In recent works we have introduced the parameter space $\mathcal{Z}_N$ of $A$-variations of the Hardy $Z$-function, $Z(t)$, whose elements are functions of the form \begin{equation} \label{eq:Z-sections} Z_N(t ; \overline{a} ) = \cos(\theta(t))+ \sum_{k=1}^{N} \frac{a_k}{\sqrt{k+1} } \cos ( \theta (t) - \ln(k+1) t), \end{equation} where $\overline{a} = (a_1,...,a_N) \in \mathbb{R}^N$. The \( A \)-philosophy advocates that studying the discriminant hypersurface forming within such parameter spaces, often reveals essential insights about the original mathematical object and its zeros. In this paper we apply the $A$-philosophy to our space $\mathcal{Z}_N$ by introducing \( \Delta_n(\overline{a} ) \) the $n$-th Gram discriminant of \( Z(t) \). We show that the Riemann Hypothesis (RH) is equivalent to the corrected Gram's law \[ (-1)^n \Delta_n(\overline{1}) > 0, \] for any $n \in \mathbb{Z}$. We further show that the classical Gram's law \( (-1)^n Z(g_n) >0\) can be considered as a first-order approximation of our corrected law. The second-order approximation of $\Delta_n (\overline{a})$ is then shown to be related to shifts of Gram points along the \( t \)-axis. This leads to the discovery of a new, previously unobserved, repulsion phenomena \[ \left| Z'(g_n) \right| > 4 \left| Z(g_n) \right|, \] for bad Gram points $g_n$ whose consecutive neighbours $g_{n \pm 1}$ are good. Our analysis of the \(A\)-variation space \(\mathcal{Z}_N\) introduces a wealth of new results on the zeros of \(Z(t)\), casting new light on classical questions such as Gram's law, the Montgomery pair-correlation conjecture, and the RH, and also unveils previously unknown fundamental properties.

In this paper, we define a q-adic factorial and we demonstrate some properties of a generalized p-adic gamma function. Also, some numerical examples have been given

We investigate the blow-up dynamics for the $L^2$ critical two-dimensional Zakharov-Kuznetsov equation \begin{equation*} \begin{cases} \partial_t u+\partial_{x_1} (\Delta u+u^3)=0, \mbox{ } x=(x_1,x_2)\in \mathbb{R}^2, \mbox{ } t \in \mathbb{R}\\ u(0,x_1,x_2)=u_0(x_1,x_2)\in H^1(\mathbb{R}^2), \end{cases} \end{equation*} with initial data $u_0$ slightly exceeding the mass of the ground state $Q$. Employing methodologies analogous to the Martel-Merle-Raphael blow-up theory for $L^2$ critical equations, more precisely for the critical NLS equation and the quintic generalized Korteweg-de Vries equation, we categorize the solution behaviors into three outcomes: asymptotic stability, finite-time blow-up, or divergence from the soliton's vicinity. The construction of the blow-up solution involves the bubbling of the solitary wave which ensures the universal behavior and stability of the blow-up.

In Part I of this series, the author has shown how to extend the framework of Riemannian geometry so as to include infinitesimals of higher than first order. The purpose of the present contribution is to initiate an investigation into the implications of higher-order differential geometry for the general theory of relativity. As we have seen, a novel concept of inertial motion is implied in the analogue of the geodesic equation when modified to include the effects of higher infinitesimals and it therefore should not come as a surprise that it has potentially observable kinematic consequences. The route we prefer to take goes through the Einstein-Hilbert action generalized to reflect the presence of higher infinitesimals and a cosmological constant. A variational principle yields an hierarchy of field equations, which reduce to the Einsteinian case at first order. In the weak-field limit, we recover the usual relativistic equation for a moving body to leading order. To exemplify the theoretical framework, we undertake a preliminary study of the Schwarzschild solution and Friedmann-Robertson-Walker cosmology in the presence of second-order terms. But the most exciting results concern the novel effects in orbital mechanics that arise when the higher-order corrections cannot be neglected. Indeed, the higher-order Riemannian geometry predicts a modification of Newtonian dynamics corresponding to the Pioneer anomaly for a spacecraft on a hyperbolic escape trajectory from the solar system and to the flyby anomaly for the differential between ingoing and outgoing asymptotic velocities of a spacecraft passing near a rotating planet -- both of which are found to agree well with sensitive empirical findings.

In Part I of the present series of papers, we adumbrate our idea of Riemannian geometry to higher order in the infinitesimals and derive expressions for the appropriate generalizations of parallel transport and the Riemannian curvature tensor. In Part II, the implications of higher-order geometry for the general theory of relativity beyond Einstein are developed. In the present Part III, we expand on the framework of Part II so as to take up the problem of field-theoretical unification. Employing the form of the Einstein-Hilbert action to higher order as proposed in Part II, we show how in nearly flat space the higher-order terms give rise to a gauge theory of Yang-Mills type. At the 2-jet level, the electroweak force emerges after imposition of gauge fixing. In fact, the proposed form of the Einstein-Hilbert action permits us to say more: we argue that the equivalence principle results in a Proca term that brings about the spontaneous symmetry breaking of the standard model. Two empirical predictions support our reasoning: first, we obtain a theoretical value for the Weinberg angle and second, we find that -- without any adjustable parameters -- the implied value of Coulomb's constant agrees well with experiment. The final section examines the 3-jet level. The same mechanism that produces Glashow-Weinberg-Salam electroweak theory at the 2-jet level eventuates in a chromodynamical force having $SU(3)$ symmetry at the 3-jet level.

We study the regularity and comparison principle for a gradient degenerate Neumann problem. The problem is a generalization of the Signorini or thin obstacle problem which appears in the study of certain singular anisotropic free boundary problems arising from homogenization. In scaling terms, the problem is critical since the gradient degeneracy and the Neumann PDE operator are of the same order. We show the (optimal) $C^{1,\frac{1}{2}}$ regularity in dimension $d=2$ and we show the same regularity result in $d\geq 3$ conditional on the assumption that the degenerate values of the solution do not accumulate. We also prove a comparison principle characterizing minimal supersolutions, which we believe will have applications to homogenization and other related scaling limits.

A systematic and comprehensive study of p-adic refinement equations and subdivision scheme associated with a finitely supported refinement mask are carried out in this paper. The Lq -convergence of the subdivision scheme is characterized in terms of the q-norm joint spectral radii of a collection of operators associated with the refinement mask. Also, the smoothness of complex-valued functions on Qp is investigated.

We present an application of recent well-posedness results in the theory of delay differential equations for ordinary differential equations arXiv:2308.04730 to a generalized population model for stem cell maturation. The weak approach using Sobolev-spaces we take allows for a larger class of initial prehistories and makes checking the requirements for well-posedness of such a model considerably easier compared to previous approaches. In fact the present approach is a possible means to guarantee that the solution manifold is not empty, which is a necessary requirement for a $C^{1}$-approach to work.

We prove the equivalence between the several notions of generalized Ricci curvature found in the literature. As an application, we characterize when the total generalized Ricci tensor is symmetric.

This paper proposes a sparse regression strategy for discovery of ordinary and partial differential equations from incomplete and noisy data. Inference is performed over both equation parameters and state variables using a statistically motivated likelihood function. Sparsity is enforced by a selection algorithm which iteratively removes terms and compares models using statistical information criteria. Large scale optimization is performed using a second-order variant of the Levenberg-Marquardt method, where the gradient and Hessian are computed via automatic differentiation. Illustrations involving canonical systems of ordinary and partial differential equations are used to demonstrate the flexibility and robustness of the approach. Accurate reconstruction of systems is found to be possible even in extreme cases of limited data and large observation noise.

We dualise the classical fact that an operad with multiplication leads to cohomology groups which form a Gerstenhaber algebra to the context of cooperads: as a result, a cooperad with comultiplication induces a homology theory that is endowed with the structure of a Gerstenhaber coalgebra, that is, it comes with a graded cocommutative coproduct which is compatible with a coantisymmetric cobracket in a dual Leibniz sense. As an application, one obtains Gerstenhaber coalgebra structures on Tor groups over bialgebras or Hopf algebras, as well as on Hochschild homology for Frobenius algebras.

Although Deligne's theorem classifies all symmetric tensor categories (STCs) with moderate growth over algebraically closed fields of characteristic zero, the classification does not extend to positive characteristic. At the forefront of the study of STCs is the search for an analog to Deligne's theorem in positive characteristic, and it has become increasingly apparent that the Verlinde categories are to play a significant role. Moreover, these categories are largely unstudied, but have already shown very interesting phenomena as both a generalization of and a departure from superalgebra and supergeometry. In this paper, we study $\mathrm{Ver}_4^+$, the simplest non-trivial Verlinde category in characteristic $2$. In particular, we classify all isomorphism classes of non-degenerate symmetric bilinear forms and non-degenerate quadratic forms and study the associated Witt semi-ring that arises from the addition and multiplication operations on bilinear forms.

We introduce a new model equation for Stokes gravity waves based on conformal transformations of Euler's equations. The local version of the model equation is relevant for dynamics of shallow water waves. It allows us to characterize the traveling periodic waves both in the case of smooth and peaked waves and to solve the existence problem exactly, albeit not in elementary functions. Spectral stability of smooth waves with respect to co-periodic perturbations is proven analytically based on the exact count of eigenvalues in a constrained spectral problem.

We consider circle patterns on closed tori equipped with complex projective structures. There is an embedding of the space of circle patterns to the Teichm\"{u}ller space of a punctured surface. Via the embedding, the Weil-Petersson symplectic form is pulled back to the space of circle patterns. We investigate its non-degeneracy. On the other hand, we also complete a conjecture that the space of circle patterns is homeomorphic to the Teichm\"{u}ller space of the closed torus.

In this work, we introduce a novel approach for solving tridiagonal Toeplitz systems with multiple right-hand sides.

We show that there are $2^{\aleph_0}$ non-isomorphic universal sofic groups. This proves a conjecture of Simon Thomas.

We determine all modular curves $X_0(N)/\langle w_d\rangle$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$, when $N$ is square-free.

Federated learning has attracted significant recent attention due to its applicability across a wide range of settings where data is collected and analyzed across disparate locations. In this paper, we study federated nonparametric goodness-of-fit testing in the white-noise-with-drift model under distributed differential privacy (DP) constraints. We first establish matching lower and upper bounds, up to a logarithmic factor, on the minimax separation rate. This optimal rate serves as a benchmark for the difficulty of the testing problem, factoring in model characteristics such as the number of observations, noise level, and regularity of the signal class, along with the strictness of the $(\epsilon,\delta)$-DP requirement. The results demonstrate interesting and novel phase transition phenomena. Furthermore, the results reveal an interesting phenomenon that distributed one-shot protocols with access to shared randomness outperform those without access to shared randomness. We also construct a data-driven testing procedure that possesses the ability to adapt to an unknown regularity parameter over a large collection of function classes with minimal additional cost, all while maintaining adherence to the same set of DP constraints.

We study homogeneous instantons on the seven dimensional Stiefel manifold V in the context of G2 and Sasakian geometry. According to the reductive decomposition of V we provide an explicit description of all invariant G2 and Sasakian structures. In particular, we characterise the invariant G2- structures inducing a Sasakian metric, among which the well known nearly parallel G2-structure (Sasaki- Einstein) is included. As a consequence, we classify the invariant connections on homogeneous principal bundles over V with gauge group U(1) and SO(3), satisfying either the G2 or the Sasakian instanton condition. Finally, we analyse the Yang Mills condition for those invariant connections.

This paper studies federated learning for nonparametric regression in the context of distributed samples across different servers, each adhering to distinct differential privacy constraints. The setting we consider is heterogeneous, encompassing both varying sample sizes and differential privacy constraints across servers. Within this framework, both global and pointwise estimation are considered, and optimal rates of convergence over the Besov spaces are established. Distributed privacy-preserving estimators are proposed and their risk properties are investigated. Matching minimax lower bounds, up to a logarithmic factor, are established for both global and pointwise estimation. Together, these findings shed light on the tradeoff between statistical accuracy and privacy preservation. In particular, we characterize the compromise not only in terms of the privacy budget but also concerning the loss incurred by distributing data within the privacy framework as a whole. This insight captures the folklore wisdom that it is easier to retain privacy in larger samples, and explores the differences between pointwise and global estimation under distributed privacy constraints.

We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity $C^1$. With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the $\mathsf{RCD}$-splitting theorem \cite{gigli2013splitting, gigli2014splitoverview}. Along the way, we establish a Bochner-Weitzenb\"ock identity which permits %incorporates both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in \cite{mondino2024equivalence}. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see \cite{mondino2024equivalence}) between distributional Ricci curvature bounds and $\mathsf{RCD}$-type bounds, using in part the stability of the variable $\mathsf{CD}$-condition under suitable limits \cite{ketterer2017variableCD}.

Let $E$ be a module of projective dimension one over a Noetherian ring $R$ and consider its Rees algebra $\mathcal{R}(E)$. We study this ring as a quotient of the symmetric algebra $\mathcal{S}(E)$ and consider the ideal $\mathcal{A}$ defining this quotient. In the case that $\mathcal{S}(E)$ is a complete intersection ring, we employ a duality between $\mathcal{A}$ and $\mathcal{S}(E)$ in order to study the Rees ring $\mathcal{R}(E)$ in multiple settings. In particular, when $R$ is a complete intersection ring defined by quadrics, we consider its module of K\"ahler differentials $\Omega_{R/k}$ and its associated tangent algebras.

This paper analyses the optimal control of infectious disease propagation using a classic susceptible-infected-recovered (SIR) model characterised by permanent immunity and the absence of available vaccines. The control is performed over a time-dependent mean reproduction number, in order to minimise the cumulative number of ever-infected individuals (recovered), under different constraints. We consider constraints on isolation measures ranging from partial lockdown to non-intervention, as well as the social and economic costs associated with such isolation, and the capacity limitations of intensive care units that limits the number of infected individuals to a maximum allowed value. We rigorously derive an optimal quarantine strategy based on necessary optimality conditions. The obtained optimal strategy is of a boundary-bang type, comprising three phases: an initial phase with no intervention, a second phase maintaining the infected population at its maximum possible value, and a final phase of partial lockdown applied over a single interval. The optimal policy is further refined by optimising the transition times between these phases. We show that these results are in excellent agreement with the numerical solution of the problem.

Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius $r, \, r \in (0,1]$, where the endpoints of the chords are drawn according to a given probability distribution on $\mathbb{S}^1$. We show that, for $r=1,$ the problem is degenerated in the sense that any continuous measure is an extremiser, and that, for $r$ sufficiently close to $1,$ the desired maximal value is strictly below the one for $r=1$ by a polynomial factor in $1-r.$ Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is $1/4$ for $r \in (0,1/2).$ Connections with other variational problems and energy minimization problems are also presented.

We present a new method of establishing a bijective correspondence - in fact, a lattice isomorphism - between action- and coaction-invariant ideals of C*-algebras and their crossed products by a fixed locally compact group. It is known that such a correspondence exists whenever the group is amenable; our results hold for any locally compact group under a natural form of coaction invariance.

We introduce a generic numerical schemes for fully nonlinear parabolic PDEs on the full domain, where the nonlinearity is convex on the Hessian of the solution. The main idea behind this paper is reduction: we reduce the fully nonlinear problem to a series of simpler semilinear ones parameterized by the diffusion term and take the maximum over the solution of those problems through a novel gradient ascend method on the diffusion coefficient that we introduced for such semilinear PDEs. The promise of this study is a step toward the proof-of-concept that for any arbitrary numerical scheme for semilinear parabolic PDEs, one can build an iterative methods based on gradient ascent to converge to the solution of a fully nonlinear problem. Therefore, investment in our effort to create more efficient semilinear PDEs leads to efficiency for fully nonlinear PDEs. We leverage the theoretical results of Soner et al. [2012] to obtain numerical schemes for fully nonlinear PDEs and to justify the convergence in our methodology. The method is tested in a numerical experiment on the fully problem of portfolio optimization under stochastic volatility model.

The Conditional Gradient (or Frank-Wolfe) method is one of the most well-known methods for solving constrained optimization problems appearing in various machine learning tasks. The simplicity of iteration and applicability to many practical problems helped the method to gain popularity in the community. In recent years, the Frank-Wolfe algorithm received many different extensions, including stochastic modifications with variance reduction and coordinate sampling for training of huge models or distributed variants for big data problems. In this paper, we present a unified convergence analysis of the Stochastic Frank-Wolfe method that covers a large number of particular practical cases that may have completely different nature of stochasticity, intuitions and application areas. Our analysis is based on a key parametric assumption on the variance of the stochastic gradients. But unlike most works on unified analysis of other methods, such as SGD, we do not assume an unbiasedness of the real gradient estimation. We conduct analysis for convex and non-convex problems due to the popularity of both cases in machine learning. With this general theoretical framework, we not only cover rates of many known methods, but also develop numerous new methods. This shows the flexibility of our approach in developing new algorithms based on the Conditional Gradient approach. We also demonstrate the properties of the new methods through numerical experiments.

We show the integrated density of states for a variety of Jacobi operators on graphs, such as the Anderson model and random hopping models on graphs with Gaussian heat kernel bounds, can be estimated from above and below in terms of the localization landscape counting function. Specific examples of these graphs include stacked and decorated lattices, graphs corresponding to band matrices, and aperiodic tiling graphs. The upper bound part of the landscape law also applies to the fractal Sierpinski gasket graph. As a consequence of the landscape law, we obtain landscape-based proofs of the Lifshitz tails in several models including random band matrix models, certain bond percolation Hamiltonians on $\mathbb{Z}^d$, and Jacobi operators on certain stacks of graphs. We also present intriguing numerical simulations exploring the behavior of the landscape counting function across various models.

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are immediate consequences of the main result. Finally, combinatorial identities involving harmonic-like numbers and other prominent sequences like hyperharmonic numbers and odd harmonic numbers are offered.

We prove two results on Fractal Uncertainty Principle (FUP) for discrete Cantor sets with large alphabets. First, we give an example of an alphabet with dimension $\delta \in (\frac12,1)$ where the FUP exponent is exponentially small as the size of the alphabet grows. Secondly, for $\delta \in (0,\frac12]$ we show that a similar alphabet has a large FUP exponent, arbitrarily close to the optimal upper bound of $\frac12-\frac\delta2$, if we dilate the Fourier transform by a factor satisfying a generic Diophantine condition. We give an application of the latter result to spectral gaps for open quantum baker's maps.

We classify the nilpotent almost abelian Lie algebras admitting complex or symplectic structures. It turns out that if a nilpotent almost abelian Lie algebra admits a complex structure, then it necessarily admits a symplectic structure. Given an even dimensional almost abelian Lie algebra, we show that it always admits a complex structure when it is 2-step nilpotent and it always admits a symplectic structure when it is $k$-step nilpotent for $k=2, \, 3$ or $4$. Several consequences of the classification theorems are obtained.

A modified form of Legendre-Gauss orthogonal direct collocation is developed for solving optimal control problems whose solutions are nonsmooth due to control discontinuities. This new method adds switch-time variables, control variables, and collocation conditions at both endpoints of a mesh interval, whereas these new variables and collocation conditions are not included in standard Legendre-Gauss orthogonal collocation. The modified Legendre-Gauss collocation method alters the search space of the resulting nonlinear programming problem and enables determining accurately the location of the nonsmoothness in the optimal control. The transformed adjoint system of the modified Legendre-Gauss collocation method is then derived and shown to satisfy a discrete form of the continuous variational necessary conditions for optimality. The method is motivated via a control-constrained triple-integrator minimum-time optimal control problem where the solution possesses a two-switch bang-bang optimal control structure. In addition, the method developed in this paper is compared with existing Gaussian quadrature collocation methods. The method developed in this paper is shown to be capable of accurately solving optimal control problems with a discontinuous optimal control.

Experimental data on the propagation of wildfires show that its short-time spread has a double semi-elliptical shape. Our main goal is to show that this shape can be accurately approximated in polar coordinates by choosing suitable parameters in Gielis superformula and, then, implemented in a geometrical model for wildfire propagation. In this model, the firefront can be determined by computing the lightlike geodesics of a specific Finsler spacetime, and we derive a concise and efficient expression of the geodesic equations in these coordinates.

A Bayesian data assimilation scheme is formulated for advection-dominated advective and diffusive evolutionary problems, based upon the Dynamic Likelihood (DLF) approach to filtering. The DLF was developed specifically for hyperbolic problems -waves-, and in this paper, it is extended via a split step formulation, to handle advection-diffusion problems. In the dynamic likelihood approach, observations and their statistics are used to propagate probabilities along characteristics, evolving the likelihood in time. The estimate posterior thus inherits phase information. For advection-diffusion the advective part of the time evolution is handled on the basis of observations alone, while the diffusive part is informed through the model as well as observations. We expect, and indeed show here, that in advection-dominated problems, the DLF approach produces better estimates than other assimilation approaches, particularly when the observations are sparse and have low uncertainty. The added computational expense of the method is cubic in the total number of observations over time, which is on the same order of magnitude as a standard Kalman filter and can be mitigated by bounding the number of forward propagated observations, discarding the least informative data.

In this work, a delay-tolerant unmanned aerial vehicle (UAV) relayed covert and secure communication framework is investigated. In this framework, a legitimate UAV serves as an aerial relay to realize communication when the direct link between the terrestrial transmitter and receiver is blocked and also acts as a friendly jammer to suppress the malicious nodes presented on the ground. Subsequently, considering the uncertainty of malicious nodes' positions, a robust fractional programming optimization problem is built to maximize energy efficiency by jointly optimizing the trajectory of the UAV, the transmit power of the transmitter, and the time-switching factor. For the extremely complicated covert constraint, Pinsker's inequality, Jensen's inequality, and the bisection search method are employed to construct a tractable shrunken one. After this, an alternate optimization-based algorithm is proposed to solve the fractional programming optimization problem. To achieve low complexity, we design the primal-dual search-based algorithm and the successive convex approximation-based algorithm, respectively, for each sub-problem. Numerical results show the effectiveness of our proposed algorithm.

In this paper we prove gap theorems in Yang-Mills theory for complete four-dimensional manifolds with positive Yamabe constant. We extend the results of Gursky-Kelleher-Streets to complete manifolds. We also describe the equality in the gap theorem in terms of the basic instanton, which is interesting even for compact manifolds.

To leverage prediction models to make optimal scheduling decisions in service systems, we must understand how predictive errors impact congestion due to externalities on the delay of other jobs. Motivated by applications where prediction models interact with human servers (e.g., content moderation), we consider a large queueing system comprising of many single server queues where the class of a job is estimated using a prediction model. By characterizing the impact of mispredictions on congestion cost in heavy traffic, we design an index-based policy that incorporates the predicted class information in a near-optimal manner. Our theoretical results guide the design of predictive models by providing a simple model selection procedure with downstream queueing performance as a central concern, and offer novel insights on how to design queueing systems with AI-based triage. We illustrate our framework on a content moderation task based on real online comments, where we construct toxicity classifiers by finetuning large language models.

We review the original approach to the Le-Murakami-Ohtsuki (LMO) invariant of closed 3-manifolds (as opposed to the later approach based on the Aarhus integral). Following the ideas of surgery presentation, we introduce a class of combinatorial structures, called Kirby structures, which we prove to yield multiplicative 3-manifold invariants. We illustrate this with the Reshetikhin-Turaev invariants. We then introduce a class of combinatorial structures, called pre-LMO structures, and prove that they give rise to Kirby structures. We show how the Kontsevich integral can be used to construct a pre-LMO structure. This yields two families of multiplicative 3-manifolds invariants $\{\Omega_n^{\mathfrak{c}}\}_{n\geq 1}$ and $\{\Omega_n^{\mathfrak{d}}\}_{n\geq 1}$. We review the elimination of redundant information in the latter family, leading to the construction of the LMO invariant. We also provide uniqueness results of some aspects of the LMO construction. The family of invariants $\{\Omega_n^{\mathfrak{c}}\}_{n\geq 1}$ is not discussed explicitly in the literature; whereas $\Omega_n^{\mathfrak{c}}$ enables one to recover $\Omega_n^{\mathfrak{d}}$ for any $n \geq 1$, we show that these invariants coincide for $n = 1$.

In this paper, we analyze the existence of algebraic and topological structures in the set of sequences that contain only a finite number of zero coordinates. Inspired by the work of Daniel Cariello and Juan B. Seoane-Sep\'ulveda, our research reveals new insights and complements their notable results beyond the classical \( \ell_p \) spaces for \( p \) in the interval from 1 to infinity, including the intriguing case where \( p \) is between 0 and 1. Our exploration employs notions such as S-lineability, pointwise lineability, and (alpha, beta)-spaceability. This investigation allowed us to verify, for instance, that the set \( F \setminus Z(F) \), where \( F \) is a closed subspace of \( \ell_p \) containing \( c_0 \), is (alpha, beta)-spaceable if and only if alpha is finite.

Causal inference in longitudinal studies is often hampered by treatment-confounder feedback. Existing methods typically assume discrete time steps or step-like data changes, which we term ``regular and irregular functional studies,'' limiting their applicability to studies with continuous monitoring data, like intensive care units or continuous glucose monitoring. These studies, which we formally term ``functional longitudinal studies,'' require new approaches. Moreover, existing methods tailored for ``functional longitudinal studies'' can only investigate static treatment regimes, which are independent of historical covariates or treatments, leading to either stringent parametric assumptions or strong positivity assumptions. This restriction has limited the range of causal questions these methods can answer and their practicality. We address these limitations by developing a nonparametric framework for functional longitudinal data, accommodating dynamic treatment regimes that depend on historical covariates or treatments, and may or may not depend on the actual treatment administered. To build intuition and explain our approach, we provide a comprehensive review of existing methods for regular and irregular longitudinal studies. We then formally define the potential outcomes and causal effects of interest, develop identification assumptions, and derive g-computation and inverse probability weighting formulas through novel applications of stochastic process and measure theory. Additionally, we compute the efficient influence curve using semiparametric theory. Our framework generalizes existing literature, and achieves double robustness under specific conditions. Finally, to aid interpretation, we provide sufficient and intuitive conditions for our identification assumptions, enhancing the applicability of our methodology to real-world scenarios.

In this article, we study maximal functions related to hypersurfaces with vanishing Gaussian curvature in $\mathbb{R}^3$. Firstly, we characterize the $L^p\rightarrow L^q$ boundedness of local maximal operators along homogeneous hypersurfaces. Moreover, weighted $L^p$-estimates are obtained for the corresponding global operators. Secondly, for a class of hypersurfaces that lack a homogeneous structure and pass through the origin, we attempt to look for other geometric properties instead of height of hypersurfaces to characterize the optimal $L^p$-boundedness of the corresponding global maximal operators.

This paper studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.

We prove sharp estimates for incidences involving planar tubes that satisfy packing conditions. We apply them to improve the estimates for the Fourier transform of fractal measures supported on planar curves.

Motivated by the works of Erd\"os, Pomerance, Wolke and Harman on the sum-of-divisor function $\sigma(n)$, we study the distribution of a special class of natural numbers closely related to (multiply) perfect numbers which we term `$(\ell;k)$-within-perfect numbers', where $\ell >1$ is a real number and $k: [1, \infty) \rightarrow (0, \infty)$ is an increasing and unbounded function.

In this article, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of a parabolic evolution equations on complex plane, in which the diffusion operator has the form \(\overline{\Box}_{\varphi} = \overline{D}\, \overline{D}^{\ast}\), where \(\overline{D} f = \bar{\partial}f + \varphi_{\bar{z}} f\), the function \(\varphi\) is smooth and subharmonic on \(\mathbb{C}\), and \(\overline{D}^{\ast}\) is the formal adjoint of \(\overline{D}\). Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich \cite{raich06} and a fixed point argument.

We study the inviscid Burgers equation on the circle $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ forced by the derivative of a Poisson point process on $\mathbb{R}\times\mathbb{T}$. We construct global solutions with mean $\theta$ simultaneously for all $\theta\in\mathbb{R}$, and in addition construct their associated global shocks (which are unique except on a countable set of $\theta$). We then show that as $\theta$ changes, the solution only changes through the movement of the global shock, and give precise formulas for this movement. This is an analogue of previous results by the author and Yu Gu in the viscous case with white-in-time forcing, which related the derivative of the solution in $\theta$ to the density of a particle diffusing in the Burgers flow.

For $n\geq 25$, we construct a smooth metric $\tilde{g}$ on the standard $n$-dimensional sphere $\mathbb{S}^n$ such that there exists a sequence of smooth metrics $\{\tilde{g}_k\}_{k\in\mathbb{N}}$ conformal to $\tilde g$ where each $\tilde g_k$ has scalar curvature $R_{\tilde{g}_k}\equiv 1$ and their volumes $\text{Vol}(\mathbb{S}^n,\tilde{g}_k)$ tend to infinity as $k$ approaches infinity.

This paper establishes a variant of Stewart's theorem (Theorem~6.4 of Stewart, {\em SIAM Rev.}, 15:727--764, 1973) for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both the Frobenius and spectral norms, whereas the new variant uses the spectral norm and any unitarily invariant norm that offer choices per convenience of particular applications and lead to sharper bounds than that straightforwardly derived from Stewart's original theorem with the help of the well-known equivalence inequalities between matrix norms. Of interest in their own right, bounds on the solution to two couple Sylvester equations are established for a few different circumstances.

Let $X=(X_t, t\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W=\{W(t,x), t\geq 0, x\in \mathbb{R}^d\}$ white in time and colored in space with correlation kernel $g(x,y)$. When $d\geq 3$, under the condition that the correlation function $g(x,y)$ is bounded above by some appropriate function $\bar{g}(x-y)$, we present the quenched and annealed Strong Law of Large Numbers and the Central Limit Theorems regarding the weighted occupation measure $\int_0^t X_s ds$ as $t\to \infty$.

For a convex set $K\subset \mathbb R^n$ and the associated anisotropic perimeter $P_K$, we establish that every $(\epsilon,\,r)$-minimizer for $P_K$ satisfies a local John property. Furthermore, we prove that a certain class of John domains, including $(\epsilon,\,r)$-minimizers close to $K$, admits a trace inequality. As a consequence, we provide a more concrete proof for a crucial step in the quantitative Wulff inequality, thereby complementing the seminal work of Figalli, Maggi, and Pratelli.

We study the nonlinear steady Boltzmann equation in the half space, with phase transition and Dirichlet boundary condition. In particular, we study the regularity of the solution to the half-space problem in the situation that the gas is in contact with its condensed phase. We propose a novel kinetic weight and establish a weighted $C^1$ estimate under the spatial domain $x\in [0,\infty)$, which is unbounded and not strictly convex. Additionally, we prove the $W^{1,p}$ estimate without any weight for $p<2$.

We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of $L$ is obtained by forming an associated dual space $X$ that is a monadic orthoframe. This is a set with an orthogonality relation and an additional binary relation satisfying certain conditions. For the MacNeille completion, $X$ is formed from the non-zero elements of $L$, and for the canonical completion, $X$ is formed from the proper filters of $L$. The corresponding completion of $L$ is then obtained as the ortholattice of bi-orthogonally closed subsets of $X$ with an additional operation defined through the binary relation of $X$. With the introduction of a suitable topology on an orthoframe, as was done by Goldblatt and Bimb\'o, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence.

This paper is concerned with the existence and qualitative properties of positive ground state solutions for the planar Schr\"odinger-Newton equation on the disc. First, we prove the existence and radial symmetry of all the positive ground state solutions by employing the symmetric decreasing rearrangement and Talenti's inequality. Next, we develop Newton's theorem and then use the contraction mapping principle to establish the uniqueness of the positive ground state solution for the Schr\"odinger-Newton equation on the disc in the two dimensional case. Finally, we show that the unique positive ground state solution converges to the trivial solution as the radius $R$ tending to infinity, which is totally different from the higher dimensional case in \cite{Guo-Wang-Yi}.

Let $G$ be a simple graph with associated diagonal matrix of vertex degrees $D(G)$, adjacency matrix $A(G)$, Laplacian matrix $L(G)$ and signless Laplacian matrix $Q(G)$. Recently, Nikiforov proposed the family of matrices $A_\alpha(G)$ defined for any real $\alpha\in [0,1]$ as $A_\alpha(G):=\alpha\,D(G)+(1-\alpha)\,A(G)$, and also mentioned that the matrices $A_\alpha(G)$ can underpin a unified theory of $A(G)$ and $Q(G)$. Inspired from the above definition, we introduce the $B_\alpha$-matrix of $G$, $B_\alpha(G):=\alpha A(G)+(1-\alpha)L(G)$ for $\alpha\in [0,1]$. Note that $ L(G)=B_0(G), D(G)=2B_{\frac{1}{2}}(G), Q(G)=3B_{\frac{2}{3}}(G), A(G)=B_1(G)$. In this article, we study several spectral properties of $ B_\alpha $-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of $ B_\alpha(G) $ is continuous on $ \alpha $. Using this, we characterize positive semidefinite $ B_\alpha $-matrices in terms of $\alpha$. As a consequence, we provide an upper bound of the independence number of $ G $. Besides, we establish some bounds for the largest and the smallest eigenvalues of $B_\alpha(G)$. As a result, we obtain a bound for the chromatic number of $G$ and deduce several known results. In addition, we present a Sachs-type result for the characteristic polynomial of a $ B_\alpha $-matrix.

We extend the range of parameters associated with the Gagliardo-Nirenberg interpolation inequalities in the fractional Coulomb-Sobolev spaces for radial functions. We also study the optimality of this newly extended range of parameters.

A semitopological group $G$ is called {\it an $n$-semitopological group}, if for any $g\in G$ with $e\not\in\overline{\{g\}}$ there is a neighborhood $W$ of $e$ such that $g\not\in W^{n}$, where $n\in\mathbb{N}$. The class of $n$-semitopological groups ($n\geq 2$) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any $n\in\mathbb{N}$. Some properties of $n$-semitopological groups are studied, and some questions about $n$-semitopological groups are posed. Some generalized metric properties of $n$-semitopological groups are discussed, which contains mainly results are that (1) each Hausdorff first-countable 2-semitopological group admits a coarser Hausdorff semi-metrizable topology and has $G_{\delta}^{\ast}$-diagonal; (2) each locally compact, Baire and $\sigma$-compact 2-semitopological group is a topological group; (3) the condensation of some kind of 2-semitopological groups topologies are given. Finally, some cardinal invariants of $n$-semitopological groups are discussed.

This paper is concerned with the propagation dynamics of time almost periodic reaction-diffusion equations. Assuming the existence of a time almost periodic traveling wave connecting two stable steady states, we focus especially on the asymptotic behavior of average wave speeds in both rapidly oscillating and slowly oscillating environments. We prove that, in the rapidly oscillating case, the average speed converges to the constant wave speed of the homogenized equation; while in the slowly oscillating case, it approximates the arithmetic mean of the constant wave speeds for a family of equations with frozen coefficients. In both cases, we provide estimates on the convergence rates showing that, in comparison to the limiting speeds, the deviations of average speeds for almost periodic traveling waves are at most linear in certain sense. Furthermore, our explicit formulas for the limiting speeds indicate that temporal variations have significant influences on wave propagation. Even in periodic environments, it can alter the propagation direction of bistable equations.

We study Hamiltonian paths and cycles in undirected graphs from an operadic viewpoint. We show that the graphical collection $\mathsf{Ham}$ encoding directed Hamiltonian paths in connected graphs admits an operad-like structure, called a contractad. Similarly, we construct the graphical collection of Hamiltonian cycles $\mathsf{CycHam}$ that forms a right module over the contractad $\mathsf{Ham}$. We use the machinery of contractad generating series for counting Hamiltonian paths/cycles for particular types of graphs.

We introduce a notion of equivariant vector bundles on schemes over semirings. We do this by considering the functor of points of a locally free sheaf. We prove that every toric vector bundle on a toric scheme $X$ over an idempotent semifield equivariantly splits as a sum of toric line bundles. We then study the equivariant Picard group $\text{Pic}_G(X)$. Finally, we prove a version of Klyachko's classification theorem for toric vector bundles over an idempotent semifield.

Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove multiple upper bounds on the expected number of vertices of a random shadow of a fixed polytope. Our bounds are in terms of various parameters in the literature including geometric diameter and edge lengths, minimal and maximal slack, maximal coordinates for lattice polytopes, and maximum absolute values of subdeterminants. For the case of geometric diameter and edge lengths, we prove lower bounds and argue that our upper and lower bounds are both tight for zonotopes.

Recent advances in {matrix-mimetic} tensor frameworks have made it possible to preserve linear algebraic properties for multilinear data analysis and, as a result, to obtain optimal representations of multiway data. Matrix mimeticity arises from interpreting tensors as operators that can be multiplied, factorized, and analyzed analogous to matrices. Underlying the tensor operation is an algebraic framework parameterized by an invertible linear transformation. The choice of linear mapping is crucial to representation quality and, in practice, is made heuristically based on expected correlations in the data. However, in many cases, these correlations are unknown and common heuristics lead to suboptimal performance. In this work, we simultaneously learn optimal linear mappings and corresponding tensor representations without relying on prior knowledge of the data. Our new framework explicitly captures the coupling between the transformation and representation using variable projection. We preserve the invertibility of the linear mapping by learning orthogonal transformations with Riemannian optimization. We provide original theory of uniqueness of the transformation and convergence analysis of our variable-projection-based algorithm. We demonstrate the generality of our framework through numerical experiments on a wide range of applications, including financial index tracking, image compression, and reduced order modeling. We have published all the code related to this work at https://github.com/elizabethnewman/star-M-opt.

We derive some maximal inequalities for the bifractional Brownian motion using comparison theorems for Gaussian processes.

Let Y be a random variable whose degenerate moment generating functions exist in some neighborhoods of the origin. The aim of this paper is to study the probabilistic degenerate Stirling numbers of the first kind associated with Y which are constructed from the degenerate cumulant generating function of Y. They are a degenerate version of the probabilistic Stirling numbers of the first kind associated with Y, which were recently introduced by Adell-Benyi. We investigate some properties, related identities, recurrence relations and explicit expressions for those numbers. In addition, we apply our results to the special cases of normal and gamma random variables.

In 2017 Tao proposed a variant Sarnak's M\"{o}bius disjointness conjecture with logarithmic averaging: For any zero entropy dynamical system $(X,T)$, $\frac{1}{\log N} \sum_{n=1} ^N \frac{f(T^n x) \mu (n)}{n}= o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. Nonetheless, all of our examples satisfy the conjecture.

The concept of pseudo q-factorization graphs was recently introduced by the last two authors as a combinatorial language which is suited for capturing certain properties of Drinfeld polynomials. Using certain known representation theoretic facts about tensor products of Kirillov Reshetikhin modules and qcharacters, combined with special topological/combinatorial properties of the underlying q-factorization graphs, the last two authors showed that, for algebras of type A, modules associated to totally ordered graphs are prime, while those associated to trees are real. In this paper, we extend the latter result. We introduce the notions of strongly real modules and that of trees of modules satisfying certain properties. In particular, we can consider snake trees, i.e., trees formed from snake modules. Among other results, we show that a certain class of such generalized trees, which properly contains the snake trees, give rise to strongly real modules.

The peaceable queens problem asks to determine the maximum number $a(n)$ such that there is a placement of $a(n)$ white queens and $a(n)$ black queens on an $n \times n$ chessboard so that no queen can capture any queen of the opposite color. In this paper, we consider the peaceable queens problem and its variant on the toroidal board. For the toroidal board, we provide new upper and lower bounds. Somewhat surprisingly, our bounds show that there is a sharp contrast in behaviour between the odd torus and the even torus. Our lower bounds are given by explicit constructions. For the upper bounds, we formulate the problem as a quadratic optimization problem with at most $70$ variables, regardless of the size of the board. We solve our quadratic program exactly using modern optimization software. Our method is quite robust. For example, with very minor changes, it also provides upper bounds for the regular board. In particular, we show that $a(n) \leq 0.1641n^2$, for all $n$. This improves on the bound $a(n) \leq 0.25n^2$ of van Bommel and MacEachern. We also provide a local search algorithm and a software implementation which converges very rapidly to solutions which appear optimal. Our algorithm is sufficiently robust that it works on both the classical and toroidal boards. For example, for the classical board, the algorithm quickly finds the so-called Ainley construction. Thus, our work provides some further evidence that the Ainley construction is indeed optimal.

False Data Injection (FDI) attacks are one of the challenges that the modern power system, as a cyber-physical system, is encountering. Designing AC FDI attacks that accurately address the physics of the power systems could jeopardize the security of power systems as they can easily bypass the traditional Bad Data Detection (BDD) algorithm. Knowing the essence of the AC FDI attack and how they can be designed gives insight about detecting the system again these attacks. Moreover, recognition of the nature of these attacks, especially when they are designed optimally, is essential for benchmarking various defensive approaches to increase the resilience of power systems. This paper presents a unified approach to demonstrate the process of designing optimal AC FDI attack. In this connection, we first define the process of designing an AC-based FDI attack that satisfies AC power flow equations. We then formulate an optimization problem to design an optimal AC FDI attack that both satisfies AC power flow equations and overloads a specific line in the system. The objective function is defined to optimize the magnitude of the attack vector in such a way that it can evade residue-based BDD approaches. The proposed approach for designing AC FDI attacks is applied to the IEEE 118-bus test case system. Various comparisons are conducted to elaborate on the impact of optimally designing AC FDI attacks on the residual for the AC state estimation algorithm. Comparing the results of optimal and non-optimal AC FDI attacks demonstrates the impact on the difficulty of detecting FDI attacks and the importance of optimally designing these attacks.

It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line {(with Dirichlet boundary conditions) is not essentially self-adjoint}: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions.

Information density and its exponential form, known as lift, play a central role in information privacy leakage measures. $\alpha$-lift is the power-mean of lift, which is tunable between the worst-case measure max-lift ($\alpha=\infty$) and more relaxed versions ($\alpha<\infty$). This paper investigates the optimization problem of the privacy-utility tradeoff where $\alpha$-lift and mutual information are privacy and utility measures, respectively. Due to the nonlinear nature of $\alpha$-lift for $\alpha<\infty$, finding the optimal solution is challenging. Therefore, we propose a heuristic algorithm to estimate the optimal utility for each value of $\alpha$, inspired by the optimal solution for $\alpha=\infty$. In proposing the algorithm, we prove and use the convexity of $\alpha$-lift with respect to the lift.

Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial $t$-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte $T$-designs. We explore when a $T$-design is also a $T'$-design where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.

We provide a self-contained introduction to Gr\"obner bases of submodules of $R[x_1, \ldots, x_n]^k$, where $R$ is a Euclidean domain, and explain how to use these bases to solve linear systems over $R[x_1, \ldots, x_n]$.

We give a result estimating the dimension of the Lie algebra of Killing vector fields on an irreducible non-trivial gradient Ricci soliton. Then we study the structure of this manifold when the maximal dimension is attained. There are local and global implications.

The aim of this paper is to give a new explicit construction of Lusztig's asymptotic algebra in affine type $\mathsf{A}$. To do so, we construct a balanced system of cell modules, prove an asymptotic version of the Plancherel Theorem and develop a relative version of the Satake Isomorphism for each two-sided Kazhdan-Lusztig cell.

Numerical data suggest that the zeros $\rho$ of the auxiliary Riemann function in the upper half-plane satisfy $\mathop{\mathrm{Re}}(\rho)<1$. We show that this is true for those zeros with $\mathop{\mathrm{Im}}(\rho)> 3.9211\dots10^{65}$. We conjecture that this is true for all of them.

In the present paper we study the geometry of plane quartics with large automorphism groups. We show results devoted to smooth plane quartics that are invariant under the action of the elementary abelian group of type $[2,2,2]$, and we study geometric properties of the smooth plane quartic having automorphism group of order $48$.

Metric dimension is an essential parameter in graph theory that aids in addressing issues pertaining to information retrieval, localization, network design, and chemistry through the identification of the least possible number of elements necessary to identify the distances between vertices in a graph uniquely. A variant of metric dimension, called the edge metric dimension focuses on distinguishing the edges in a graph $G$, with a vertex subset. The minimum possible number of vertices in such a set is denoted as $\dim_E(G)$. This paper presents the precise edge metric dimension of silicate networks.

This paper focuses on Majority Dynamics in sparse graphs, in particular, as a tool to study internal cuts. It is known that, in Majority Dynamics on a finite graph, each vertex eventually either comes to a fixed state, or oscillates with period two. The empirical evidence acquired by simulations suggests that for random odd-regular graphs, approximately half of the vertices end up oscillating with high probability. We notice a local symmetry between oscillating and non-oscillating vertices, that potentially can explain why the fraction of the oscillating vertices is concentrated around $\frac{1}{2}$. In our simulations, we observe that the parts of random odd-regular graph under Majority Dynamics with high probability do not contain $\lceil \frac{d}{2} \rceil$-cores at any timestep, and thus, one cannot use Majority Dynamics to prove that internal cuts exist in odd-regular graphs almost surely. However, we suggest a modification of Majority Dynamics, that yields parts with desired cores with high probability.

We study the index theory of curved Bernstein-Gelfand-Gelfand (BGG) sequences in parabolic geometry and their role in $K$-homology and noncommutative geometry. The BGG-sequences fit into $K$-homology, and we solve their index problem. We provide a condition for when the BGG-complex on the flat parabolic geometry $G/P$ of a semisimple Lie group $G$ fits into $G$-equivariant $K$-homology by means of Heisenberg calculus. For higher rank Lie groups, we prove a no-go theorem showing that the approach fails.

We propose a new random process to construct the eigenvectors of some random operators which make a short and clean connection with the resolvent. In this process the center of localization has to be chosen randomly.

We present a complete classification of simply-connected pluriclosed manifolds with parallel Bismut torsion, extending previously known results in the literature. Consequently, we also establish a splitting theorem for compact manifolds that are both pluriclosed with parallel Bismut torsion and Calabi-Yau with torsion.

In these notes we propose and analyze an inertial type method for obtaining stable approximate solutions to nonlinear ill-posed operator equations. The method is based on the Levenberg-Marquardt (LM) iteration. The main obtained results are: monotonicity and convergence for exact data, stability and semi-convergence for noisy data. Regarding numerical experiments we consider: i) a parameter identification problem in elliptic PDEs, ii) a parameter identification problem in machine learning; the computational efficiency of the proposed method is compared with canonical implementations of the LM method.

The quality of underground space information data has become a major problem endangering the safety of underground space. After research and analysis, we found that the current high accuracy information data remote detection methods are limited to the detection of overground spaces objects, and are not applicable to the detection of various information data in underground space. In this paper, we analyze the spectral properties of the fractional-order differential (FDO) operator, and establish mathematical model of remote transmission and high-precision detection of information data, which realizes the functions of high-precision and remote detection of information data. By fusing the information data to detect the mathematical model in a long distance and with high accuracy, A mathematical model has been established to improve the quality of underground spatial information data. Through the application in engineering practice, the effectiveness of this method in underground space information data detection is verified.

In this paper, we prove the non-vanishing conjecture for cotangent bundles on isotrivial elliptic surfaces. Combined with the result by H\"{o}ring and Peternell, it completely solves the question for surfaces with Kodaira dimension at most $1$.

We consider a system of binary interacting chains describing the dynamics of a group of $N$ components that, at each time unit, either send some signal to the others or remain silent otherwise. The interactions among the chains are encoded by a directed Erd\"os-R\'enyi random graph with unknown parameter $ p \in (0, 1) .$ Moreover, the system is structured within two populations (excitatory chains versus inhibitory ones) which are coupled via a mean field interaction on the underlying Erd\"os-R\'enyi graph. In this paper, we address the question of inferring the connectivity parameter $p$ based only on the observation of the interacting chains over $T$ time units. In our main result, we show that the connectivity parameter $p$ can be estimated with rate $N^{-1/2}+N^{1/2}/T+(\log(T)/T)^{1/2}$ through an easy-to-compute estimator. Our analysis relies on a precise study of the spatio-temporal decay of correlations of the interacting chains. This is done through the study of coalescing random walks defining a backward regeneration representation of the system. Interestingly, we also show that this backward regeneration representation allows us to perfectly sample the system of interacting chains (conditionally on each realization of the underlying Erd\"os-R\'enyi graph) from its stationary distribution. These probabilistic results have an interest in its own.

Deriving exact density functions for Gibbs point processes has been challenging due to their general intractability, stemming from the intractability of their normalising constants/partition functions. This paper offers a solution to this open problem by exploiting a recent alternative representation of point process densities. Here, for a finite point process, the density is expressed as the void probability multiplied by a higher-order Papangelou conditional intensity function. By leveraging recent results on dependent thinnings, exact expressions for generating functionals and void probabilities of locally stable point processes are derived. Consequently, exact expressions for density/likelihood functions, partition functions and posterior densities are also obtained. The paper finally extends the results to locally stable Gibbsian random fields on lattices by representing them as point processes.

In this article we study the convex hull spanned by the union of trajectories of a standard planar Brownian motion, and an independent standard planar Brownian bridge. We find exact values of the expectation of perimeter and area of such a convex hull. As an auxiliary result, that is of interest in its own right, we provide an explicit shape of the probability density function of a random variable that represents the time when combined maximum of a standard one-dimensional Brownian motion, and an independent standard one-dimensional Brownian bridge is attained. At the end, we generalize our results to the case of multiple independent standard planar Brownian motions and Brownian bridges.

We elaborate on a problem raised by Schmidt in 1967 which generalizes the theory of classical Diophantine approximation to subspaces of $\R^n$. We consider Diophantine exponents for linear subspaces of $\R^n$ which generalize the irrationality measure for real numbers. We prove here that we have no smooth relations among some functions associated to these exponents. To establish this result, we construct subspaces for which we are able to compute the exponents.

In this paper, we explore a distributed setting, where a user seeks to compute a linearly-separable Boolean function of degree $M$ from $N$ servers, each with a cache size $M$. Exploiting the fundamental concepts of sensitivity and influences of Boolean functions, we devise a novel approach to capture the interplay between dataset placement across servers and server transmissions and to determine the optimal solution for dataset placement that minimizes the communication cost. In particular, we showcase the achievability of the minimum average joint sensitivity, $\frac{N}{2^{M-1}}$, as a measure for the communication cost.

For a large class of cohomology theories, we prove that refined unramified cohomology is canonically isomorphic to the hypercohomology of a natural truncated complex of Zariski sheaves. This generalizes a classical result of Bloch and Ogus and solves a conjecture of Kok and Zhou.

We show that for any finitely presented group $G$, there is a simply connected closed 4-manifold containing an infinite family of topologically isotopic but smoothly inequivalent 2-links whose 2-link group is $G$. We also show that, if $G$ satisfies the necessary topological conditions, these 2-links have nullhomotopic components.

We establish the existence of quasi-periodic traveling wave solutions for the $\beta$-plane equation on $\mathbb{T}^2$ with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the frequency of oscillations of the external force. Due to the presence of small divisors, the proof relies on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. To our knowledge, this is the first instance of constructing quasi-periodic solutions for a quasilinear PDE in dimensions greater than one, with a 1-smoothing dispersion relation that is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by preserving the traveling-wave structure, the conservation of momentum and by implementing normal form methods for the linearized system with sublinear dispersion relation in higher space dimension.

We propose an isogeometric approach to model the deformation of active thin films using layered, nonlinear, Kirchhoff Love shells. Isogeometric Collocation and Galerkin formulations are employed to discretize the electrophysiological and mechanical sub-problems, respectively, with the possibility to adopt different element and time-step sizes. Numerical tests illustrate the capabilities of the active stress based approach to effectively simulate the contraction of thin films in both quasi-static and dynamic conditions.

Starting from a doubly infinite sequence of complex numbers, the aim of this paper is to extend certain Markov inequalities for the determinant of Hankel matrices and the zeros of the corresponding orthogonal polynomials on the real line (A. Markov in Notes of the Imperial Academy of Sciences, St. Petersburg, 74 (Appendix n. 2) (1894) 1-30. English translation, by J. Shohat, Duke Math. J. 7 (1940), 85-96) to the Toeplitz case, where the central role is played by CD kernels and paraorthogonal polynomials on the unit circle. In particular, we consider the case in which the starting sequence is a two-sided P\'olya frequency sequence.

We present bounds between different widths of convex subsets of Banach spaces, including Gelfand and Bernstein widths. Using this, and some relations between widths and minimal errors, we obtain bounds on the maximal gain of adaptive and randomized algorithms over non-adaptive, deterministic ones for approximating linear operators on convex sets. Our results also apply to the approximation of embeddings into the space of bounded functions based on function evaluations, i.e., to sampling recovery in the uniform norm. We conclude with a list of open problems.

In Parts I and II of this series of papers, three new methods for the computation of eigenvalues of singular pencils were developed: rank-completing perturbations, rank-projections, and augmentation. It was observed that a straightforward structure-preserving adaption for symmetric pencils was not possible and it was left as an open question how to address this challenge. In this Part III, it is shown how the observed issue can be circumvented by using Hermitian perturbations. This leads to structure-preserving analogues of the three techniques from Parts I and II for Hermitian pencils (including real symmetric pencils) as well as for related structures. It is an important feature of these methods that the sign characteristic of the given pencil is preserved. As an application, it is shown that the resulting methods can be used to solve systems of bivariate polynomials.

An anticode ${\bf C} \subset {\bf F}_q^n$ with the diameter $\delta$ is a code in ${\bf F}_q^n$ such that the distance between any two distinct codewords in ${\bf C}$ is at most $\delta$. The famous Erd\"{o}s-Kleitman bound for a binary anticode ${\bf C}$ of the length $n$ and the diameter $\delta$ asserts that $$|{\bf C}| \leq \Sigma_{i=0}^{\frac{\delta}{2}} \displaystyle{n \choose i}.$$ In this paper, we give an antiGriesmer bound for $q$-ary projective linear anticodes, which is stronger than the above Erd\"{o}s-Kleitman bound for binary anticodes. The antiGriesmer bound is a lower bound on diameters of projective linear anticodes. From some known projective linear anticodes, we construct some linear codes with optimal or near optimal minimum distances. A complementary theorem constructing infinitely many new projective linear $(t+1)$-weight code from a known $t$-weight linear code is presented. Then many new optimal or almost optimal few-weight linear codes are given and their weight distributions are determined. As a by-product, we also construct several infinite families of three-weight binary linear codes, which lead to $l$-strongly regular graphs for each odd integer $l \geq 3$.

We consider the 1d nonlinear Schr\"odinger equation (NLS) on the torus with initial data distributed according to the Gaussian measure with covariance operator $(1 - \Delta)^{-s}$, where $\Delta$ is the Laplace operator. We prove that the Gaussian measures are quasi-invariant along the flow of (NLS) for the full range $s > \frac{3}{2}$. This improves a previous result obtained by Planchon, Tzvetkov and Visciglia (in 2019), where the quasi-invariance is proven for $s=2k$, for all integers $k\geq 1$. In our approach, to prove the quasi-invariance, we directly establish an explicit formula for the Radon-Nikodym derivative $G_s(t,.)$ of the transported measures, which is obtained as the limit of truncated Radon-Nikodym derivatives $G_{s,N}(t,.)$ for transported measures associated with a truncated system. We also prove that the Radon-Nikodym derivatives belong to $L^p$, $p>1$, with respect to $H^1(\mathbb{T})$-cutoff Gaussian measures, relying on the introduction of weighted Gaussian measures produced by a normal form reduction, following a recent work by Sun and Tzvetkov (in 2023). Additionally, we prove that the truncated densities $G_{s,N}(t,.)$ converges to $G_s(t,.)$ in $L^p$ (with respect to the $H^1(\mathbb{T})$-cutoff Gaussian measures).

In 1976, Delsarte introduced the notion of $q$-analogs of designs, and $q$-analogs of graphs were introduced recently by M. Braun et al. In this paper, we extend that study by giving a method for constructing transitive regular $q$-analogs of graphs. Further, we illustrate the method by giving some examples. Additionally, we introduced the notion of $q$-analogs of quasi-strongly regular graphs and give examples of transitive $q$-analogs of quasi-strongly regular graphs coming from spreads.

In this paper the total positivity of quasi-Riordan arrays is investigated with use of the sequence characterization of quasi-Riordan arrays. Due to the correlation between quasi-Riordan arrays and Riordan arrays, this study is an in-depth discussion of the total positivity of Riordan arrays.

Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a spanning tree of $G$. The leaf degree of a vertex $v\in V(T)$ is defined as the number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum leaf degree among all the vertices of $T$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal degree matrix of $G$. Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of $G$. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. In this paper, we investigate the connection between the spanning tree and the signless Laplacian spectral radius of $G$, and put forward a sufficient condition based upon the signless Laplacian spectral radius to guarantee that a graph $G$ contains a spanning tree with leaf degree at most $k$. Finally, we construct some extremal graphs to claim all the bounds obtained in this paper are sharp.

Due to the uncontrolled and complex real-life radio propagation environments, Claude Shannon's information theory of communications describes fundamental limits to state-of-the-art 5G radio access network (RAN) capacity, with respect to fixed radio resource usage. Fortunately, recent research has found that a holographic metasurface-based new physical layer architecture may hold the key to overcome these fundamental limits of current mobile networks under a new paradigm, smart wireless environment (SWE), where the long-standing challenge of mobile communications, fading channel hostility, may be solved, leading to a step-change boost in network performance and user experience. Despite recent research activities in SWE, the best way to implement it as a network operator remains an open challenge. In this industrial review, we adopt a novel yet realistic mobile channel stabilisation perspective for network operators to understand this paradigm shift. More specifically, we provide a technical analysis of the synergy between key next-gen mobile network enablers, e.g., holographic metasurface, wireless sensing, and machine intelligence, as well as of how this synergy leads to a robust future RAN architecture. Against the as yet unclear theoretical boundaries and low technology readiness level (TRL) of SWE enhanced telecommunications, we conclude by identifying critical challenges in future commercial deployments.

Let $\mathbb{R}^{+}=[0, \infty)$ and let $\mathbf{End}_{\mathbb{R}^+}$ be the set of all endomorphisms of the monoid $(\mathbb{R}^+, \vee)$. The set $\mathbf{End}_{\mathbb{R}^+}$ is a monoid with respect to the operation of the function composition $g \circ f$. It is shown that $g : \mathbb{R}^+ \to \mathbb{R}^+$ is pseudometric-preserving iff $g \in \mathbf{End}_{\mathbb{R}^+}$. In particular, a function $f : \mathbb{R}^+ \to \mathbb{R}^+$ is ultrametric-preserving iff it is an endomorphism of $(\mathbb{R}^+,\vee)$ with kelnel consisting only the zero point. We prove that a given $\mathbf{A} \subseteq \mathbf{End}_{\mathbb{R}^+}$ is a submonoid of $(\mathbf{End}, \circ)$ iff there is a class $\mathbf{X}$ of pseudoultrametric spaces such that $\mathbf{A}$ coincides with the set of all functions which preserve the spaces from $\mathbf{X}$. An explicit construction of such $\mathbf{X}$ is given.

In the recent work [arXiv:2308.03216], Coghi and Maurelli proved pathwise uniqueness of solutions to the vorticity form of stochastic 2D Euler equation, with Kraichnan transport noise and initial data in $L^1\cap L^p$ for $p>3/2$. The aim of this note is to remove the constraint on $p$, showing that pathwise uniqueness holds for all $L^1\cap L^p$ initial data with arbitrary $p>1$.

In this paper we consider examples of positive generalized Wiener functions and we establish a large deviation principle for the generalized multiple intersection local time of the multidimensional Brownian motion.

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of the Gelfand--Phillips property and the (coarse) Gelfand--Phillips property of order $p$ of Banach spaces, we introduce and study different types of the Gelfand--Phillips property of order $(p,q)$ (the $GP_{(p,q)}$ property) and the coarse Gelfand--Phillips property of order $p$ in the realm of all locally convex spaces. We compare these classes and show that they are stable under taking direct product, direct sums and closed subspaces. It is shown that any locally convex space is a quotient space of a locally convex space with the $GP_{(p,q)}$ property. Characterizations of locally convex spaces with the introduced Gelfand--Phillips type properties are given.

The two-phase horizontally periodic quasistationary Stokes flow in $\mathbb{R}^2$, describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, which is parameterized as the graph of a function $f=f(t)$, is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function $f$. Based on abstract parabolic theory, it is proven that the problem is well-posed in all subcritical spaces $\mathrm{H}^r(\mathbb{S})$, $r\in(3/2,2)$. Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.

Biological pattern formation is one of the most intriguing phenomena in nature. Simplest examples of such patterns are represented by travelling waves and stationary periodic patterns which occur during various biological processes including morphogenesis and population dynamics. Formation of these patterns in populations of motile microorganisms such as Dictyostelium discoideum and E. coli have been shown in a number of experimental studies. Conditions for formation of various types of patterns are commonly addressed in mathematical studies of dynamical systems containing diffusive and advection terms. In this work, we do mathematical study of spatio-temporal patterns forming in growing population of chemotactically active bacteria. In particular, we perform linear analysis to find conditions for formation of stationary periodic patterns, and nonlinear (Fourier) analysis to find characteristics, such as amplitude and wavelength, of these patterns. We verify our analytical results by means of numerical simulations.

Let $ G_1 \circledast G_2$,$ G_1 \sqcupdot G_2 $ and $ G_1 \sqcupplus G_2$ denote the total corona, $Q$-vertex corona and $Q$-edge corona of two graphs $ G_1$ and $ G_2 $, respectively. In this paper, we compute the $A_\alpha$-spectrum of $ G_1 \circledast G_2$,$ G_1 \sqcupdot G_2 $ and $ G_1 \sqcupplus G_2$ for regular graphs $ G_1$ and $ G_2$. As an application, we construct infinitely many pairs of $A_\alpha$-cospectral graphs.

Bilevel optimization, a well-established field for modeling hierarchical decision-making problems, has recently intersected with sustainability studies and practices, resulting in a series of works focusing on bilevel optimization problems involving multiple decision makers with diverse economic, environmental, and social objectives. This survey offers a comprehensive overview of sustainable bilevel optimization applications. First, we introduce the main concepts related to the nature of bilevel optimization problems and present some typical mathematical formulations for bilevel pricing problems that cover the majority of the collected applications. Then, we review the most relevant works published in sustainable bilevel optimization, giving a classification based on the application domains and their association with well-known operations research problems, while briefly discussing the proposed solution methodologies. We survey applications on transportation and logistics, production planning and manufacturing, water, waste, and agriculture management, supply chains, and disaster prevention and response. Finally, we outline a list of open questions and opportunities for future research in this domain.

We develop a second-order accurate central scheme for the two-dimensional hyperbolic system of in-homogeneous conservation laws. The main idea behind the scheme is that we combine the well-balanced deviation method with the Kurganov-Tadmor (KT) scheme. The approach satisfies the well-balanced property and retains the advantages of KT scheme: Riemann-solver-free and the avoidance of oversampling on the regions between Riemann-fans. The scheme is implemented and applied to a number of numerical experiments for the Euler equations with gravitational source term and the results are non-oscillatory. Based on the same idea, we construct a semi-discrete scheme where we combine the above two methods and illustrate the maximum principle.

We consider convergence properties of the long-term behaviors with respect to the coefficient of the stochastic term for a nonautonomous stochastic $p$-Laplacian lattice equation with multiplicative noise. First, the upper semi-continuity of pullback random $(\ell^2,\ell^q)$-attractor is proved for each $q\in[1,+\infty)$. Then, a convergence result of the time-dependent invariant sample Borel probability measures is obtained in $\ell^2$. Next, we show that the invariant sample measures satisfy a stochastic Liouville type equation and a termwise convergence of the stochastic Liouville type equations is verified. Furthermore, each family of the invariant sample measures is turned out to be a sample statistical solution, which hence also fulfills a convergence consequence.

Ising machines (IM) have recently been proposed as unconventional hardware-based computation accelerators for solving NP-hard problems. In this work, we present a model for a time-multiplexed IM based on the nonlinear oscillations in a delay line-based resonator and numerically study the effects that the circuit parameters, specifically the compression gain $\beta_r$ and frequency nonlinearity $\beta_i$, have on the IM solutions. We find that the likelihood of reaching the global minimum -- the global minimum probability (GMP) -- is the highest for a certain range of $\beta_r$ and $\beta_i$ located near the edge of the synchronization region of the oscillators. The optimal range remains unchanged for all tested coupling topologies and network connections. We also observe a sharp transition line in the ($\beta_i, \beta_r$) space above which the GMP falls to zero. In all cases, small variations in the natural frequency of the oscillators do not modify the results, allowing us to extend this model to realistic systems.

In this paper, we obtain the exact blow-up profiles of solutions of the Keller-Segel-Patlak system in the space with dimensions $N\ge 3$, which solves an open problem proposed by P. Souplet and M. Winkler in 2019. To establish this achievement, we develop the zero number argument for nonlinear equations with unbounded coefficients and construct a family of auxiliary backward self-similar solutions through nontrivial ODE analysis.

We study the asymptotic properties of a stochastic model for the induction equations of the magnetic field in a three dimensional periodic domain. The turbulent velocity field driving the electromotive force on the magnetic field is modeled by a noise white in time. For this model we rigorously take a scaling limit leading to a deterministic model. While in case of isotropic turbulence this produces an additional dissipation in the limit model which influences also the decay rate of the Magnetic field in the stochastic model, the case of turbulence devoloped in a preferential direction allows us to find a dynamo effect.

We propose a model order reduction approach to speed up the computation of seismograms, i.e. the solution of the seismic wave equation evaluated at a receiver location, for different model parameters. Our approach achieves a reduction of the unknowns by a factor of approximately 1000 for various numerical experiments for a 2D subsurface model of Groningen, the Netherlands, even if the wave speeds of the subsurface are relatively varied. Moreover, using parallel computing, the reduced model can approximate the (time domain) seismogram in a lower wall clock time than an implicit Newmark-beta method. To realize this reduction, we exploit the fact that seismograms are low-pass filtered for the observed seismic events by considering the Laplace-transformed problem in frequency domain. Therefore, we can avoid the high frequencies that would require many reduced basis functions to reach the desired accuracy and generally make the reduced order approximation of wave problems challenging. Instead, we can prove for our ansatz that for a fixed subsurface model the reduced order approximation converges exponentially fast in the frequency range of interest in the Laplace domain. We build the reduced model from solutions of the Laplace-transformed problem via a (Proper Orthogonal Decomposition-)Greedy algorithm targeting the construction of the reduced model to the time domain seismograms; the latter is achieved by using an a posteriori error estimator that does not require computing any time domain counterparts. Finally, we show that we obtain a stable reduced model thus overcoming the challenge that standard model reduction approaches do not necessarily yield a stable reduced model for wave problems.

Erd\H{o}s and Graham found it conceivable that the best $n$-term Egyptian underapproximation of almost every positive number for sufficiently large $n$ gets constructed in a greedy manner, i.e., from the best $(n-1)$-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero. [This note solves Problem 206 on Bloom's website "Erd\H{o}s problems".]

We prove that the topology on the density space with respect to a unital C*-algebra and a faithful induced by the C*-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and C*-norm when the C*-algebra is assumed to be finite dimensional. Finally, we provide an example of when the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.

This paper deals with the application of probabilistic time integration methods to semi-explicit partial differential-algebraic equations of parabolic type and its semi-discrete counterparts, namely semi-explicit differential-algebraic equations of index 2. The proposed methods iteratively construct a probability distribution over the solution of deterministic problems, enhancing the information obtained from the numerical simulation. Within this paper, we examine the efficacy of the randomized versions of the implicit Euler method, the midpoint scheme, and exponential integrators of first and second order. By demonstrating the consistency and convergence properties of these solvers, we illustrate their utility in capturing the sensitivity of the solution to numerical errors. Our analysis establishes the theoretical validity of randomized time integration for constrained systems and offers insights into the calibration of probabilistic integrators for practical applications.

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three. As a byproduct of our rationality constructions we get estimates on the number of their rational points over a number field, and a class of quadro-cubic Cremona correspondences of even dimensional projective spaces.

In this article, we consider several local conditions under which linear mappings on algebras act like Lie n-centralizers and we study these linear mappings, Lie n-centralizers and n-commuting linear maps.

We call an order $O$ in a quadratic field $K$ odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve $E$ over the field $C$ of complex numbers with CM odd (resp. even) if its endomorphism ring $End(E)$ is an odd (resp. even) order in the corresponding imaginary quadratic field. Suppose that $j(E)$ is a real number and let us consider the set $J(R,E)$ of all $j(E')$ where $E'$ is any elliptic curve that enjoys the following properties. 1) $E'$ is isogenous to $E$; 2) $j(E')$ is a real number; 3) $E'$ has the same parity as $E$. We prove that the closure of $J(R,E)$ in the set $R$ of real numbers is the closed semi-infinite interval $(-\infty,1728]$ (resp. the whole $R$) if $E$ is odd (resp. even). This paper was inspired by a question of Jean-Louis Colliot-Th\'el\`ene and Alena Pirutka about the distribution of $j$-invariants of certain elliptic curves of CM type.

It is shown that for any compact Lie group $G$ (odd or even dimensional), the tangent bundle $TG$ admits a left-invariant integrable almost complex structure, where the Lie group structure on $TG$ is the natural one induced from $G$. The aforementioned complex structure on $TG$ is inspired by Samelson's construction for even dimensional compact Lie groups.

We consider a class of infinite-dimensional singular stochastic control problems. These can be thought of as spatial monotone follower problems and find applications in spatial models of production and climate transition. Let $(D,\mathcal{M},\mu)$ be a finite measure space and consider the Hilbert space $H:=L^2(D,\mathcal{M},\mu; \mathbb{R})$. Let then $X$ be an $H$-valued stochastic process on a suitable complete probability space, whose evolution is determined through an SPDE driven by a self-adjoint linear operator $\mathcal{A}$ and affected by a cylindrical Brownian motion. The evolution of $X$ is controlled linearly via an $H$-valued control consisting of the direction and the intensity of action, a real-valued nondecreasing right-continuous stochastic process, adapted to the underlying filtration. The goal is to minimize a discounted convex cost-functional over an infinite time-horizon. By combining properties of semiconcave functions and techniques from viscosity theory, we first show that the value function of the problem $V$ is a $C^{1,Lip}(H)$-viscosity solution to the corresponding dynamic programming equation, which here takes the form of a variational inequality with gradient constraint. Then, by allowing the decision maker to choose only the intensity of the control and requiring that the given control direction $\hat{n}$ is an eigenvector of the linear operator $\mathcal{A}$, we establish that the directional derivative $V_{\hat{n}}$ is of class $C^1(H)$, hence a second-order smooth-fit principle in the controlled direction holds for $V$. This result is obtained by exploiting a connection to optimal stopping and combining results and techniques from convex analysis and viscosity theory.

We study the infinite-horizon distributionally robust (DR) control of linear systems with quadratic costs, where disturbances have unknown, possibly time-correlated distribution within a Wasserstein-2 ambiguity set. We aim to minimize the worst-case expected regret-the excess cost of a causal policy compared to a non-causal one with access to future disturbance. Though the optimal policy lacks a finite-order state-space realization (i.e., it is non-rational), it can be characterized by a finite-dimensional parameter. Leveraging this, we develop an efficient frequency-domain algorithm to compute this optimal control policy and present a convex optimization method to construct a near-optimal state-space controller that approximates the optimal non-rational controller in the $\mathit{H}_\infty$-norm. This approach avoids solving a computationally expensive semi-definite program (SDP) that scales with the time horizon in the finite-horizon setting.

We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal $\mathfrak{gl}_2$ algebra on relaxed Verma modules, and $q$-hypergeometric opers with apparent singularities. We introduce alongside the notion of apparent singularities for linear $q$-difference operators and discuss some of their properties. We also touch on a generalization to $\mathfrak{gl}_n$.

Due to the utilization of large antenna arrays at base stations (BSs) and the operations of wireless communications in high frequency bands, mobile terminals often find themselves in the near-field of the array aperture. In this work, we address the signal processing challenges of integrated near-field localization and communication in uplink transmission of an integrated sensing and communication (ISAC) system, where the BS performs joint near-field localization and signal detection (JNFLSD). We show that JNFLSD can be formulated as a matrix factorization (MF) problem with proper structures imposed on the factor matrices. Then, leveraging the variational inference (VI) and unitary approximate message passing (UAMP), we develop a low complexity Bayesian approach to MF, called UAMP-MF, to handle a generic MF problem. We then apply the UAMP-MF algorithm to solve the JNFLSD problem, where the factor matrix structures are fully exploited. Extensive simulation results are provided to demonstrate the superior performance of the proposed method.

We construct a Banach space $X$ for which the set of norm-attaining functionals $NA(X,\mathbb{R})$ does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on $X$, no other element of the segment between them attains its norm. Equivalently, the intersection of $NA(X,\mathbb{R})$ with the unit sphere of a two-dimensional subspace of $X^*$ contains, at most, four elements. In terms of proximinality, we show that for every closed subspace $Z$ of $X$ of codimension two, at most four element of the unit sphere of $X/Z$ has a representative of norm-one. We further relate this example with an open problem on norm-attaining operators.

In this paper, we treat $\mathscr{D}$-modules on the basic affine space $G/U$ and their global sections for a semisimple complex algebraic group $G$. Our aim is to prepare basic results about large non-irreducible modules for the branching problem and harmonic analysis of reductive Lie groups. A main tool is a formula given by Bezrukavnikov--Braverman--Positselskii. The formula is about a product of functions and their Fourier transforms on $G/U$ like Capelli's identity. Using the formula, we give a generalization of the Beilinson--Bernstein correspondence. We show that the global sections of holonomic $\mathscr{D}$-modules are also holonomic using the formula. As a consequence, we give a large algebra action on the $\mathfrak{u}$-cohomologies $H^i(\mathfrak{u}; V)$ of a $\mathfrak{g}$-module $V$ when $V$ is realized as a holonomic $\mathscr{D}$-module. We consider affinity of the supports of the $\mathfrak{t}$-modules $H^i(\mathfrak{u}; V)$.

This work investigates the optimal selection of the last $m$th success in a sequence of $n$ independent Bernoulli trials. We propose a threshold strategy that is $\varepsilon$-optimal under minimal assumptions about the monotonicity of the trials' success probabilities. This new strategy ensures stopping at most one step earlier than the optimal rule. Specifically, the new threshold coincides with the point where the sum of success probabilities in the remaining trials equals $m$. We show that the underperformance of the new rule, in comparison to the optimal one, is of the order $O(n^{-2})$ in the case of a Karamata-Stirling success profile with parameter $\theta > 0$ where $p_k = \theta / (\theta + k - 1)$ for the $k$th trial. We further leverage the classical weak convergence of the number of successes in the trials to a Poisson random variable to derive the asymptotic solution of the stopping problem. Finally, we present illustrative results highlighting the close performance between the two rules.

We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers [32, 33, 19, 16, 17].

In this work we apply Riemann-Hilbert problem to study two families of orthogonal polynomials on the unit circle, named of modified Jacobi and Bessel. We derive first and second order differential equations, for these families of orthogonal polynomials and functions of second kind, as well as nonlinear equations that the Verblunsky coefficients satisfy.

The Fredholm integral equations of the first kind is a typical ill-posed problem, so that it is usually difficult to obtain its analytical minimal-norm solution. This paper gives a closed-form minimal-norm solution for the degenerate kernel equations based on the H-HK formulation. Furthermore, it has been shown that the structure of solutions to degenerate kernel equations and matrix equations are consistent. Subsequently, the obtained results are extended to non-degenerate integral equations. Finally, the validity and applicability of the proposed method are demonstrated by some examples.

In this article, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By constructing a new locally constrained inverse curvature flow, we obtain the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space. This generalizes a theorem of Brendle-Guan-Li \cite{BGL} for convex closed hypersurfaces in hyperbolic space.

We formulate an {\it effective cone conjecture} for klt Calabi--Yau pairs $(X,\Delta)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,\Delta)$. Assuming the existence of good minimal models in dimension $\dim(X)$, known to hold in dimension up to $3$, we prove that the effective cone conjecture for $(X,\Delta)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,\Delta)$. As an application, we show that the movable cone conjecture unconditionally holds for the smooth Calabi--Yau threefolds introduced by Schoen and studied by Namikawa, Grassi and Morrison. We also show that for such a Calabi--Yau threefold $X$, all of its minimal models, apart from $X$ itself, have rational polyhedral nef cones.

In this paper we compute the integral Chow ring of the moduli stack $\mathcal{R}_2$ of Prym pairs of genus 2 with integral coefficients.

This paper studies permutation statistics that count occurrences of patterns. Their expected values on a product of $t$ permutations chosen randomly from $\Gamma \subseteq S_{n}$, where $\Gamma$ is a union of conjugacy classes, are considered. Hultman has described a method for computing such an expected value, denoted $\mathbb{E}_{\Gamma}(s,t)$, of a statistic $s$, when $\Gamma$ is a union of conjugacy classes of $S_{n}$. The only prerequisite is that the mean of $s$ over the conjugacy classes is written as a linear combination of irreducible characters of $S_{n}$. Therefore, the main focus of this article is to express the means of pattern-counting statistics as such linear combinations. A procedure for calculating such expressions for statistics counting occurrences of classical and vincular patterns of length 3 is developed, and is then used to calculate all these expressions. The results can be used to compute $\mathbb{E}_{\Gamma}(s,t)$ for all the above statistics, and for all functions on $S_{n}$ that are linear combinations of them.

A well known procedure to get quantum hydrodynamical models for charge transport is to resort to the Wigner equations and deduce the hierarchy of the moment equations as in the semiclassical approach. If one truncates the moment hierarchy to a finite order, the resulting set of balance equations requires some closure assumption because the number of unknowns exceed the number of equations. In the classical and semiclassical kinetic theory a sound approach to get the desired closure relations is that based on the Maximum Entropy Principle (MEP) [13] (see[20] for charge transport in semiconductors). In [9] a quantum MEP hydrodynamical model has been devised for charge transport in the parabolic band approximation by introducing quantum correction based on the equilibrium Wigner function [30]. An extension to electron moving in pristine graphene has been obtained in [29]. Here we present a quantum hydrodynamical model which is valid for a general energy band considering a closure of the moment system deduced by the Wigner equation resorting to a quantum version of MEP. Explicit formulas for quantum correction at order \hbar^2 are obtained with the aid of the Moyal calculus for silicon and graphene removing the limitation that the quantum corrections are based on the equilibrium Wigner function as in [9, 29]. As an application, quantum correction to the mobilities are deduced.

In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_\varepsilon(u):=\int_\Sigma \frac{1}{2}|du|^2_h\ \,dvol_h +\frac{1}{4\varepsilon^2}(1-|u|^2)^2\,dvol_h=\int_{\Sigma}e_{\varepsilon}(u)$. Our first main result is a precise point-wise estimate for $e_\varepsilon(u_k)$ in the regions where compactness fails, which also implies the $L^{2,1}$ quantization in the bubbling process. Our second main result consists in applying the method developed in a previous joint paper with T. Rivi\`ere to study the upper-semi-continuity of the extended Morse index to sequences of critical points of $E_{\epsilon}$: given a sequence of critical points $u_{\varepsilon_k}:\Sigma\to \mathbb{R}^{n+1}$ of $E_\varepsilon$ that converges in the bubble tree sense to a harmonic map $u_\infty\in W^{1,2}(\Sigma,{S}^{n})$ and bubbles $v^i_{\infty}:\mathbb{R}^2\to {S}^{n}$, we show that the extended Morse indices of the maps $v^i,u_\infty$ control the extended Morse index of the sequence $u_{\varepsilon_k}$ for $k$ large enough.

In this note, the first-order Dickson polynomials are introduced through a particular case of the expression of the trace of the $n^{th}$ power of a matrix in terms of powers of the trace and determinant of the matrix itself. The technique relies on the Cayley-Hamilton theorem and its application to the derivation of formulas due to Carlitz and to second-order Dickson polynomials is straightforward. Finally, generalization of Dickson polynomials over finite fields and multivariate Dickson polynomials are evoked as potential avenues of investigation in the same framework.

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in $\mathbb{P}^4 (\mathbb{F}_{q^2})$ has at most $d(q^5+q^2) + q^3 + 1$ points in common with a threefold of degree $d$ defined over $\mathbb{F}_{q^2}$. He proved the conjecture for $d=2$. In this paper, we show that the conjecture is true for $d = 3$ and $q \ge 7$.

Determining the weight distributions of the projective Reed-Muller codes is a very hard problem and has been studied extensively in the literature. In this article, we provide an alternative proof of the second weight of the projective Reed-Muller codes $\PRM (d, m)$ where $m \ge 3$ and $3 \le d \le \frac{q+3}{2}$. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on $d$. Furthermore, we compute the second weight of $\PRM (d, 2)$ for $3 \le d \le q-1$. Furthermore, we give an upper bound for the third weight of $\PRM(d, 2)$.

We use variations on Lax type operators to find explicit formulas for certain elements of finite $W$-algebras. These give a complete set of generators for all finite $W$-algebras of types B,C,D for which the Dynkin grading is even.

In this paper, we study the impact of the existence of multiple IRSs in a homogeneous wireless network, in which all BSs, users (U), and IRSs are spatially distributed by an independent homogeneous PPP, with density $\lambda_{{\rm BS}}\rm{[BS/m^2]}$, $\lambda_{{\rm U}}\rm{[U/m^2]}$, and $\lambda_{{\rm IRS}}\rm{[IRS/m^2]}$, respectively. We utilize a uniformly random serving strategy for BS and IRS to create stochastic symmetry in the network. We analyze the performance of the network and study the effect of the existence of the IRS on the network performance. To this end, for a typical user in the system, we derive analytical upper and lower bounds on the expectation of the power (second statistical moment) of the desired signal and the interference caused by BSs and other users. After that, we obtain analytical upper bounds on the decay of the probability of the power of the desired signal and the interference for the typical user (which results in a lower bound for the cumulative distribution function (CDF)). Moreover, we derive upper bounds on the decay of the probability of the capacity of one typical user, which results in a lower bound for the outage probability. In the numerical results, we observe that the numerical calculation of the power of the desired signal and the interference is near the derived lower bounds and we show that the increment of the parameter ${(\lambda_{\rm IRS})}$ causes increment in powers of both the desired and interference signals. We also observe that the increment of the parameter ${\lambda_{\rm IRS}}$ causes the decrement of outage probability.

Positivstellens{\"a}tze are a group of theorems on the positivity of involution algebras over $\mathbb{R}$ or $\mathbb{C}$. One of the most well-known Positivstellensatz is the solution to Hilbert's 17th problem given by E. Artin, which asserts that a real polynomial in $n$ commutative variables is nonnegative on real affine space if and only if it is a sum of fractional squares. Let $m$ and $n$ be two positive integers. For the free group $F_n$ generated by $n$ letters, and a symmetric polynomial $b$ with variables in $F_n$ and with $n$-by-$n$ complex matrices coefficients, we use real algebraic geometry to give a new proof showing that $b$ is a sum of Hermitian squares if and only if $b$ is mapped to a positive semidefinite matrix under any finitely dimensional unitary representation of $F_n$.

We study objects whose definition is linked to partial orders. An algebra over the operad of posets is a set whose endomorphisms contain the endomorphisms of posets. Notable examples of such algebras include the zeta values, order polytopes, and shuffle series. The language of operads provides us with tools to better understand the combinatorics of these objects. In these notes we describe a non-trivial example of a suboperad ``The Wixarika posets'', alongside its associated algebras. This example is rich enough to showcase the particularities of the field, without delving into technicalities. An old version of these notes were used to give a minicourse in Govt. College University Lahore in Pakistan 2022, and at the BIRS-CMO workshop ``6th meeting of the Mexican Mathematicians in the World'' 2024.

We develop a new parallel algorithm for minimizing Lipschitz, convex functions with a stochastic subgradient oracle. The total number of queries made and the query depth, i.e., the number of parallel rounds of queries, match the prior state-of-the-art, [CJJLLST23], while improving upon the computational depth by a polynomial factor for sufficiently small accuracy. When combined with previous state-of-the-art methods our result closes a gap between the best-known query depth and the best-known computational depth of parallel algorithms. Our method starts with a ball acceleration framework of previous parallel methods, i.e., [CJJJLST20, ACJJS21], which reduce the problem to minimizing a regularized Gaussian convolution of the function constrained to Euclidean balls. By developing and leveraging new stability properties of the Hessian of this induced function, we depart from prior parallel algorithms and reduce these ball-constrained optimization problems to stochastic unconstrained quadratic minimization problems. Although we are unable to prove concentration of the asymmetric matrices that we use to approximate this Hessian, we nevertheless develop an efficient parallel method for solving these quadratics. Interestingly, our algorithms can be improved using fast matrix multiplication and use nearly-linear work if the matrix multiplication exponent is 2.

Facility location problems have been a major research area of interest in the last several decades. In particular, uncapacitated location problems (ULP) have enormous applications. Variations of ULP often appear, especially as large-scale subproblems in more complex combinatorial optimization problems. Although many researchers have studied different versions of ULP (e.g., uncapacitated facility location problem (UCFLP) and p-Median problem), most of these authors have considered small to moderately sized problems. In this paper, we address the ULP and provide a fast adaptive meta-heuristic for large-scale problems. The approach is based on critical event memory tabu search. For the diversification component of the algorithm, we have chosen a procedure based on a sequencing problem commonly used for traveling salesman-type problems. The efficacy of this approach is evaluated across a diverse range of benchmark problems sourced from the Internet, with a comprehensive comparison against four prominent algorithms in the literature. The proposed adaptive critical event tabu search (ACETS) demonstrates remarkable effectiveness for large-scale problems. The algorithm successfully solved all problems optimally within a short computing time. Notably, ACETS discovered three best new solutions for benchmark problems, specifically for Asymmetric 500A-1, Asymmetric 750A-1, and Symmetric 750B-4, underscoring its innovative and robust nature.

Reconfigurable intelligent surfaces (RISs) have emerged as a promising technology to enhance the performance of sixth-generation (6G) and beyond communication systems. The passive nature of RISs and their large number of reflecting elements pose challenges to the channel estimation process. The associated complexity further escalates when the channel coefficients are fast-varying as in scenarios with user mobility. In this paper, we propose an extended channel estimation framework for RIS-assisted multiple-input multiple-output (MIMO) systems based on a convolutional neural network (CNN) integrated with an autoregressive (AR) predictor. The implemented framework is designed for identifying the aging pattern and predicting enhanced estimates of the wireless channels in correlated fast-fading environments. Insightful simulation results demonstrate that our proposed CNN-AR approach is robust to channel aging, exhibiting a high-precision estimation accuracy. The results also show that our approach can achieve high spectral efficiency and low pilot overhead compared to traditional methods.

Though achieving marvelous progress in various scenarios, existing semantic communication frameworks mainly consider single-input single-output Gaussian channels or Rayleigh fading channels, neglecting the widely-used multiple-input multiple-output (MIMO) channels, which hinders the application into practical systems. One common solution to combat MIMO fading is to utilize feedback MIMO channel state information (CSI). In this paper, we incorporate MIMO CSI into system designs from a new perspective and propose the learnable CSI fusion semantic communication (LCFSC) framework, where CSI is treated as side information by the semantic extractor to enhance the semantic coding. To avoid feature fusion due to abrupt combination of CSI with features, we present a non-invasive CSI fusion multi-head attention module inside the Swin Transformer. With the learned attention masking map determined by both source and channel states, more robust attention distribution could be generated. Furthermore, the percentage of mask elements could be flexibly adjusted by the learnable mask ratio, which is produced based on the conditional variational interference in an unsupervised manner. In this way, CSI-aware semantic coding is achieved through learnable CSI fusion masking. Experiment results testify the superiority of LCFSC over traditional schemes and state-of-the-art Swin Transformer-based semantic communication frameworks in MIMO fading channels.

We prove that for any initial data on a genus zero spectral curve the corresponding correlation differentials of topological recursion are KP integrable. As an application we prove KP integrability of partition functions associated via ELSV-type formulas to the $r$-th roots of the twisted powers of the log canonical bundles.

We show that under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. This in particular leads to a considerable strengthening of our previous work on the Shafarevich conjecture. A key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group of Hodge modules, and to play this off against an improvement of the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi.

The study of marriage dynamics and of strategies to reduce the likelihood of divorce has been an important research area for many years. Gottman's research on successful marriages revealed three matched interaction styles: conflict-avoiding, validating, and volatile. There has, however, been little progress in explaining how couples develop these styles of interaction and why failure to do so leads to failed marriages. In this paper, we show that these interaction styles arise as solutions to an optimal control problem where the couples jointly maximize a common goal. The validating style arises when the benefit from achieving joint happiness is balanced by the emotional cost of adopting a particular style. The ubiquitous conflict-avoider style arises naturally when the couple does not care about the cost. The volatile style is not an optimal solution, but volatile marriages may still be successful for couples with highly positive natural dispositions. The problem of the spouses having different goals in marriage is relevant to marriage repair, and this problem will be studied in the next paper using differential game theory.

In a recent work, Bo'az Klartag showed that, given a convex body with minimal volume product, its isotropic constant is related to its volume product. As a consequence, he obtained that a strong version of the slicing conjecture implies Mahler's conjecture. In this work, we extend these geometrical results to the realm of log-concave functions. In this regard, the functional analogues of the projective perturbations of the body are the log-Laplace perturbations of the function. The differentiation along these transformations is simplified thanks to the known properties of the log-Laplace transform. Moreover, we show that achieving such an analogous result requires the consideration of the suitable version of the isotropic constant, notably the one incorporating the entropy. Finally, an investigation into the equivalences between the functional and geometrical strong forms of the slicing conjecture is provided.

We study the structure of adic curves over an affinoid field of arbitrary rank. In particular, quite analogously to Berkovich geometry we classify points on curves, prove a semistable reduction theorem in the version of Ducros' triangulations, define associated curve skeletons and prove that they are deformational retracts in a suitable sense. An important new technical tool is an appropriate compactification of ordered groups that we call the ranger compactification. Intervals of rangers are then used to define metric structures and construct deformational retractions.

A $\mathbf{GL}$-variety is a (typically infinite dimensional) variety modeled on the polynomial representation theory of the general linear group. In previous work, we studied these varieties in characteristic 0. In this paper, we obtain results in positive characteristic: for example, we prove a version of Chevalley's theorem on constructible sets. We give an application of our theory to strength of polynomials.

This paper introduces heat semigroups of topological Markov chains and Cuntz-Krieger algebras by means of spectral noncommutative geometry. Using recent advances on the logarithmic Dirichlet Laplacian on Ahlfors regular metric-measure spaces, we construct spectral triples on Cuntz-Krieger algebras from singular integral operators. These spectral triples exhaust K-homology and for Cuntz algebras we can compute their heat operators explicitly as Riesz potential operators. We also describe their isometry group in terms of the automorphism group of the underlying directed graph and prove that the Voiculescu noncommutative topological entropy vanishes on isometries.

For positive integers $n\ge s> r$, the Tur\'an function $T(n,s,r)$ is the smallest size of an r-graph with n vertices such that every set of s vertices contains at least one edge. Also, define the Tur\'an density $t(s,r)$ as the limit of $T(n,s,r)/ {n\choose r}$ as $n\to\infty$. The question of estimating these parameters received a lot of attention after it was first raised by Tur\'an in 1941. A trivial lower bound is $t(s,r)\ge 1/{s\choose s-r}$. In the early 1990s, de Caen conjectured that $r\cdot t(r+1,r)\to\infty$ as $r\to\infty$. We disprove this conjecture by showing more strongly that for every integer $R\ge1$ there is $\mu_R$ (in fact, $\mu_R$ can be taken to grow as $(1+o(1))\, R\ln R$) such that $t(r+R,r)\le (\mu_R+o(1))/ {r+R\choose R}$ as $r\to\infty$, that is, the trivial lower bound is tight for every $R$ up to a multiplicative constant $\mu_R$.

For an index set $\Gamma$ and a cardinal number $\kappa$ the $\Sigma_{\kappa}$-product of real lines $\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$ consist of all elements of $\mathbb{R}^{\Gamma}$ with $<\kappa$ nonzero coordinates. A compact space is $\kappa$-Corson if it can be embedded into $\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$ for some $\Gamma$. We also consider a class of compact spaces wider than the class of $\omega$-Corson compact spaces, investigated by Nakhmanson and Yakovlev as well as Marciszewski, Plebanek and Zakrzewski called $NY$ compact spaces. For a Tychonoff space $X$, let $C_{p}(X)$ be the space of real continuous functions on the space $X$, endowed with the pointwise convergence topology. We present here a characterisation of $\kappa$-Corson compact spaces $K$ for regular, uncountable cardinal numbers $\kappa$ in terms of function spaces $C_{p}(K)$, extending a theorem of Bell and Marciszewski and a theorem of Pol. We also prove that classes of $NY$ compact spaces and $\omega$-Corson compact spaces $K$ are preserved by linear homeomorphisms of function spaces $C_{p}(K)$.

We prove existence and uniqueness of complex Hodge structures on modular functors, under the assumptions that their conformal blocks are semisimple as representations of mapping class groups and that the associated categories are ribbon. These $2$ assumptions are satisfied by modular functors coming from Lie algebras, and it is an open conjecture that they are always satisfied. The proof is based on the non-abelian Hodge correspondence and Ocneanu rigidity. Given a modular functor, we explain how its Hodge numbers fit into a Frobenius algebra and the Chern characters of its Hodge decompositions into a new cohomological field theory (CohFT). In the case of $\mathrm{SU}(2)$ modular functors of level $2$ times an odd number, we give explicit formulas for all Hodge numbers, in any genus $g$.

In this article, we discuss the existence and asymptotically autonomous robustness (AAR) (almost surely) of random attractors for 3D stochastic globally modified Navier-Stokes equations (SGMNSE) on Poincar\'e domains (which may be bounded or unbounded). Our aim is to investigate the existence and AAR of random attractors for 3D SGMNSE when the time-dependent forcing converges to a time-independent function under the perturbation of linear multiplicative noise as well as additive noise. The main approach is to provide a way to justify that, on some uniformly tempered universe, the usual pullback asymptotic compactness of the solution operators is uniform across an infinite time-interval $(-\infty,\tau]$. The backward uniform ``tail-smallness'' and ``flattening-property'' of the solutions over $(-\infty,\tau]$ have been demonstrated to achieve this goal. To the best of our knowledge, this is the first attempt to establish the existence as well as AAR of random attractors for 3D SGMNSE on unbounded domains.

We introduce a new concept, the APN-defect, which can be thought of as measuring the distance of a given function $G:\mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n}$ to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity) $G$ using the so-called difference squares. We describe the relations between the APN-defect and other recent concepts of similar nature. Upper and lower bounds for the values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. Its exact values in some cases are also calculated. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending on $n$ is evaluated and the implications are discussed. In the forthcoming second part of this work we further examine modifications of the inverse function. We also study modifications of classes of functions of low uniformity over infinitely many extensions of $\mathbb{F}_{2^n}$. We present quantitative results on their differential behaviour, especially in connection with their APN-defects.

In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper, for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson's rank conjectures. Specifically, we give a new proof of Ramanujan's congruences $p(5n+4)\equiv 0\pmod 5$ and $p(7n+5)\equiv 0\pmod 7$.

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of Campos, Jenssen, Michelen, and Sahasrabudhe in the $\ell_2$ norm. Our main tools are the graph-theoretic result used in the $\ell_2$ construction and recent progress on the Bourgain slicing problem.

We study the real Rel orbits of some translation surfaces in the stratum $\mathcal{H}(1,1)$. Specifically, surfaces that are tremors of the locus of branched double covers of tori. We give necessary and sufficient conditions on tremors of a surface so that the real Rel orbit is recurrent. As a consequence, we are able to provide explicit examples of trajectories of real Rel that are not recurrent.

The rank of a tiling's return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first \v Cech cohomology $\check H^1(\Omega)$ gives upper and lower bounds for the size of the return module. For all sufficiently large patches, the rank of the return module is at most the same as the rank of the cohomology. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(\Omega)$. Therefore, for generic tile shapes and sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(\Omega)$.

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there is a set $\mathcal{G}\subset (0,\infty)$ with density $1$ at infinity such that for every $V\in \mathcal{G}$ there is a unique isoperimetric set of volume $V$ in $M$, and its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface that satisfies the previous assumptions, and with the following property: there are arbitrarily large and diverging intervals $I_n\subset (0,\infty)$ such that isoperimetric sets with volumes $V\in I_n$ exist, but they are neither unique nor they have strictly volume preserving stable boundaries. The proof relies on a set of new ideas, as the present setting goes beyond the range of applicability of the methods based on the implicit function theorem, and no symmetry is assumed.

In the present paper, we investigate a conjecture of J\"urgen Herzog. Let $S$ be a local regular ring with residue field $K$ or a positively graded $K$-algebra, $I\subset S$ be a perfect ideal of grade two, and let $R=S/I$ with canonical module $\omega_R$. Herzog conjectured that the canonical trace $\text{tr}(\omega_R)$ is obtained by specialization from the generic case of maximal minors. We prove this conjecture in several cases, and present a criterion that guarantees that the canonical trace specializes under some additional assumptions. As the final conclusion of all of our results, we classify the nearly Gorenstein monomial ideals of height two.

Motivated by the work of Bryant on constant mean curvature (CMC) $1$-immersions of surfaces into the hyperbolic space H^3 and after the results of Tarantello (2023), we pursue a possible parametrization for the moduli space of (CMC) 1-immersions of a surface S (closed, orientable and of genus >1) into hyperbolic 3-manifolds. Those immersions enter as "critical" object in our analysis. In fact, they can be attained only as limits of the (CMC) c-immersions (as c tends to 1), obtained in Huang-Lucia-Tarantello (2022), for |c|<1. However, such passage to the limit can be prevented by possible blow-up phenomena, so that the pullback metrics of the (CMC) c-immersions may yield (at the limit) to a singular metric with conical singularities at finitely many points (the blow-up points). In case of genus g=2, blow up can occur only at a single point, and in Tarantello (2023) it was shown how it could be prevented and the passage to the limit ensured in terms of the Kodaira map. In this note we sharpen this result and for genus g=2, we obtaina condition (we believe sharp) which involves only the Kodaira map on the six Weierstrass points. In addition we tackle the case of higher genus, where multiple blow-up points occur. In this case, we need to identify a suitable replacement of the Kodaira map, now defined on the space of non-zero effective divisors. More importantly, we need to improve in a substantial way the asymptotic analysis of Tarantello (2023) limited to the case of "blow-up" with minimal mass. In this direction we give a contribution which best applies to the case of genus g=3, but also provides a relevant step and a convincing indication on what should happen in the general case.

Change of numeraire is a classical tool in mathematical finance. Campi-Laachir-Martini established its applicability to martingale optimal transport. We note that the results of Campi-Laachir-Martini extend to the case of weak martingale transport. We apply this to shadow couplings, continuous time martingale transport problems in the framework of Huesmann-Trevisan and in particular to establish the correspondence between stretched Brownian motion with its geometric counterpart. Note: We emphasize that we learned about the geometric stretched Brownian motion gSBM (defined in PDE terms) in a presentation of Loeper \cite{Lo23} before our work on this topic started. We noticed that a change of numeraire transformation in the spirit of \cite{CaLaMa14} allows for an alternative viewpoint in the weak optimal transport framework. We make our work public following the publication of Backhoff-Loeper-Obloj's work \cite{BaLoOb24} on arxiv.org. The article \cite{BaLoOb24} derives gSBM using PDE techniques as well as through an independent probabilistic approach which is close to the one we give in the present article.

Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff dimension there is no such restriction. This thesis focuses on a family of dimensions parameterised by $\theta \in (0,1)$, called the intermediate dimensions, which are defined by requiring that $\mbox{diam}(U) \leq (\mbox{diam}(V))^{\theta}$ for all sets $U,V$ in the cover. We begin by generalising the intermediate dimensions to allow for greater refinement in how the relative sizes of the covering sets are restricted. These new dimensions can recover the interpolation between Hausdorff and box dimension for compact sets whose intermediate dimensions do not tend to the Hausdorff dimension as $\theta \to 0$. We also use a Moran set construction to prove a necessary and sufficient condition, in terms of Dini derivatives, for a given function to be realised as the intermediate dimensions of a set. We proceed to prove that the intermediate dimensions of limit sets of infinite conformal iterated function systems are given by the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This applies to sets defined using continued fraction expansions, and has applications to dimensions of projections, fractional Brownian images, and general H\"older images. Finally, we determine a formula for the intermediate dimensions of all self-affine Bedford-McMullen carpets. The functions display features not witnessed in previous examples, such as having countably many phase transitions. We deduce that two carpets have equal intermediate dimensions if and only if the multifractal spectra of the corresponding uniform Bernoulli measures coincide.

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\qtq{on}\R\times\R^N, \] with $\alpha=\tfrac{4-2b}{N-2}$, $N=\{3,4,5\}$ and $0<b\leq \min\Big\{\tfrac{6-N}{2},\tfrac{4}{N}$\Big\}. This paper establishes global well-posedness and scattering for the non-radial energy-critical case in $\dot{H}^1(\R^N)$. It extends the previous research by Murphy and the first author \cite{GM}, which focused on the case $(N,\alpha,b)=(3,2,1)$. The novelty here, beyond considering higher dimensions, lies in our assumption of the condition $\sup_{t\in I}\|\nabla u(t)\|_{L^2}<\|\nabla Q\|_{L^2}$, which is weaker than the condition stated in \cite{Guzman}. Consequently, if a solution has energy and kinetic energy less than the ground state $Q$ at some point, then the solution is global and scatters. Moreover, we show scattering for the defocusing case. On the other hand, in this work, we also investigate the blow-up issue with nonradial data for $N\geq 3$ in $H^1(\mathbb{R}^N)$. This implies that our result holds without classical assumptions such as spherically symmetric data or $|x|u_0 \in L^2(\mathbb{R}^N)$. \ \noindent Mathematics Subject Classification. 35A01, 35QA55, 35P25.

We study the expressivity of ReLU neural networks in the setting of a binary classification problem from a topological perspective. Recently, empirical studies showed that neural networks operate by changing topology, transforming a topologically complicated data set into a topologically simpler one as it passes through the layers. This topological simplification has been measured by Betti numbers, which are algebraic invariants of a topological space. We use the same measure to establish lower and upper bounds on the topological simplification a ReLU neural network can achieve with a given architecture. We therefore contribute to a better understanding of the expressivity of ReLU neural networks in the context of binary classification problems by shedding light on their ability to capture the underlying topological structure of the data. In particular the results show that deep ReLU neural networks are exponentially more powerful than shallow ones in terms of topological simplification. This provides a mathematically rigorous explanation why deeper networks are better equipped to handle complex and topologically rich data sets.

In studies of quantum squeezing, the emphasis is typically placed more on specific squeezed states and their evolution rather than on the dynamical operations that could simultaneously squeeze a broader range of quantum states, regardless of their initial configuration. We explore new developments in this area, facilitated by gently acting external fields which might induce squeezing of the canonical observables $q$ and $p$ in charged particles. The extensive research in this field has yielded many valuable insights, raising the question of whether there is still room for significant new contributions to our understanding. Nonetheless, we present some exactly solvable instances of this problem, observed in symmetric evolution intervals. These intervals allow for the explicit determination of the temporal dependence of external fields necessary to generate the required evolution operators. Our findings are linked with a straightforward application of Toeplitz matrices, offering a more accessible description of the problem compared to the frequently employed Ermakov-Milne invariants.

This paper delves into recent advancements in Quantum Reinforcement Learning (QRL), particularly focusing on non-commutative environments, which represent uncharted territory in this field. Our research endeavors to redefine the boundaries of decision-making by introducing formulations and strategies that harness the inherent properties of quantum systems. At the core of our investigation characterization of the agent's state space within a Hilbert space ($\mathcal{H}$). Here, quantum states emerge as complex superpositions of classical state introducing non-commutative quantum actions governed by unitary operators, necessitating a reimagining of state transitions. Complementing this framework is a refined reward function, rooted in quantum mechanics as a Hermitian operator on $\mathcal{H}$. This reward function serves as the foundation for the agent's decision-making process. By leveraging the quantum Bellman equation, we establish a methodology for maximizing expected cumulative reward over an infinite horizon, considering the entangled dynamics of quantum systems. We also connect the Quantum Bellman Equation to the Degree of Non Commutativity of the Environment, evident in Pure Algebra. We design a quantum advantage function. This ingeniously designed function exploits latent quantum parallelism inherent in the system, enhancing the agent's decision-making capabilities and paving the way for exploration of quantum advantage in uncharted territories. Furthermore, we address the significant challenge of quantum exploration directly, recognizing the limitations of traditional strategies in this complex environment.

Graph Convolutional Networks (GCN) are Graph Neural Networks where the convolutions are applied over a graph. In contrast to Convolutional Neural Networks, GCN's are designed to perform inference on graphs, where the number of nodes can vary, and the nodes are unordered. In this study, we address two important challenges related to GCNs: i) oversmoothing; and ii) the utilization of node relational properties (i.e., heterophily and homophily). Oversmoothing is the degradation of the discriminative capacity of nodes as a result of repeated aggregations. Heterophily is the tendency for nodes of different classes to connect, whereas homophily is the tendency of similar nodes to connect. We propose a new strategy for addressing these challenges in GCNs based on Transfer Entropy (TE), which measures of the amount of directed transfer of information between two time varying nodes. Our findings indicate that using node heterophily and degree information as a node selection mechanism, along with feature-based TE calculations, enhances accuracy across various GCN models. Our model can be easily modified to improve classification accuracy of a GCN model. As a trade off, this performance boost comes with a significant computational overhead when the TE is computed for many graph nodes.

A natural definition for instanton density operator in lattice QCD has been long desired. We show this problem is, and has to be, resolved by higher category theory. The problem is resolved by refining at a conceptual level the Yang-Mills theory on lattice, in order to recover the homotopy information in the continuum, which would have been lost if we put the theory on lattice in the traditional way. The refinement needed is a generalization -- through the lens of higher category theory -- of the familiar process of Villainization that captures winding in lattice XY model and Dirac quantization in lattice Maxwell theory. The apparent difference is that Villainization is in the end described by principal bundles, hence familiar, but more general topological operators can only be captured on the lattice by more flexible structures beyond the usual group theory and fibre bundles, hence the language of categories becomes natural and necessary. The key structure we need for our particular problem is called multiplicative bundle gerbe, based upon which we can construct suitable structures to naturally define the 2d Wess-Zumino-Witten term, 3d skyrmion density operator and 4d hedgehog defect for lattice $S^3$ (pion vacua) non-linear sigma model, and the 3d Chern-Simons term, 4d instanton density operator and 5d Yang monopole defect for lattice $SU(N)$ Yang-Mills theory. In a broader perspective, higher category theory enables us to rethink more systematically the relation between continuum quantum field theory and lattice quantum field theory. We sketch a proposal towards a general machinery that constructs the suitably refined lattice degrees of freedom for a given non-linear sigma model or gauge theory in the continuum, realizing the desired topological operators on the lattice.

We present exact solutions describing dynamics of two algebraic solitons in the massive Thirring model. Each algebraic soliton corresponds to a simple embedded eigenvalue in the Kaup--Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. By coalescence of speeds of the two algebraic solitons, we find a new solution for an algebraic double-soliton which corresponds to a double embedded eigenvalue. We show that the double-soliton attains the double mass of a single soliton and describes a slow interaction of two identical algebraic solitons.

In this paper, we are aiming to propose a novel mathematical model that studies the dynamics of synaptic damage in terms of concentrations of toxic neuropeptides/neurotransmitters during neurotransmission processes. Our primary objective is to employ Wardrop's first and second principles within a neural network of the brain. In order to comprehensively incorporate Wardrop's first and second principles into the neural network of the brain, we introduce two novel concepts: \textit{neuropeptide's (neurotransmitter's) equilibrium} and \textit{synapses optimum}. The \textit{neuropeptide/neurotransmitter equilibrium} refers to \textit{a distribution of toxic neuropeptides/neurotransmitters that leads to uniform damage across all synaptic links}. Meanwhile, \textit{synapses optimum} is \textit{the most desirable distribution of toxic neuropeptides/neurotransmitters that minimizes the cumulative damage experienced by all synapses}. In the context of a neural network within the brain, an analogue of the price of anarchy is \textit{the price of cognition} which is \textit{the most unfavorable ratio between the overall impairment caused by toxic neuropeptide's (neurotransmitter's) equilibrium in comparison to the optimal state of synapses (synapses optimum)}. To put it differently, \textit{the price of cognition} measures \textit{the loss of cognitive ability resulting from increased concentrations of toxic neuropeptides/neurotransmitters}. Additionally, a replicator equation is proposed within this framework that leads to the establishment of the synapses optimum during the neurotransmission process.

Before we attempt to learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: multiple observation processes of the system. We thus determine (a) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (b) what information is unrelated to these common observables, and therefore particular to each observation process, and not contributing to the desired function. Any data-driven function approximation technique can subsequently be used to learn the input-output relation, from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular ``twists'' of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations of one process to entire level sets of measurements of the process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures ``now'', while the second observation process measures ``in the future'', the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.

Many multi-agent systems in practice are decentralized and have dynamically varying dependencies. There has been a lack of attempts in the literature to analyze these systems theoretically. In this paper, we propose and theoretically analyze a decentralized model with dynamically varying dependencies called the Locally Interdependent Multi-Agent MDP. This model can represent problems in many disparate domains such as cooperative navigation, obstacle avoidance, and formation control. Despite the intractability that general partially observable multi-agent systems suffer from, we propose three closed-form policies that are theoretically near-optimal in this setting and can be scalable to compute and store. Consequentially, we reveal a fundamental property of Locally Interdependent Multi-Agent MDP's that the partially observable decentralized solution is exponentially close to the fully observable solution with respect to the visibility radius. We then discuss extensions of our closed-form policies to further improve tractability. We conclude by providing simulations to investigate some long horizon behaviors of our closed-form policies.

In this paper, we propose a local squared Wasserstein-2 (W_2) method to solve the inverse problem of reconstructing models with uncertain latent variables or parameters. A key advantage of our approach is that it does not require prior information on the distribution of the latent variables or parameters in the underlying models. Instead, our method can efficiently reconstruct the distributions of the output associated with different inputs based on empirical distributions of observation data. We demonstrate the effectiveness of our proposed method across several uncertainty quantification (UQ) tasks, including linear regression with coefficient uncertainty, training neural networks with weight uncertainty, and reconstructing ordinary differential equations (ODEs) with a latent random variable.

We describe how to calculate standard errors for A/B tests that include clustered data, ratio metrics, and/or covariate adjustment. We may do this for power analysis/sample size calculations prior to running an experiment using historical data, or after an experiment for hypothesis testing and confidence intervals. The different applications have a common framework, using the sample variance of certain residuals. The framework is compatible with modular software, can be plugged into standard tools, doesn't require computing covariance matrices, and is numerically stable. Using this approach we estimate that covariate adjustment gives a median 66% variance reduction for a key metric, reducing experiment run time by 66%.

We lay the groundwork for a UV-complete formulation of the Euclidean Jackiw-Teitelboim two-dimensional models of quantum gravity when the boundary lengths are finite, emphasizing the discretized approach. The picture that emerges is qualitatively new. For the disk topology, the problem reduces to counting so-called self-overlapping curves, that are closed loops that bound a distorted disk, with an appropriate multiplicity. We build a matrix model that does the correct counting. The theories in negative, zero and positive curvatures have the same UV description but drastically different macroscopic properties. The Schwarzian theory emerges in the limit of very large and negative cosmological constant in the negative curvature model, as an effective theory valid on distance scales much larger than the curvature length scale. In positive curvature, we argue that large geometries are ubiquitous and that the theory exists only for positive cosmological constant. Our discussion is pedagogical and includes a review of several relevant topics.

The transformer architecture has prevailed in various deep learning settings due to its exceptional capabilities to select and compose structural information. Motivated by these capabilities, Sanford et al. proposed the sparse token selection task, in which transformers excel while fully-connected networks (FCNs) fail in the worst case. Building upon that, we strengthen the FCN lower bound to an average-case setting and establish an algorithmic separation of transformers over FCNs. Specifically, a one-layer transformer trained with gradient descent provably learns the sparse token selection task and, surprisingly, exhibits strong out-of-distribution length generalization. We provide empirical simulations to justify our theoretical findings.

Individual species may experience diverse outcomes, from prosperity to extinction, in an ecological community subject to external and internal variations. Despite the wealth of theoretical results derived from random matrix ensembles, a theoretical framework still remains to be developed to understand species-level dynamical heterogeneity within a given community, hampering real-world ecosystems' theoretical assessment and management. Here, we consider empirical plant-pollinator mutualistic networks, additionally including all-to-all intragroup competition, where species abundance evolves under a Lotka-Volterra-type equation. Setting the strengths of competition and mutualism to be uniform, we investigate how individual species persist or go extinct under varying the interaction strengths. By employing bifurcation theory in tandem with numerical continuation, we elucidate transcritical bifurcations underlying species extinction and demonstrate that the Hopf bifurcation of unfeasible equilibria and degenerate transcritical bifurcations give rise to multistability, i.e., the coexistence of multiple attracting feasible equilibria. These bifurcations allow us to partition the parameter space into different regimes, each with distinct sets of extinct species, offering insights into how interspecific interactions generate one or multiple extinction scenarios within an ecological network.

A simple and intuitive method for feature selection consists of choosing the feature subset that maximizes a nonparametric measure of dependence between the response and the features. A popular proposal from the literature uses the Hilbert-Schmidt Independence Criterion (HSIC) as the nonparametric dependence measure. The rationale behind this approach to feature selection is that important features will exhibit a high dependence with the response and their inclusion in the set of selected features will increase the HSIC. Through counterexamples, we demonstrate that this rationale is flawed and that feature selection via HSIC maximization can miss critical features.

A researcher collecting data from a randomized controlled trial (RCT) often has access to an auxiliary observational dataset that may be confounded or otherwise biased for estimating causal effects. Common modeling assumptions impose restrictions on the outcome mean function - the conditional expectation of the outcome of interest given observed covariates - in the two datasets. Running examples from the literature include settings where the observational dataset is subject to outcome-mediated selection bias or to confounding bias taking an assumed parametric form. We propose a succinct framework to derive the efficient influence function for any identifiable pathwise differentiable estimand under a general class of restrictions on the outcome mean function. This uncovers surprising results that with homoskedastic outcomes and a constant propensity score in the RCT, even strong parametric assumptions cannot improve the semiparametric lower bound for estimating various average treatment effects. We then leverage double machine learning to construct a one-step estimator that achieves the semiparametric efficiency bound even in cases when the outcome mean function and other nuisance parameters are estimated nonparametrically. The goal is to empower a researcher with custom, previously unstudied modeling restrictions on the outcome mean function to systematically construct causal estimators that maximially leverage their assumptions for variance reduction. We demonstrate the finite sample precision gains of our estimator over existing approaches in extensions of various numerical studies and data examples from the literature.

Mixed Integer Linear Programming (MILP) is a fundamental tool for modeling combinatorial optimization problems. Recently, a growing body of research has used machine learning to accelerate MILP solving. Despite the increasing popularity of this approach, there is a lack of a common repository that provides distributions of similar MILP instances across different domains, at different hardness levels, with standardized test sets. In this paper, we introduce Distributional MIPLIB, a multi-domain library of problem distributions for advancing ML-guided MILP methods. We curate MILP distributions from existing work in this area as well as real-world problems that have not been used, and classify them into different hardness levels. It will facilitate research in this area by enabling comprehensive evaluation on diverse and realistic domains. We empirically illustrate the benefits of using Distributional MIPLIB as a research vehicle in two ways. We evaluate the performance of ML-guided variable branching on previously unused distributions to identify potential areas for improvement. Moreover, we propose to learn branching policies from a mix of distributions, demonstrating that mixed distributions achieve better performance compared to homogeneous distributions when there is limited data and generalize well to larger instances.

Given a graph $G(V,E)$, a vertex subset S of G is called an open packing in G if no pair of distinct vertices in S have a common neighbour in G. The size of a largest open packing in G is called the open packing number of G and is denoted by $\rho^o(G)$. It would be interesting to note that the open packing number is a lower bound for the total domination number in graphs with no isolated vertices [Henning and Slater, 1999]. Given a graph G and a positive integer k, the decision problem OPEN PACKING tests whether G has an open packing of size at least k. The optimization problem MAX-OPEN PACKING takes a graph G as input and finds the open packing number of G. It is known that OPEN PACKING is NP-complete on split graphs (i.e., the class of $\{2K_2,C_4,C_5\}$-free graphs) [Ramos et al., 2014]. In this work, we complete the study on the complexity (P vs NPC) of OPEN PACKING on H-free graphs for every graph H with at least three vertices by proving that OPEN PACKING is (i) NP-complete on $K_{1,3}$-free graphs and (ii) polynomial time solvable on $(P_4\cup rK_1)$-free graphs for every $r\geq 1$. In the course of proving (ii), we show that for every $t\in {2,3,4}$ and $r\geq 1$, if G is a $(P_t\cup rK_1)$-free graph, then $\rho^o(G)$ is bounded above by a linear function of r. Moreover, we show that OPEN PACKING parameterized by solution size is W[1]-complete on $K_{1,3}$-free graphs and MAX-OPEN PACKING is hard to approximate within a factor of $n^{(\frac{1}{2}-\delta)}$ for any $\delta>0$ on $K_{1,3}$-free graphs unless P=NP. Further, we prove that OPEN PACKING is (a) NP-complete on $K_{1,4}$-free split graphs and (b) polynomial time solvable on $K_{1,3}$-free split graphs. We prove a similar dichotomy result on split graphs with degree restrictions on the vertices in the independent set of the clique-independent set partition of the split graphs.

This work proposes a provably convergent and low complexity optimization algorithm for the maximization of the secrecy energy efficiency in the uplink of a wireless network aided by a Reconfigurable Intelligent Surface (RIS), in the presence of an eavesdropper. The mobil users' transmit powers and the RIS reflection coefficients are optimized. Numerical results show the performance of the proposed methods and compare the use of active and nearly-passive RISs from an energy-efficient perspective.

End-to-end topological learning using 1-parameter persistence is well-known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique called GRIL. We establish a theoretical foundation of differentiating GRIL producing D-GRIL. We show that D-GRIL can be used to learn a bifiltration function on standard benchmark graph datasets. Further, we exhibit that this framework can be applied in the context of bio-activity prediction in drug discovery.

This paper is the initial part of a comprehensive study of spacetimes that admit the canonical forms of Killing tensor in General Relativity. Our scope is to derive either new exact solutions of Einstein's equations or to determine the hidden symmetries of the already known ones. In this preliminary work we first introduce the canonical forms of Killing tensor. Subsequently, we employ the integrability conditions of each canonical form along with the Einstein field equations (in vacuum with {\Lambda}) and the Bianchi identities in an attempt to create a solvable yet overdetermined system of equations. Finally, we obtain multiple special algebraic solutions according to the Petrov classification (D, III, N, O). The latter becomes possible since our analysis is embodied with the usage of the Newman-Penrose formalism of null tetrads.

We prove that the intrinsic geometry of compact cross-sections of an extremal horizon in four-dimensional Einstein-Maxwell theory must admit a Killing vector field or is static. This implies that any such horizon must be an extremal Kerr-Newman horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant.

Any four-dimensional Supersymmetric Quantum Field Theory with eight supercharges can be associated to a monoidal category of BPS line defects. Any Coulomb vacuum of such a theory can be conjecturally associated to an "algebra of BPS particles'', exemplified by certain Cohomological Hall Algebras. We conjecture the existence of a monoidal functor from the category of line defects to a certain category of bimodules for the BPS Algebra in any Coulomb vacuum. We describe images of simple objects under the conjectural functor and study their monoidal structure in examples. As we vary the choice of vacuum, we expect the collection of functors associated to any given theory to capture the full information of the original monoidal category of lines.

Barrier crossing is a widespread phenomenon across natural and engineering systems. While an abundant cross-disciplinary literature on the topic has emerged over the years, the stochastic underpinnings of the process are yet to be fully understood. We fill this knowledge gap by considering a diffusing particle and presenting a stochastic definition of Brownian motion in the presence of a permeable barrier. This definition relies on reflected Brownian motion and on the crossing events being Poisson processes subordinated by the local time of the underlying motion at the barrier. Within this paradigm we derive the exact expression for the distribution of the number of crossings, and find an experimentally measurable statistical definition of permeability. We employ Feynman-Kac theory to derive and solve a set of governing birth-death diffusion equations and extend them to when barrier permeability is asymmetric. As an application we study a system of infinite, identical and periodically placed asymmetric barriers for which we derive analytically effective transport parameters. This periodic arrangement induces an effective drift at long times whose magnitude depends on the difference in the permeability on either side of the barrier as well as on their absolute values. As the asymmetric permeabilities act akin to localised ``ratchet'' potentials that break spatial symmetry and detailed balance, the proposed arrangement of asymmetric barriers provides an example of a noise-induced drift without the need to time-modulate any external force or create temporal correlations on the motion of a diffusing particle.

This thesis revolves around an area of computer science called "semantics". We work with operational semantics, equational theories, and denotational semantics. The first contribution of this thesis is a study of the commutativity of effects through the prism of monads. Monads are the generalisation of algebraic structures such as monoids, which have a notion of centre: the centre of a monoid is made of elements which commute with all others. We provide the necessary and sufficient conditions for a monad to have a centre. We also detail the semantics of a language with effects that carry information on which effects are central. Moreover, we provide a strong link between its equational theories and its denotational semantics. The second focus of the thesis is quantum computing, seen as a reversible effect. Physically permissible quantum operations are all reversible, except measurement; however, measurement can be deferred at the end of the computation, allowing us to focus on the reversible part first. We define a simply-typed reversible programming language performing quantum operations called "unitaries". A denotational semantics and an equational theory adapted to the language are presented, and we prove that the former is complete. Furthermore, we study recursion in reversible programming, providing adequate operational and denotational semantics to a Turing-complete, reversible, functional programming language. The denotational semantics uses the dcpo enrichment of rig join inverse categories. This mathematical account of higher-order reasoning on reversible computing does not directly generalise to its quantum counterpart. In the conclusion, we detail the limitations and possible future for higher-order quantum control through guarded recursion.

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

Living organisms rely on internal models of the world to act adaptively. These models cannot encode every detail and hence need to compress information. From a cognitive standpoint, information compression can manifest as a distortion of latent representations, resulting in the emergence of representations that may not accurately reflect the external world or its geometry. Rate-distortion theory formalizes the optimal way to compress information, by considering factors such as capacity limitations, the frequency and the utility of stimuli. However, while this theory explains why the above factors distort latent representations, it does not specify which specific distortions they produce. To address this question, here we systematically explore the geometry of the latent representations that emerge in generative models that operate under the principles of rate-distortion theory ($\beta$-VAEs). Our results highlight that three main classes of distortions of internal representations -- prototypization, specialization, orthogonalization -- emerge as signatures of information compression, under constraints on capacity, data distributions and tasks. These distortions can coexist, giving rise to a rich landscape of latent spaces, whose geometry could differ significantly across generative models subject to different constraints. Our findings contribute to explain how the normative constraints of rate-distortion theory distort the geometry of latent representations of generative models of artificial systems and living organisms.

The Coordinate Ascent Variational Inference scheme is a popular algorithm used to compute the mean-field approximation of a probability distribution of interest. We analyze its random scan version, under log-concavity assumptions on the target density. Our approach builds on the recent work of M. Arnese and D. Lacker, \emph{Convergence of coordinate ascent variational inference for log-concave measures via optimal transport} [arXiv:2404.08792] which studies the deterministic scan version of the algorithm, phrasing it as a block-coordinate descent algorithm in the space of probability distributions endowed with the geometry of optimal transport. We obtain tight rates for the random scan version, which imply that the total number of factor updates required to converge scales linearly with the condition number and the number of blocks of the target distribution. By contrast, available bounds for the deterministic scan case scale quadratically in the same quantities, which is analogue to what happens for optimization of convex functions in Euclidean spaces.

The scope of this manuscript is to review some recent developments in statistics for discretely observed semimartingales which are motivated by applications for financial markets. Our journey through this area stops to take closer looks at a few selected topics discussing recent literature. We moreover highlight and explain the important role played by some classical concepts of probability and statistics. We focus on three main aspects: Testing for jumps; rough fractional stochastic volatility; and limit order microstructure noise. We review jump tests based on extreme value theory and complement the literature proposing new statistical methods. They are based on asymptotic theory of order statistics and the R\'{e}nyi representation. The second stage of our journey visits a recent strand of research showing that volatility is rough. We further investigate this and establish a minimax lower bound exploring frontiers to what extent the regularity of latent volatility can be recovered in a more general framework. Finally, we discuss a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices and its probabilistic and statistical foundation.

End-to-end visual communication systems typically optimize a trade-off between channel bandwidth costs and signal-level distortion metrics. However, under challenging physical conditions, this traditional discriminative communication paradigm often results in unrealistic reconstructions with perceptible blurring and aliasing artifacts, despite the inclusion of perceptual or adversarial losses for optimizing. This issue primarily stems from the receiver's limited knowledge about the underlying data manifold and the use of deterministic decoding mechanisms. To address these limitations, this paper introduces DiffCom, a novel end-to-end generative communication paradigm that utilizes off-the-shelf generative priors and probabilistic diffusion models for decoding, thereby improving perceptual quality without heavily relying on bandwidth costs and received signal quality. Unlike traditional systems that rely on deterministic decoders optimized solely for distortion metrics, our DiffCom leverages raw channel-received signal as a fine-grained condition to guide stochastic posterior sampling. Our approach ensures that reconstructions remain on the manifold of real data with a novel confirming constraint, enhancing the robustness and reliability of the generated outcomes. Furthermore, DiffCom incorporates a blind posterior sampling technique to address scenarios with unknown forward transmission characteristics. Extensive experimental validations demonstrate that DiffCom not only produces realistic reconstructions with details faithful to the original data but also achieves superior robustness against diverse wireless transmission degradations. Collectively, these advancements establish DiffCom as a new benchmark in designing generative communication systems that offer enhanced robustness and generalization superiorities.

This paper studies the robust Hankel recovery problem, which simultaneously removes the sparse outliers and fulfills missing entries from the partial observation. We propose a novel non-convex algorithm, coined Hankel Structured Newton-Like Descent (HSNLD), to tackle the robust Hankel recovery problem. HSNLD is highly efficient with linear convergence, and its convergence rate is independent of the condition number of the underlying Hankel matrix. The recovery guarantee has been established under some mild conditions. Numerical experiments on both synthetic and real datasets show the superior performance of HSNLD against state-of-the-art algorithms.

We derive the general conditions for fully-nonlinear symmetry-integrable second-order evolution equations and their first-order recursion operators. We then apply the established Propositions to find links between a class of fully-nonlinear third-order symmetry-integrable evolution equations and fully-nonlinear second-order symmetry-integrable evolution equations.

In 1991, Gelfand and Retakh embodied the idea of a noncommutative Dieudonne determinant in the case of RTT algebra, namely, they found a representation of the quantum determinant of RTT algebra in the form of a product of principal quasi-determinants. In this note we construct an analogue of the above statement for the RE-algebra corresponding to the Drinfeld R-matrix for the order $n=2,3$. Namely, we have found a family of quasi-determinants that are principal with respect to the antidiagonal, commuting among themselves, whose product turns out to be the quantum determinant of this algebra. This family generalizes the construction of integrals of the full Toda system due to Deift et al. for the quantum case of RE-algebras. In our opinion, this result also clarifies the role of RE-algebras as a quantum homogeneous spaces and can be used to construct effective quantum field theories with a boundary.

This report presents the GPU acceleration of implicit kinetic meshfree methods using modified LU-SGS algorithms. The meshfree scheme is based on the least squares kinetic upwind method (LSKUM). In the existing matrix-free LU-SGS approaches for kinetic meshfree methods, the products of split flux Jacobians and increments in conserved vectors are approximated by increments in the split fluxes. In our modified LU-SGS approach, the Jacobian vector products are computed exactly using algorithmic differentiation (AD). The implicit GPU solvers with exact and approximate computation of the Jacobian vector products are applied to the standard test cases for two-dimensional inviscid flows. Numerical results have shown that the GPU solvers with the exact computation of the Jacobian vector products are computationally more efficient and yield better convergence rates than the solvers with approximations to the Jacobian vector products. Benchmarks are presented to assess the performance of implicit GPU solvers compared to the explicit GPU solver and the implicit serial LSKUM solver.

Recent advancements in neural network design have given rise to the development of Kolmogorov-Arnold Networks (KANs), which enhance speed, interpretability, and precision. This paper presents the Fractional Kolmogorov-Arnold Network (fKAN), a novel neural network architecture that incorporates the distinctive attributes of KANs with a trainable adaptive fractional-orthogonal Jacobi function as its basis function. By leveraging the unique mathematical properties of fractional Jacobi functions, including simple derivative formulas, non-polynomial behavior, and activity for both positive and negative input values, this approach ensures efficient learning and enhanced accuracy. The proposed architecture is evaluated across a range of tasks in deep learning and physics-informed deep learning. Precision is tested on synthetic regression data, image classification, image denoising, and sentiment analysis. Additionally, the performance is measured on various differential equations, including ordinary, partial, and fractional delay differential equations. The results demonstrate that integrating fractional Jacobi functions into KANs significantly improves training speed and performance across diverse fields and applications.

Generative models based on dynamical transport of measure, such as diffusion models, flow matching models, and stochastic interpolants, learn an ordinary or stochastic differential equation whose trajectories push initial conditions from a known base distribution onto the target. While training is cheap, samples are generated via simulation, which is more expensive than one-step models like GANs. To close this gap, we introduce flow map matching -- an algorithm that learns the two-time flow map of an underlying ordinary differential equation. The approach leads to an efficient few-step generative model whose step count can be chosen a-posteriori to smoothly trade off accuracy for computational expense. Leveraging the stochastic interpolant framework, we introduce losses for both direct training of flow maps and distillation from pre-trained (or otherwise known) velocity fields. Theoretically, we show that our approach unifies many existing few-step generative models, including consistency models, consistency trajectory models, progressive distillation, and neural operator approaches, which can be obtained as particular cases of our formalism. With experiments on CIFAR-10 and ImageNet 32x32, we show that flow map matching leads to high-quality samples with significantly reduced sampling cost compared to diffusion or stochastic interpolant methods.

We study the relaxation of a highly collisional, ultracold but nondegenerate gas of polar molecules. Confined within a harmonic trap, the gas is subject to fluid-gaseous coupled dynamics that lead to a breakdown of first-order hydrodynamics. An attempt to treat these higher-order hydrodynamic effects was previously made with a Gaussian ansatz and coarse-graining model parameter [R. R. W. Wang & J. L. Bohn, Phys. Rev. A 108, 013322 (2023)], leading to an approximate set of equations for a few collective observables accessible to experiments. Here we present substantially improved reduced-order models for these same observables, admissible beyond previous parameter regimes, discovered directly from particle simulations using the WSINDy algorithm (Weak-form Sparse Identification of Nonlinear Dynamics). The interpretable nature of the learning algorithm enables estimation of previously unknown physical quantities and discovery of model terms with candidate physical mechanisms, revealing new physics in mixed collisional regimes. Our approach constitutes a general framework for data-driven model identification leveraging known physics.

We propose a new transformer-based image and video tokenizer with Binary Spherical Quantization (BSQ). BSQ projects the high-dimensional visual embedding to a lower-dimensional hypersphere and then applies binary quantization. BSQ is (1) parameter-efficient without an explicit codebook, (2) scalable to arbitrary token dimensions, and (3) compact: compressing visual data by up to 100$\times$ with minimal distortion. Our tokenizer uses a transformer encoder and decoder with simple block-wise causal masking to support variable-length videos as input. The resulting BSQ-ViT achieves state-of-the-art visual reconstruction quality on image and video reconstruction benchmarks with 2.4$\times$ throughput compared to the best prior methods. Furthermore, by learning an autoregressive prior for adaptive arithmetic coding, BSQ-ViT achieves comparable results on video compression with state-of-the-art video compression standards. BSQ-ViT also enables masked language models to achieve competitive image synthesis quality to GAN- and diffusion-based methods.