In this paper, some existence results for sign-changing critical points of locally Lipschitz functionals in real Banach space are obtained by the method combining the invariant sets of descending ow method with a quantitative deformation. First we assume the locally Lipschitz functionals to be outwardly directed on the the boundary of some closed convex sets of the real Banach space. By using the relation between the critical points on the Banach space and those of the closed convex sets, we construct a quantitative deformation lemma, and then we obtain some linking type of critical points theorems. These theoretical results can be applied to the study of the existence of sign-changing solutions for differential inclusion problems. In contrast with the related results in the literatures, the main results of this paper relax the requirement that the functional being of C1 continuous to locally Lipschitz.

One of the most important problems in the studying of frames and its extensions is the invariance of these systems under perturbation. The current paper is concerned with the invariance of Modular biframes for operators under some class of closed range operators.

Let $K$ be a local field of residue characteristic $p>0$. We explain how to compute the semistable reduction of $K$-curves $Y$ equipped with a degree-$p$ morphism from $Y$ to the projective line. This includes the reduction at $p$ of superelliptic curves of degree $p$, but our approach is not limited to Galois covers. We give particular attention to the reduction of plane quartics at $p=3$, which case is implemented in SageMath. We use the language of non-archimedean analytic geometry in the sense of Berkovich. A key tool is the different function of Cohen, Temkin, and Trushin.

The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton-Watson process with migration introduced in [Yanev & Mitov (1980) C. R. Acad. Bulg. Sci. 33(4):473-475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

Electric and magnetic waveguides are considered in planar Dirac materials like graphene as well as their classical version for relativistic particles of zero mass and electric charge. In order to solve the Dirac-Weyl equation analytically, we have assumed the displacement symmetry of the system along a direction. In these conditions we have examined the rest of symmetries relevant each type, magnetic or electric system, which will determine their similarities and differences. We have worked out waveguides with square profile in detail to show up some of the most interesting features also in quantum and classical complementary contexts. All the results have been visualized along a series of representative graphics showing explicitly the main properties for both types of waveguides.

We classify connected \'etale algebras $A$'s in multiplicity-free modular fusion categories $\mathcal B$'s with $\text{rank}(\mathcal B)\le9$. We also identify categories $\mathcal B_A$'s of right $A$-modules. The results have physical applications in constraining renormalization group flows. As demonstration, we study massive renormalization group flows from non-unitary minimal models to predict ground state degeneracies and prove spontaneous $\mathcal B$-symmetry breaking.

In four-dimensional conformal field theory, the numbers a and c are defined as coefficients of particular terms in the operator product expansion (OPE) of the energy-momentum tensor. With supersymmetry there are relations between these coefficients and mixed R-symmetry and gravitational anomalies. In this paper we prove a relationship between these coefficients and anomalies to holomorphic reparametrization symmetry at the level of the holomorphic twist.

High peak-to-average power ratio (PAPR) is one of the main factors limiting cell coverage for cellular systems, especially in the uplink direction. Discrete Fourier transform spread orthogonal frequency-domain multiplexing (DFT-s-OFDM) with spectrally-extended frequency-domain spectrum shaping (FDSS) is one of the efficient techniques deployed to lower the PAPR of the uplink waveforms. In this work, we propose a machine learning-based framework to determine the FDSS filter, optimizing a tradeoff between the symbol error rate (SER), the PAPR, and the spectral flatness requirements. Our end-to-end optimization framework considers multiple important design constraints, including the Nyquist zero-ISI (inter-symbol interference) condition. The numerical results show that learned FDSS filters lower the PAPR compared to conventional baselines, with minimal SER degradation. Tuning the parameters of the optimization also helps us understand the fundamental limitations and characteristics of the FDSS filters for PAPR reduction.

Organizing physics has been a long-standing preoccupation of applied category theory, going back at least to Lawvere. We contribute to this research thread by noticing that Hamiltonian mechanics and gradient descent depend crucially on a consistent choice of transformation -- which we call a reaction structure -- from the cotangent bundle to the tangent bundle. We then construct a compositional theory of reaction structures. Reaction-based systems offer a different perspective on composition in physics than port-Hamiltonian systems or open classical mechanics, in that reaction-based composition does not create any new constraints that must be solved for algebraically. The technical contributions of this paper are the development of symmetric monoidal categories of open energy-driven systems and open differential equations, and a functor between them, functioning as a "functorial semantics" for reaction structures. This approach echoes what has previously been done for open games and open gradient-based learners, and in fact subsumes the latter. We then illustrate our theory by constructing an $n$-fold pendulum as a composite of $n$-many pendula.

We endow the cohomology of configuration spaces of a manifold with a product arising from superposing configurations. We prove that, under the scanning isomorphism, this product corresponds to the cup-product of the section space of the standard scanning bundle of the manifold.

This paper revisits the identity detection problem under the current grant-free protocol in massive machine-type communications (mMTC) by asking the following question: for stable identity detection performance, is it enough to permit active devices to transmit preambles without any handshaking with the base station (BS)? Specifically, in the current grant-free protocol, the BS blindly allocates a fixed length of preamble to devices for identity detection as it lacks the prior information on the number of active devices $K$. However, in practice, $K$ varies dynamically over time, resulting in degraded identity detection performance especially when $K$ is large. Consequently, the current grant-free protocol fails to ensure stable identity detection performance. To address this issue, we propose a two-stage communication protocol which consists of estimation of $K$ in Phase I and detection of identities of active devices in Phase II. The preamble length for identity detection in Phase II is dynamically allocated based on the estimated $K$ in Phase I through a table lookup manner such that the identity detection performance could always be better than a predefined threshold. In addition, we design an algorithm for estimating $K$ in Phase I, and exploit the estimated $K$ to reduce the computational complexity of the identity detector in Phase II. Numerical results demonstrate the effectiveness of the proposed two-stage communication protocol and algorithms.

Laurent Phenomenon algebras, first introduced by Lam and Pylyavskyy, are a generalization of cluster algebras that still possess many salient features of cluster algebras. Linear Laurent Phenomenon algebras, defined by Lam and Pylyavskyy, are a subclass of Laurent Phenomenon algebras whose structure is given by the data of a directed graph. The main result of this paper is that the cluster monomials of a linear Laurent Phenomenon algebra form a linear basis, conjectured by Lam and Pylyavskyy and analogous to a result for cluster algebras by Caldero and Keller.

Convergence of stochastic integrals driven by Wiener processes $W_n$, with $W_n \to W$ almost surely in $C_t$, is crucial in analyzing SPDEs. Our focus is on the convergence of the form $\int_0^T V_n\, \mathrm{d} W_n \to \int_0^T V\, \mathrm{d} W$, where $\{V_n\}$ is bounded in $L^p(\Omega \times [0,T];X)$ for a Banach space $X$ and some finite $p > 2$. This is challenging when $V_n$ converges to $V$ weakly in the temporal variable. We supply convergence results to handle stochastic integral limits when strong temporal convergence is lacking. A key tool is a uniform mean $L^1$ time translation estimate on $V_n$, an estimate that is easily verified in many SPDEs. However, this estimate alone does not guarantee strong compactness of $(\omega,t)\mapsto V_n(\omega,t)$. Our findings, especially pertinent to equations exhibiting singular behavior, are substantiated by establishing several stability results for stochastic transport equations and conservation laws.

In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for discrete iterated integrals. They imply the cyclic sum formula for multiple polylogarithms.

In this paper, we construct the ADHM quiver representations and the corresponding sheaves as the mirror objects of formal deformations of the framed immersed Lagrangian sphere decorated with flat bundles. More generally, framed double quivers of Nakajima are constructed as localized mirrors of framed Lagrangian immersions in dimension two. This produces a localized mirror functor to the dg category of modules over the framed preprojective algebra. For affine ADE quivers in specific multiplicities, the corresponding (unframed) Lagrangian immersions are homological tori, whose moduli of stable deformations are asymptotically locally Euclidean (ALE) spaces. We show that framed stable Lagrangian branes are transformed into monadic complexes of framed torsion-free sheaves over the ALE spaces. A main ingredient is the notion of framed Lagrangian immersions. Moreover, it is important to note that the deformation space of a Lagrangian immersion with more than one component is stacky. Using the formalism of quiver algebroid stacks, we find isomorphisms between the moduli of stable Lagrangian immersions and that of special Lagrangian fibers of an SYZ fibration in the affine $A_n$ cases.

We investigate the lifespan of solutions to a specific variant of the semilinear wave equation, which incorporates weighted nonlinearity $$ u_{tt}-u_{xx} =|x|^\alpha |u|^p, \quad\mbox{for}\;\;\; (t,x)\in (0,\infty)\times\mathbb{R}, $$ where $p>1$, $\alpha\in\mathbb{R}$. We explore the behavior of solutions for small initial data, considering the influence of weighted nonlinearities on the lifespan.

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex is adjacent to a vertex in $S$. The total domination number $\gamma_t(G)$ is the minimum cardinality of a total dominating set of $G$. The total domination subdivision number $\mbox{sd}_{\gamma_t}(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to increase the total domination number. Haynes et al. (Discrete Math. 286 (2004) 195--202) have given a constructive characterization of trees whose total domination subdivision number is~$3$. In this paper, we give new characterizations of trees whose total domination subdivision number is 3.

We present a structure preserving PINN for solving a series of time dependent PDEs with periodic boundary. Our method can incorporate the periodic boundary condition as the natural output of any deep neural net, hence significantly improving the training accuracy of baseline PINN. Together with mini-batching and other PINN variants (SA-PINN, RBA-PINN, etc.), our structure preserving PINN can even handle stiff PDEs for modeling a wide range of convection-diffusion and reaction-diffusion processes. We demonstrate the effectiveness of our PINNs on various PDEs from Allen Cahn, Gray Scott to nonlinear Schrodinger.

We show how to convert the generating series of interpolated multiple zeta values, or multiple $t$ values, with repeating blocks of length 1 into hypergeometric series. Then we invoke creative telescoping on their generating functions, in some known cases for illustration, and in some apparently new cases, reducing them to polynomials in Riemann zeta values. The new evaluations, including $ \zeta^{1/2}(\{\bar2\}^n,3) $, $ \zeta^\star(\{1,3\}^n,1,2) $ and $ t^{1/2}(2,\{1\}^n,2) $, resolve some questions raised elsewhere, and seem to be non-trivial using other methods.

Andrews and Keith recently produced a general Schmidt type partition theorem using a novel interpretation of Stockhofe's bijection, which they used to find new $q$-series identities. This includes an identity for a trivariate 2-colored partition generating function. In this paper, their Schmidt type theorem is further generalized akin to how Franklin classically extended Glaisher's theorem. As a consequence, we obtain a companion to Andrews and Keith's 2-colored identity for overpartitions. These identities appear to be special cases of a much more general result.

Every Boolean bent function $f$ can be written either as a concatenation $f=f_1||f_2$ of two complementary semi-bent functions $f_1,f_2$; or as a concatenation $f=f_1||f_2||f_3||f_4$ of four Boolean functions $f_1,f_2,f_3,f_4$, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation $f$ (not) belong to the completed Maiorana-McFarland class $\mathcal{M}^\#$? In this article, we answer this question completely by providing a full characterization of the structure of $\mathcal{M}$-subspaces for the concatenation of the form $f=f_1||f_2$ and $f=f_1||f_2||f_3||f_4$, which allows us to specify the necessary and sufficient conditions so that $f$ is outside $\mathcal{M}^\#$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside $\mathcal{M}^\#$ in the special case when $f=g||h||g||(h+1)$, where $g$ and $h$ are bent functions.

The false confidence theorem establishes that, for any data-driven, precise-probabilistic method for uncertainty quantification, there exists (non-trivial) false hypotheses to which the method tends to assign high confidence. This raises concerns about the reliability of these widely-used methods, and shines new light on the consonant belief function-based methods that are provably immune to false confidence. But an existence result alone is insufficient. Towards a partial answer to the title question, I show that, roughly, complements of convex hypotheses are afflicted by false confidence.

A graded Artinian algebra $A$ has the Weak Lefschetz Property if there exists a linear form $\ell$ such that the multiplication map by $\ell:[A]_i\to [A]_{i+1}$ has maximum rank in every degree. The linear forms satisfying this property form a Zariski-open set; its complement is called the non-Lefschetz locus of $A$. In this paper, we investigate analogous questions for degree-two forms rather than lines. We prove that any complete intersection $A=k[x_1,x_2,x_3]/(f_1,f_2,f_3)$, with $\text{char } k=0$, has the Strong Lefschetz Property at range $2$, i.e. there exists a linear form $\ell\in [R]_1$, such that the multiplication map $\times \ell^2: [M]_i\to [M]_{i+2}$ has maximum rank in each degree. Then we focus on the forms of degree 2 such that $ \times C: [A]_i\to [A]_{i+2}$ fails to have maximum rank in some degree $i$. The main result shows that the non-Lefschetz locus of conics for a general complete intersection $A=k[x_1,x_2,x_3]/(f_1,f_2,f_3)$ has the expected codimension as a subscheme of $\mathbb{P}^5$. The hypothesis of generality is necessary. We include examples of monomial complete intersections in which the non-Lefschetz locus of conics has different codimension. To extend a similar result to the first cohomology modules of rank $2$ vector bundles over $\mathbb{P}^2$, we explore the connection between non-Lefschetz conics and jumping conics. The non-Lefschetz locus of conics is a subset of the jumping conics. Unlike the case of the lines, this can be proper when $\mathcal{E}$ is semistable with first Chern class even.

The classical method of Rothe proves existence and uniqueness theorems for initial boundary value problems for parabolic equations using the explicit finite difference scheme with respect to time. In this method, an elliptic boundary value problem is investigated on each time step. On the other hand, time dependent experimental data are always collected on discrete time grids, and the grid step size cannot be arranged to be infinitely small. The same is true for numerical studies. Therefore, it makes an applied sense to consider both unique continuation problems and coefficient inverse problems for parabolic equations, which are written in the form of finite differences with respect to time and without allowing the grid step size to tend to zero. This leads to a boundary value problem for a coupled system of elliptic equations with both Dirichlet and Neumann boundary data, which is somewhat similar to the Rothe's method. Dissimilarities are named as well. Two long standing open questions are addressed within this framework. A specific applied example of monitoring epidemics is discussed. In particular, a numerical method for this problem is constructed and its global convergence analysis is provided.

The study of dynamical systems on complex networks is of paramount importance in engineering, given that many natural and artificial systems find a natural embedding on discrete topologies. For instance, power grids, chemical reactors and the brain, to name a few, can be modeled through the network formalism by considering elementary units coupled via the links. In recent years, scholars have developed numerical and theoretical tools to study the stability of those coupled systems when subjected to perturbations. In such framework, it was found that asymmetric couplings enhance the possibilities for such systems to become unstable. Moreover, in this scenario the polynomials whose stability needs to be studied bear complex coefficients, which makes the analysis more difficult. In this work, we put to use a recent extension of the well-known Routh-Hurwitz stability criterion, allowing to treat the complex coefficient case. Then, using the Brusselator model as a case study, we discuss the stability conditions and the regions of parameters when the networked system remains stable.

We apply the better-behaved GKZ hypergeometric systems to study toric Calabi-Yau Deligne-Mumford stacks and their Hori-Vafa mirrors given by affine hypersurfaces in algebraic tori. We show the equality between A-model and B-model central charges, in terms of period integrals and hypergeometric series respectively. This settles a conjecture of Hosono, which could also be considered as a generalization of the Gamma conjecture for local mirror symmetry.

In this paper, we consider the problem of minimum-time optimal control for a dynamical system with initial state uncertainties and propose a sequential convex programming (SCP) solution framework. We seek to minimize the expected terminal (mission) time, which is an essential capability for planetary exploration missions where ground rovers have to carry out scientific tasks efficiently within the mission timelines in uncertain environments. Our main contribution is to convert the underlying stochastic optimal control problem into a deterministic, numerically tractable, optimal control problem. To this end, the proposed solution framework combines two strategies from previous methods: i) a partial model predictive control with consensus horizon approach and ii) a sum-of-norm cost, a temporally strictly increasing weighted-norm, promoting minimum-time trajectories. Our contribution is to adopt these formulations into an SCP solution framework and obtain a numerically tractable stochastic control algorithm. We then demonstrate the resulting control method in multiple applications: i) a closed-loop linear system as a representative result (a spacecraft double integrator model), ii) an open-loop linear system (the same model), and then iii) a nonlinear system (Dubin's car).

We have developed novel algorithms for optimizing nonsmooth, nonconvex functions in which the nonsmoothness is caused by nonsmooth operators presented in the analytical form of the objective. The algorithms are based on encoding the active branch of each nonsmooth operator such that the active smooth component function and its code can be extracted at any given point, and the transition of the solution from one smooth piece to another can be detected via tracking the change of active branches of all the operators. This mechanism enables the possibility of collecting the information about the sequence of active component functions encountered in the previous iterations (i.e., the component transition information), and using it in the construction of a current local model or identification of a descent direction in a very economic and effective manner. Based on this novel idea, we have developed a trust-region method and a joint gradient descent method driven by the component information for optimizing the encodable piecewise-smooth, nonconvex functions. It has further been shown that the joint gradient descent method using a technique called proactive component function accessing can achieve a linear rate of convergence if a so-called multi-component Polyak-Lojasiewicz inequality and some other regularity conditions hold at a neighborhood of a local minimizer.

In a recent work [Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 2024], Gastel and Neff introduced an interesting system from a geometrically nonlinear flat cosserat micropolar model and established interior regularity in the critical dimension. Motived by this work, in this article, we establish both interior regularity and sharp $L^p$ regularity for their system in supercritical dimensions.

In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-\'Emery Ricci curvature. In particular, we give a sharp Willmore-like inequality in shrinking gradient Ricci solitons. These results can be regarded as generalizations of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality in weighted manifolds. As applications, we derive some isoperimetric type inequalities under certain existence assumptions of isoperimetric regions in weighted manifolds.

The design of precoding plays a crucial role in achieving a high downlink sum-rate in multiuser multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems. In this correspondence, we propose a deep learning based joint CSI feedback and multiuser precoding method in frequency division duplex systems, aiming at maximizing the downlink sum-rate performance in an end-to-end manner. Specifically, the eigenvectors of the CSI matrix are compressed using deep joint source-channel coding techniques. This compression method enhances the resilience of the feedback CSI information against degradation in the feedback channel. A joint multiuser precoding module and a power allocation module are designed to adjust the precoding direction and the precoding power for users based on the feedback CSI information. Experimental results demonstrate that the downlink sum-rate can be significantly improved by using the proposed method, especially in scenarios with low signal-to-noise ratio and low feedback overhead.

In this paper, we derive decay rates of the solutions to the incompressible Navier-Stokes equations and Hall-magnetohydrodynamic equations. We first improve the decay rate of weak solutions of these equations by refining the Fourier splitting method with initial data in the space of pseudo-measures. We also deal with these equations with initial data in Lei-Lin spaces and find decay rates of solutions in Lei-Lin spaces.

Let $K$ be a complete non-archimedean valuation field of characteristic $0$, with non-trivial valuation, equipped with (possibly multiple) commuting bounded derivations. We prove a decomposition theorem for finite differential modules over $K$, where decompositions regarding the extrinsic subsidiary $\partial$-generic radii of convergence in the sense of Kedlaya-Xiao. Our result is a refinement of a previous decomposition theorem due to Kedlaya and Xiao. As a key step in the proof, we prove a decomposition theorem in a stronger form in the case where $K$ is equipped with a single derivation. To achieve this goal, we construct an object $f_{0*}L_0$ representing the usual dual functor and study some filtrations of $f_{0*}L_0$, which is used to construct the direct summands appearing in our decomposition theorem.

Let $(\mathcal{F},S)$ be a foliated surface of general type with reduced singularities over the complex number. We establish the Noether type inequalities for $(\mathcal{F},S)$. Namely, we prove that $\mathrm{vol}(\mathcal{F}) \geq p_g(\mathcal{F})-2$, and that $\mathrm{vol}(\mathcal{F}) \geq 2p_g(\mathcal{F})-4$ if moreover the surface $S$ is also of general type. Examples show that both of the Noether type inequalities are sharp.

We prove that Bourgeois' contact structures on $M \times \mathbb{T}^{2}$ determined by the supporting open books of a contact manifold $(M, \xi)$ are always tight. The proof is based on a contact homology computation leveraging holomorphic foliations and Kuranishi structures.

Nonlinear complexity is an important measure for assessing the randomness of sequences. In this paper we investigate how circular shifts affect the nonlinear complexities of finite-length binary sequences and then reveal a more explicit relation between nonlinear complexities of finite-length binary sequences and their corresponding periodic sequences. Based on the relation, we propose two algorithms that can generate all periodic binary sequences with any prescribed nonlinear complexity.

Let $E \subseteq \mathbb{R}^n$ be a union of line segments and $F \subseteq \mathbb{R}^n$ the set obtained from $E$ by extending each line segment in $E$ to a full line. Keleti's \textit{line segment extension conjecture} posits that the Hausdorff dimension of $F$ should equal that of $E$. Working in $\mathbb{R}^2$, we use effective methods to prove a strong packing dimension variant of this conjecture, from which the generalized Kakeya conjecture for packing dimension immediately follows. This is followed by several doubling estimates in higher dimensions and connections to related problems.

We consider periodic homogenisation problems for gPAM, $\Phi^{4}_{3}$ and modified KPZ equations. Using Littlewood-Paley blocks, we establish para-products and commutator estimates for generalised Besov spaces associated with self-adjoint elliptic operators. With this, we show that, the homogenisation and renormalisation procedures commute in aforementioned models under suitable assumptions.

Interviews run on people, programs run on operating systems, voting schemes run on voters, games run on players. Each of these is an example of the abstraction pattern runs on matter. Pattern determines the decision tree that governs how a situation can unfold, while matter responds with decisions at each juncture. In this article, we will give a straightforward and concrete construction of the free monad monad for $(\mathbf{Poly}, \mathbin{\triangleleft}, \mathcal{y})$, the category of polynomial functors with the substitution monoidal product. Although the free monad has been well-studied in other contexts, the construction we give is streamlined and explicitly illustrates how the free monad represents terminating decision trees. We will also explore the naturally arising interaction between the free monad and cofree comonad. Again, while the interaction itself is known, the perspective we take is the free monad as a module over the cofree comonad. Lastly, we will give four applications of the module action to interviews, computer programs, voting, and games. In each example, we will see how the free monad represents pattern, the cofree comonad represents matter, and the module action represents runs on.

Deep learning techniques have shown significant potential in many applications through recent years. The achieved results often outperform traditional techniques. However, the quality of a neural network highly depends on the used training data. Noisy, insufficient, or biased training data leads to suboptimal results. We present a hybrid method that combines deep learning with iterated graph Laplacian and show its application in acoustic impedance inversion which is a routine procedure in seismic explorations. A neural network is used to obtain a first approximation of the underlying acoustic impedance and construct a graph Laplacian matrix from this approximation. Afterwards, we use a Tikhonov-like variational method to solve the impedance inversion problem where the regularizer is based on the constructed graph Laplacian. The obtained solution can be shown to be more accurate and stable with respect to noise than the initial guess obtained by the neural network. This process can be iterated several times, each time constructing a new graph Laplacian matrix from the most recent reconstruction. The method converges after only a few iterations returning a much more accurate reconstruction. We demonstrate the potential of our method on two different datasets and under various levels of noise. We use two different neural networks that have been introduced in previous works. The experiments show that our approach improves the reconstruction quality in the presence of noise.

Sequences with low aperiodic autocorrelation sidelobes have been extensively researched in literatures. With sufficiently low integrated sidelobe level (ISL), their power spectrums are asymptotically flat over the whole frequency domain. However, for the beam sweeping in the massive multi-input multi-output (MIMO) broadcast channels, the flat spectrum should be constrained in a passband with tunable bandwidth to achieve the flexible tradeoffs between the beamforming gain and the beam sweeping time. Motivated by this application, we construct a family of sequences termed the generalized step-chirp (GSC) sequence with a closed-form expression, where some parameters can be tuned to adjust the bandwidth flexibly. In addition to the application in beam sweeping, some GSC sequences are closely connected with Mow's unified construction of sequences with perfect periodic autocorrelations, and may have a coarser phase resolution than the Mow sequence while their ISLs are comparable.

We investigate the notion of subsystem in the framework of spectral triple as a generalized notion of noncommutative submanifold. In the case of manifolds, we consider several conditions on Dirac operators which turn embedded submanifolds into isometric submanifolds. We then suggest a definition of spectral subtriple based on the notion of submanifold algebra and the already existing notions of Riemannian, isometric, and totally geodesic morphisms. We have shown that our definitions work at least in some relevant almost commutative examples.

We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regular $s$-summable spectral triples $(\mathcal{A}, \mathcal{H}, D)$, and an asymptotic expansion of $\mathrm{Tr}(P e^{-t(D+V)^2})$ as $t \downarrow 0$, where $P$ and $V$ belong to the algebra generated by $\mathcal{A}$ and $D$, and $V$ is bounded and self-adjoint.

An $\ell$-vertex-ranking of a graph $G$ is a colouring of the vertices of $G$ with integer colours so that in any connected subgraph $H$ of $G$ with diameter at most $\ell$, there is a vertex in $H$ whose colour is larger than that of every other vertex in $H$. The $\ell$-vertex-ranking number, $\chi_{\ell-\mathrm{vr}}(G)$, of $G$ is the minimum integer $k$ such that $G$ has an $\ell$-vertex-ranking using $k$ colours. We prove that, for any fixed $d$ and $\ell$, every $d$-degenerate $n$-vertex graph $G$ satisfies $\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/(\ell+1)}\log n)$ if $\ell$ is even and $\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/\ell}\log n)$ if $\ell$ is odd. The case $\ell=2$ resolves (up to the $\log n$ factor) an open problem posed by \citet{karpas.neiman.ea:on} and the cases $\ell\in\{2,3\}$ are asymptotically optimal (up to the $\log n$ factor).

This study introduces a computational approach leveraging physics-informed neural networks (PINNs) for the efficient computation of arterial blood flows, particularly focusing on solving the incompressible Navier-Stokes equations by using the domain decomposition technique. Unlike conventional computational fluid dynamics methods, PINNs offer advantages by eliminating the need for discretized meshes and enabling the direct solution of partial differential equations (PDEs). In this paper, we propose the weighted Extended Physics-Informed Neural Networks (WXPINNs) and weighted Conservative Physics-Informed Neural Networks (WCPINNs), tailored for detailed hemodynamic simulations based on generalized space-time domain decomposition techniques. The inclusion of multiple neural networks enhances the representation capacity of the weighted PINN methods. Furthermore, the weighted PINNs can be efficiently trained in parallel computing frameworks by employing separate neural networks for each sub-domain. We show that PINNs simulation results circumvent backflow instabilities, underscoring a notable advantage of employing PINNs over traditional numerical methods to solve such complex blood flow models. They naturally address such challenges within their formulations. The presented numerical results demonstrate that the proposed weighted PINNs outperform traditional PINNs settings, where sub-PINNs are applied to each subdomain separately. This study contributes to the integration of deep learning methodologies with fluid mechanics, paving the way for accurate and efficient high-fidelity simulations in biomedical applications, particularly in modeling arterial blood flow.

In this short note, we prove that the one-dimensional Kronecker sequence $i\alpha \bmod 1, i=0,1,2,\ldots,$ is quasi-uniform if and only if $\alpha$ is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (1965).

In this paper, we consider a Hele-Shaw model that describes tumor growth subject to nutrient supply. This model was recently studied in \cite{feng2022tumor} via asymptotic analysis. Our contributions are twofold: Firstly, we provide a rigorous derivation of this Hele-Shaw model by taking the incompressible limit of the porous medium reaction-diffusion equation, which solidifies the mathematical foundations of the model. Secondly, from a bifurcation theory perspective, we prove the existence of non-symmetric traveling wave solutions to the model, which reflect the intrinsic boundary instability in tumor growth dynamics.

In this article we develop a graphical calculus for stable invariants of Riemannian manifolds akin to the graphical calculus for Rozansky-Witten invariants for hyperk\"ahler manifolds; based on interpreting trivalent graphs with colored edges as stably invariant polynomials on the space of algebraic curvature tensors. In this graphical calculus we describe explicitly the Pfaffian polynomials central to the Theorem of Chern-Gau{\ss}-Bonnet and the normalized moment polynomials calculating the moments of sectional curvature considered as a random variable on the Gra{\ss}mannian of planes. Eventually we illustrate the power of this graphical calculus by deriving a curvature identity for compact Einstein manifolds of dimensions greater than 2 involving the Euler characteristic, the third moment of sectional curvature and the $L^2$--norm of the covariant derivative of the curvature tensor. A model implementation of this calculus for the computer algebra system Maxima is available for download under this http URL

This paper explores the distributed broadcast problem within the context of network communications, a critical challenge in decentralized information dissemination. We put forth a novel hypergraph-based approach to address this issue, focusing on minimizing the number of broadcasts to ensure comprehensive data sharing among all network users. A key contribution of our work is the establishment of a general lower bound for the problem using the min-cut capacity of hypergraphs. Additionally, we present the distributed broadcast for quasi-trees (DBQT) algorithm tailored for the unique structure of quasi-trees, which is proven to be optimal. This paper advances both network communication strategies and hypergraph theory, with implications for a wide range of real-world applications, from vehicular and sensor networks to distributed storage systems.

In this paper, we consider the well-posedness theory of two-dimensional compressible subsonic jet flows for steady full Euler system with general vorticity. Inspired by the analysis in arXiv:2006.05672, we show that the stream function formulation for such system admits a variational structure. Then the existence and uniqueness of a smooth subsonic jet flow can be established by the variational method developed by Alt, Caffarelli and Friedman. Furthermore, the far fields behavior of the flow and the existence of a critical upstream pressure are also obtained.

We compute the quadratic Euler characteristic of the symmetric powers of a smooth, projective curve over any field $k$ that is not of characteristic two, using the Motivic Gauss-Bonnet Theorem of Levine-Raksit. As an application, we show over a field of characteristic zero that the power structure on the Grothendieck-Witt ring introduced by Pajwani-P\'al computes the compactly supported $\mathbb{A}^1$-Euler characteristic of symmetric powers for all curves.

We investigate in this work families $(u_\epsilon)_{\epsilon >0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schr\"odinger equations of the following type: $$\Delta_g u_\epsilon + h_\epsilon u_\epsilon = |u_{\epsilon}|^{p_\epsilon-2} u_\epsilon $$ in a closed manifold $(M,g)$, where $h_\epsilon$ converges to $h$ in $C^1(M)$. Assuming that $(u_\epsilon)_{\epsilon >0}$ blows-up as \emph{a single sign-changing bubble}, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between $h$, the geometry of $(M,g)$ and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of $u_\epsilon$.

Integrated sensing and communication (ISAC) is expected to play a prominent role among emerging technologies in future wireless communications. In particular, a communication radar coexistence system is degraded significantly by mutual interference. In this work, given the advantages of promising reconfigurable intelligent surface (RIS), we propose a simultaneously transmitting and reflecting RIS (STAR-RIS)-assisted radar coexistence system where a STAR-RIS is introduced to improve the communication performance while suppressing the mutual interference and providing full space coverage. Based on the realistic conditions of correlated fading, and the presence of multiple user equipments (UEs) at both sides of the RIS, we derive the achievable rates at the radar and the communication receiver side in closed forms in terms of statistical channel state information (CSI). Next, we perform alternating optimization (AO) for optimizing the STAR-RIS and the radar beamforming. Regarding the former, we optimize the amplitudes and phase shifts of the STAR-RIS through a projected gradient ascent algorithm (PGAM) simultaneously with respect to the amplitudes and phase shifts of the surface for both energy splitting (ES) and mode switching (MS) operation protocols. The proposed optimization saves enough overhead since it can be performed every several coherence intervals. This property is particularly beneficial compared to reflecting-only RIS because a STAR-RIS includes the double number of variables, which require increased overhead. Finally, simulation results illustrate how the proposed architecture outperforms the conventional RIS counterpart, and show how the various parameters affect the performance. Moreover, a benchmark full instantaneous CSI (I-CSI) based design is provided and shown to result in higher sum-rate but also in large overhead associated with complexity.

In this paper, the event-triggered resilient filtering problem is investigated for a class of two-dimensional systems with asynchronous-delay under binary encoding-decoding schemes with probabilistic bit flips. To reduce unnecessary communications and computations in complex network systems, alleviate network energy consumption, and optimize the use of network resources, a new event-triggered mechanism is proposed, which focuses on broadcasting necessary measurement information to update innovation only when the event generator function is satisfied. A binary encoding-decoding scheme is used in the communication process to quantify the measurement information into a bit stream, transmit it through a memoryless binary symmetric channel with a certain probability of bit flipping, and restore it at the receiver. In order to utilize the delayed decoded measurement information that a measurement reconstruction approach is proposed. Through generating space equivalence verification, it is found that the reconstructed delay-free decoded measurement sequence contains the same information as the original delayed decoded measurement sequence. In addition, resilient filter is utilized to accommodate possible estimation gain perturbations. Then, a recursive estimator framework is presented based on the reconstructed decoded measurement sequence. By means of the mathematical induction technique, the unbiased property of the proposed estimator is proved. The estimation gain is obtained by minimizing an upper bound on the filtering error covariance. Subsequently, through rigorous mathematical analysis, the monotonicity of filtering performance with respect to triggering parameters is discussed.

The escalating process of urbanization has raised concerns about incidents arising from overcrowding, necessitating a deep understanding of large human crowd behavior and the development of effective crowd management strategies. This study employs computational methods to analyze real-world crowd behaviors, emphasizing self-organizing patterns. Notably, the intersection of two streams of individuals triggers the spontaneous emergence of striped patterns, validated through both simulations and live human experiments. Addressing a gap in computational methods for studying these patterns, previous research utilized the pattern-matching technique, employing the Nelder-Mead Simplex algorithm for fitting a two-dimensional sinusoidal function to pedestrian coordinates. This paper advances the pattern-matching procedure by introducing Simulated Annealing as the optimization algorithm and employing a two-dimensional square wave for data fitting. The amalgamation of Simulated Annealing and the square wave significantly enhances pattern fitting quality, validated through statistical hypothesis tests. The study concludes by outlining potential applications of this method across diverse scenarios.

We consider Hecke symmetries on a 3-dimensional vector space with the associated R-symmetric algebra isomorphic to the polynomial algebra $k[x_1,x_2,x_3]$ twisted by an automorphism. The main result states that any such a Hecke symmetry is itself a twist of a Hecke symmetry with the associated R-symmetric algebra isomorphic to $k[x_1,x_2,x_3]$. This allows us to describe equivalence classes of such Hecke symmetries.

Given a connection $A$ on a $SU(2)$-bundle $P$ over $\mathbb{R}^4$ with finite Yang-Mills energy $YM(A)$ and nonzero curvature $F_A(0)$ at the origin, and given $\rho>0$ small enough, we construct a new connection $\hat A$ on a bundle $\hat P$ of different Chern class ($|c_2(A)-c_2(\hat A)|=8\pi^2$), in such a way that $\hat A$ is gauge equivalent to $A$ in $\mathbb{R}^4\setminus B_\rho(0)$ and $$YM(\hat A)\le YM(A)+8\pi^2-\varepsilon_0\rho^4|F_A(0)|^2$$ for a universal constant $\varepsilon_0>0$.

We study algebraic independence problem for the Taylor coefficients of the Anderson-Thakur series arisen as deformation series of positive characteristic multiple zeta values (abbreviated as MZV's). These Taylor coefficients are simply specialization of hyperderivatives of the Anderson-Thakur series. We consider the prolongation of t-motives associated with MZV's, and then determine the dimension of the t-motivic Galois groups in question under certain hypothesis. By using Papanikolas' theory, it enables us to obtain the desired algebraic independence result.

In this paper, we consider two critical aspects of security in the \textit{distributed computing (DC)} model: \textit{secure data shuffling} and \textit{secure coded computing}. It is imperative that any external entity overhearing the transmissions does not gain any information about the \textit{intermediate values (IVs)} exchanged during the shuffling phase of the DC model. Our approach ensures IV confidentiality during data shuffling. Moreover, each node in the system must be able to recover the IVs necessary for computing its output functions but must also remain oblivious to the IVs associated with output functions not assigned to it. We design secure DC methods and establish achievable limits on the tradeoffs between the communication and computation loads to contribute to the advancement of secure data processing in distributed systems.

We study the existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes (NPNS) system in $\mathbb{R}^N, N\geq 3$. We obtain a global in-time strong solution without any smallness assumptions on the initial data.

We discuss exponential decay in $L^p(R^N)$, $1\leq p\leq \infty$,of solutions of a fractional Schr\"odinger parabolic equation with a locally uniformly integrable potential. The exponential type of the semigroup of solutions is considered and its independence in of $1\leq p\leq \infty$ is addressed. We characterise a large class of potentials for which solutions decay exponentially.

In this paper, we mainly study the necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators on weighted Bergman spaces over a tubular domains by using the Carlson measures on tubular domains. We also give some related results about Carlson measures.

In this paper we find a decomposition of higher order Lipschitz functions into the traces of a polymonogenic function and solve a related Riemann-Hilbert problem. Our approach lies in using a cliffordian Cauchy-type operator, which behaves as an involution operator on higher order Lipschitz spaces. The result obtained is a multidimensional sharpened version of the Hardy decomposition of H\"older continuous functions on a simple closed curve in the complex plane.

We consider the inverse problem of determining the metric-measure structure of collapsing manifolds from local measurements of spectral data. In the part I of the paper, we proved the uniqueness of the inverse problem and a continuity result for the stability in the closure of Riemannian manifolds with bounded diameter and sectional curvature in the measured Gromov-Hausdorff topology. In this paper we show that when the collapse of dimension is $1$-dimensional, it is possible to obtain quantitative stability of the inverse problem for Riemannian orbifolds. The proof is based on an improved version of the quantitative unique continuation for the wave operator on Riemannian manifolds by removing assumptions on the covariant derivatives of the curvature tensor.

In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(\mathbb{Z}/N\mathbb{Z})^{\times}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.

In this paper, we study the tangent space to a sub-Finsler manifold in the measured Gromov-Hausdorff sense. We prove that the metric tangent is described by the nilpotent approximation, generalizing the sub-Riemannian result. Additionally, we study the blow-up of a measure on a sub-Finsler manifold. We identify the new notion of bounded measure which ensures that, in the limit, the blow-up is a scalar multiple of the Lebesgue measure. Our results have applications in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler manifolds. In particular, we show the failure of the CD condition in equiregular sub-Finsler manifolds with growth vector (2,3), equipped with a bounded measure.

The main goal of this work is to provide sample-path estimates for the solution of slowly time-dependent SPDEs perturbed by a cylindrical fractional Brownian motion. Our strategy is similar to the approach by Berglund and Nader for space-time white noise. However, the setting of fractional Brownian motion does not allow us to use any martingale methods. Using instead optimal estimates for the probability that the supremum of a Gaussian process exceeds a certain level, we derive concentration estimates for the solution of the SPDE, provided that the Hurst index $H$ of the fractional Brownian motion satisfies $H>\frac14$. As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any $H\in(0,1)$.

The aim of this article is to detect the ascent and descent of weighted composition operators on Lorentz spaces. We investigate the conditions on the measurable transformation $T$ and the complex-valued measurable function $u$ defined on measure space $(X, \mathcal{A}, \mu)$ that cause the weighted composition operators on Lorentz space $L(p, q)$, $1 < p \leq \infty, 1 \leq q \leq \infty$ to have finite or infinite ascent (descent). We also give a number of examples in support of our findings.

This paper focuses on the Bregman divergence defined by the reciprocal function, called the inverse divergence. For the loss function defined by the monotonically increasing function $f$ and inverse divergence, the conditions for the statistical model and function $f$ under which the estimating equation is unbiased are clarified. Specifically, we characterize two types of statistical models, an inverse Gaussian type and a mixture of generalized inverse Gaussian type distributions, to show that the conditions for the function $f$ are different for each model. We also define Bregman divergence as a linear sum over the dimensions of the inverse divergence and extend the results to the multi-dimensional case.

The present paper mainly considers the representation type of the enveloping algebra of monomial algebra. Let $A$ be a monomial algebra and $A^e= A\otimes_{\mathrm{l}\!\mathrm{k}} A^{\mathrm{op}}$ its enveloping algebra. It is shown that $A^e$ is representation-finite if and only if $A \cong \pmb{A}_n/\mathrm{rad}^2 \pmb{A}_n$, where $\pmb{A}_n$ is the path algebra $\mathrm{l}\!\mathrm{k}\mathcal{Q}$ with $\mathcal{Q} = 1 \longrightarrow 2 \longrightarrow \cdots \longrightarrow n$. Moreover, we show that the number of all isoclasses of indecomposable $(\pmb{A}_n/ \mathrm{rad}^2\pmb{A}_n)^e$-modules is $\frac{4}{3}n^3 + n^2-\frac{7}{3}n+1$, and classify all indecomposable modules over $(\pmb{A}_n/ \mathrm{rad}^2\pmb{A}_n)^e$. Finally, the Clebsch-Gordon problem over $(\pmb{A}_n/ \mathrm{rad}^2\pmb{A}_n)^e$ is studied.

We prove effective finiteness results concerning polynomial values of the sums $$ b^k +\left(a+b\right)^k + \cdots + \left(a\left(x-1\right) + b\right)^k $$ and $$ b^k - \left(a+b\right)^k + \left(2a+b\right)^k - \ldots + (-1)^{x-1} \left(a\left(x-1\right) + b\right)^k , $$ where $a \neq 0,b, k$ are given integers with $\gcd(a,b)=1$ and $k \geq 2$.

To efficiently tackle parametrized multi and/or large scale problems, we propose an adaptive localized model order reduction framework combining both local offline training and local online enrichment with localized error control. For the latter, we adapt the residual localization strategy introduced in [Buhr, Engwer, Ohlberger, Rave, SIAM J. Sci. Comput., 2017] which allows to derive a localized a posteriori error estimator that can be employed to adaptively enrich the reduced solution space locally where needed. Numerical experiments demonstrate the potential of the proposed approach.

We review properties of confluent functions and the closely related Laguerre polynomials, and determine their bilinear integrals. As is well-known, these integrals are convergent only for a limited range of parameters. However, when one uses the generalized integral they can be computed essentially without restricting the parameters. This gives the (generalized) Gram matrix of Laguerre polynomials. If the parameters are not negative integers, then Laguerre polynomials are orthogonal, or at least pseudo-orthogonal in the case of generalized integrals. For negative integer parameters, the orthogonality relations are more complicated.

We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian pair correlations need not be uniformly distributed, contrary to what was till recently believed.

In this paper, we study CR maps between hyperquadrics and Winkelmann hypersurfaces. Based on a previous study on the CR Ahlfors derivative of Lamel-Son and a recent result of Huang-Lu-Tang-Xiao on CR maps between hyperquadrics, we prove that a transversal CR map from a hyperquadric into a hyperquadric or a Winkelmann hypersurface extends to a local holomorphic isometric embedding with respect to certain K\"ahler metrics if and only if the Hermitian part of its CR Ahlfors derivative vanishes on an open set of the source. Our proof is based on relating the geometric rank of a CR map into a hyperquadric and its CR Ahlfors derivative.

In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.

In this work, we present an efficient way to decouple the multicontinuum problems. To construct decoupled schemes, we propose Implicit-Explicit time approximation in general form and study them for the fine-scale and coarse-scale space approximations. We use a finite-volume method for fine-scale approximation, and the nonlocal multicontinuum (NLMC) method is used to construct an accurate and physically meaningful coarse-scale approximation. The NLMC method is an accurate technique to develop a physically meaningful coarse scale model based on defining the macroscale variables. The multiscale basis functions are constructed in local domains by solving constraint energy minimization problems and projecting the system to the coarse grid. The resulting basis functions have exponential decay properties and lead to the accurate approximation on a coarse grid. We construct a fully Implicit time approximation for semi-discrete systems arising after fine-scale and coarse-scale space approximations. We investigate the stability of the two and three-level schemes for fully Implicit and Implicit-Explicit time approximations schemes for multicontinuum problems in fractured porous media. We show that combining the decoupling technique with multiscale approximation leads to developing an accurate and efficient solver for multicontinuum problems.

Flow interaction between a plain-fluid region in contact with a porous layer attracted significant attention from modelling and analysis sides due to numerous applications in biology, environment and industry. In the most widely used coupled model, fluid flow is described by the Stokes equations in the free-flow domain and Darcy's law in the porous medium, and complemented by the appropriate interface conditions. However, traditional coupling concepts are restricted, with a few exceptions, to one-dimensional flows parallel to the fluid-porous interface. In this work, we use an alternative approach to model interaction between the plain-fluid domain and porous medium by considering a transition zone, and propose the full- and hybrid-dimensional Stokes-Brinkman-Darcy models. In the first case, the equi-dimensional Brinkman equations are considered in the transition region, and the appropriate interface conditions are set on the top and bottom of the transition zone. In the latter case, we perform a dimensional model reduction by averaging the Brinkman equations in the normal direction and using the proposed transmission conditions. The well-posedness of both coupled problems is proved, and some numerical simulations are carried out in order to validate the concepts.

Large tensors are frequently encountered in various fields such as computer vision, scientific simulations, sensor networks, and data mining. However, these tensors are often too large for convenient processing, transfer, or storage. Fortunately, they typically exhibit a low-rank structure that can be leveraged through tensor decomposition. Despite this, performing large-scale tensor decomposition can be time-consuming. Sketching is a useful technique to reduce the dimensionality of the data. In this study, we introduce a novel two-sided sketching method based on the $t$-product decomposition and the discrete cosine transformation. We conduct a thorough theoretical analysis to assess the approximation error of the proposed method. Specifically, we enhance the algorithm with power iteration to achieve more precise approximate solutions. Extensive numerical experiments and comparisons on low-rank approximation of color images and grayscale videos illustrate the efficiency and effectiveness of the proposed approach in terms of both CPU time and approximation accuracy.

We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\mathcal{O}(n \log n)$ time. In particular, for a Toeplitz matrix $\mathbf{\Sigma}$ with values $\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$ where $S(\omega)$ is piecewise smooth, we give an approximation $\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$, where $\mathbf{\mathcal{F}}$ is the DFT matrix, $\mathbf{D}$ is diagonal, and the matrices $\mathbf{U}$ and $\mathbf{V}$ are in $\mathbb{C}^{n \times r}$ with $r \ll n$. Studying these matrices in the context of time series, we offer a theoretical explanation of this structure and connect it to existing spectral-domain approximation frameworks. We then give a complete discussion of the numerical method for assembling the approximation and demonstrate its efficiency for improving Whittle-type likelihood approximations, including dramatic examples where a correction of rank $r = 2$ to the standard Whittle approximation increases the accuracy from $3$ to $14$ digits for a matrix $\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$. The method and analysis of this work applies well beyond time series analysis, providing an algorithm for extremely accurate direct solves with a wide variety of symmetric Toeplitz matrices. The analysis employed here largely depends on asymptotic expansions of oscillatory integrals, and also provides a new perspective on when existing spectral-domain approximation methods for Gaussian log-likelihoods can be particularly problematic.

We derive and analyze numerical methods for weak approximation of underdamped (kinetic) Langevin dynamics in bounded domains. First-order methods are based on an Euler-type scheme interlaced with collisions with the boundary. To achieve second order, composition schemes are derived based on decomposition of the generator into collisional drift, impulse, and stochastic momentum evolution. In a deterministic setting, this approach would typically lead to first-order approximation, even in symmetric compositions, but we find that the stochastic method can provide second-order weak approximation with a single gradient evaluation, both at finite times and in the ergodic limit. We provide theoretical and numerical justification for this observation using model problems and compare and contrast the numerical performance of different choices of the ordering of the terms in the splitting scheme.

We consider the viscous incompressible fluids in a three-dimensional horizontally periodic domain bounded below by a fixed smooth boundary and above by a free moving surface. The fluid dynamics are governed by the Navier-Stokes equations with the effect of gravity and surface tension on the free surface. We develop a global well-posedness theory by a nonlinear energy method in low regular Sobolev spaces with several techniques, including: the horizontal energy-dissipation estimates, a new tripled bootstrap argument inspired by Guo and Tice [Arch. Ration. Mech. Anal.(2018)]. Moreover, the solution decays asymptotically to the equilibrium in an exponential rate.

In a world increasingly awash with data, the need to extract meaningful insights from data has never been more crucial. Functional Data Analysis (FDA) goes beyond traditional data points, treating data as dynamic, continuous functions, capturing ever-changing phenomena nuances. This article introduces FDA, merging statistics with real-world complexity, ideal for those with mathematical skills but no FDA background.

This paper studies Schauder theory to transmission problems modelled by fully nonlinear uniformly elliptic equations of second order. We focus on operators F that fails to be concave or convex in the space of symmetric matrices. In a first scenario, it is considered that F enjoys a small ellipticity aperture. In our second case, we study regularity results where the convexity of the superlevel (or sublevel) sets is verified, implying that the operator F is quasiconcave (or quasiconvex).

We introduce two models of industrial drying - a simplified one-equation model, and a detailed three-equation model. The purpose of the simplified model is rigorous validation of numerical methods for PDE-constrained optimal control. The purpose of the detailed model is to be able to predict and control the behaviour of an industrial disk drier. For both models, we introduce a fully validated numerical method to compute the optimal source term to maintain the outlet temperature as close as possible to the set-point temperature. By performing simulations using realistic parameters for industrial driers, we illustrate potential applications of the method.

The space-air-ground integrated network (SAGIN) is a pivotal architecture to support ubiquitous connectivity in the upcoming 6G era. Inter-operator resource and service sharing is a promising way to realize such a huge network, utilizing resources efficiently and reducing construction costs. Given the rationality of operators, the configuration of resources and services in SAGIN should focus on both the overall system performance and individual benefits of operators. Motivated by emerging symbiotic communication facilitating mutual benefits across different radio systems, we investigate the resource and service sharing in SAGIN from a symbiotic communication perspective in this paper. In particular, we consider a SAGIN consisting of a ground network operator (GNO) and a satellite network operator (SNO). Specifically, we aim to maximize the weighted sum rate (WSR) of the whole SAGIN by jointly optimizing the user association, resource allocation, and beamforming. Besides, we introduce a sharing coefficient to characterize the revenue of operators. Operators may suffer revenue loss when only focusing on maximizing the WSR. In pursuit of mutual benefits, we propose a mutual benefit constraint (MBC) to ensure that each operator obtains revenue gains. Then, we develop a centralized algorithm based on the successive convex approximation (SCA) method. Considering that the centralized algorithm is difficult to implement, we propose a distributed algorithm based on Lagrangian dual decomposition and the consensus alternating direction method of multipliers (ADMM). Finally, we provide extensive numerical simulations to demonstrate the effectiveness of the two proposed algorithms, and the distributed optimization algorithm can approach the performance of the centralized one.

Let $W$ be a standard Brownian motion with $W_0 = 0$ and let $b\colon[0,\infty) \to \mathbb{R}$ be a continuous function with $b(0) > 0$. In this article, we look at the classical First Passage Time (FPT) problem, i.e., the question of determining the distribution of $\tau := \inf \{ t\in [0,\infty)\colon W_t \geq b(t) \}.$ More specifically, we revisit the method of images, which we feel has received less attention than it deserves. The main observation of this approach is that the FPT problem is fully solved if a measure $\mu$ exists such that \begin{align*} \int_{(0,\infty)} \exp\left(-\frac{\theta^2}{2t}+\frac{\theta b(t)}{t}\right)\mu(d\theta)=1, \qquad t\in(0,\infty). \end{align*} The goal of this article is to lay the foundation for answering the still open question of the existence and characterisation of such a measure $\mu$ for a given curve $b$. We present a new duality approach that allows us to give sufficient conditions for the existence. Moreover, we introduce a very efficient algorithm for approximating the representing measure $\mu$ and provide a rigorous theoretical foundation.

Let $G$ be an affine algebraic group over an algebraically closed field of positive characteristic. Recent work of Hardesty, Nakano, and Sobaje gives necessary and sufficient conditions for the existence of so-called mock injective $G$-modules, that is, modules which are injective upon restriction to all Frobenius kernels of $G$. In this paper, we give analogous results for contramodules, including showing that the same necessary and sufficient conditions on $G$ guarantee the existence of mock-projective contramodules. In order to do this we first develop contramodule analogs to many well-known (co)module constructions.

For a pair $(G,\mathcal{P})$ consisting of a finitely generated group and finite collection of subgroups, we introduce a simplicial $G$-complex $\mathcal{K}(G,\mathcal{P})$ called the coset intersection complex. We prove that the quasi-isometry type and the homotopy type of $\mathcal{K}(G,\mathcal{P})$ are quasi-isometric invariants of the group pair $(G,\mathcal{P})$. Classical properties of $\mathcal{P}$ in $G$ correspond to topological or geometric properties of $\mathcal{K}(G,\mathcal{P})$, such as having finite height, having finite width, being almost malnormal, admiting a malnormal core, or having thickness of order one. As applications, we obtain that a number of algebraic properties of $\mathcal{P}$ in $G$ are quasi-isometry invariants of the pair $(G,\mathcal{P})$. For a certain class of right-angled Artin groups and their maximal parabolic subgroups, we show that the complex $\mathcal{K}(G,\mathcal{P})$ is quasi-isometric to the Deligne complex; in particular, it is hyperbolic.

In a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the singular part is assumed to have much stronger singularities and the same uniqueness theorem is proved. By applying the time-dependent method invented by Enss and Weder in 1995, the high-velocity limit for a wider class of the scattering operator with stronger singularities also uniquely determines uniquely the interactions of a multi-dimensional system.

Osburn, Sahu and Straub introduced the numbers: \begin{align*} A_n^{(r,s,t)}=\sum_{k=0}^n{n\choose k}^r{n+k\choose k}^s{2k\choose n}^t, \end{align*} for non-negative integers $n,r,s,t$ with $r\ge 2$, which includes two kinds of Ap\'ery numbers and four kinds of Ap\'ery-like numbers as special cases, and showed that the numbers $\{A_n^{(r,s,t)}\}_{n\ge 0}$ satisfy the Gauss congruences of order $3$. We establish an extension of Osburn--Sahu--Straub congruence through Bernoulli numbers, which is one step deep congruence of the Gauss congruence for $A_n^{(r,s,t)}$.

In this article we study the Honda-Tate theory for log abelian varieties over an fs log point $S=(\mathrm{Spec}(\mathbf{k}),M_S)$ for $\mathbf{k}=\mathbb{F}_q$ a finite field, generalizing the classical Honda-Tate theory for abelian varieties over $\mathbf{k}$. For the standard log point $S$, we give a complete description of the isogeny classes of such log abelian varieties using Weil $q$-numbers of weight 0,1, and 2. In the general case where $M_S$ admits a global chart $P\to\mathbf{k}$ with $P=\mathbb{N}^k$, we also give a complete description of simple isogeny classes of log abelian varieties over $S$ in terms of rational points in generalized simplices.

In this paper, we are concerned with the H\"older regularity for solutions of the nonlocal evolutionary equation $$ \partial_t u+(-\Delta_p)^s u = 0. $$ Here, $(-\Delta_p)^s$ is the fractional $p$-Laplacian, $0<s<1$ and $1<p<2$. We establish H\"older regularity with explicit H\"older exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained H\"older exponents are almost sharp. Our results complement the previous results for the superquadratic case when $p\geq 2$.

Over fourty years ago, the physicist Polyakov proposed a bold framework for string theory, in which the problem was reduced to the study of certain "random surfaces". He further made the tantalising suggestion that this theory could be explicitly solved. Recent breakthroughs from the last fifteen years have not only given a concrete mathematical basis for this theory but also verified some of its most striking predictions, as well as Polyakov's original vision. This theory, now known in the mathematics literature either as Liouville quantum gravity or Liouville conformal field theory, is based on a remarkable combination of ideas coming from different fields, above all probability and geometry. This book is intended to be an introduction to these developments assuming as few prerequisites as possible.

We show that the maximal shifts in the minimal free resolution of the quotients of a polynomial ring by a monomial ideal are subadditive as a function of the homological degree. This answers a question that has received some attention in recent years. To do so, we define and study a new model for the homology of posets, given by the so called synor complex. We also introduce an Eilenberg-Zilber type shuffle product on the simplicial chain complex of lattices. Combining these concepts we prove that the existence of a non-zero homology class for a lattice forces certain non-zero homology classes in lower intervals. This result then translates into properties of the minimal free resolution. In particular, it implies a generalization of the original question.

This paper examines the application of adaptive mesh refinement (AMR) in the field of numerical weather prediction (NWP). We implement and assess two distinct AMR approaches and evaluate their performance through standard NWP benchmarks. In both cases, we solve the fully compressible Euler equations, fundamental to many non-hydrostatic weather models. The first approach utilizes oct-tree cell-based mesh refinement coupled with a high-order discontinuous Galerkin method for spatial discretization. In the second approach, we employ level-based AMR with the finite difference method. Our study provides insights into the accuracy and benefits of employing these AMR methodologies for the multi-scale problem of NWP. Additionally, we explore essential properties including their impact on mass and energy conservation. Moreover, we present and evaluate an AMR solution transfer strategy for the tree-based AMR approach that is simple to implement, memory-efficient, and ensures conservation for both flow in the box and sphere. Furthermore, we discuss scalability, performance portability, and the practical utility of the AMR methodology within an NWP framework -- crucial considerations in selecting an AMR approach. The current de facto standard for mesh refinement in NWP employs a relatively simplistic approach of static nested grids, either within a general circulation model or a separately operated regional model with loose one-way synchronization. It is our hope that this study will stimulate further interest in the adoption of AMR frameworks like AMReX in NWP. These frameworks offer a triple advantage: a robust dynamic AMR for tracking localized and consequential features such as tropical cyclones, extreme scalability, and performance portability.

We address online estimation of microbial growth dynamics in bioreactors from measurements of a fluorescent reporter protein synthesized along with microbial growth. We consider an extended version of standard growth models that accounts for the dynamics of reporter synthesis. We develop state estimation from sampled, noisy measurements in the cases of known and unknown growth rate functions. Leveraging conservation laws and regularized estimation techniques, we reduce these nonlinear estimation problems to linear time-varying ones, and solve them via Kalman filtering. We establish convergence results in absence of noise and show performance on noisy data in simulation.

We investigate obstruction classes of moduli spaces of sheaves on K3 surfaces. We extend previous results by Caldararu, explicitly determining the obstruction class and its order in the Brauer group. Our main theorem establishes a short exact sequence relating the Brauer group of the moduli space to that of the underlying K3 surface. This provides a criterion for when the moduli space is fine, generalising well-known results for K3 surfaces. Additionally, we explore applications to Ogg-Shafarevich theory for Beauville-Mukai systems. Furthermore, we investigate birational equivalences of Beauville-Mukai systems on elliptic K3 surfaces, presenting a complete characterisation of such equivalences.

We build a one-to-one correspondence between $3$-dimensional (generic) crystabelline representations of the absolute Galois group of $\mathbb{Q}_p$ and certain locally analytic representations of $\mathrm{GL}_3(\mathbb{Q}_p)$. We show that the correspondence can be realized in spaces of $p$-adic automorphic representations.

Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a new way of deriving coefficients of implicit Runge-Kutta methods. This approach based on repeated integrals yields both new and well-known Butcher's tableaux. We discuss the properties of newly derived methods and compare them with standard collocation implicit Runge-Kutta methods in a series of numerical experiments. In particular, we observe higher accuracy and higher experimental order of convergence of some newly derived methods.

A realization or linearization is a triple, $(A,b,c)$, consisting of a $d-$tuple, $A= (A _1, \cdots, A_d )$, $d\in \mathbb{N}$, of bounded linear operators on a separable, complex Hilbert space, $\mathcal{H}$, and vectors $b,c \in \mathcal{H}$. Any such realization defines a (uniformly) analytic non-commutative (NC) function in an open neighbourhood of the origin, $0:= (0, \cdots , 0)$, of the NC universe of $d-$tuples of square matrices of any fixed size via the formula $h(X) = I \otimes b^* ( I \otimes I _{\mathcal{H}} - \sum X_j \otimes A_j ) ^{-1} I \otimes c$. It is well-known that an NC function has a finite-dimensional realization if and only if it is a non-commutative rational function that is defined at $0$. Such finite realizations contain valuable information about the NC rational functions they generate. By considering more general, infinite-dimensional realizations we study, construct and characterize more general classes of uniformly analytic NC functions. In particular, we show that an NC function, $h$, is (uniformly) entire, if and only if it has a jointly compact and quasinilpotent realization. Restricting our results to one variable shows that an analytic Taylor-MacLaurin series extends globally to an entire or meromorphic function if and only if it has a realization whose component operator is compact and quasinilpotent, or compact, respectively. This then motivates our definition of the set of global uniformly meromorphic NC functions as the (universal) skew field (of fractions) generated by NC rational expressions in the (semi-free ideal) ring of NC functions with jointly compact realizations.

We study a space-time Brownian motion with drift B(t)=(t_0+t,y_0+W(t)+t) killed at the moving boundary of the cone {(t,x):0<x<t}. This article determines the parabolic Martin boundary and all harmonic functions associated with this process. To that end, the asymptotics of Green's functions are determined along all directions. We also find the exit probabilities at the edges, the probability of remaining in the cone forever and the laws of the exit point and exit time. From this, we derive an explicit formula for the transition kernel of the process. These results arise from two different methods initially introduced to study random walks. An analytical approach, developed in the 1970s by Malyshev and based on the steepest descent method on a Riemann surface, is used to determine the asymptotics of the Green's functions. A recursive compensation approach, inspired by the method developed in the 1990s by Adan, Wessels and Zijm, is used to determine the harmonic functions.

This is the first in a sequence of articles exploring the relationship between commutative algebras and $E_\infty$-algebras in characteristic $p$ and mixed characteristic. In this paper we lay the groundwork by defining a new class of cohomology operations over $\mathbb F_p$ called cotriple products, generalising Massey products. We compute the secondary cohomology operations for a strictly commutative dg-algebra and the obstruction theories these induce, constructing several counterexamples to characteristic 0 behaviour, one of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We construct some families of higher cotriple products and comment on their behaviour. Finally, we distingush a subclass of cotriple products that we call higher Steenrod operation and conclude with our main theorem, which says that $E_\infty$-algebras can be rectified if and only if the higher Steenrod operations vanish coherently.

This paper addresses the construction and analysis of a class of domain decomposition methods for the iterative solution of the quasi-static Biot problem in three-field formulation. The considered discrete model arises from time discretization by the implicit Euler method and space discretization by a family of strongly mass-conserving methods exploiting $H^{div}$-conforming approximations of the solid displacement and fluid flux fields. For the resulting saddle-point problem, we construct monolithic overlapping domain decomposition (DD) methods whose analysis relies on a transformation into an equivalent symmetric positive definite system and on stable decompositions of the involved finite element spaces under proper problem-dependent norms. Numerical results on two-dimensional test problems are in accordance with the provided theoretical uniform convergence estimates for the two-level multiplicative Schwarz method.

The weighted essentially non-oscillatory {technique} using a stencil of $2r$ points (WENO-$2r$) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of $r+1$ nodes. The result is an interpolant of order $2r$ at the smooth parts and order $r+1$ when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.

In this paper we propose a new generalized cyclic symmetric structure in the factor matrices of polyadic decompositions of matrix multiplication tensors for non-square matrix multiplication to reduce the number of variables in the optimization problem and in this way improve the convergence.

A tensor invariant is defined on a para quaternionic contact manifold in terms of the curvature and torsion of the canonical para quaternionic connection involving derivatives up to third order of the contact form. This tensor, called para quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry, the Chern-Moser tensor in CR geometry, the para contact curvature in para CR geometry and to the quaternionic contact conformal curvature in quaternionic contact geometry. It is shown that a para quaternionic contact manifold is locally para quaternionic contact conformal to the standard flat para quaternionic contact structure on the para quaternionic Heisenberg group, or equivalently, to the standard para 3-Sasakian structure on the para quaternionic pseudo-sphere iff the para quaternionic contact conformal curvature vanishes.

In this paper we prove that for non-negative measurable functions $f$, \begin{align*} I_\alpha f \in BMO(\mathbb{R}^n) \text{ if and only if } I_\alpha f \in BMO^\beta(\mathbb{R}^n) \text{ for } \beta \in (n-\alpha,n]. \end{align*} Here $I_\alpha$ denotes the Riesz potential of order $\alpha$ and $BMO^\beta$ represents the space of functions of bounded $\beta$-dimensional mean oscillation.

Matlis duality for modules over commutative rings gives rise to the notion of Matlis reflexivity. It is shown that Matlis reflexive modules form a Krull-Schmidt category. For noetherian rings the absence of infinite direct sums is a characteristic feature of Matlis reflexivity. This leads to a discussion of objects that are extensions of artinian by noetherian objects. Classifications of Matlis reflexive modules are provided for some small examples.

This paper presents a novel approach for distributed model predictive control (MPC) for piecewise affine (PWA) systems. Existing approaches rely on solving mixed-integer optimization problems, requiring significant computation power or time. We propose a distributed MPC scheme that requires solving only convex optimization problems. The key contribution is a novel method, based on the alternating direction method of multipliers, for solving the non-convex optimal control problem that arises due to the PWA dynamics. We present a distributed MPC scheme, leveraging this method, that explicitly accounts for the coupling between subsystems by reaching agreement on the values of coupled states. Stability and recursive feasibility are shown under additional assumptions on the underlying system. Two numerical examples are provided, in which the proposed controller is shown to significantly improve the CPU time and closed-loop performance over existing state-of-the-art approaches.

We introduce the notion of paraquaternionic contact structures (pqc structures), which turns out to be a generalization of the para 3-Sasakian geometry. We derive a distinguished linear connection preserving the pqc structure. Its torsion tensor is expressed explicitly in terms of the structure tensors. We define pqc-Einstein manifolds and show that para 3-Sasakian spaces are precisely pqc manifolds, which are pqc-Einstein. We introduce the paraquaternionic Heisenberg qroup and show that it is the flat model of the pqc geometry.

Recently, in order to formulate a categorical version of the local Langlands correspondence, several authors have constructed moduli spaces of $\mathbf{Z}[1/p]$-valued L-parameters for $p$-adic groups. The connected components of these spaces over various $\mathbf{Z}[1/p]$-algebras $R$ are conjecturally related to blocks in categories of $R$-representations of $p$-adic groups. Dat-Helm-Kurinczuk-Moss described the components when $R$ is an algebraically closed field and gave a conjectural description when $R = \overline{\mathbf{Z}}[1/p]$. In this paper, we prove a strong form of this conjecture applicable to any integral domain $R$ over $\overline{\mathbf{Z}}[1/p]$.

In this paper, we establish the first explicit and non-asymptotic global convergence analysis of the BFGS method when deployed with an inexact line search scheme that satisfies the Armijo-Wolfe conditions. We show that BFGS achieves a global convergence rate of $(1-\frac{1}{\kappa})^k$ for $\mu$-strongly convex functions with $L$-Lipschitz gradients, where $\kappa=\frac{L}{\mu}$ denotes the condition number. Furthermore, if the objective function's Hessian is Lipschitz, BFGS with the Armijo-Wolfe line search achieves a linear convergence rate only determined by the line search parameters and independent of the condition number. These results hold for any initial point $x_0$ and any symmetric positive definite initial Hessian approximation matrix $B_0$, although the choice of $B_0$ affects the iteration count required to attain these rates. Specifically, we show that for $B_0 = LI$, the rate of $O((1-\frac{1}{\kappa})^k)$ appears from the first iteration, while for $B_0 = \mu I$, it takes $d\log \kappa$ iterations. Conversely, the condition number-independent linear convergence rate for $B_0 = LI$ occurs after $O\left(\kappa\left(d +\frac{M \sqrt{f(x_0)-f(x_*)}}{\mu^{3/2}}\right)\right)$ iterations, whereas for $B_0 = \mu I$, it holds after $O\left(\frac{M \sqrt{f(x_0)-f(x_*)}}{\mu^{3/2}}\left(d\log \kappa + \kappa\right)\right)$ iterations. Here, $d$ denotes the dimension of the problem, $M$ is the Lipschitz parameter of the Hessian, and $x_*$ denotes the optimal solution. We further leverage these global linear convergence results to characterize the overall iteration complexity of BFGS when deployed with the Armijo-Wolfe line search.

It is well known that the Young lattice is the Bratelli diagram of the symmetric groups expressing how irreducible representations restrict from $S_N$ to $S_{N-1}$. In 1988, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was realized as the Bratelli diagram of a family of algebras by Okada in 1994. In this paper, we realize the Okada algebra and its associated monoid using a labeled version of Temperley-Lieb arc-diagrams. We prove in full generality that the dimension of the Okada algebra is $n!$. In particular, we interpret a natural bijection between permutations and labeled arc-diagrams as an instance of Fomin's Robinson-Schensted correspondence for the Young-Fibonacci lattice. We prove that the Okada monoid is aperiodic and describe its Green relations. Lifting those results to the algebra allows us to construct a cellular basis of the Okada algebra. }

We show that for all homogeneous polynomials $ f_{m}$ of degree $m$, in $d$ variables, and each $j = 1, \dots , d$, we have \begin{equation*} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}% ^{d-1}\right) } \geq \frac{\pi ^{2}}{4\left( m+ d + 1 \right)^{2}} \left \langle f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}^{d-1}\right) }. \end{equation*} This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem, when the data are given by entire functions of order sufficiently low on nonhyperbolic quadratic hypersurfaces.

We consider nonparametric statistical inference on a periodic interaction potential $W$ from noisy discrete space-time measurements of solutions $\rho=\rho_W$ of the nonlinear McKean-Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to $W$ give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities $\bar \rho$ towards $\rho_W$. We further show that if the initial condition $\phi$ is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer the potential $W$ itself at convergence rates $N^{-\theta}$ for appropriate $\theta>0$, where $N$ is the number of measurements. The exponent $\theta$ can be taken to approach $1/2$ as the regularity of $W$ increases corresponding to `near-parametric' models.

We continue to study (see arXiv:2401.08618, https://doi.org/10.48550/arXiv.2401.08618) a renewal equation $\phi(t)=\frak F\phi_t$ proposed in [C. Barril et al., J. Math. Biology, https://doi.org/10.1007/s00285-024-02084-x] to model trees growth. This time we are considering the case when the per capita reproduction rate $\beta(x)$ is a non-monotone (unimodal) function of tree's height $x$. Note that the height of some species of trees can impact negatively seed viability, in a kind of autogamy depression. Similarly to previous works, it is also assumed that the growth rate $g(x)$ of an individual of height $x$ is a strictly decreasing function. Here we analyse the connection between dynamics of the associated one-dimensional map $F(b)= {\frak F}b,$ $b \in {\mathbb R}_+$, and the delayed (hence infinite-dimensional) model $\phi(t)=\frak F\phi_t$. Our key observation is that this model is of monotone positive feedback type since $F$ is strictly increasing on ${\mathbb R}_+$ independently on the monotonicity properties of $\beta$.

In 1977 Diaconis and Graham proved two inequalities relating different measures of disarray in permutations, and asked for a characterization of those permutations for which equality holds in one of these inequalities. Such a characterization was first given in 2013. Recently, another characterization was given by Woo, using a topological link in $\mathbb R^3$ that can be associated to the cycle diagram of a permutation. We show that Woo's characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham's inequality is directly related to the Euler characteristic of the associated link. This connection provides a new proof of the original result of Diaconis and Graham. We also characterize permutations with a fixed discrepancy in terms of their associated links and find that the stabilized-interval-free permutations are precisely those whose associated links are nonsplit.

In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,\eta)$ of order $m\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure $\gamma \Lambda$ of underlying Poisson point process $\eta$. The main result are new concentration bounds of the form \[ \mathbb{P}(|F_m ( f , \eta) -\mathbb{E} F_m ( f , \eta)| \ge t)\leq 2\exp(-I(\gamma,t)), \] where $I(\gamma,t)$ satisfies $I(\gamma,t)=\Theta(t^{1\over m}\log t)$ as $t\to\infty$ and $\gamma$ is fixed. The function $I(\gamma,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $\Lambda$. One of the key ingredients of the proof are fine bounds for the centred moments of $F_m(f,\eta)$. We discuss the optimality of obtained bounds and consider a number of applications related to Gilbert graphs and Poisson hyperplane processes in constant curvature spaces.

In the present work, we prove that, if $A(\cdot)$ is a compact almost automorphic matrix and the system $$x'(t) = A(t)x(t)\, ,$$ possesses an exponential dichotomy with Green function $G(\cdot, \cdot)$, then its associated system $$y'(t) = B(t)y(t)\, ,$$ where $B(\cdot) \in H(A)$ (the hull of $A(\cdot)$) also possesses an exponential dichotomy. Moreover, the Green function $G(\cdot, \cdot)$ is compact Bi-almost automorphic in $\mathbb{R}^2$, this implies that $G(\cdot, \cdot)$ is $\Delta_2$ - like uniformly continuous, where $\Delta_2$ is the principal diagonal of $\mathbb{R}^2$, an important ingredient in the proof of invariance of the compact almost automorphic function space under convolution product with kernel $G(\cdot, \cdot)$. Finally, we study the existence of a positive compact almost automorphic solution of non-autonomous differential equations of biological interest having non-linear harvesting terms and mixed delays.

Boolean networks can be viewed as functions on the set of binary strings of a given length, described via logical rules. They were introduced as dynamic models into biology, in particular as logical models of intracellular regulatory networks involving genes, proteins, and metabolites. Since genes can have several modes of action, depending on their expression levels, binary variables are often not sufficiently rich, requiring the use of multi-valued networks instead. The steady state analysis of Boolean networks is computationally complex, and increasing the number of variable values beyond $2$ adds substantially to this complexity, and no general methods are available beyond simulation. The main contribution of this paper is to give an algorithm to compute the steady states of a multi-valued network that has a complexity that, in many cases, is essentially the same as that for the case of binary values. Our approach is based on a representation of multi-valued networks using multi-valued logic functions, providing a biologically intuitive representation of the network. Furthermore, it uses tools to compute lattice points in rational polytopes, tapping a rich area of algebraic combinatorics as a source for combinatorial algorithms for Boolean network analysis. An implementation of the algorithm is provided.

Determining the Shannon capacity of graphs is a long-standing open problem in information theory, graph theory and combinatorial optimization. Over decades, a wide range of upper and lower bound methods have been developed to analyze this problem. However, despite tremendous effort, even small instances of the problem have remained open. In recent years, a new dual characterization of the Shannon capacity of graphs, asymptotic spectrum duality, has unified and extended known upper bound methods and structural theorems. In this paper, building on asymptotic spectrum duality, we develop a new theory of graph distance, that we call asymptotic spectrum distance, and corresponding limits (reminiscent of, but different from, the celebrated theory of cut-norm, graphons and flag algebras). We propose a graph limit approach to the Shannon capacity problem: to determine the Shannon capacity of a graph, construct a sequence of easier to analyse graphs converging to it. (1) We give a very general construction of non-trivial converging sequences of graphs (in a family of circulant graphs). (2) We construct Cauchy sequences of finite graphs that do not converge to any finite graph, but do converge to an infinite graph. We establish strong connections between convergence questions of finite graphs and the asymptotic properties of Borsuk-like infinite graphs on the circle. (3) We observe that all best-known lower bound constructions for Shannon capacity of small odd cycles can be obtained from a "finite" version of the graph limit approach. We develop computational and theoretical aspects of this approach and use these to obtain a new Shannon capacity lower bound for the fifteen-cycle. The theory of asymptotic spectrum distance applies not only to Shannon capacity of graphs; indeed, we will develop it for a general class of mathematical objects and their asymptotic properties.

Gromov-Witten invariants arise in the topological A-model as counts of worldsheet instantons. On the A-side, these invariants can be computed for a Fano or semi-Fano toric variety using generating functions associated to the toric divisors. On the B-side, the same invariants can be computed from the periods of the mirror. We utilize scattering diagrams (aka wall structures) in the Gross-Siebert mirror symmetry program to extend the calculation of Gromov-Witten invariants to non-Fano toric varieties. Following the work of Carl-Pumperla-Siebert, we compute corrected mirror superpotentials $\vartheta_1(\mathbb{F}_m)$ and their periods for the Hirzebruch surfaces $\mathbb{F}_m$ with $m \ge 2$.

We investigate the order of the Tate--Shafarevich group of abelian varieties modulo rational squares. Our main result shows that every square-free natural number appears as the non square-free part of the Tate--Shafarevich group of some abelian variety, thereby validating a conjecture of W. Stein.

The universal Drinfeld-Yetter algebra is an associative algebra whose co-Hochschild cohomology controls the existence of quantization functors of Lie bialgebras, such as the renowned one due to Etingof and Kazhdan. It was initially introduced by Enriquez and later re-interpreted by Appel and Toledano Laredo as an algebra of endomorphisms in the colored PROP of a Drinfeld-Yetter module over a Lie bialgebra. In this paper, we provide an explicit formula for its structure constants in terms of certain diagrams, which we term Drinfeld-Yetter looms.

In this paper we study rectifying submanifolds of a Riemannian manifold endowed with an anti-torqued vector field. For this, we first determine a necessary and sufficient condition for the ambient space to admit such a vector field. Then we characterize submanifolds for which an anti-torqued vector field is always assumed to be tangent or normal. A similar characterization is also done in the case of the torqued vector fields. Finally, we obtain that the rectifying submanifolds with anti-torqued axis are the warped products whose warping function is a first integration of the conformal scalar of the axis.

Transportation matrices are $m\times n$ non-negative matrices whose row sums and row columns are equal to, or dominated above with given integral vectors $R$ and $C$. Those matrices belong to a convex polytope whose extreme points have been previously characterized. In this article, a more general set of non-negative transportation matrices is considered, whose row sums are bounded by two integral non-negative vectors $R_{min}$ and $R_{max}$ and column sums are bounded by two integral non-negative vectors $C_{min}$ and $C_{max}$. It is shown that this set is also a convex polytope whose extreme points are then fully characterized.

This paper develops structure-preserving, oscillation-eliminating discontinuous Galerkin (OEDG) schemes for ideal magnetohydrodynamics (MHD), as a sequel to our recent work [Peng, Sun, and Wu, OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws, 2023]. The schemes are based on a locally divergence-free (LDF) oscillation-eliminating (OE) procedure to suppress spurious oscillations while maintaining many of the good properties of original DG schemes, such as conservation, local compactness, and optimal convergence rates. The OE procedure is built on the solution operator of a novel damping equation -- a simple linear ordinary differential equation (ODE) whose exact solution can be exactly formulated. Because this OE procedure does not interfere with DG spatial discretization and RK stage update, it can be easily incorporated to existing DG codes as an independent module. These features make the proposed LDF OEDG schemes highly efficient and easy to implement.In addition, we present a positivity-preserving (PP) analysis of the LDF OEDG schemes on Cartesian meshes via the optimal convex decomposition technique and the geometric quasi-linearization (GQL) approach. Efficient PP LDF OEDG schemes are obtained with the HLL flux under a condition accessible by the simple local scaling PP limiter.Several one- and two-dimensional MHD tests confirm the accuracy, effectiveness, and robustness of the proposed structure-preserving OEDG schemes.

We introduce a detection algorithm for SAGBI basis in polynomial rings, analogous to a Gr\"obner basis detection algorithm previously proposed by Gritzmann and Sturmfels. We also present two accompanying software packages named SagbiGbDetection for Macaulay2 and Julia. Both packages allow the user to find one or more term orders for which a set of input polynomials form either Gr\"obner basis for the ideal they generate or a SAGBI basis for the subalgebra. Additionally, we investigate the computational complexity of homogeneous SAGBI detection and apply our implementation to several novel examples.

We propose a test problem for Navier-Stokes solvers based on the flow around a cylinder. We choose a range of Reynolds numbers for which the flow is time-dependent but can be characterized as essentially two-dimensional. The test problem requires accurate resolution of chaotic dynamics over a long time interval. It also requires the use of a relatively large computational domain, part of which is curved, and it requires evaluation of derivatives of the solution and pressure on the curved boundary. We review the performance of different finite element methods for the proposed range of Reynolds numbers. These tests indicate that some of the most established methods do not capture the correct behavior.

We discuss a complementary asymptotic analysis of the so called minimal random walk. More precisely, we present a version of the almost sure central limit theorem as well as a generalization of the recently proposed quadratic strong laws. In addition, alternative demonstrations of the functional limit theorems will be supplied based on a P\'olya urn scheme instead of a martingale approach.

We show that the directed landscape is a black noise in the sense of Tsirelson and Vershik. As a corollary, we show that for any microscopic system in which the height profile converges in law to the directed landscape, the driving noise is asymptotically independent of the height profile. This decoupling result provides one answer to the question of what happens to the driving noise in the limit under the KPZ scaling, and illustrates a type of noise sensitivity for systems in the KPZ universality class. Such decoupling and sensitivity phenomena are not present in the intermediate-disorder or weak-asymmetry regime, thus illustrating a contrast from the weak KPZ scaling regime. Along the way, we prove a strong mixing property for the directed landscape on a bounded time interval under spatial shifts, with a mixing rate $\alpha(N)\leq Ce^{-dN^3}$ for some $C,d>0$.

We give a complete list of the points in the spectrum $$\mathcal{Z}=\{\inf_{(x,y)\in\Lambda,xy\neq0}{\left\vert xy\right\vert},\,\text{$\Lambda$ is a unimodular rational lattice of $\mathbb{R}^2$}\}$$ above $\frac{1}{3}.$ We further show that the set of limit points of $\mathcal{Z}$ with values larger than $\frac{1}{3},$ is equal to the set $\{\frac{2m}{\sqrt{9m^2-4}+3m},\text{ where $m$ is a Markoff number}\}$.

We consider the cubic Schr\"odinger equation posed on product spaces subject to a generic Diophantine condition. Our analysis shows that the small-amplitude solutions undergo modified scattering to an effective dynamics governed by some interactions that do not amplify the Sobolev norms. This is in sharp contrast with the infinite energy cascade scenario observed by Hani--Pausader--Tzvetkov--Visciglia in the absence of Diophantine conditions.

We present successive convexification, a real-time-capable solution method for nonconvex trajectory optimization, with continuous-time constraint satisfaction and guaranteed convergence, that only requires first-order information. The proposed framework combines several key methods to solve a large class of nonlinear optimal control problems: (i) exterior penalty-based reformulation of the path constraints; (ii) generalized time-dilation; (iii) multiple-shooting discretization; (iv) $\ell_1$ exact penalization of the nonconvex constraints; and (v) the prox-linear method, a sequential convex programming (SCP) algorithm for convex-composite minimization. The reformulation of the path constraints enables continuous-time constraint satisfaction even on sparse discretization grids and obviates the need for mesh refinement heuristics. Through the prox-linear method, we guarantee convergence of the solution method to stationary points of the penalized problem and guarantee that the converged solutions that are feasible with respect to the discretized and control-parameterized optimal control problem are also Karush-Kuhn-Tucker (KKT) points. Furthermore, we highlight the specialization of this property to global minimizers of convex optimal control problems, wherein the reformulated path constraints cannot be represented by canonical cones, i.e., in the form required by existing convex optimization solvers. In addition to theoretical analysis, we demonstrate the effectiveness and real-time capability of the proposed framework with numerical examples based on popular optimal control applications: dynamic obstacle avoidance and rocket landing.

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provides a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They displayed an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.

The main contribution of our paper is to introduce a number of multivariate quantum fidelities and show that they satisfy several desirable properties that are natural extensions of those of the Uhlmann and Holevo fidelities. We propose three variants that reduce to the average pairwise fidelity for commuting states: average pairwise $z$-fidelities, the multivariate semi-definite programming (SDP) fidelity, and a multivariate fidelity inspired by an existing secrecy measure. The second one is obtained by extending the SDP formulation of the Uhlmann fidelity to more than two states. All three of these variants satisfy the following properties: (i) reduction to multivariate classical fidelities for commuting states, (ii) the data-processing inequality, (iii) invariance under permutations of the states, (iv) its values are in the interval $[0,1]$; they are faithful, that is, their values are equal to one if and only if all the states are equal, and they satisfy orthogonality, that is their values are equal to zero if and only if the states are mutually orthogonal to each other, (v) direct-sum property, (vi) joint concavity, and (vii) uniform continuity bounds under certain conditions. Furthermore, we establish inequalities relating these different variants, indeed clarifying that all these definitions coincide with the average pairwise fidelity for commuting states. Lastly, we introduce another multivariate fidelity called multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. We also show that it satisfies most of the desirable properties listed above, it is a function of a multivariate log-Euclidean divergence, and has an operational interpretation in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis.

We present a variational quantum algorithm for solving combinatorial optimization problems with linear-depth circuits. Our algorithm uses an ansatz composed of Hamiltonian generators designed to control each term in the target combinatorial function, along with parameter updates following a modified version of quantum imaginary time evolution. We evaluate this ansatz in numerical simulations that target solutions to the MAXCUT problem. The state evolution is shown to closely mimic imaginary time evolution, and its optimal-solution convergence is further improved using adaptive transformations of the classical Hamiltonian spectrum, while resources are minimized by pruning optimized gates that are close to the identity. With these innovations, the algorithm consistently converges to optimal solutions, with interesting highly-entangled dynamics along the way. This performant and resource-minimal approach is a promising candidate for potential quantum computational advantages on near-term quantum computing hardware.

We introduce an assessment procedure for interactive segmentation models. Based on concepts from Bayesian Experimental Design, the procedure measures a model's understanding of point prompts and their correspondence with the desired segmentation mask. We show that Oracle Dice index measurements are insensitive or even misleading in measuring this property. We demonstrate the use of the proposed procedure on three interactive segmentation models and subsets of two large image segmentation datasets.

Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is an important problem in online algorithms and mobile computing that has been studied for several decades, and which now is essentially resolved in its original formulation. In this paper, we demonstrate that what appears to be an innocuous modification of the problem actually leads to a drastic (exponential) reduction of the competitive ratio. Specifically, we present an algorithm that is $O(\log^2 w)$-competitive for traversing unweighted layered graphs of width $w$. Our technique is based on a simple entropic regularizer, which evolves as the agent progresses in the layered graph. Our algorithm is randomized and simply maintains that at all layers, the probability distribution of the position of the mobile agent maximizes the entropic regularizer.

We investigate the proof theory of regular expressions with fixed points, construed as a notation for (omega-)context-free grammars. Starting with a hypersequential system for regular expressions due to Das and Pous, we define its extension by least fixed points and prove soundness and completeness of its non-wellfounded proofs for the standard language model. From here we apply proof-theoretic techniques to recover an infinitary axiomatisation of the resulting equational theory, complete for inclusions of context-free languages. Finally, we extend our syntax by greatest fixed points, now computing omega-context-free languages. We show the soundness and completeness of the corresponding system using a mixture of proof-theoretic and game-theoretic techniques.

We present a velocity-based Monte Carlo fluid solver that overcomes the limitations of its existing vorticity-based counterpart. Because the velocity-based formulation is more commonly used in graphics, our Monte Carlo solver can be readily extended with various techniques from the fluid simulation literature. We derive our method by solving the Navier-Stokes equations via operator splitting and designing a pointwise Monte Carlo estimator for each substep. We reformulate the projection and diffusion steps as integration problems based on the recently introduced walk-on-boundary technique [Sugimoto et al. 2023]. We transform the volume integral arising from the source term of the pressure Poisson equation into a form more amenable to practical numerical evaluation. Our resulting velocity-based formulation allows for the proper simulation of scenes that the prior vorticity-based Monte Carlo method [Rioux-Lavoie and Sugimoto et al. 2022] either simulates incorrectly or cannot support. We demonstrate that our method can easily incorporate advancements drawn from conventional non-Monte Carlo methods by showing how one can straightforwardly add buoyancy effects, divergence control capabilities, and numerical dissipation reduction methods, such as advection-reflection and PIC/FLIP methods.

In this chapter, we investigate the spectrum measurement results in Asia-Pacific region. Then the spectrum sharing policy in the Asia-Pacific region is reviewed in details, where the national projects and strategies on spectrum refarming and spectrum sharing in China, Japan, Singapore, India, Korea and Australia are investigated. Then we introduce the spectrum sharing test-bed that is developed in China, which is a cognitive radio enabled TD-LTE test-bed utilizing TVWS. This chapter provides a brief introduction of the spectrum sharing mechanism and policy of Asia-Pacific region.

In this paper, we analyze the impact of data freshness on remote inference systems, where a pre-trained neural network infers a time-varying target (e.g., the locations of vehicles and pedestrians) based on features (e.g., video frames) observed at a sensing node (e.g., a camera). One might expect that the performance of a remote inference system degrades monotonically as the feature becomes stale. Using an information-theoretic analysis, we show that this is true if the feature and target data sequence can be closely approximated as a Markov chain, whereas it is not true if the data sequence is far from Markovian. Hence, the inference error is a function of Age of Information (AoI), where the function could be non-monotonic. To minimize the inference error in real-time, we propose a new "selection-from-buffer" model for sending the features, which is more general than the "generate-at-will" model used in earlier studies. In addition, we design low-complexity scheduling policies to improve inference performance. For single-source, single-channel systems, we provide an optimal scheduling policy. In multi-source, multi-channel systems, the scheduling problem becomes a multi-action restless multi-armed bandit problem. For this setting, we design a new scheduling policy by integrating Whittle index-based source selection and duality-based feature selection-from-buffer algorithms. This new scheduling policy is proven to be asymptotically optimal. These scheduling results hold for minimizing general AoI functions (monotonic or non-monotonic). Data-driven evaluations demonstrate the significant advantages of our proposed scheduling policies.

Differentially private federated learning is crucial for maintaining privacy in distributed environments. This paper investigates the challenges of high-dimensional estimation and inference under the constraints of differential privacy. First, we study scenarios involving an untrusted central server, demonstrating the inherent difficulties of accurate estimation in high-dimensional problems. Our findings indicate that the tight minimax rates depends on the high-dimensionality of the data even with sparsity assumptions. Second, we consider a scenario with a trusted central server and introduce a novel federated estimation algorithm tailored for linear regression models. This algorithm effectively handles the slight variations among models distributed across different machines. We also propose methods for statistical inference, including coordinate-wise confidence intervals for individual parameters and strategies for simultaneous inference. Extensive simulation experiments support our theoretical advances, underscoring the efficacy and reliability of our approaches.

$U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are constructed by contractions of complex tensors of order $r+q$, also denoted $(r,q)$. These tensors transform under $r$ fundamental representations of the unitary group $U(N)$ and $q$ fundamental representations of the orthogonal group $O(N)$. Therefore, $U(N)^{\otimes r} \otimes O(N)^{\otimes q}$ invariants are tensor model observables endowed with a tensor field of order $(r,q)$. We enumerate these observables using group theoretic formulae, for arbitrary tensor fields of order $(r,q)$. Inspecting lower-order cases reveals that, at order $(1,1)$, the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order $(r,q)$, the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For $r>1$, the number of invariants matches the number of ($q$-dependent) weighted equivalence classes of branched covers of the 2-sphere with $r$ branched points. At $r=1$, the counting maps to the enumeration of branched covers of the 2-sphere with $q+3$ branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.

The paper presents an analytical theory quantitatively describing the heterogeneous combustion of nonvolatile (metal) particles in the diffusion-limited regime. It is assumed that the particle is suspended in an unconfined, isobaric, quiescent gaseous mixture and the chemisorption of the oxygen takes place evenly on the particle surface. The exact solution of the particle burn time is derived from the conservation equations of the gas-phase described in a spherical coordinate system with the utilization of constant thermophysical properties, evaluated at a reference film layer. This solution inherently takes the Stefan flow into account. The approximate expression of the time-dependent particle temperature is solved from the conservation of the particle enthalpy by neglecting the higher order terms in the Taylor expansion of the product of the transient particle density and diameter squared. Coupling the solutions for the burn time and time-dependent particle temperature provides quantitative results when initial and boundary conditions are specified. The theory is employed to predict the burn time and temperature of micro-sized iron particles, which are then compared with measurements, as the first validation case. The theoretical burn time agrees with the experiments almost perfectly at both low and high oxygen levels. The calculated particle temperature matches the measurements fairly well at relatively low oxygen mole fractions, whereas the theory overpredict the particle peak temperature due to the neglect of evaporation and the possible transition of the combustion regime.

We introduce an extension of classical cellular automata (CA) to arbitrary labeled graphs, and show that FO logic on CA orbits is equivalent to MSO logic. We deduce various results from that equivalence, including a characterization of finitely generated groups on which FO model checking for CA orbits is undecidable, and undecidability of satisfiability of a fixed FO property for CA over finite graphs. We also show concrete examples of FO formulas for CA orbits whose model checking problem is equivalent to the domino problem, or its seeded or recurring variants respectively, on any finitely generated group. For the recurring domino problem, we use an extension of the FO signature by a relation found in the well-known Garden of Eden theorem, but we also show a concrete FO formula without the extension and with one quantifier alternation whose model checking problem does not belong to the arithmetical hierarchy on group Z^2.

Identifying partial differential equations (PDEs) from data is crucial for understanding the governing mechanisms of natural phenomena, yet it remains a challenging task. We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs from limited prior knowledge automatically. Our method automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes, demonstrating its robustness in handling noisy and non-uniformly distributed data. We also test the algorithm's performance on datasets consisting solely of random noise to simulate scenarios with severely compromised data quality. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases. We highlight the potential of combining statistical methods, machine learning, and dynamical systems theory to automatically discover governing equations from collected data, streamlining the scientific modeling process.

Inspired by a recent work of Dubrovin [7], for each simple Lie algebra $\mathfrak{g}$, we introduce an infinite family of pairwise commuting ODEs and define their $\tau$-functions. We show that these $\tau$-functions can be identified with the $\tau$-functions for the Drinfeld--Sokolov hierarchy of $\mathfrak{g}$-type. Explicit examples for $\mathfrak{g}=A_1$ and $A_2$ are provided, which are connected to the KdV hierarchy and the Boussinesq hierarchy respectively.

Higher-dimensional automata, i.e., pointed labeled precubical sets, are a powerful combinatorial-topological model for concurrent systems. In this paper, we show that for every (nonempty) connected polyhedron there exists a shared-variable system such that the higher-dimensional automaton modeling the state space of the system has the homotopy type of the polyhedron.

We consider the problem of regularized Poisson Non-negative Matrix Factorization (NMF) problem, encompassing various regularization terms such as Lipschitz and relatively smooth functions, alongside linear constraints. This problem holds significant relevance in numerous Machine Learning applications, particularly within the domain of physical linear unmixing problems. A notable challenge arises from the main loss term in the Poisson NMF problem being a KL divergence, which is non-Lipschitz, rendering traditional gradient descent-based approaches inefficient. In this contribution, we explore the utilization of Block Successive Upper Minimization (BSUM) to overcome this challenge. We build approriate majorizing function for Lipschitz and relatively smooth functions, and show how to introduce linear constraints into the problem. This results in the development of two novel algorithms for regularized Poisson NMF. We conduct numerical simulations to showcase the effectiveness of our approach.

In this paper, we show how mixed-integer conic optimization can be used to combine feature subset selection with holistic generalized linear models to fully automate the model selection process. Concretely, we directly optimize for the Akaike and Bayesian information criteria while imposing constraints designed to deal with multicollinearity in the feature selection task. Specifically, we propose a novel pairwise correlation constraint that combines the sign coherence constraint with ideas from classical statistical models like Ridge regression and the OSCAR model.

Continuity equations associated to continuous-time Markov processes can be considered as Euclidean Schr\"odinger equations, where the non-hermitian quantum Hamiltonian $\bold{H}={\bold{div}}{\bold J}$ is naturally factorized into the product of the divergence operator ${\bold {div}}$ and the current operator ${\bold J}$. For non-equilibrium Markov jump processes in a space of $N$ configurations with $M$ links and $C=M-(N-1)\geq 1$ independent cycles, this factorization of the $N \times N$ Hamiltonian ${\bold H}={\bold I}^{\dagger}{\bold J}$ involves the incidence matrix ${\bold I}$ and the current matrix ${\bold J}$ of size $M \times N$, so that the supersymmetric partner ${\hat{\bold H}}= {\bold J}{\bold I}^{\dagger}$ governing the dynamics of the currents living on the $M$ links is of size $M \times M$. To better understand the relations between the spectral decompositions of these two Hamiltonians $\bold{H}={\bold I}^{\dagger}{\bold J}$ and ${\hat {\bold H}} ={\bold J}{\bold I}^{\dagger}$ with respect to their bi-orthogonal basis of right and left eigenvectors that characterize the relaxation dynamics towards the steady state and the steady currents, it is useful to analyze the properties of the Singular Value Decompositions of the two rectangular matrices ${\bold I}$ and ${\bold J} $ of size $M \times N$ and the interpretations in terms of discrete Helmholtz decompositions. This general framework concerning Markov jump processes can be adapted to non-equilibrium diffusion processes governed by Fokker-Planck equations in dimension $d$, where the number $N$ of configurations, the number $M$ of links and the number $C=M-(N-1)$ of independent cycles become infinite, while the two matrices ${\bold I}$ and ${\bold J}$ become first-order differential operators acting on scalar functions to produce vector fields.

This paper aims to explore the evolution of image denoising in a pedagological way. We briefly review classical methods such as Fourier analysis and wavelet bases, highlighting the challenges they faced until the emergence of neural networks, notably the U-Net, in the 2010s. The remarkable performance of these networks has been demonstrated in studies such as Kadkhodaie et al. (2024). They exhibit adaptability to various image types, including those with fixed regularity, facial images, and bedroom scenes, achieving optimal results and biased towards geometry-adaptive harmonic basis. The introduction of score diffusion has played a crucial role in image generation. In this context, denoising becomes essential as it facilitates the estimation of probability density scores. We discuss the prerequisites for genuine learning of probability densities, offering insights that extend from mathematical research to the implications of universal structures.

In the study of quantum state transfer, one is interested in being able to transmit a quantum state with high fidelity within a quantum spin network. In most of the literature, the state of interest is taken to be associated with a standard basis vector; however, more general states have recently been considered. Here, we consider a general linear combination of two vertex states, which encompasses the definitions of pair states and plus states in connected weighted graphs. A two-state in a graph $X$ is a quantum state of the form $\mathbf{e}_u+s\mathbf{e}_v$, where $u$ and $v$ are two vertices in $X$ and $s$ is a non-zero real number. If $s=-1$ or $s=1$, then such a state is called a pair state or a plus state, respectively. In this paper, we investigate quantum state transfer between two-states, where the Hamiltonian is taken to be the adjacency, Laplacian or signless Laplacian matrix of the graph. By analyzing the spectral properties of the Hamiltonian, we characterize strongly cospectral two-states built from strongly cospectral vertices. This allows us to characterize perfect state transfer (PST) between two-states in complete graphs, cycles and hypercubes. We also produce infinite families of graphs that admit strong cospectrality and PST between two-states that are neither pair nor plus states. Using singular values and singular vectors, we show that vertex PST in the line graph of $X$ implies PST between the plus states formed by corresponding edges in $X$. Furthermore, we provide conditions such that the converse of the previous statement holds. As an application, we characterize strong cospectrality and PST between vertices in line graphs of trees, unicyclic graphs and Cartesian products.

When solving the Hamiltonian path problem it seems natural to be given additional precedence constraints for the order in which the vertices are visited. For example one could decide whether a Hamiltonian path exists for a fixed starting point, or that some vertices are visited before another vertex. We consider the problem of finding a Hamiltonian path that observes all precedence constraints given in a partial order on the vertex set. We show that this problem is $\mathsf{NP}$-complete even if restricted to complete bipartite graphs and posets of height 2. In contrast, for posets of width $k$ there is an $\mathcal{O}(k^2 n^k)$ algorithm for arbitrary graphs with $n$ vertices. We show that it is unlikely that the running time of this algorithm can be improved significantly, i.e., there is no $f(k) n^{o(k)}$ time algorithm under the assumption of the Exponential Time Hypothesis. Furthermore, for the class of outerplanar graphs, we give an $\mathcal{O}(n^2)$ algorithm for arbitrary posets.

For a fixed graph $H$, the $H$-SUBGRAPH HITTING problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of $H$ as a subgraph. This problem can be seen as a generalization of VERTEX COVER, which corresponds to the case $H = K_2$. We initiate a study of $H$-SUBGRAPH HITTING from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class $C$. Our main contribution is to identify graph parameters that, when $H$-SUBGRAPH HITTING is parameterized by the vertex-deletion distance to a class $C$ where any of these parameters is bounded, and assuming standard complexity assumptions and that $H$ is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if $H$ is a clique. These new graph parameters are inspired by the notion of $C$-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit $H$ as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

We investigate the well-posedness of the characteristic initial-boundary value problem for the Einstein equations in Bondi-like coordinates (including Bondi, double-null and affine). We propose a definition of strong hyperbolicity of a system of partial differential equations of any order, and show that the Einstein equations in Bondi-like coordinates in their second-order form used in numerical relativity do not meet it, in agreement with results of Giannakopoulos et al for specific first-order reductions. In the principal part, frozen coefficient approximation that one uses to examine hyperbolicity, we explicitly construct the general solution to identify the solutions that obstruct strong hyperbolicity. Independently, we present a first-order symmetric hyperbolic formulation of the Einstein equations in Bondi gauge, linearised about Schwarzschild, thus completing work by Frittelli. This establishes an energy norm ($L^2$ in the metric perturbations and selected first and second derivatives), in which the initial-boundary value problem, with initial data on an outgoing null cone and boundary data on a timelike cylinder or an ingoing null cone, is well-posed, thus verifying a conjecture by Giannakopoulos et al. Unfortunately, our method does not extend to the pure initial-value problem on a null cone with regular vertex.

We propose a notion of lift for quantum CSS codes, inspired by the geometrical construction of Freedman and Hastings. It is based on the existence of a canonical complex associated to any CSS code, that we introduce under the name of Tanner cone-complex, and over which we generate covering spaces. As a first application, we describe the classification of lifts of hypergraph product codes (HPC) and demonstrate the equivalence with the lifted product code (LPC) of Panteleev and Kalachev, including when the linear codes, factors of the HPC, are Tanner codes. As a second application, we report several new non-product constructions of quantum CSS codes, and we apply the prescription to generate their lifts which, for certain selected covering maps, are codes with improved relative parameters compared to the initial one.

This work introduces a new method for selecting the number of components in finite mixture models (FMMs) using variational Bayes, inspired by the large-sample properties of the Evidence Lower Bound (ELBO) derived from mean-field (MF) variational approximation. Specifically, we establish matching upper and lower bounds for the ELBO without assuming conjugate priors, suggesting the consistency of model selection for FMMs based on maximizing the ELBO. As a by-product of our proof, we demonstrate that the MF approximation inherits the stable behavior (benefited from model singularity) of the posterior distribution, which tends to eliminate the extra components under model misspecification where the number of mixture components is over-specified. This stable behavior also leads to the $n^{-1/2}$ convergence rate for parameter estimation, up to a logarithmic factor, under this model overspecification. Empirical experiments are conducted to validate our theoretical findings and compare with other state-of-the-art methods for selecting the number of components in FMMs.

In this work we give an efficient construction of unitary $k$-designs using $\tilde{O}(k\cdot poly(n))$ quantum gates, as well as an efficient construction of a parallel-secure pseudorandom unitary (PRU). Both results are obtained by giving an efficient quantum algorithm that lifts random permutations over $S(N)$ to random unitaries over $U(N)$ for $N=2^n$. In particular, we show that products of exponentiated sums of $S(N)$ permutations with random phases approximately match the first $2^{\Omega(n)}$ moments of the Haar measure. By substituting either $\tilde{O}(k)$-wise independent permutations, or quantum-secure pseudorandom permutations (PRPs) in place of the random permutations, we obtain the above results. The heart of our proof is a conceptual connection between the large dimension (large-$N$) expansion in random matrix theory and the polynomial method, which allows us to prove query lower bounds at finite-$N$ by interpolating from the much simpler large-$N$ limit. The key technical step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-$N$ expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension $N$.

This study delves into the auto-ignition temperature of n-heptane and ethanol mixtures within a counterflow flame configuration under low strain rate, with a particular focus on the impact of ethanol blending on heat release rates. Employing the sensitivity analysis method inspired by Zurada's sensitivity approach for neural network, this study identifies the group of critical species influencing the heat release rate. Further analysis concentration change reveals the intricate interactions among these various radicals across different temperature zones. It is found that, in n-heptane dominant mixtures, inhibition of low-temperature chemistry (LTC) caused by additional ethanol, impacts heat release rate at high temperature zone through diffusion effect of specific radicals such as CH2O, C2H4, C3H6 and H2O2. For ethanol-dominant mixtures, an increase in heat release rate was observed with higher ethanol fraction. Further concentration change analysis elucidated it is primarily attributed to the decomposition of ethanol and its subsequent reactions. This research underscores the significance of incorporating both chemical kinetics and species diffusion effects when analyzing the counterflow configuration of complex fuel mixtures.

Quantum systems typically reach thermal equilibrium rather quickly when coupled to a thermal environment. The usual way of bounding the speed of this process is by estimating the spectral gap of the dissipative generator. However the gap, by itself, does not always yield a reasonable estimate for the thermalization time in many-body systems: without further structure, a uniform lower bound on it only constrains the thermalization time to grow polynomially with system size. Here, instead, we show that for a large class of geometrically-2-local models of Davies generators with commuting Hamiltonians, the thermalization time is much shorter than one would na\"ively estimate from the gap: at most logarithmic in the system size. This yields the so-called rapid mixing of dissipative dynamics. The result is particularly relevant for 1D systems, for which we prove rapid thermalization with a system size independent decay rate only from a positive gap in the generator. We also prove that systems in hypercubic lattices of any dimension, and exponential graphs, such as trees, have rapid mixing at high enough temperatures. We do this by introducing a novel notion of clustering which we call "strong local indistinguishability" based on a max-relative entropy, and then proving that it implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour commuting models. This has consequences for the rate of thermalization towards Gibbs states, and also for their relevant Wasserstein distances and transportation cost inequalities. Along the way, we show that several measures of decay of correlations on Gibbs states of commuting Hamiltonians are equivalent, a result of independent interest. At the technical level, we also show a direct relation between properties of Davies and Schmidt dynamics, that allows to transfer results of thermalization between both.

In the rapidly advancing domain of quantum optimization, the confluence of quantum algorithms such as Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) with robust optimization methodologies presents a cutting-edge frontier. Although it seems natural to apply quantum algorithms when facing uncertainty, this has barely been approached. In this paper we adapt the aforementioned quantum optimization techniques to tackle robust optimization problems. By leveraging the inherent stochasticity of quantum annealing and adjusting the parameters and evaluation functions within QAOA, we present two innovative methods for obtaining robust optimal solutions. These heuristics are applied on two use cases within the energy sector: the unit commitment problem, which is central to the scheduling of power plant operations, and the optimization of charging electric vehicles (EVs) including electricity from photovoltaic (PV) to minimize costs. These examples highlight not only the potential of quantum optimization methods to enhance decision-making in energy management but also the practical relevance of the young field of quantum computing in general. Through careful adaptation of quantum algorithms, we lay the foundation for exploring ways to achieve more reliable and efficient solutions in complex optimization scenarios that occur in the real-world.