coarse-to-fine predictor, the coefficients are then adjusted to account for the prediction from a lower-resolution to a higher-resolution, which is also optimized in a data-driven manner.

In previous work arXiv:2407.13556, we proposed an extended phase space structure at null infinity accommodating large gauge symmetries for sub$^n$-leading soft theorems in Yang-Mills, via dressing fields arising in the Stueckelberg procedure. Here, we give an explicit boundary action controlling the dynamics of these fields. This allows for a derivation from first principles of the associated charges, together with an explicit renormalization procedure when taking the limit to null and spatial infinity, matching with charges proposed in previous work. Using the language of fibre bundles, we relate the existence of Stueckelberg fields to the notion of extension/reduction of the structure group of a principal bundle, thereby deriving their transformation rules in a natural way, thus realising them as Goldstone-like objects. Finally, this allows us to give a geometric picture of the gauge transformation structure at the boundary, via a loop group coming from formal expansions in the coordinate transversal to the boundary.

We introduce the Fourier Learning Machine (FLM), a neural network (NN) architecture designed to represent a multidimensional nonharmonic Fourier series. The FLM uses a simple feedforward structure with cosine activation functions to learn the frequencies, amplitudes, and phase shifts of the series as trainable parameters. This design allows the model to create a problem-specific spectral basis adaptable to both periodic and nonperiodic functions. Unlike previous Fourier-inspired NN models, the FLM is the first architecture able to represent a complete, separable Fourier basis in multiple dimensions using a standard Multilayer Perceptron-like architecture. A one-to-one correspondence between the Fourier coefficients and amplitudes and phase-shifts is demonstrated, allowing for the translation between a full, separable basis form and the cosine phase--shifted one. Additionally, we evaluate the performance of FLMs on several scientific computing problems, including benchmark Partial Differential Equations (PDEs) and a family of Optimal Control Problems (OCPs). Computational experiments show that the performance of FLMs is comparable, and often superior, to that of established architectures like SIREN and vanilla feedforward NNs.

Data-driven acceleration of scientific computing workflows has been a high-profile aim of machine learning (ML) for science, with numerical simulation of transient partial differential equations (PDEs) being one of the main applications. The focus thus far has been on methods that require classical simulations to train, which when combined with the data-hungriness and optimization challenges of neural networks has caused difficulties in demonstrating a convincing advantage against strong classical baselines. We consider an alternative paradigm in which the learner uses a classical solver's own data to accelerate it, enabling a one-shot speedup of the simulation. Concretely, since transient PDEs often require solving a sequence of related linear systems, the feedback from repeated calls to a linear solver such as preconditioned conjugate gradient (PCG) can be used by a bandit algorithm to online-learn an adaptive sequence of solver configurations (e.g. preconditioners). The method we develop, PCGBandit, is implemented directly on top of the popular open source software OpenFOAM, which we use to show its effectiveness on a set of fluid and magnetohydrodynamics (MHD) problems.

The advent of fault-tolerant quantum computing (FTQC) promises to tackle classically intractable problems. A key milestone is solving the Navier-Stokes equations (NSE), which has remained formidable for quantum algorithms due to their high input-output overhead and nonlinearity. Here, we establish a full-stack framework that charts a practical pathway to a quantum advantage for large-scale NSE simulation. Our approach integrates a spectral-based input/output algorithm, an explicit and synthesized quantum circuit, and a refined error-correction protocol. The algorithm achieves an end-to-end exponential speedup in asymptotic complexity, meeting the lower bound for general quantum linear system solvers. Through symmetry-based circuit synthesis and optimized error correction, we reduce the required logical and physical resources by two orders of magnitude. Our concrete resource analysis demonstrates that solving NSE on a $2^{80}$-grid is feasible with 8.71 million physical qubits (at an error rate of $5 \times 10^{-4}$) in 42.6 days -- outperforming a state-of-the-art supercomputer, which would require over a century. This work bridges the gap between theoretical quantum speedup and the practical deployment of high-performance scientific computing.

We construct symmetry topological field theories (SymTFTs) using the sandwich construction of Pulmann-Ševera-Valach that manifest the centre symmetries of Chern-Simons theory and Yang-Mills theory as well as general relativity in the MacDowell-Mansouri formulation. The 'filling' of the sandwich is an AKSZ sigma model whose target space is a Weil algebra, augmented with discrete degrees of freedom given by a choice of topological boundary condition.

Using the Bordered Floer theory of Lipshitz-Ozsváth-Thurston we prove that the $(p,q)$-cables of any non-trivial knots are not Heegaard Floer homologically thin. Using the proof and a theorem of Zemke, we find a larger set of satellite knots which is a proper superset of the set of all cable knots, having the same property.

We use Volterra-Hamilton systems theory and their associated cost functional to study the population dynamics and productive processes of coral reefs in recovery from bleaching and show that the cost of \mbox{production} remains the same after the process. The geometrical KCC-invariants are determined for the model proposed to describe the renewed symbiotic interaction between coral and algae.

Given a grid diagram for a knot or link K in $S^3$, we construct a filtered spectrum whose homology is the knot Floer homology of K. We conjecture that the filtered homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions.

The $p^\infty$-fine Selmer group of an elliptic curve $E$ over a number field $F$ is a subgroup of the classical $p^\infty$-Selmer group of $E$ over $F$. Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group of $E$ over a $p$-adic Lie extension of a number field is intricately related to some deep questions in classical Iwasawa theory; for example, Iwasawa's classical $\mu$-invariant vanishing conjecture. In this article, we study the properties of the $p^\infty$-fine Selmer group of an elliptic curve over certain $p$-adic Lie extensions of a number field. We also define and discuss $p^\infty$-fine Selmer group of an elliptic curve over function fields of characteristic $p$ and also of characteristic $\ell \neq p.$ We relate our study with a conjecture of Jannsen.

This paper almost classifies the maximal subgroups of $E_7(q)$ for general $q$ a power of a prime $p$. Only four potential maximal subgroups are missing: $PSL_2(7)$ (unknown for $p\neq 2,3,7$), $PSL_2(8)$ ($p=2$) and $PSL_2(9)=A_6$ ($p\neq 2,3$). In addition, there is one issue with the precise structure with the positive-dimensional maximal subgroup of type $A_2$. We are able to give a complete determination of the maximal subgroups for $E_7(q)$ for $q$ an arbitrary power of $3$ and $q=4$.

In this paper we study categorical properties of the category of abelian hypergroups that leads to the notion of hyper (almost) preadditive and hyper (almost) abelian categories. Our goal is to create a path towards a general theory of homological algebra for hyper-algebras. This is a first attempt to achieve this goal. We hope to improve the definitions and results, and provide more examples soon.

We give a direct and self-contained proof that if $H$ is a Hopf algebra and $A\subset H$ is a right coideal subalgebra such $A$ is a direct summand in $H$ as an $A$-bimodule, then $H$ is faithfully flat as a left and right $A$-module.

Skorokhod's J1 and M1 topologies are standard tools in proving limit theorems for stochastic processes. Motivated by applications, we extend these topologies so that they are capable of describing the convergence of a sequence of functions that are not all defined on the same domain. Traditionally, the J1 and M1 topologies are defined using time changes. Instead, we base our definitions on the point of view that the graph of a cadlag function can naturally be viewed as a compact set that is equipped with a total order. The distance between two graphs is then measured by matching points on one graph with points on the other graph in a way that respects the total order. We treat the J1 and M1 topologies in a unified framework and simplify the existing theory. We introduce a space of paths, elements of which are cadlag functions defined on an arbitrary closed subset of the real line. We show that this space is Polish and derive compactness criteria. Specialised to functions that are all defined on the same domain, this yields new proofs of known results.

We construct a monoidal model structure on the category of all curved coalgebras and show that it is Quillen equivalent, via the extended bar-cobar adjunction, to another model structure we construct on the category of curved algebras. When the coalgebras under consideration are conilpotent and the algebras are dg, i.e. uncurved, this corresponds to the ordinary dg Koszul duality of Positselski and Keller-Lefèvre. As an application we construct global noncommutative moduli spaces for flat connections on vector bundles, holomorphic structures on almost complex vector bundles, dg modules over a dg algebra, objects in a dg category, and others.

We construct an $\mathfrak{sl}_2$-action on equivariant $\mathfrak{gl}_N$-link homologies. As a consequence we obtain an action of $\mathfrak{sl}_2$ on these homologies as well as a $p$-DG structures for $p$ a prime number. We explore topological applications of these structures.

A sequence of integers of the form $\lfloor n^{\alpha}\rfloor$ $(n=1,2,\ldots)$ for some fixed non-integral $\alpha>1$ is called a Piatetski-Shapiro sequence, where $\lfloor x\rfloor$ denotes the integer part of $x$. Let $\mathrm{PS}(\alpha)$ denote the set of all those terms. In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\in \mathrm{PS}(\alpha)^3$ for almost every $\alpha>3$. Furthermore, we show that $\mathrm{PS}(\alpha)$ has only finitely many arithmetic progressions of length $3$ for almost every $\alpha>10$. In addition, we estimate upper bounds for the Hausdorff dimension of the set of $\alpha\in [s,t]$ such that $y=a_1x_1+\cdots +a_nx_n$ has infinitely many solutions on $\mathrm{PS}(\alpha)$.

It will be proved that a $k$-clique in the $1$-skeleton of either the order polytope or the chain polytope corresponds to the $(k-1)$-face, which is a simplex, in each polytope. These results generalize the known explicit descriptions of edges and triangular $2$-faces of each polytope.

We show that a discrete harmonic function which is bounded on a large portion of a periodic planar graph is constant. A key ingredient is a new unique continuation result for the weighted graph Laplacian. The proof relies on the structure of level sets of discrete harmonic functions, using arguments as in Bou-Rabee--Cooperman--Dario (2023) which exploit the fact that, on a planar graph, the sub- and super-level sets cannot cross over each other. In the special case of the square lattice this yields a new, geometric proof of the Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin (2017).

We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the boundedness of dual iterates. We show that DPALM can produce a (near) $\vareps$-KKT point within $O(\vareps^{-2})$ outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of $\widetilde{\mathcal{O}}\left(\varepsilon^{-2.5} \right)$ to produce an $\varepsilon$-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is $\widetilde{\mathcal{O}}\left(\varepsilon^{-3} \right)$ to produce a near $\varepsilon$-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.

We study independent long-range percolation on $\mathbb{Z}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\|x-y\|>1$ are connected by an edge is proportional to $\frac{\beta}{\|x-y\|^s}$, where $\beta>0$ and $s> 0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\geq 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the probability that the origin and $ne_1$ are connected by a path of length $k$ in the critical case $s=2d$. We use these results to determine the Hausdorff dimension of the critical long-range percolation metric that was recently constructed by Ding, Fan, and Huang [14].

In 2018, the longest vector problem (LVP) and the closest vector problem (CVP) in $p$-adic lattices were introduced. These problems are closely linked to the orthogonalization process. In this paper, we first prove that every $p$-adic lattice has an orthogonal basis and give definition to the successive maxima and the escape distance, as the $p$-adic analogues of the successive minima and the covering radius in Euclidean lattices. Then, we present deterministic polynomial time algorithms to perform the orthogonalization process, solve the LVP and solve the CVP with an orthogonal basis of the whole vector space. Finally, we conclude that orthogonalization and the CVP are polynomially equivalent.

We confirm a conjecture of Fox, Pach, and Suk, that for every $d>0$, there exists $c>0$ such that every $n$-vertex graph of VC-dimension at most $d$ has a clique or stable set of size at least $n^c$. This implies that, in the language of model theory, every graph definable in NIP structures has a clique or anti-clique of polynomial size, settling a conjecture of Chernikov, Starchenko, and Thomas. Our result also implies that every two-colourable tournament satisfies the tournament version of the Erdős-Hajnal conjecture, which completes the verification of the conjecture for six-vertex tournaments. The result extends to uniform hypergraphs of bounded VC-dimension as well. The proof method uses the ultra-strong regularity lemma for graphs of bounded VC-dimension proved by Lovász and Szegedy and the method of iterative sparsification introduced by the authors in an earlier paper.

In this paper, we simulate diffusion bridges by using an approximation of the Wiener-chaos expansion (WCE), or a Fourier-Hermite expansion, for a related diffusion process. Indeed, we consider the solution of stochastic differential equations, and we apply the WCE to a particular representation of the diffusion bridge. Thus, we obtain a method to simulate the proposal diffusion bridges that is fast and that in every attempt constructs a diffusion bridge, which means there are no rejection rates. The method presented in this work could be very useful in statistical inference. We validate the method with a simple Ornstein-Uhlenbeck process. We apply our method to three examples of SDEs and show the numerical results.

In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations, wherein physically admissible solutions can be discerned. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. This article focuses on small $BV$ solutions to the barotropic Euler equation in one spatial dimension. Our investigation demonstrates that these solutions are inviscid limits for solutions to the associated compressible Navier-Stokes equation. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values.

We give a bound on the number of weighted real forms of a complex variety with finite automorphism group, where the weight is the inverse of the number of automorphisms of the real form. We give another bound involving the Sylow 2-subgroup and as an application we give bounds on real forms of plane curves.

We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze-Weinstein, integral $p$-adic Hodge theory, and an integral refinement of a theorem of Tate on the Galois cohomology of non-archimedean fields.

Storing digital data in synthetic DNA faces challenges in ensuring data reliability in the presence of edit errors--deletions, insertions, and substitutions--that occur randomly during various stages of the storage process. Current limitations in DNA synthesis technology also impose the use of short DNA sequences, highlighting the particular need for short edit-correcting codes. Motivated by these factors, we introduce a systematic code designed to correct random edits while adhering to typical length constraints in DNA storage. We evaluate its performance both theoretically and through simulations, and assess its integration within a DNA storage framework, revealing promising results.

We study the approximation by a semi-discrete finite-volume scheme of the Gross-Pitaevskii equation with time-dependent potential in two dimensions, performing a two-point flux approximation scheme in space. We rigorously analyze the error bounds relying on discrete uniform Sobolev inequalities. We finally perform some numerical simulations to investigate convergence error.

Let $G$ be a $p$-adic analytic pro-$p$ group of dimension $d$. We produce an approximate series which descends regularly in strata and whose terms deviate from the lower $p$-series in a uniformly bounded way. This brings to light a new set of rational invariants, canonically associated to $G$, that yield the aforementioned uniform bound and that restrict the possible values for the Hausdorff dimensions of closed subgroups of $G$ with respect to the lower $p$-series. In particular, the Hausdorff spectrum of $G$ with respect to the lower $p$-series is discrete and consists of at most $2^d$ rational numbers.

A graph $H$ is said to be an induced minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. Equivalently, $H$ is an induced minor of $G$ if there exists an induced minor model of $H$ in $G$, that is, a collection of pairwise disjoint subsets of vertices of $G$ labeled by the vertices of $H$, each inducing a connected subgraph in $G$, such that two vertices of $H$ are adjacent if and only if there is an edge in $G$ between the corresponding subsets. In this paper, we investigate structural properties of induced minor models, including bounds on treewidth and chromatic number of the subgraphs induced by minimal induced minor models. It is known that for some graphs $H$, testing whether a given graph $G$ contains $H$ as an induced minor is an NP-complete problem. Nevertheless, as algorithmic applications of our structural results, we make use of recent developments regarding tree-independence number to show that if $H$ is the $4$-wheel, the $5$-vertex complete graph minus an edge, or a complete bipartite graph $K_{2,q}$, then there is a polynomial-time algorithm to find in a given graph $G$ an induced minor model of $H$ in $G$, if there is one. We also develop an alternative polynomial-time algorithm for recognizing graphs that do not contain $K_{2,3}$ as an induced minor, which revolves around the idea of detecting the induced subgraphs whose presence is forced when the input graph contains $K_{2,3}$ as an induced minor, using the so-called shortest path detector. It turns out that all these induced subgraphs are Truemper configurations.

We introduce a formalism of infinite, linearly ordered products in general groups. Using this, we define infinite compositions in certain groups of formal power series such as transseries. We show that such groups can sometimes be represented as infinite, linearly ordered, semidirect products of ordered Abelian groups.

This paper studies a variant of Horrocks-type criteria for vector bundles mainly through a syzygy theoretic approach. Starting with explaining various proofs of the splitting criteria for ACM and Buchsbaum bundles, we are going to give new sights of an investigation of quasi-Buchsbaum bundles. Not only our result characterizes the null-correlation bundle on ${\mathbb P}^n$, but also our approach classifies quasi-Buchsbaum bundles on ${\mathbb P}^3$ with simple cohomologies in terms of standard system of parameters and the corresponding skew-symmetric matrices towards a study of the structure of quasi-Buchsbaum modules.

We study the Milnor fiber boundary for hyperplane arrangements in $\mathbb{C}^3$. This is one of the examples of non-isolated surface singularities, which are studied by Némethi--Szilárd. In this paper, we compute the first homology group of the Milnor fiber boundary for a generic arrangement, which gives an affirmative answer to the conjecture of Suciu. Also, we give an example of an arrangement with $n$ hyperplanes, whose torsion part in the Milnor fiber boundary homology contains a direct summand other than $\mathbb{Z}_{n}$, for certain value of $n$.

We consider a loosely coupled, non-iterative Robin-Robin coupling method proposed and analyzed in [J. Numer. Math., 31(1):59--77, 2023] for a parabolic-parabolic interface problem and prove estimates for the discrete time derivatives of the scalar field in different norms. When the interface is flat and perpendicular to two of the edges of the domain we prove error estimates in the $H^2$-norm. Such estimates are key ingredients to analyze a defect correction method for the parabolic-parabolic interface problem. Numerical results are shown to support our findings.

We prove global well-posedness for the cubic nonlinear Schrödinger equation with nonlinearity concentrated on a homogeneous Poisson process.

This paper revisits the set membership identification for linear control systems and establishes its convergence rates under relaxed assumptions on (i) the persistent excitation requirement and (ii) the system disturbances. In particular, instead of assuming persistent excitation exactly, this paper adopts the block-martingale small-ball condition enabled by randomly perturbed control policies to establish the convergence rates of SME with high probability. Further, we relax the assumptions on the shape of the bounded disturbance set and the boundary-visiting condition. Our convergence rates hold for disturbances bounded by general convex sets, which bridges the gap between the previous convergence analysis for general convex sets and the existing convergence rate analysis for $\ell_\infty$ balls. Further, we validate our convergence rates by several numerical experiments. This manuscript contains supplementary content in the Appendix.

We conjecture results about the moments of mixed derivatives of the Riemann zeta function, evaluated at the non-trivial zeros of the Riemann zeta function. We do this in two different ways, both giving us the same conjecture. In the first, we find asymptotics for the moments of derivatives of the characteristic polynomials of matrices in the Circular Unitary Ensemble. In the second, we consider the hybrid model approach first proposed by Gonek, Hughes and Keating.

It is shown that if $A$ is a Borel subset of the first Heisenberg group contained in either a vertical subgroup or a horizontal plane, then vertical projections almost surely do not decrease the Hausdorff dimension of $A$, with respect to the Korányi metric. For general Borel sets, it is shown that if $\dim A >2 $, then vertical projections of $A$ almost surely have dimension at least $(4+\dim A)/3$. This improves the known bound in the general case when $2 < \dim A < 13/5$. The horizontal case and the general case both rely on a variable coefficient local smoothing inequality.

This paper solves the first problem of the open problems in topos theory posted by William Lawvere, which asks the existence of a Grothendieck topos that has proper class many quotient topoi. This paper concretely constructs such Grothendieck topoi, including the presheaf topos of the free monoid generated by countably infinite elements $\mathbf{PSh}(M_\omega)$. Utilizing the combinatorics of the classifying topos of the theory of inhabited objects and considering pairing functions, the problem is reduced to a theorem of Vopěnka, Pultr, and Hedrl\'{ı}n, which states that any set admits a rigid relational structure.

For any $2 \le n < \omega$, we introduce a forcing poset using generalized promises which adds a normal $n$-splitting subtree to a $(\ge \! n)$-splitting normal Aronszajn tree. Using this forcing poset, we prove several consistency results concerning finitely splitting subtrees of Aronszajn trees. For example, it is consistent that there exists an infinitely splitting Suslin tree whose topological square is not Lindelöf, which solves an open problem due to Marun. For any $2 < n < \omega$, it is consistent that every $(\ge \! n)$-splitting normal Aronszajn tree contains a normal $n$-splitting subtree, but there exists a normal infinitely splitting Aronszajn tree which contains no $(< \! n)$-splitting subtree. To show the latter consistency result, we prove a forcing iteration preservation theorem related to not adding new small-splitting subtrees of Aronszajn trees.

A dual version of a conjecture by Woodall asserts that, in a planar digraph, the length of a shortest dicycle equals the maximum number of pairwise disjoint feedback arc sets. We verify this conjecture for the case where the underlying graph is a 3-tree or a partial 3-tree with girth $3$. Additionally, we show that every 3-tree has a feedback arc set of size at most~$m/3-1$, where~$m$ is the number of arcs of the digraph, and this bound is tight. We further establish an upper bound on the size of a minimum feedback arc set in $k$-trees. Finally, we discuss some open problems and conjectures.

We give a list of $113$ holomorphic eta-quotients of integral weight ($66$ of which are primitive) and provide a uniform closed formula for their Fourier coefficients $c(l)$ where $l\equiv1\bmod{m}$ with some fixed $m\mid24$. The proof involves Wohlfahrt's extension of Hecke operators and a dimension formula for spaces of modular forms of general multiplier system. We further provide the expansions of these eta-quotients as linear combinations of standard Eisenstein series.

In this paper we consider a colouring version of the general position problem. The \emph{$\gp $-chromatic number} is the smallest number of colours needed to colour the vertices of the graph such that each colour class has the no-three-in-line property. We determine bounds on this colouring number in terms of the diameter, general position number, size, chromatic number, cochromatic number and total domination number and prove realisation results. We also determine the $\gp $-chromatic number of several graph classes, including Kneser graphs $K(n,2)$, line graphs of complete graphs, complete multipartite graphs, block graphs and Cartesian products. Finally, we show that the $\gp $-colouring problem is NP-complete.

We study the computation of the rate-distortion-perception function (RDPF) for discrete memoryless sources subject to a single-letter average distortion constraint and a perception constraint belonging to the family of $f$-divergences. In this setting, the RDPF forms a convex programming problem for which we characterize optimal parametric solutions. We employ the developed solutions in an alternating minimization scheme, namely Optimal Alternating Minimization (OAM), for which we provide convergence guarantees. Nevertheless, the OAM scheme does not lead to a direct implementation of a generalized Blahut-Arimoto (BA) type of algorithm due to implicit equations in the iteration's structure. To overcome this difficulty, we propose two alternative minimization approaches whose applicability depends on the smoothness of the used perception metric: a Newton-based Alternating Minimization (NAM) scheme, relying on Newton's root-finding method for the approximation of the optimal solution of the iteration, and a Relaxed Alternating Minimization (RAM) scheme, based on relaxing the OAM iterates. We show, by deriving necessary and sufficient conditions, that both schemes guarantee convergence to a globally optimal solution. We also provide sufficient conditions on the distortion and perception constraints, which guarantee that the proposed algorithms converge exponentially fast in the number of iteration steps. We corroborate our theoretical results with numerical simulations and establish connections with existing results.

In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset with its spectrum, which is a compact T_1 topological space. In this paper, we focus on the case where such finite sets have a graph structure and the relations belong to a given graph category. We relate topological properties of the spectrum to combinatorial properties of the graph categories involved. We then utilise this to exhibit elementary combinatorial constructions of well-known continua as Fraïssé limits of finite graphs in categories with relational morphisms.

For any commutative finite flat group scheme, Grothendieck constructed an embedding into some smooth group scheme. This embedding is called the Grothendieck resolution. Let $p$ be a prime number and $n$ a positive integer. In connection with the normal basis problems in the framework of group schemes proposed by Suwa and the author, we consider the Grothendieck resolution for a certain finite flat commutative group scheme of order $p^{n}$ over an $\Bbb{F}_{p}$-algebra.

In this paper, we introduce a spatial model for dormancy in random environment via a two-type branching random walk in continuous-time, where individuals can switch between dormant and active states through spontaneous switching independent of the random environment. However, the branching mechanism is governed by a random environment which dictates the branching rates. We consider three specific choices for random environments composed of particles: (1) a Bernoulli field of immobile particles, (2) one moving particle, and (3) a Poisson field of moving particles. In each case, the particles of the random environment can either be interpreted as catalysts, accelerating the branching mechanism, or as traps, aiming to kill the individuals. The different between active and dormant individuals is defined in such a way that dormant individuals are protected from being trapped, but do not participate in migration or branching. We quantify the influence of dormancy on the growth resp. survival of the population by identifying the large-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the parabolic Anderson model via the Feynman- Kac formula. Especially, the quantitative investigation of the role of dormancy is done by extending the Parabolic Anderson model to a two-type random walk.

There is a well studied notion of GIT-stability for coherent systems over curves, which depends on a real parameter $\alpha$. For generated coherent systems, there is a further notion of stability derived from Mumford's definition of linear stability for varieties in projective space. Let $\alpha_S$ be close to zero and $\alpha_L \gg 0$. We show that a generated coherent system which is $\alpha_S$-stable and linearly stable is $\alpha_L$-stable, and give examples showing that without further assumptions, there are no other implications between these three types of stability. We observe that several of the systems constructed have stable dual span bundle, including systems which are not $\alpha$-semistable for any value of $\alpha$. We use this to prove a case of Butler's conjecture for systems of type $(2, d, 5)$.

We study conjugacy of formal derivations on fields of generalised power series in characteristic 0. Casting the problem of Poincaré resonance in terms of asymptotic differential algebra, we give conditions for conjugacy of parabolic flat log-exp transseries, flat grid-based transseries, logarithmic transseries, power series with exponents and coefficients in an ordered field, and formal Puiseux series.

The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are applied to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety of skew braces to each of the three above-mentioned subvarieties. The corresponding central extensions of skew braces are characterized in purely algebraic terms, leading to some new results, such as an explicit Stallings-Stammbach exact sequence associated with any exact sequence of skew braces, and a new result concerning central series.

The bipartite matching problem is widely applied in the field of transportation; e.g., to find optimal matches between supply and demand over time and space. Recent efforts have been made on developing analytical formulas to estimate the expected matching distance in bipartite matching with randomly distributed vertices in two- or higher-dimensional spaces, but no accurate formulas currently exist for one-dimensional problems. This paper presents a set of closed-form formulas, without curve-fitting, that can provide accurate average distance estimates for one-dimensional random bipartite matching problems (RBMP). We first focus on a lattice case and propose a new method that relates the corresponding matching distance to the area size between a random walk path and the x-axis. This result directly leads to a straightforward closed-form formula for balanced RBMPs. For unbalanced RBMPs on a lattice, we first analyze the properties of an unbalanced random walk that can be related to balanced RBPMs after optimally removing a subset of unmatched points, and then derive a set of approximate formulas. Additionally, we build upon an optimal point removal strategy to derive a set of recursive formulas that can provide more accurate estimates. Then, we extend the results to three problem variants, including RBMPs with periodic boundaries, uniformly distributed points, and arbitrary-length line. Last, we shift our focus to regular networks, and use the one-dimensional results as building blocks to derive RBMP formulas. To verify the accuracy of the proposed formulas, a set of Monte-Carlo simulations are generated for a variety of matching problems settings. Results indicate that our proposed formulas provide quite accurate distance estimations for one-dimensional line segments and networks under a variety of conditions.

Using the group structure of the state space of $q-$state models, a new definition of contour for long-range spin-systems in $\Z^d$ ($d\geq 2$), and a multidimensional version of Fröhlich-Spencer contours, we prove phase transition for a class of ferromagnetic long-range systems which includes the Clock and Potts models. Our arguments work for the entire region of exponents of regular power-law interactions, namely $\alpha > d$, and for any $q \geq 2$. As an application, we prove phase transition for Potts models with decaying fields when the field decays fast enough and in the presence of a random external field.

A set of integers $A$ is non-averaging if there is no element $a$ in $A$ which can be written as an average of a subset of $A$ not containing $a$. We show that the largest non-averaging subset of $\{1, \ldots, n\}$ has size $n^{1/4+o(1)}$, thus solving the Erdős-Straus problem. We also determine the largest size of a non-averaging set in a $d$-dimensional box for any fixed $d$. Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.

Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable methodologies, for such a significant portion of real systems, have been historically scarce. Around the year 2000, the backstepping approach to PDE control began to offer not only a less abstract alternative to PDE control techniques replicating optimal and spectrum assignment techniques of the 1960s, but also enabled the methodologies of adaptive and nonlinear control, matured in the 1980s and 1990s, to be extended from ODEs to PDEs, allowing feedback synthesis for physical and engineering systems that are uncertain, nonlinear, and infinite-dimensional. The PDE backstepping literature has grown in its nearly a quarter century of development to many hundreds of papers and nearly a dozen books. This survey aims to facilitate the entry, for a new researcher, into this thriving area of overwhelming size and topical diversity. Designs of controllers and observers, for parabolic, hyperbolic, and other classes of PDEs, in one and more dimensions (in box and spherical geometries), with nonlinear, adaptive, sampled-data, and event-triggered extensions, are covered in the survey. The lifeblood of control are technology and physics. The survey places a particular emphasis on applications that have motivated the development of the theory and which have benefited from the theory and designs: applications involving flows, flexible structures, materials, thermal and chemically reacting dynamics, energy (from oil drilling to batteries and magnetic confinement fusions), and vehicles.

In this paper we study the derivation of nonlinear bending models for prestrained elastic plates from three-dimensional non-linear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the $\Gamma$-limits as the thickness $h\ll1$ and the size of the material microstructure $\e\ll1$ vanish. In the regime $\e\ll h$ we show that the consecutive and simultaneous limit are equivalent, and also analyze the rate of convergence. In contrast, we observe that there are several different limit models in the case $h\ll \e$.

We provide a complete classification, in the language of weak-combinatorics, of minimal plus-one generated line arrangements in the complex projective plane with double and triple intersection points.

In this paper we study semi-discrete and fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation with a logarithmic potential. Specifically we consider linear finite elements discretising space and backward Euler time discretisation. Our analysis relies on a specific geometric assumption on the evolution of the surface. Our main results are $L^2_{H^1}$ error bounds for both the semi-discrete and fully discrete schemes, and we provide some numerical results.

It is conjectured since long that each smooth convex body $\mathbf{P}\subset \mathbb{R}^n$ has a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $\mathbf{P}$. The conjecture is proven for $n=2,3,4$. We treat the same problem for convex polytopes in $\mathbb{R}^3$ and prove that each generic polytope has a point in its interior with at least $10$ normals to the boundary. This bound is exact: there exists a tetrahedron with no more than $10$ normals emanating from a point in its interior. The proof is based on piecewise linear analog of Morse theory, analysis of bifurcations, and some combinatorial tricks.

We consider a finite group $G$ acting on a manifold $M$. For any equivariant Morse function, which is a generic condition, there does not always exist an equivariant metric $g$ on $M$ such that the pair $(f,g)$ is Morse-Smale. Here, the pair $(f,g)$ is called Morse-Smale if the descending and ascending manifolds intersect transversely. The best possible metrics $g$ are those that make the pair $(f,g)$ stably Morse-Smale. A diffeomorphism $\phi: M \to M$ is a reflection, if $\phi^2 = \operatorname{id}$ and the fixed point set of $\phi$ forms a codimension-one submanifold (with $M \setminus M^{\operatorname{fix}}$ not necessarily disconnected). In this note, we focus on the special case where the group $G = \{\operatorname{id}, \phi\}$. We show that the condition of being stably Morse-Smale is generic for metrics $g$. Given a stably Morse-Smale pair, we introduce a canonical equivariant Thom-Smale-Witten complex by counting certain broken trajectories. This has applications to the case when we have a manifold with boundary and when the Morse function has critical points on the boundary. We provide an alternative definition of the Thom-Smale-Witten complexes, which are quasi-isomorphic to those defined by Kronheimer and Mrowka. We also explore the case when $G$ is generated by multiple reflections. As an example, we compute the Thom-Smale-Witten complex of an upright higher-genus surface by counting broken trajectories.

We study the maximal correlation coefficient $R(X,Y)$ between two stochastic processes $X$ and $Y$. In the case when $(X,Y)$ is a random walk, we find $R(X,Y)$ using the Csáki-Fischer identity and the lower semicontinuity of the map $\text{Law}(X,Y) \to R(X,Y)$. When $(X,Y)$ is a two-dimensional Lévy process, we express $R(X,Y)$ in terms of the Lévy measure of the process and the covariance matrix of the diffusion part of the process. Consequently, for a two-dimensional $\alpha$-stable random vector $(X,Y)$ with $0<\alpha<2$, we express $R(X,Y)$ in terms of $\alpha$ and the spectral measure $\tau$ of the $\alpha$-stable distribution. We also establish analogs and extensions of the Dembo-Kagan-Shepp-Yu inequality and the Madiman-Barron inequality.

This paper investigates the construction and analysis of permutation codes under the Chebyshev distance. Direct product group permutation (DPGP) codes, independently introduced by Kløve et al. and Tamo et al., represent the best-known class of permutation codes in terms of both size and minimum distance, while also allowing for algebraic and efficient encoding and decoding. In contrast, this study focuses on recursively extended permutation (REP) codes, proposed by Kløve et al. as a recursive alternative. We analyze the properties of REP codes and prove that, despite their distinct construction principles, optimal REP codes achieve exactly the same size and minimum distance as the best DPGP codes under the Chebyshev metric. This surprising equivalence uncovers a deep connection between two structurally dissimilar code families and establishes REP codes as a structurally flexible yet equally powerful alternative to DPGP codes. In addition, we present efficient encoding and decoding algorithms for REP codes, including a sequential encoder with $O(n \log n)$ complexity and a bounded-distance decoder with $O(n \log^2 n)$ complexity.

A Polish group $G$ has the generic point property if any minimal $G$-flow admits a comeager orbit, or equivalently if the universal minimal flow (UMF) does. The class $\mathsf{GPP}$ of such Polish groups is a proper extension of the class $\sf{PCMD}$ of Polish groups with metrizable UMF. Motivated by analogous results for $\mathsf{PCMD}$, we define and explore a robust generalization of $\sf{GPP}$ which makes sense for all topological groups, thus defining the class $\mathsf{TMD}$ of topological groups with tractable minimal dynamics. These characterizations yield novel results even for $\mathsf{GPP}$; for instance, a Polish group is in $\mathsf{GPP}$ iff its UMF has no points of first countability. Motivated by work of Kechris, Pestov, and Todorčević that connects topological dynamics and structural Ramsey theory, we state and prove an abstract KPT correspondence which characterizes the class $\mathsf{TMD}$ and shows that $\mathsf{TMD}$ is $\Delta_1$ in the Lévy hierarchy. We then develop set-theoretic methods which allow us to apply forcing and absoluteness arguments to generalize numerous results about $\mathsf{GPP}$ to all of $\mathsf{TMD}$. We also apply these new set-theoretic methods to first generalize parts of Glasner's structure theorem for minimal, metrizable tame flows to the non-metrizable setting, and then to prove the revised Newelski conjecture regarding definable NIP groups. We conclude by discussing some tantalizing connections between definable NIP groups and $\mathsf{TMD}$ groups.

Using the \texttt{WeylModules} \textsf{GAP} Package, we compute structural information about certain Weyl modules for type $G_2$ in characteristic $2$. This gives counterexamples to two conjectures stated by S.~Donkin in 1990. It also illustrates capabilities of the package, which can in principle be applied to Weyl modules for any simple, simply-connected algebraic group in any characteristic, subject of course to time and space limitations of computational resources.

We design and investigate a variety of multigrid solvers for high-order local discontinuous Galerkin methods applied to elliptic interface and multiphase Stokes problems. Using the template of a standard multigrid V-cycle, we consider a variety of element-wise block smoothers, including Jacobi, multi-coloured Gauss-Seidel, processor-block Gauss-Seidel, and with special interest, smoothers based on sparse approximate inverse (SAI) methods. In particular, we develop SAI methods that: (i) balance the smoothing of velocity and pressure variables in Stokes problems; and (ii) robustly handles high-contrast viscosity coefficients in multiphase problems. Across a broad range of two- and three-dimensional test cases, including Poisson, elliptic interface, steady-state Stokes, and unsteady Stokes problems, we examine a multitude of multigrid smoother and solver combinations. In every case, there is at least one approach that matches the performance of classical geometric multigrid algorithms, e.g., 4 to 8 iterations reduce the residual by 10 orders of magnitude. We also discuss their relative merits with regard to simplicity, robustness, computational cost, and parallelisation.

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify ''genus zero pairs'' of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.

Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic twists over $K$, the only known counterexamples (necessarily of dimension $\geq 4$) are not geometrically simple. We show that, for every prime $p\equiv 13 \pmod{24}$, there exists a pair of geometrically simple abelian varieties of dimension $p-1$ over $\mathbb{Q}$ that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory.

In $1941$ Dushnik and Miller introduced the concept of dimension of a poset. In $2020$ Bhavale and Waphare introduced the concept of an RC-lattice as a lattice in which all the reducible elements are lying on a chain. In this paper, we obtain the dimension of adjunct sum of two lattices. We obtain a bound on the dimension of a dismantlable lattice in terms of its nullity. We also prove that the dimension of an RC-lattice on $n$ elements is at the most three. Consequently, we prove that an RC-lattice is non planar if and only if its dimension is three.

This article shows that the capacity region of a 2-users weak Gaussian interference channel is achieved using Gaussian code-books. The approach relies on traversing the boundary in incremental steps. Starting from a corner point with Gaussian code-books, and relying on calculus of variation, it is shown that the end point in each step is achieved using Gaussian code-books. Optimality of Gaussian code-books is first established by limiting the random coding to independent and identically distributed scalar (single-letter) samples. Then, it is shown that the optimum solution for vector inputs coincides with the single-letter case. It is also shown that the maximum number of phases needed to realize the gain due to power allocation over time is two. It is also established that the solution to the Han-Kobayashi achievable rate region, with single letter Gaussian random code-books, achieves the optimum boundary.

In this article, we study the higher-power moments of restricted divisor functions. In order to establish our main results, we prove a more general result pertaining to the distribution of solutions to certain multiplicative Diophantine equations.

We continue the study of the geometry and topology of compact submanifolds of arbitrary codimension in space forms that satisfy a pinching condition involving the length of the second fundamental form and the mean curvature. Our primary focus is on four-dimensional submanifolds, where both the results obtained and the methods employed differ substantially and are considerably more intricate than in higher dimensions. This study relies critically on concepts from four-dimensional geometry, the theory of Riemannian manifolds with nonnegative isotropic curvature, and the Bochner technique, each playing an essential role. The results are sharp and extend previous work by several authors, without imposing additional assumptions on either the mean curvature or the fundamental group of the submanifold.

Many real-world problems, such as those with fairness constraints, involve complex expectation constraints and large datasets, necessitating the design of efficient stochastic methods to solve them. Most existing research focuses on cases with no {constraint} or easy-to-project constraints or deterministic constraints. In this paper, we consider nonconvex nonsmooth stochastic optimization problems with expectation constraints, for which we build a novel exact penalty model. We first show the relationship between the penalty model and the original problem. Then on solving the penalty problem, we present a single-loop SPIDER-type stochastic subgradient method, which utilizes the subgradients of both the objective and constraint functions, as well as the constraint function value at each iteration. Under certain regularity conditions (weaker than Slater-type constraint qualification or strong feasibility assumed in existing works), we establish an iteration complexity result of $O(\epsilon^{-4})$ to reach a near-$\epsilon$ stationary point of the penalized problem in expectation, matching the lower bound for such tasks. Building on the exact penalization, an $(\epsilon,\epsilon)$-KKT point of the original problem is obtained. For a few scenarios, our complexity of either the {objective} sample subgradient or the constraint sample function values can be lower than the state-of-the-art results by a factor of $\epsilon^{-2}$. Moreover, on solving two fairness-constrained problems and a multi-class Neyman-Pearson classification problem, our method is significantly (up to 466 times) faster than the state-of-the-art algorithms, including switching subgradient method and inexact proximal point methods.

Let $\mathrm{Mp}(2n)$ be the metaplectic group over a local field $F \supset \mathbb{Q}_p$ defined by an additive character of $F$ of conductor $4\mathfrak{o}_F$. Gan-Savin ($p \neq 2$) and Takeda-Wood ($p=2$) obtained an equivalence between the Bernstein block of $\mathrm{Mp}(2n)$ containing the even (resp. odd) Weil representation and the Iwahori-spherical block of the split $\mathrm{SO}(2n+1)$ (resp. its non-split inner form), by giving an isomorphism between Hecke algebras. We revisit this equivalence from an endoscopic perspective. It turns out that the L-parameters of irreducible representations are preserved, whilst the difference between characters of component groups is governed by symplectic local root numbers.

We give an explicit characterization of the solid rings of the fourth type, finishing a work by Bousfield and Kan in the 1970s. As a consequence of our methods, we give criteria to find the core of a ring and explicitly compute the core of some rings.

Reconfigurable intelligent surfaces (RIS) is regarded as a key enabler of wave/analog-domain beamforming, processing, and computing in future wireless communication systems. Recently, Beyond Diagonal RIS (BD-RIS) has been proposed as a generalization of conventional RIS, offering enhanced design flexibility thanks to the presence of tunable impedances that connect RIS elements. However, increased interconnections lead to high circuit complexity, which poses a significant practical challenge. In this paper, we address the fundamental open question: What is the class of BD-RIS architectures that achieves the optimal performance in a RIS-aided multiuser multi-input multi-output (MIMO) system? By modeling BD-RIS architectures using graph theory, we identify a class of BD-RIS architectures that achieves the optimal performance--matching that of fully-connected RIS--while maintaining low circuit complexity. Our result holds for a broad class of performance metrics, including the commonly used sum channel gain/sum-rate/energy efficiency maximization, transmit power minimization, and the information-theoretic capacity region. The number of tunable impedances in the proposed class is ${O}(N\min\{D,N/2\})$, where $N$ denotes the number of RIS elements and $D$ is the degree of freedom of the multiuser MIMO channel, i.e., the minimum between the number of transmit antennas and the total number of received antennas across all users. Since $D$ is much smaller than $N$ in practice, the complexity scales as ${O}(ND)$, which is substantially lower than the ${O}(N^2)$ complexity of fully-connected RIS. We further introduce two novel BD-RIS architectures--band-connected RIS and stem-connected RIS--and show that they belong to the optimal architecture class under certain conditions. Simulation results validate the optimality and enhanced performance-complexity tradeoff of our proposed architectures.

We give a sufficient condition for the order function of a conditionally convergent ordinary Dirichlet series to be linear when it is non-trivial. This is in terms of a ``negative order'' generalisation of the abscissae of summability introduced by Bohr. It is shown that the condition is necessary in the presence of subconvexity and a functional equation, and we discuss the broader case.

We consider a $2$-dimensional autonomous system subject to a $1$-periodic perturbation, i.e. $$ \dot{\vec{x}}=\vec{f}(\vec{x})+\epsilon\vec{g}(t,\vec{x},\epsilon),\quad \vec{x}\in\Omega .$$ We assume that for $\epsilon=0$ there is a trajectory $\vec{\gamma}(t)$ homoclinic to the origin which is a critical point: in this context Melnikov theory provides a sufficient condition for the insurgence of a chaotic pattern when $\epsilon \ne 0$. In this paper we show that for any line $\Xi$ transversal to $\{\vec{\gamma}(t) \mid t \in \mathbb{R} \}$ and any $\tau \in [0,1]$ we can find a set $\Sigma^+(\Xi,\tau)$ of initial conditions giving rise to a pattern chaotic just in the future, located in $\Xi$ at $t=\tau$. Further diam$(\Sigma^+(\Xi,\tau)) \le \epsilon^{(1+\nu)/ \underline{\sigma}}$ where $\underline{\sigma}>0$ is a constant and $\nu>0$ is a parameter that can be chosen as large as we wish. The same result holds true for the set $\Sigma^-(\Xi,\tau)$ of initial conditions giving rise to a pattern chaotic just in the past. In fact all the results are developed in a piecewise-smooth context, assuming that $\vec{0}$ lies on the discontinuity curve $\Omega^0$: we recall that in this setting chaos is not possible if we have sliding phenomena close to the origin. This paper can also be considered as the first part of the project to show the existence of classical chaotic phenomena when sliding close to the origin is not present.

In this paper, we provide a comprehensive analysis of involutive quantales, with a particular focus on quantic frames. We extend the axiomatic foundations of quantale-enriched topological spaces to include closure under the anti-homomorphic involution, facilitating a balanced topologization of the spectrum of unital $C^*$-algebras that encompasses both closed right and left ideals through the concept of quantic frames. Specifically, certain subspaces of pure states are identified as strongly Hausdorff separated quantale-enriched involutive topological spaces.

We find necessary and sufficient conditions for high-order persistence of resonant caustics in perturbed circular billiards. The main tool is a perturbation theory based on the Bialy-Mironov generating function for convex billiards. All resonant caustics with period $q$ persist up to order $\lceil q/n \rceil -1$ under any polynomial deformation of the circle of degree $n$.

The use of non-orthogonal signals has several benefits over orthogonal signals in multi-coded communications. We provide a novel, theoretical study of non-orthogonal signaling to expand the applicability of these schemes. Motivated by a class of multi-carrier spread spectrum systems, this paper presents a thorough symbol error rate analysis of the broad class of multi-code signaling methods when they make use of codes which are not necessarily orthogonal. Our analysis is also extended to the case where the code set includes the negative of each code vector, i.e., an extension to biorthogonal signaling. Moreover, it is shown that the symbol error rate results derived in this paper reduce to those available in the literature when the multi-codes are orthogonal or have equal correlation between vectors. Additionally, we show how Monte Carlo integration can be used to evaluate the integrals in the error probability calculation and derive low complexity upper bounds on the error probabilities. We show that by combining these techniques, the error probability can be efficiently computed across the full SNR regime. Finally, we use the upper bound of the error probability to develop some analytical insights about the impacts of non-orthogonality among the code vectors on the symbol error probability.

A regular separable first-countable countably compact space is called a Nyikos space. In this paper, we give a partial solution to an old problem of Nyikos by showing that each locally compact Nyikos inverse topological semigroup is compact. Also, we show that a topological semigroup $S$ that contains a dense inverse subsemigroup is a topological inverse semigroup, provided (i) $S$ is compact, or (ii) $S$ is countably compact and sequential. The latter result solves a problem of Banakh and Pastukhova and provides the automatic continuity of inversion in certain compact-like inverse semigroups.

Suppose $\{X_{t}:t\ge 0\}$ is a supercritical superprocess on a Luzin space $E$, with a non-local branching mechanism and probabilities $\mathbb{P}_{\delta_{x}}$, when initiated from a unit mass at $x\in E$. By ``supercritical", we mean that the first moment semigroup of $X_{t}$ exhibits a Perron-Frobenius type behaviour characterized by an eigentriplet $(\lambda_{1},\varphi,\widetilde{\varphi})$, where the principal eigenvalue $\lambda_{1}$ is greater than $0$. Under a second moment condition, we prove that $X_{t}$ satisfies a law of large numbers. The main purpose of this paper is to further investigate the fluctuations of the linear functional $\mathrm{e}^{-\lambda_{1}t}\langle f,X_{t}\rangle$ around the limit given by the law of large numbers. To this end, we introduce a parameter $\epsilon(f)$ for a bounded measurable function $f$, which determines the exponent term of the decay rate for the first moment of the fluctuation. Qualitatively, the second-order behaviour of $\langle f,X_{t}\rangle$ depends on the sign of $\epsilon(f)-\lambda_{1}/2$. We prove that, for a suitable test function $f$, the fluctuation of the associated linear functional exhibits distinct asymptotic behaviours depending on the magnitude of $\epsilon(f)$: If $\epsilon(f)\ge \lambda_{1}/2$, the fluctuation converges in distribution to a Gaussian limit under appropriate normalization; If $\epsilon(f)<\lambda_{1}/2$, the fluctuation converges to an $L^{2}$ limit with a larger normalization factor. In particular, when the test function is chosen as the right eigenfunction $\varphi$, we establish a functional central limit theorem. As an application, we consider a multitype superdiffusion in a bounded domain. For this model, we derive limit theorems for the fluctuations of arbitrary linear functionals.

In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= \lambda w(x)|u|^{s-2}u+\theta B\left(x,u\right) \quad \text{in } \Omega,\quad u= 0 \quad\text{on } \partial \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial\Omega$, $1<p<s<q<N$, $\frac{q}{p}<1+\frac{1}{N}$, $0\leq a(\cdot)\in C^{0,1}(\overline{\Omega})$, $\lambda$, $\theta$ are real parameters, $w$ is a suitable weight and $B\colon \overline{\Omega}\times \mathbb{R}\to\mathbb{R}$ is given by \begin{align*} B(x,t) :=b_0(x)|t|^{p^*-2}t+b(x)|t|^{q^*-2}t, \end{align*} where $r^*:=Nr/(N-r)$ for $r\in\{p,q\}$. Here the right-hand side combines the effect of a critical term given by $B(\cdot,\cdot)$ and a sandwich-type perturbation with exponent $s \in (p,q)$. Under different values of the parameters $\lambda$ and $\theta$, we prove the existence and multiplicity of solutions to the problem above. For this, we mainly exploit different variational methods combined with topological tools, like a new concentration-compactness principle, a suitable truncation argument and the Krasnoselskii's genus theory, by considering very mild assumptions on the data $a(\cdot)$, $b_0(\cdot)$ and $b(\cdot)$.

This paper presents a novel approach for steering the state of a stochastic control-affine system to a desired target within a finite time horizon. Our method leverages the time-reversal of diffusion processes to construct the required feedback control law. Specifically, the control law is the so-called score function associated with the time-reversal of random state trajectories that are initialized at the target state and are simulated backwards in time. A neural network is trained to approximate the score function, enabling applicability to both linear and nonlinear stochastic systems. Numerical experiments demonstrate the effectiveness of the proposed method across several benchmark examples.

Reliability-based design optimization (RBDO) is a methodology for designing systems and components under the consideration of probabilistic uncertainty. In practical engineering, the number of input data is often limited, which can damage the validity of the optimal results obtained by RBDO. Confidence-based design optimization (CBDO) has been proposed to account for the uncertainty of the input distribution. However, this approach faces challenges, computational cost and accuracy when dealing with highly nonlinear performance constraints. In this paper, we consider the compliance minimization problem of truss structures with uncertain external forces. Armed with the advanced risk measure, conditional Value-at-Risk (CVaR), we formulate a bi-objective optimization problem for the worst-case expected value and the worst-case CVaR of compliance, which allows us to account for the tail risk of performance functions not addressed in CBDO. Employing kernel density estimation for estimation of the input distribution allows us to eliminate the need for modeling the input distribution. We show that this problem reduces to a second-order cone programming when assigning either uniform kernel or triangular kernel. Finally, through numerical experiments, we obtain the Pareto front for the bi-objective optimization problem of the worst-case expected value and CVaR of compliance of truss structures, and confirm the changes in the Pareto solutions.

Heterogeneity in federated learning (FL) is a critical and challenging aspect that significantly impacts model performance and convergence. In this paper, we propose a novel framework by formulating heterogeneous FL as a hierarchical optimization problem. This new framework captures both local and global training processes through a bilevel formulation and is capable of the following: (i) addressing client heterogeneity through a personalized learning framework; (ii) capturing the pre-training process on the server side; (iii) updating the global model through nonstandard aggregation; (iv) allowing for nonidentical local steps; and (v) capturing clients' local constraints. We design and analyze an implicit zeroth-order FL method (ZO-HFL), equipped with nonasymptotic convergence guarantees for both the server-agent and the individual client-agents, and asymptotic guarantees for both the server-agent and client-agents in an almost sure sense. Notably, our method does not rely on standard assumptions in heterogeneous FL, such as the bounded gradient dissimilarity condition. We implement our method on image classification tasks and compare with other methods under different heterogeneous settings.

Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since first-order model checking is fixed-parameter tractable when a witness certifying small twin-width is provided. However, the behavior of twin-width in specific graph classes, particularly cubic graphs, remains poorly understood. While cubic graphs are known to have unbounded twin-width, no explicit cubic graph of twin-width greater than 4 is known. This paper explores this phenomenon in regular and near-regular graph classes. We show that extremal graphs of bounded degree and high twin-width are asymmetric, partly explaining their elusiveness. Additionally, we establish bounds for circulant and d-degenerate graphs, and examine strongly regular graphs, which exhibit similar behavior to cubic graphs. Our results include determining the twin-width of Johnson graphs over 2-sets, and cyclic Latin square graphs.

The study of stochastic variational principles involves the problem of constructing fixed-endpoint and adapted variations of semimartingales. We provide a detailed construction of variations of semimartingales that are not only fixed at deterministic endpoints, but also fixed at first entry times and first exit times for charts in a manifold. We prove a stochastic version of the fundamental lemma of calculus of variations in the context of these variations. Using this framework, we provide a generalization of the stochastic Hamilton-Pontryagin principle in local coordinates to arbitrary noise semimartingales. For the corresponding global form of the stochastic Hamilton-Pontryagin principle, we introduce a novel approach to global variational principles by restricting to semimartingales obtained as solutions of Stratonovich equations.

In this article, we study Sobolev homeomorphisms and composition operators on homogeneous Lie groups. We prove that a measurable homeomorphism $\varphi: \Omega \to\widetilde{\Omega}$ belongs to the Sobolev space $L^{1}_{q}(\Omega; \widetilde{\Omega})$, $1\leq q < \infty$, if and only if $\varphi$ generates a bounded composition operator on Sobolev spaces.

We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces emerges naturally in this context by requiring the special set-theoretic Yang-Baxter algebra to be a Hopf algebra and a quasi-triangular bialgebra after twisting. The fundamental representation of the universal R-matrix yields the familiar set-theoretic (combinatorial) solutions of the Yang-Baxter equation. We then apply the same Drinfel'd twist to the gl_n Yangian after introducing the augmented Yangian. We show that the augmented Yangian is also a Hopf algebra and we also obtain its twisted version.

Two pertinent questions for any support theory of a monoidal triangulated category are whether it is functorial and if the tensor product property holds. To this end, we consider the complete prime spectrum of an essentially small monoidal triangulated category, which we show is universal among support data satisfying the tensor product property, even if it is empty. The complete prime spectrum is functorial and parametrizes radical thick tensor-ideals, a noncommutative analogue of Balmer's reconstruction theorem. We give criteria for when induced maps on complete prime spectra are injective or surjective, and determine the complete prime spectrum for crossed product categories. Finally, we determine the universal functorial support theory for monoidal triangulated categories coinciding with the Balmer spectrum on braided monoidal triangulated categories.

Let $G$ a locally compact group and $(\Phi,\Psi)$ be a complementary pair of Young functions. Let $A_\Phi(G)$ be the Orlicz analogue of the classical Figà-Talamanca Herz algebra $A_p(G).$ In this article, we establish a necessary and sufficient condition for the equality $\Lambda(A_\Phi(G)^{\ast\ast}) = A_\Phi(G)$ to hold, where $\Lambda(A_\Phi(G)^{\ast\ast})$ denotes the topological center of the double dual of $A_\Phi(G)$ when equipped with the first Arens product. Furthermore, we prove several results concerning the semi-simplicity of the Banach algebras $A_\Phi(G)^{\ast\ast}$ and $UCB_\Psi(\widehat{G})^\ast.$

We prove that if $A$ is a subset of those primes which are congruent to $1 \pmod{3}$ such that the relative density of $A$ in this residue class is larger than $\frac{1}{2},$ then every sufficiently large odd integer $n$ which satisfies $n \equiv 0 \pmod{3}$ can be written as a sum of three primes from $A.$ Moreover the threshold of $\frac{1}{2}$ for the relative density is best possible.

We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $\mu$-Polyak-Łojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.

Let $G$ be a finite abelian group of order $n$, and for each $a\in G$ and integer $1\le h\le n$ let $\mathcal{F}_a(h)$ denote the family of all $h$-element subsets of $G$ whose sum is $a$. A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families $\mathcal{F}_a(h)$ (as $a$ ranges over $G$) become asymptotically equal as $n\rightarrow \infty$ when $h=\left\lfloor\frac{n}{2}\right\rfloor$. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every $4\leq h\leq \left\lfloor\frac{n}{2}\right\rfloor+1$.

The classical three-peg Tower of Hanoi puzzle admits the well-known closed-form solution \(M(3,n)=2^n-1\), but the general case with \(p \geq 4\) pegs has remained an open problem for over a century. In this paper, we present a complete unified closed-form formula for the minimum number of moves \(M(p,n)\) required to solve the Tower of Hanoi problem for any number of pegs \(p\) and disks \(n\). Building upon our previous work, which introduced Menon's Conjecture \(M(p,n)=4n-2p+1\) for the restricted domain \(p-1 \leq n \leq \frac{p(p-1)}{2}\), we now extend this result to all values of \(n\). Our key insight is that the solution naturally partitions into regimes governed by simplex number boundaries, with each regime exhibiting linear growth with slopes that are successive powers of two. This leads to the unified expression: \[ M(p,n) = 2^{i+1}n - \sum_{k=0}^{i} 2^k \binom{p+k-2}{k}, \quad i = \min \left\{ j \geq 0 : n \leq \binom{p-1+j}{j+1} \right\}. \] We show how this formula recovers classical results, including \(M(3,n)=2^n-1\) and the four-peg Reve's Puzzle solution, as special cases. Computational verification across more than 1,750 test instances confirms perfect agreement with Frame-Stewart values. Furthermore, this closed-form reduces the complexity of determining \(M(p,n)\) from \(O(pn^2)\) to \(O(p\log n)\), providing both theoretical unification and practical computational improvements.

We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $\alpha(\sigma,0,1)$ with $\sigma\in\mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $\gamma=1$ for $\sigma\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $\gamma = 4/3$ for a general $\sigma\in\mathbb{R}$. In the present paper, we obtain the threshold $\gamma=1$ in $H^N(N>13/2)$, hence improving the above results when $\sigma$ is a rational number. The nonlinear inviscid damping for velocity $u^2_{\neq}$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order $\nu^{-1/3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}.

In this paper, we propose a feedback control strategy to protect vulnerable areas from wildfires. We consider a system of coupled partial differential equations (PDEs) that models heat propagation and fuel depletion in wildfires and study two cases. First, when the wind velocity is known, we design a Neumann-type boundary controller guaranteeing that the temperature of some protected region converges exponentially, in the $L^2$ norm, to the ambient temperature. Second, when the wind velocity is unknown, we design an adaptive Neumann-type boundary controller guaranteeing the asymptotic convergence, in the $L^2$ norm, of the temperature of the protected region to the ambient temperature. In both cases, the controller acts along the boundary of the protected region and relies solely on temperature measurements along that boundary. Our results are supported by numerical simulations.

Fast Fourier Transform (FFT) libraries are widely used for evaluating discrete convolutions. Most FFT implementations follow some variant of the Cooley-Tukey framework, in which the transform is decomposed into butterfly operations and index-reversal permutations. While butterfly operations dominate the floating-point operation count, the memory access patterns induced by index-reversal permutations significantly degrade the FFT's arithmetic intensity. When performing discrete convolution, the three sets of index-reversal permutations which occur in FFT-based implementations using Cooley-Tukey frameworks cancel out, thus paving the way to implementations free of any permutation. To the best of our knowledge, such permutation-free variants of FFT-based discrete convolution are not commonly used in practice, making such kernels worth investigating. Here, we look into such permutation-avoiding convolution procedures for multi-dimensional cases within a general radix Cooley-Tukey framework. We perform numerical experiments to benchmark the algorithms presented against state-of-the-art FFT-based convolution implementations. Our results suggest that developers of FFT libraries should consider supporting permutation-avoiding convolution kernels.

This paper investigates the entropy production rate and time-reversibility for general jump diffusions (Lévy processes) on $\mathbb{R}^n$. We first formulate the entropy production rate and explore its associated thermodynamic relations for jump diffusions. Subsequently, we derive the entropy production rate using the relative entropy between the forward and time-reversed path measures for stationary jump diffusions via the Girsanov transform. Furthermore, we establish the equivalence among time-reversibility, zero entropy production rate, detailed balance condition, and the gradient structure for stationary jump diffusions.

Let $S$ be a smooth irreducible curve defined over $\overline{\mathbb{Q}}$, let $\mathcal{A}$ be an abelian scheme over $S$ and $\mathcal{C}$ a curve inside $\mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In this paper we prove that the set of points in which $\mathcal{C}$ intersects proper flat subgroup schemes of $\mathcal{A}$ tangentially is finite. The crucial case of elliptic curves already follows from a result by Corvaja, Demeio, Masser and Zannier: in this case we provide an alternative proof using the Pila-Zannier method. Such a proof may lead to an effective result using an effective point-counting theorem. This fits in the framework of the so-called problems of unlikely intersections, and can be seen as a variation of the relative Pink conjecture for abelian varieties.

Leighton's Graph Covering Theorem states that if two finite graphs have the same universal covering tree, then they also have a common finite degree cover. Bass and Kulkarni gave an alternative proof of this fact using tree lattices. We give an example of two graphs that admit a common finite cover which can not be obtained using tree lattice techniques. If two groups embed as finite index subgroups, we say they are co-commensurable. Our example comes from an explicit commensuration that cannot be induced by a co-commensuration. Next we state and prove a general theorem that gives necessary and sufficient conditions for when a commensuration can be induced by a co-commensuration. The developed machinery is then used to show that normal virtual retracts are virtual direct summands, answering a question of Merladet and Minasyan. In an appendix, applications to commensurating graphs of groups, biautomaticity, and hereditary conjugacy separability are given.

We introduce manifolds with kinks, a class of manifolds with possibly singular boundary that notably contains manifolds with smooth boundary and corners. We derive the asymptotic behavior of the Graph Laplace operator with Gaussian kernel and its deterministic limit on these spaces as bandwidth goes to zero. We show that this asymptotic behavior is determined by the inward sector of the tangent space and, as special cases, we derive its behavior near interior and singular points. Lastly, we show the validity of our theoretical results using numerical simulation.

We prove that solutions to linear kinetic equations in a half-space with absorbing boundary conditions decay for large times like $t^{-\frac{1}{2}-\frac{d}{4}}$ in a weighted $\sfL^{2}$ space and like $t^{-1-\frac{d}{2}}$ in a weighted $\sfL^{\infty}$ space, i.e. faster than in the whole space and in agreement with the decay of solutions to the heat equation in the half-space with Dirichlet conditions. The class of linear kinetic equations considered includes the linear relaxation equation, the kinetic Fokker-Planck equation and the Kolmogorov equation associated with the time-integrated spherical Brownian motion.

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an exceptional character $\chi$ with $L(1,\chi)=o\left((\log D)^{-5}\right)$, for any $q<x^{\frac 23-\varepsilon}$, the asymptotic $$\psi(x,q,a)=\frac{\psi(x)}{\phi(q)}\left(1-\chi\left(\frac{aD}{(D,q)}\right)+o(1)\right)$$ holds for almost all $a$ with $(a,q)=1$. We also find that for any fixed $a$, the above holds for almost all $q<x^{\frac 23-\varepsilon}$ with $(a,q)=1$. Previous prime equidistribution results under the assumption of Siegel zeroes (by Friedlander-Iwaniec and the current author) have found that the above asymptotic holds either for all $a$ and $q$ or on average over a range of $q$ (i.e. for the Elliott-Halberstam conjecture), but only under the assumption that $q<x^{\theta}$ where $\theta=\frac{30}{59}$ or $\frac{16}{31}$, respectively.

In previous work we computed the number $C_n(q)$ of ideals of codimension $n$ of the algebra ${\mathbb{F}}_q[x,y,x^{-1}, y^{-1}]$ of two-variable Laurent polynomials over a finite field: it turned out that $C_n(q)$ is a palindromic polynomial of degree $2n$ in $q$, divisible by $(q-1)^2$. The quotient $P_n(q) = C_n(q)/(q-1)^2$ is a palindromic polynomial of degree $2n-2$. For each $n\geq 1$ let ${\overline{P}}_n(X) \in {\mathbb{Z}}[X]$ be the degree $n-1$ polynomial such that ${\overline{P}}_n(q+q^{-1}) = P_n(q)/q^{n-1}$. In this note we show that for any integer $N$ the integer value ${\overline{P}}_n(N)$ is close to the value at $N$ of the degree $n-1$ polynomial $F_{n-1}(X) = 1 + \sum_{k=1}^{n-1} \, {\overline{T}}_k(X)$, which is a sum of monic versions ${\overline{T}}_k(X)$ of Chebyshev polynomials of the first kind. We give a precise formula for ${\overline{P}}_n(X)$ as a linear combination of $F_k(X)$'s, each appearance of the latter being parametrized by an odd divisor of $n$. As a consequence, ${\overline{P}}_n(X) = F_{n-1}(X)$ if and only if $n$ is a power of $2$. We exhibit similar formulas for $C_n(q)$.

In 1973, Calderón proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem, we proved that a $2 \times 2$ psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calderón's result for $m=2$, and left the sos (sum-of-squares) rank problem of $m \times 2$ biquadratic forms for $m \ge 3$ to further exploration. In this paper, we show that a $3 \times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. For $m \ge 4$, we make a conjecture that an $m \times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms.

In 2003, Benoit Cloitre entered a family of sequences in the OEIS that we call hiccup sequences. We collect the various claims, observations, and proofs of properties of these sequences that have been entered in the OEIS over the years, and present a unified approach, inspired by a remarkable theorem of Bosma, Dekking, and Steiner.

We present a new version of the Friedman-Magidor theorem: for every measurable cardinal $\kappa$ and $\tau\leq\kappa^{++}$, there exists a forcing extension $V\subseteq V[G]$ such that any normal measure $U\in V$ on $\kappa$ has exactly $\tau$ distinct lifts in $V[G]$, and every normal measure on $\kappa$ in $V[G]$ arises as such a lift. This version differs from the original Friedman-Magidor theorem in several notable ways. First, the new technique does not involve forcing over canonical inner models or rely on any fine-structural tools or assumptions, allowing it to be applied in the realm of large cardinals beyond the current reach of the inner model program. Second, in the case where $\tau\leq \kappa^+$, all lifts of a normal measure $U\in V$ on $\kappa$ to $V[G]$ have the same ultrapower. Finally, the technique generalizes to a version of the Friedman-Magidor theorem for extenders. An additional advantage is that the forcing used is notably simple, relying only on nonstationary support product forcing.

This represents a talk given at the International Conference for Basic Science, July 2025. We review the theory of canonical bases of quantum groups and its relation with the theory of total positivity.

Based on the Wronski determinant, we propose the construction of linearly independent orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of the functions under consideration, that satisfies minimal properties. Two applications are considered, including solutions to ordinary differential equations and the construction of basis functions. We also present a conjecture that connects the latter two concepts, which leads to what we call the Wronski basis.

We study the $\mathrm{C}^2$ estimates for $p$-Hessian equations with general left-hand and right-hand terms on closed Riemannian manifolds of dimension $n$. To overcome the constraints of closed manifolds, we advance a new kind of "subsolution", called pseudo-solution, which generalizes "$\mathcal{C}$-subsolution" to some extent and is well-defined for fully general $p$-Hessian equations. Based on pseudo-solutions, we prove the $\mathrm{C}^1$ estimates for general $p$-Hessian equations, and the corresponding second-order estimates when $p\in\{2, n-1, n\}$, under sharp conditions -- we don't impose curvature restrictions, convexity conditions or "MTW condition" on our main results. Some other conclusions related to a priori estimates and different kinds of "subsolutions" are also given, including estimates for "semi-convex" solutions and when there exists a pseudo-solution.

Given a finite group action on a smooth manifold, we study the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? Birman-Hilden and Maclachlan-Harvey proved the answer is "yes" for most surfaces. By contrast, we give a general criterion in higher dimensions under which there are many equivariant diffeomorphisms which are isotopic but not equivariantly isotopic. Examples satisfying this criterion include branched covers of split links and "stabilized" branched covers. We prove the result by constructing an invariant valued in the homology of a certain infinite cover of the manifold. We give applications to outer automorphism groups of free products and to group actions on manifolds which fiber over the circle.

Given $d \in {\bf N}, \lambda >0$, the random connection model in a region $A \subseteq {\bf R}^d$ is a graph with vertex set given by a homogeneous Poisson point process of intensity $\lambda $ in $A$, with an edge placed between each pair $x,y$ of vertices with probability $\phi(\|x-y\|)$, where $\phi: {\bf R}_+ \to [0,1]$ is a nonincreasing finite-range connection function. We show that if $d \geq 3$ and $\lambda$ is strictly supercritical for $A = {\bf R}^d$, then the model remains supercritical if it is restricted to a region $A$ of the form ${\bf R}^2 \times [-K/2,K/2]^{d-2}$, provided $K$ is sufficiently large. This is a continuum analogue of a well-known result of Grimmett and Marstrand for lattice percolation. We prove this by adapting Grimmett and Marstrand's original proof; Faggionato and Hartarsky have also proved this recently by other means.

Given any generically étale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.

In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a generalization of parking functions known as vector parking functions or $\boldsymbol{u}$-parking functions, in which a nonnegative integer capacity is given to each parking spot in the street. With certain restrictions on $\boldsymbol{u}$, we enumerate outcomes of $\boldsymbol{u}$-parking functions with a fixed set of lucky cars or with a fixed number of lucky cars. We also count outcomes according to which spots contain lucky cars, and give formulas for enumerating $\boldsymbol{u}$-parking functions themselves according to their set of lucky cars.

Preliminary version of a course in univariate real analysis, with 14 chapters and 1 appendix (Chapters 1-8 complete at present). 1. Infinite sums. Real numbers; 2. Limits of sequences and subsequences; 3. Arithmetic of limits. AK series; 4. Infinite series. Elementary functions; 5. Limits of functions. Asymptotic notation; 6. Continuous functions; 7. Derivatives; 8. Applications of mean value theorems; 9. Taylor polynomials and series. Real analytic functions; 10. Primitives of uniformly continuous functions; 11. Newton integral. Primitives of rational functions; 12. Riemann integral. Transcendence of the number e; 13. Riemann integral. Henstock--Kurzweil integral; 14. More applications of Riemann integral; and A. Solutions to exercises.

Given a smooth curve with nonzero curvature $\Sigma\subset \mathbb{R}^2$, let $E_{\Sigma}$ denote the associated Fourier extension operator. For both general compact curves and the parabola, we characterize the pairs $(p,q)\in [1,\infty]^2$ for which the estimates $\|E_{\Sigma}f\|_{L^q(\Omega)}\leq C\|f\|_{L^p(\Sigma)}$ and $(\mathcal{R}(|E_{\Sigma}f|^{q}))^{\frac{1}{q}}\leq C\|f\|_{L^p(\Sigma)}$ hold, where $\Omega$ is a strip in $\mathbb{R}^2$ and $\mathcal{R}$ denotes the Radon transform. This work continues the study of mass concentration of $x\mapsto E_{\Sigma}f(x)$ near lines in $\mathbb{R}^2$, initiated by Bennett and Nakamura and later extended by Bennett, Nakamura, and the second author, where expressions of the form $(\mathcal{R}(|E_{\Sigma}f|^{2}))^{\frac{1}{2}}$ were studied.

A path system in a graph $G$ is a collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We show that the number of consistent path systems on $n$ vertices is $n^{\frac{n^2}{2}(1-o(1))}$, whereas the number of consistent path systems which are realizable as the unique geodesics w.r.t. some metric is only $2^{\Theta(n^2)}$. In addition, these insights allow us to improve known bounds on the face-count of the metric cone and shed new light on enumerating maximum-VC-classes.

Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}. $$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when $a$ and $b$ are consecutive terms of sequences having the Fibonacci recurrence and arbitrary initial terms.

We derive some rather general, but complicated, formulae to compute the survival function and the first passage time distribution of the $n^\text{th}$ coordinate of a many-body stochastic process in the presence of a killing barrier. First we will study the case of two coordinates and then we will generalize the results to three or more coordinates. Even if the results are difficult to implement, we will provide examples of their use applying them to a physical system, the single file diffusion, and to the financial problem of pricing a $n^\text{th}$-to-default credit default swap ($n^\text{th}$-CDS)

Lie algebroids, singular foliations, and Dirac structures are closely related objects. We examine the relation between their pullbacks under maps satisfying a constant rank or transversality assumption. A special case is given by blowdown maps. In that case, we also establish the relation between the blowup of a Lie algebroid and its singular foliation.

This paper completes the classification of nets of conics containing at least one double line in $\mathrm{PG}(2,q)$ for $q$ even. This classification contributes to the classification of partially symmetric tensors in $\mathbb{F}_q^3 \otimes S^2 \mathbb{F}_q^3$, $q$ even. The proof is obtained using geometric and combinatorial properties of the Veronese surface in 5-dimensional projective space over the finite field of even order. In particular, the orbits of planes in $\mathrm{PG}(5,q)$ that intersect the nucleus plane of the Veronese surface in at least one point are classified. As a result, it is shown that there are exactly $18$ equivalence classes of nets in $\mathrm{PG}(2,q)$, $q$ even, containing at least one double line, $9$ of which have an empty base.

The stability number of a graph $G$, denoted as $\alpha(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lovász's theta number and expanded through lift-and-project hierarchies as well as sums of squares (SOS) relaxations, provide powerful tools for approximating $\alpha(G)$. We build upon the copositive formulation of $\alpha(G)$ and introduce a novel SDP-based hierarchy of inner approximations to the copositive cone COP$_n$, which is derived from structured SOS representations. This hierarchy preserves essential structural properties that are missing in existing approaches, offers an SDP feasibility formulation at each level despite its non-convexity, and converges finitely to $\alpha(G)$. Our results include examples of graph families that require at least $\alpha(G) - 1$ levels for related hierarchies, indicating the tightness of the de Klerk-Pasechnik conjecture. Notably, on those graph families, our hierarchy achieves $\alpha(G)$ in a single step.

This paper proves that every periodic automorphism of a closed hyperbolic surface S sends some curve to a nearly disjoint curve. In particular, periodic maps cannot have the property that every curve fills with its image, so no such map can give a positive answer to a question of Wright. This paper also answers a question of Schleimer about irreducible periodic surface maps.

We prove that any vector bundle computing the rank-two Clifford index of a smooth projective algebraic curve is linearly semistable. We also identify conditions under which such bundles become linearly stable, thereby addressing a question posed by A. Castorena, G. H. Hitching and E. Luna in the rank-two case. Furthermore, we demostrate that in certain special cases, this property is equivalent to the (semi)stability of the associated Lazarsfeld-Mukai bundles. This yields a positive answer, in specific cases, to a generalized version of a conjecture proposed by Mistretta and Stoppino. We also study the moduli space $S_0(n,d,5)$ of generated $\alpha$-stable coherent systems of type $(n,d,5)$ for small values of $\alpha$ and $n=2,3$. We show that a general element of an irreducible component of $X \subseteq S_0(2,d,5)$ or $X \subseteq S_0(3,d,5)$ is linearly stable whenever $2\delta_2 \leq d \leq \frac{3g}{2}$. As an application of this, we prove that Butler's conjecture holds non-trivially for coherent systems of type $(2,d,5)$ within the given range for $d$.

We define a bicategory $\mathbf{2TDX}$ whose 1-cells provide a categorification of transducers, computational devices extending finite-state automata with output capabilities. This bicategory is a mathematically interesting object: its objects are categories $\mathcal{A},\mathcal{B},\dots$ and its 1-cells $(\mathcal{Q}, t) : \mathcal{A} \to \mathcal{B}$ consist of a category $\mathcal{Q}$ of `states', and a profunctor $$ t : \mathcal{A} \times \mathcal{Q}^\text{op}\times\mathcal{Q} \times (\mathcal{B}^*)^\text{op} \to \mathbf{Set} $$ where $\mathcal{B}^*$ denotes the free monoidal category over $\mathcal{B}$. Extending $t$ to $\mathcal{A}^*$ in a canonical way, to each `word' $\underline a$ in $\mathcal{A}^*$ one attaches an endoprofunctor over the category $\mathcal{Q}$ of states, enriched over presheaves on $\mathcal{B}^*$. We discuss a number of other characterizations of the hom-category $\mathbf{2TDX}(\mathcal{A},\mathcal{B})$; we establish a Kleisli-like universal property for $\mathbf{2TDX}(\mathcal{A},\mathcal{B})$ and explore the connection of $\mathbf{2TDX}$ to other bicategories of computational models, such as Bob Walters' bicategory of `circuits'; it is convenient to regard $\mathbf{2TDX}$ as the loose bicategory of a double category $\mathbb{D}\mathbf{TDX}$: the bicategory (resp., double category) of profunctors is naturally contained in the bicategory (resp., double category) $\mathbf{2TDX}$ (resp., $\mathbb{D}\mathbf{TDX}$); we study the completeness and cocompleteness properties of $\mathbb{D}\mathbf{TDX}$, the existence of companions and conjoints, and we sketch how monads, adjunctions, and other structures/properties that naturally arise from the definition work in $\mathbb{D}\mathbf{TDX}$.

We show the singular ideal in a non-Hausdorff étale groupoid C*-algebra is zero if and only if every unit is contained, at the level of group representation theory, in the collection of subgroups of the unit's isotropy group obtained as limit sets of nets in the "Hausdorff part" of the unit space. This is achieved through a study of the interplay between the Hausdorff cover and restriction maps on C*-algebras of groupoids to reductions by closed locally invariant subsets, which we show are compressible to *-homomorphisms and therefore have many of the same properties. We also prove a simpler algebraic characterization of zero singular ideal that holds whenever the isotropy group C*-algebras satisfy a certain ideal intersection property. We prove this property holds for all direct limits of virtually torsion free solvable groups.

Let $V_r(\mathbb{A}^n)$ denote the Stiefel variety ${\rm GL}_n/{\rm GL}_{n-r}$ over a field. There is a natural projection $p: V_{r+\ell}(\mathbb{A}^n) \to V_r(\mathbb{A}^n)$. The question of whether this projection admits a section was asked by M. Raynaud in 1968. We focus on the case of $r \ge 2$ and provide examples of triples $(r,n,\ell)$ for which a section does not exist. Our results produce examples of stably free modules that do not have free summands of a given rank. To this end, we also construct a splitting of $V_2(\mathbb{A}^n)$ in the motivic stable homotopy category over a field, analogous to the classical stable splitting of the Stiefel manifolds due to I. M. James.

In this article, we develop a new hyperbolic model governing the first-order dynamics of a thin film flow under the influence of gravity and solute transport. The obtained system turns out to be a non-symmetric Keyfitz-Kranzer type system. We find an entire class of convex entropies in the regions where the system remains strictly hyperbolic. By including delta shocks, we prove the existence of unique solutions of the Riemann problem. We analyze their stability with respect to the perturbation of the initial data and to the gravity and surface tension parameters. Moreover, we discuss the large time behaviour of the solutions of the perturbed Riemann problem and prove that the initial Riemann states govern it. Thus, we confirm the structural stability of the Riemann solutions under the perturbation of initial data. Finally, we validate our analytical results with well-established numerical schemes for this new system of conservation laws.

We study the shipper-side design of large-scale inbound transportation networks, motivated by Renault's global supply chain. We introduce the Shipper Transportation Design Problem, which integrates consolidation, routing, and regularity constraints, and propose a tailored Iterated Local Search (ILS) metaheuristic. The algorithm combines large-neighborhood search with MILP-based perturbations and exploits bundle-specific decompositions and giant container bounds to obtain scalable lower bounds and effective benchmarks. Computational experiments on real industrial data show that the ILS achieves an average gap of 7.9% to the best available lower bound on world-scale instances with more than 700,000 commodities and 1,200,000 arcs, delivering solutions showing a potential of 23.2% cost reduction compared to the Renault-based benchmark. To our knowledge, this is the first approach to solve shipper-side transportation design problems at such scale. Our analysis further yields managerial insights: accurate bin-packing models are essential for realistic consolidation, highly regular plans offer the best balance between cost and operational stability, and outsourcing is only attractive in low-volume contexts, while large-scale networks benefit from in-house planning.

We extend the theory of regularity structures [Hai14] to allow processes belonging to locally $m$-convex topological algebras. This extension includes processes in the locally $C^{*}$-algebras of [CHP25] used to localise singular stochastic partial differential equations involving fermions, as well as processes in Banach algebras such as infinite-dimensional semicircular\circular Brownian motion, and more generally the $q$-Gaussians of [BS91, BKS97, Boż99]. A new challenge we encounter in the $q$-Gaussian setting with $q \in (-1,1)$ are noncommutative renormalisation estimates where we must estimate operators in homogeneous $q$-Gaussian chaoses with arbitrary operator insertions. We introduce a new Banach algebra norm on $q$-Gaussian operators that allows us to control such insertions; we believe this construction could be of independent interest.

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum weight independent set, can be solved in time n^{O(k)} if the input n-vertex graph is given together with a tree decomposition of independence number k. Yolov, in [SODA 2018], gave an algorithm that, given an n-vertex graph G and an integer k, in time n^{O(k^3)} either constructs a tree decomposition of G whose independence number is O(k^3) or correctly reports that the tree-independence number of G is larger than k. In this paper, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time 2^{O(k^2)} n^{O(k)} and either outputs a tree decomposition of G with independence number at most $8k$, or determines that the tree-independence number of G is larger than k. This implies 2^{O(k^2)} n^{O(k)}-time algorithms for various problems, like maximum weight independent set, parameterized by the tree-independence number k without needing the decomposition as an input. Assuming Gap-ETH, an n^{\Omega(k)} factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k \ge 4 it is NP-hard to decide if a given graph has the tree-independence number at most k.

In neuroimaging, extensive post-processing of resting-state functional MRI (rfMRI) data is necessary for its application and investigation in relation to brain-behavior associations. Such post-processing is used to derive brain representations, lower dimensional feature sets used for brain-behavior association studies. A brain representation involves a choice of dimension reduction (a parcellation into regions or networks) and a choice of feature type, such as spatial topography, connectivity matrix, amplitude. However, widespread variability in rfMRI brain representations has hindered both reproducibility and knowledge accumulation across the field. Brain representation choice effects measurements of inter-subject variability, which muddies the comparison and integration of findings. We leveraged persistent homology on the subject-space topologies induced by 34 different brain representations to enable direct comparison of brain representations in the context of individual differences. Our findings reveal the importance of considering feature type when comparing results derived from different brain representations, suggesting best practices for assessing the replicability and generalizability of brain-behavior research in rfMRI data.

We prove that counting the analytic Brouwer degree of rational coefficient polynomial maps in $\operatorname{Map}(\mathbb C^d, \mathbb C^d)$ -- presented in degree-coefficient form -- is hard for the complexity class $\operatorname{\sharp P}$, in the following sense: if there is a randomized polynomial time algorithm that counts the Brouwer degree correctly for a good fraction of all input instances (with coefficients of bounded height where the bound is an input to the algorithm), then $\operatorname{P}^{\operatorname{\sharp P}} =\operatorname{BPP}$.

This paper provides a semidefinite programming hierarchy based on state polynomial optimization to determine the existence of quantum codes with given parameters. The hierarchy is complete, in the sense that a $(\!(n, K, {\delta})\!)_2$ code exists if and only if every level of the hierarchy is feasible. It is not limited to stabilizer codes and thus is applicable generally. While the machinery is formally dimension-free, we restrict it to qubit codes through quasi-Clifford algebras. We derive the quantum analog of a range of classical results: first, from an intermediate level a Lovász bound for self-dual quantum codes is recovered. Second, a symmetrization of a minor variation of this Lovász bound recovers the quantum Delsarte bound. Third, a symmetry reduction using the Terwilliger algebra leads to semidefinite programming bounds of size $O(n^4)$. With this we give an alternative proof that there is no $(\!(7, 1, 4)\!)_2$ quantum code, and show that $(\!(8, 9, 3)\!)_2$ and $(\!(10, 5, 4)\!)_2$ codes do not exist.

Models based on recursive adaptive partitioning such as decision trees and their ensembles are popular for high-dimensional regression as they can potentially avoid the curse of dimensionality. Because empirical risk minimization (ERM) is computationally infeasible, these models are typically trained using greedy algorithms. Although effective in many cases, these algorithms have been empirically observed to get stuck at local optima. We explore this phenomenon in the context of learning sparse regression functions over $d$ binary features, showing that when the true regression function $f^*$ does not satisfy Abbe et al. (2022)'s Merged Staircase Property (MSP), greedy training requires $\exp(\Omega(d))$ to achieve low estimation error. Conversely, when $f^*$ does satisfy MSP, greedy training can attain small estimation error with only $O(\log d)$ samples. This dichotomy mirrors that of two-layer neural networks trained with stochastic gradient descent (SGD) in the mean-field regime, thereby establishing a head-to-head comparison between SGD-trained neural networks and greedy recursive partitioning estimators. Furthermore, ERM-trained recursive partitioning estimators achieve low estimation error with $O(\log d)$ samples irrespective of whether $f^*$ satisfies MSP, thereby demonstrating a statistical-computational trade-off for greedy training. Our proofs are based on a novel interpretation of greedy recursive partitioning using stochastic process theory and a coupling technique that may be of independent interest.

Magic State Distillation (MSD) has been a research focus for fault-tolerant quantum computing due to the need for non-Clifford resource in gaining quantum advantage. Although many of the MSD protocols so far are based on stabilizer codes with transversal $T$ gates, there exists quite several protocols that don't fall into this class. Here we propose a method to map MSD protocols to iterative dynamical systems under the framework of stabilizer reduction. With the proposed mapping, we are able to analyze the performance of MSD protocols using techniques from dynamical systems theory, easily simulate the distillation process of input states under arbitrary noise and visualize it using flow diagram. We apply our mapping to common MSD protocols for $\ket{T}$ state and find some interesting properties: The $[[15, 1, 3]]$ code may distill states corresponding to $\sqrt{T}$ gate and the $[[5, 1, 3]]$ code can distill the magic state corresponding to the $T$ gate. Besides, we examine the exotic MSD protocols that may distill into other magic states proposed in [Eur. Phys. J. D 70, 55 (2016)] and identify the condition for distillable magic states. We also study new MSD protocols generated by concatenating different codes and numerically demonstrate that concatenation can generate MSD protocols with various magic states. By concatenating efficient codes with exotic codes, we can reduce the overhead of the exotic MSD protocols. We believe our proposed method will be a useful tool for simulating and visualization MSD protocols for canonical MSD protocols on $\ket{T}$ as well as other unexplored MSD protocols for other states.

Recent research has revealed that the CRT symmetry for fermions exhibits a fractionalization distinct from the $\mathbb{Z}_2^{\mathcal{C}}\times\mathbb{Z}_2^{\mathcal{R}}\times\mathbb{Z}_2^{\mathcal{T}}$ for scalar bosons. In fact, the CRT symmetry for fermions can be extended by internal symmetries such as fermion parity, thereby forming a group extension of the $\mathbb{Z}_2$ direct product. Conventionally, a Majorana fermion is defined by one Dirac fermion with trivial charge conjugation. However, when the spacetime dimension $d+1=5,6,7\bmod8$, the real dimension of Majorana fermion (dim$_{\mathbb{R}}\chi_{\mathcal{C}\ell(d,0)}$) aligns with the real dimension of Dirac fermion (dim$_{\mathbb{R}}\psi_{\mathcal{C}\ell(d)}$), rather than being half, which necessitates the introduction of a symplectic Majorana fermion, defined by two Dirac fermions with trivial charge conjugation. To include these two types of Majorana fermions, we embed the theory in $n_{\mathbb{R}}$ and define the Majorana fermion field as a representation of the real Clifford algebra with 8-fold periodicity. Within the Hamiltonian formalism, we identify the 8-fold CRT-internal symmetry groups across general dimensions. Similarly, Dirac fermion field is defined as a representation of the complex Clifford algebra with 2-fold periodicity. Interestingly, we discover that the CRT-internal symmetry groups exhibit an 8-fold periodicity that is distinct from that of the complex Clifford algebra. In certain dimensions where distinct mass terms can span a mass manifold, the CRT-internal symmetries can act non-trivially upon this mass manifold. Employing domain wall reduction method, we are able to elucidate the relationships between symmetries across different dimensions.

We study sums of independent and identically distributed random velocities in special relativity. We show that the resulting one-dimensional velocity distributions are not only stable under relativistic velocity addition but define a genuinely new class of stochastic processes--relativistic Lévy processes. Given a system, this allows identifying distinct relativistic regimes in terms of the distribution's concavity at the origin and the probability of measuring relativistic velocities. These features provide a protocol to assess the relevance of stochastic relativistic effects in actual experiments. As supporting evidence, we find agreement with previous results about heavy-ion diffusion and show that our findings are consistent with the distribution of momentum deviations observed in measurements of antiproton cooling.

We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude $q_o>0$ as $x\to \pm \infty$ by implementing numerically the corresponding Inverse Scattering Transform (IST). The Riemann--Hilbert problem associated to the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift-Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the $(x,t)$-plane depending on whether $|x/(2t)| < q_o$ or $|x/(2t)| > q_o$, as $t \to \infty$.

Operator learning is the approximation of operators between infinite dimensional Banach spaces using machine learning approaches. While most progress in this area has been driven by variants of deep neural networks such as the Deep Operator Network and Fourier Neural Operator, the theoretical guarantees are often in the form of a universal approximation property. However, the existence theorems do not guarantee that an accurate operator network is obtainable in practice. Motivated by the recent kernel-based operator learning framework, we propose a random feature operator learning method with theoretical guarantees and error bounds. The random feature method can be viewed as a randomized approximation of a kernel method, which significantly reduces the computation requirements for training. We provide a generalization error analysis for our proposed random feature operator learning method along with comprehensive numerical results. Compared to kernel-based method and neural network methods, the proposed method can obtain similar or better test errors across benchmarks examples with significantly reduced training times. An additional advantages it that our implementation is simple and does require costly computational resources, such as GPU.

We study Yang-Lee zeros in the thermodynamic limit of the 2D nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the Numerical Linked Cluster Expansion (NLCE) equipped with Exact Enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T \ll T_c$, the results suggest that two vertical lines at $\pm h_c(T)$ in the complex field plane (i.e two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is expected for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.

We provide a decision-theoretic framework for dealing with uncertainty in quantum mechanics. This uncertainty is two-fold: on the one hand there may be uncertainty about the state the quantum system is in, and on the other hand, as is essential to quantum mechanical uncertainty, even if the quantum state is known, measurements may still produce an uncertain outcome. In our framework, measurements therefore play the role of acts with an uncertain outcome and our simple decision-theoretic postulates ensure that Born's rule is encapsulated in the utility functions associated with such acts. This approach allows us to uncouple (precise) probability theory from quantum mechanics, in the sense that it leaves room for a more general, so-called imprecise probabilities approach. We discuss the mathematical implications of our findings, which allow us to give a decision-theoretic foundation to recent seminal work by Benavoli, Facchini and Zaffalon, and we compare our approach to earlier and different approaches by Deutsch and Wallace.

Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us inequalities to obtain lower bounds on the number of axioms. In this paper, we extend this result to higher-order equational theories. More precisely, we consider simply typed lambda calculus with product and unit types and study sets of equations between lambda terms. Then, we define homology groups of the given equational theory and show that a lower bound on the number of equations can be computed from the homology groups.

We investigate the collective dynamics of multivortex assemblies in a two dimensional (2D) toroidal fluid film of distinct curvature and topology. The incompressible and inviscid nature of the fluid allows a Hamiltonian description of the vortices, along with a self-force of geometric origin, arising from the standard Kirchhoff-Routh regularization procedure. The Hamiltonian dynamics is constructed in terms of $q$-digamma functions $\Psi_q(z)$, closely related to the Schottky-Klein prime function known to arise in multiply connected domains. We show the fundamental motion of the two-vortex system and identify five classes of geodesics on the torus for the special case of a vortex dipole, along with subtle distinctions from vortices in quantum superfluids. In multivortex assemblies, we observe that a randomly initialized cluster of vortices of the same sign and strength (chiral cluster) remains geometrically confined on the torus, while undergoing an overall drift along the toroidal direction, exhibiting collective dynamics. A cluster of fast and slow vortices also show the collective toroidal drift, with the fast ones predominantly occupying the core region and the slow ones expelled to the periphery of the revolving cluster. Vortex clusters of mixed sign but zero net circulation (achiral cluster) show unconfined dynamics and scatter all over the surface of the torus. A chiral cluster with an impurity in the form of a single vortex of opposite sign also show similar behavior as a pure chiral cluster, with occasional ``jets" of dipoles leaving and re-entering the revolving cluster. The work serves as a step towards analysis of vortex clusters in models that incorporate harmonic velocities in the Hodge decomposition.

Common variance estimation methods for weighted average treatment effects (WATEs) in observational studies include nonparametric bootstrap and model-based, closed-form sandwich variance estimation. However, the computational cost of bootstrap increases with the size of the data at hand. Besides, some replicates may exhibit random violations of the positivity assumption even when the original data do not. Sandwich variance estimation relies on regularity conditions that may be structurally violated. Moreover, the sandwich variance estimation is model-dependent on the propensity score model, the outcome model, or both; thus it does not have a unified closed-form expression. Recent studies have explored the use of wild bootstrap to estimate the variance of the average treatment effect on the treated (ATT). This technique adopts a one-dimensional, nonparametric, and computationally efficient resampling strategy. In this article, we propose a "post-weighting" bootstrap approach as an alternative to the conventional bootstrap, which helps avoid random positivity violations in replicates and improves computational efficiency. We also generalize the wild bootstrap algorithm from ATT to the broader class of WATEs by providing new justification for correctly accounting for sampling variability from multiple sources under different weighting functions. We evaluate the performance of all four methods through extensive simulation studies and demonstrate their application using data from the National Health and Nutrition Examination Survey (NHANES). Our findings offer several practical recommendations for the variance estimation of WATE estimators.