We determine the semisimplicity criterion for even partition algebras over the complex field. Specifically we prove that the even/2-tonal partition algebras $P_n^2(δ)$ over $\mathbb{C}$ are semisimple for all $n$ if and only if parameter $δ\not\in \mathbb{N}_0$ .

Let $A \bowtie^{f,g} (J,J')$ be the bi--amalgamation of a commutative ring $A$ with $(B,C)$ along the ideals $(J,J')$ with respect to the ring homomorphisms $(f,g)$. In this article, we study the basic homological properties of the bi--amalgamated algebra construction. We first calculate the dimension and depth of the bi--amalgamated algebra under fairly general circumstances and derive necessary and sufficient conditions for Cohen--Macaulayness in terms of maximal and big Cohen--Macaulay modules of $A$. Furthermore, we characterize the Gorenstein property of the bi--amalgamated algebra through the canonical modules of $f(A)+J$ and $g(A)+J'$. We apply our results to the theory of curve singularities by constructing Gorenstein algebroid curves through bi--amalgamated and amalgamated algebras. We also give a brief remark concerning the universally catenary property of $A\bowtie^{f,g}(J,J')$.

In this article, we study orientably-regular maps of Euler characteristic $-2p^2$ and classify those that admit a group of orientation-preserving automorphisms of order $10p^2$, where $p$ is a prime number. Along the way, we classify all compact Riemann surfaces (or complex algebraic curves) of genus $1+p^2$ endowed with a group of conformal automorphisms of order $5p^2$.

This work continues the author's earlier work (2026, Studia Mathematica) on Picard's problem: is every meromorphic function on a complete noncompact Kähler manifold with nonnegative Ricci curvature necessarily a constant, if it avoids 3 distinct values? In that prior work, a positive answer was obtained under a growth condition for non-parabolic manifolds. In this paper, we give a full solution to the non-parabolic case by removing this growth condition via a global Green function approach. For the parabolic case, to overcome the obstacle arising from the absence of a positive global Green function, we introduce a heat kernel approach to Nevanlinna theory. Based on it, we develop a Carlson-Griffiths theory, which gives the first systematic result in Nevanlinna theory for parabolic Kähler manifolds. As a direct application, we confirm the parabolic case of Picard's problem under a weak growth condition.

We prove that the relative p-adic monodromy theorem holds over a dense open subset. Moreover, we establish the equivalence of the following two statements: the local constancy of the Newton polygon function associated with a de Rham local system around rank-1 points, and the relative p-adic monodromy theorem near rank-1 points. We demonstrate how to extend the relative p-adic monodromy conjecture from the neighborhood of rank-1 points to the entire interiors of Newton partitions.

Suppose we observe data from a distribution $P$ and we wish to test the composite null hypothesis that $P\in\mathscr P$ against a composite alternative $P\in \mathscr Q\subseteq \mathscr P^c$. Herbert Robbins and coauthors pointed out around 1970 that, while no batch test can have a level $α\in(0,1)$ and power equal to one, sequential tests can be constructed with this fantastic property. Since then, and especially in the last decade, a plethora of sequential tests have been developed for a wide variety of settings. However, the literature has not yet provided a clean and general answer as to when such power-one sequential tests exist. This paper provides a remarkably general sufficient condition (that we also prove is not necessary). Focusing on i.i.d. laws in Polish spaces without any further restriction, we show that there exists a level-$α$ sequential test for any weakly compact $\mathscr P$, that is power-one against $\mathscr P^c$ (or any subset thereof). We show how to aggregate such tests into an $e$-process for $\mathscr P$ that increases to infinity under $\mathscr P^c$. We conclude by building an $e$-process that is asymptotically relatively growth rate optimal against $\mathscr P^c$, an extremely powerful result.

We study the Hitchin morphism for higher dimensional varieties and show that, for a certain class of varieties which we call r-small, the set-theoretic image of the Hitchin morphism from the Dolbeault moduli space coincides with the spectral base. In other words, a stronger version of the conjecture of Chen and Ngô holds for this class of varieties, which includes K-trivial varieties. As part of the proof, we slightly modify the construction of spectral covers to obtain normal spectral covers.

We study the large-$N$ behavior of random matrix tuples $Y^N = (Y_1^N,\dots,Y_d^N)$ with joint density proportional to $e^{-N^2 V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1/N^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S^{N}_t)_{t \geq 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction.

This paper presents a safety-critical centralized nonlinear model predictive control (NMPC) framework for cooperative payload transportation by two quadrupedal robots. The interconnected robot-payload system is modeled as a discrete-time nonlinear differential-algebraic system, capturing the coupled dynamics through holonomic constraints and interaction wrenches. To ensure safety in complex environments, we develop a control barrier function (CBF)-based NMPC formulation that enforces collision avoidance constraints for both the robots and the payload. The proposed approach retains the interaction wrenches as decision variables, resulting in a structured DAE-constrained optimal control problem that enables efficient real-time implementation. The effectiveness of the algorithm is validated through extensive hardware experiments on two Unitree Go2 platforms performing cooperative payload transportation in cluttered environments under mass and inertia uncertainty and external push disturbances.

We study Frobenius algebras of operator fields and introduce a novel notion of duality for them. We show that, under the assumption that the operator fields forming the Frobenius algebra are mutual symmetries, the operator fields in the dual Frobenius algebra are also mutual symmetries. This result allows one to construct new infinite-dimensional integrable systems of hydrodynamic type starting from a given one. As the main application, we solve the long-standing Eisenhart--Stäckel problem for any Segre characteristic and in arbitrary dimension: namely, we describe all nondegenerate finite-dimensional integrable systems whose integrals are quadratic in momenta such that the corresponding $(1,1)$-tensors commute as operator fields.

Let $M$ be the $n$-square matrix partitioned into $\ell^2$ blocks $b_{ij}$ according to some partition $P=\{C_{1},\dots,C_{\ell}\}$ of index set $\{1,\dots,n\}$. The quotient matrix $Q=(q_{ij})$ is a $k$-square matrix, with $\ell \leq k \leq n-1$, where $(ij)$-th entry is the average row sum (or column sum) of the corresponding block $b_{ij}$ in $M$. The partition $P$ is said to be \emph{equitable} if row sum of each block $b_{ij}$ is constant. In this case, the matrix $Q$ is referred to as the \emph{equitable quotient matrix} of $M$, and the spectrum of $Q$ is the subset of the spectrum of parent matrix $M$. We characterize some classes of matrices such that their equitable quotient matrix $Q$ contains all the distinct eigenvalues of $M$, thereby information can be obtained form the smallest matrix $Q$ without actually analyzing the parent matrix $M.$ We present necessary and the sufficient conditions for distinct eigenvalue of $M$ contained in the spectrum of of $Q$ in terms of eigenspaces. We end up article with some applications, where distinct eigenvalues of a parent matrix can be completely encoded by quotient matrix.

We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ Hölder regularity at the singular time. This regularity differs from the $C^{1/3}$ Hölder regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system.

Neural network based methods have emerged as a promising paradigm for scientific computing, yet they face critical bottlenecks in high frequency function approximation and partial differential equation (PDE) solving.

We study worst-case signal compression under an $\ell^2$ energy constraint, with coordinate-dependent quantization precisions. The compression problem is reduced to counting lattice points in a diagonal ellipsoid. Under balanced precision profiles, we obtain explicit, dimension-dependent upper bounds on the logarithmic codebook size. The analysis refines Landau's classical lattice point estimates using uniform Bessel bounds due to Olenko and explicit Abel summation.

In this work, I focus on a coupled system of nonlinear Choquard equations in dimension 4, characterized by critical nonlocal nonlinearities and a small repulsive interspecies interaction. I prove the existence of a new class of multi-bubble segregated solutions. Specifically, I construct solutions where the first component concentrates as a radial positive ground state, while the second component exhibits a blow-up behaviour, concentrating at k points arranged as the vertices of a regular polygon. The proof relies on a sophisticated finite-dimensional reduction method, bridging the gap between the theory of competitive systems and critical nonlocal equations. My results show that the presence of nonlocal terms preserves the qualitative segregation patterns typically observed in local Schrodinger systems.

Let $X$ be an integrable real random variable with mean zero and two-sided sub-Gaussian tail $\mathbb{P}(|X|>t)\le 2e^{-t^{2}/2}$ for all $t\ge 0$. We determine the smallest constant $c_\star$ such that $X$ is dominated in convex order by $c_\star G$, where $G$ is standard normal. Equivalently, $c_\star^2$ is the sharp one-dimensional convex sub-Gaussian comparison constant appearing in the \emph{Optimization Constants in Mathematics} repository~\cite{optimization-constants-repo}. We show that $c_\star$ is given by an explicit system of one-dimensional equations and is attained by an extremal distribution that saturates the tail constraint. Numerically, $c_\star \approx 2.30952$ (so $c_\star^2 \approx 5.33386$). We also determine the analogous sharp constant under a two-sided sub-exponential tail bound, with convex domination by a scaled Laplace law. Finally, we record two higher-dimensional consequences: a sequential tensorization principle for multivariate convex domination, and a dimension-free Gaussian comparator for the cone generated by convex ridge functions (the linear convex order).

We show the existence of planar domains with one hole for which the first non-trivial Neumann eigenfunction has a closed nodal line fully contained inside the domain. This is optimal, as it is known since Pleijel's 1956 result that the nodal line cannot be closed on simply-connected planar domains. A part of the proof is based on the study of convergence of eigenvalues and eigenfunctions of graph-like domains towards metric graphs. We improve the known results of convergence of eigenfunctions, by showing a strong transversal convergence.

In this paper, we develop a characteristic set (CS)-based method for deriving full-rank equivalence conditions of symbolic matrices over the binary field. Such full-rank conditions are of fundamental importance for many linear coding problems in communication and information theory. Building on the developed CS-based method, we present an algorithm called Binary Characteristic Set for Full Rank (BCSFR), which efficiently derives the full-rank equivalence conditions as the zeros of a series of characteristic sets. In other words, the BCSFR algorithm can characterize all feasible linear coding schemes for certain linear coding problems (e.g., linear network coding and distributed storage coding), where full-rank constraints are imposed on several symbolic matrices to guarantee decodability or other properties of the codes. The derived equivalence conditions can be used to simplify the optimization of coding schemes, since the intractable full-rank constraints in the optimization problem are explicitly characterized by simple triangular-form equality constraints.

We describe the semigroup of the Lipschitz saturation of a complex analytic toric singularity in arbitrary dimension. We give a necessary and sufficient condition for a monomial in the normalization to belong to the Lipschitz saturation, in terms of Newton polyhedra and lattice conditions, and deduce a finite algorithm to compute it. We also show that, in dimension greater than two, Campillo's notion of presaturation differs from the Lipschitz saturation, even for complex singularities.

We prove a motivic version of the Poisson formula on the adelic points of a split algebraic torus and apply it to the study of the motivic height zeta function of split projective toric varieties, in the context of the motivic Manin-Peyre principle.

Using our previous work we give a tropical formula for disk potentials for Lagrangian tori in almost toric four-manifolds, that is, fibrations by Lagrangian tori with only toric and focus-focus singularities, generalizing results of Mikhalkin for holomorphic spheres in the projective plane. As examples, we directly compute potentials for Lagrangian tori in del Pezzo surfaces equipped with monotone symplectic forms. These formulas were established in the monotone case by different methods in Pascaleff-Tonkonog, and investigated from the point of view of the Gross-Siebert program in Carl-Pumperla-Siebert, Bardwell-Evans--Cheung--Hong--Lin and also Lau-Lee-Lin.

Dynamic fading channels are modeled at two fundamentally different levels of abstraction. At the physical layer, the standard representation is a correlated Gaussian process, such as the dB-domain signal power in log-normal shadow fading. At the link layer, the dominant abstraction is the Gilbert-Elliott (GE) two-state Markov chain, which compresses the channel into a binary ``decodable or not'' sequence with temporal memory. Both models are ubiquitous, yet practitioners who need GE parameters from an underlying Gaussian fading model must typically simulate the mapping or invoke continuous-time level-crossing approximations that do not yield discrete-slot transition probabilities in closed form. This paper provides an exact, closed-form bridge. By thresholding the Gaussian process at discrete slot boundaries, we derive the GE transition probabilities via Owen's $T$-function for any threshold, reducing to an elementary arcsine identity when the threshold equals the mean. The formulas depend on the covariance kernel only through the one-step correlation coefficient $ρ= K(D)/K(0)$, making them applicable to any stationary Gaussian fading model. The bridge reveals how kernel smoothness governs the resulting link-layer dynamics: the GE persistence time grows linearly in the correlation length $T_c$ for a smooth (squared-exponential) kernel but only as $\sqrt{T_c}$ for a rough (exponential/Ornstein--Uhlenbeck) kernel. We further quantify when the first-order GE chain is a faithful approximation of the full binary process and when it is not, reconciling two diagnostics, the one-step Markov gap and the run-length total-variation distance, that can trend in opposite directions. Monte Carlo simulations validate all theoretical predictions.

Entropy stable discontinuous Galerkin (DG) methods display improved robustness for problems with shocks, turbulence, and under-resolved features by enforcing an entropy inequality. Such methods have traditionally relied on entropy conservative (EC) fluxes that are computationally expensive to evaluate. An alternative approach for enforcing an entropy inequality is through a minimally dissipative ``entropy correction" artificial viscosity. We review how to construct such an artificial viscosity formulation and extend this approach to multiple types of viscosity (e.g., viscosity and thermal diffusivity). We determine simple analytical expressions for optimal viscosity parameters. We compare this to the case of a single monolithic viscosity parameter for different 1D and 2D problems, and show that the proposed method allows users to more precisely target specific physical phenomena while retaining robustness for general problem settings.

This paper presents a systematic observer design methodology for a class of port-Hamiltonian (pH) systems with state-dependent input matrices. Such systems can model a wide range of electromechanical systems, including magnetic levitation systems, MEMS devices, and electro-active polymer actuators such as DEA actuators, HASEL actuators, etc. In these applications, state-dependent input matrices naturally arise when the system is modeled under quasi-static electrical assumptions. An LPV polytopic embedding framework, together with LMI-based synthesis conditions, is proposed. The nonlinear error dynamics are represented as a convex combination of linear vertex systems using an integral mean value representation, which enables systematic computation of the observer gains that ensures exponential convergence. Both constant-gain and gain-scheduled observers are derived. Numerical results demonstrate the effectiveness of the proposed observer, with the gain-scheduled design achieving a significant increase in the maximum certifiable decay rate compared with constant-gain approaches, thereby reducing conservatism.

The propagation of the transverse electric (TE) and transverse magnetic (TM) waves in an effectively two-dimensional (2D) isotropic medium is described by Bergmann's equation of acoustics. We develop a dynamical formulation of the stationary scattering of these waves and explore its application in the study of the low-frequency behavior of the scattering data. Specifically, we introduce a suitable notion of fundamental transfer matrix for TE and TM waves in 2D. This is an integral operator $\widehat{\mathbf{M}}$ that carries the information about the scattering properties of the medium and admits a Dyson series expansion involving a non-Hermitian Hamiltonian operator. For situations where the inhomogeneities of the medium are confined to a layer of thickness $\ell$, we use the Dyson series for $\widehat{\mathbf{M}}$ to construct the series expansion of the scattering amplitude in powers of $k\ell$, where $k$ is the incident wavenumber. We derive analytic expressions for the leading- and next-to-leading-order terms of this series, verify the effectiveness of their application to a class of exactly solvable models, and use them to study low-frequency invisibility. In particular, we develop a low-frequency cloaking scheme which is applicable for both TE and TM waves. Our results have immediate applications in the study of low-frequency scattering of acoustic waves in a 2D fluid as these waves are also described by Bergmann's equation.

Within this research, two combinatorial bijections using Young diagrams were studied. The first is a special case of a bijective correspondence between two classes of combinatorial objects. Its proof, based on Young diagrams, establishes equinumerosity and provides an explicit constructive mapping. The second is a generalization to any natural d, preserving bijectivity. It shows the combinatorial structure remains stable under changes in the parameter, with Young diagrams serving as a universal language. A notable and non-obvious aspect of this generalization is the symmetry revealed in the construction. Intuitively, it was not evident that one could consider not only the natural order of residues but also any permutation of them.

We study the persistent homology of the offset filtration generated by the range of a planar Brownian motion with constant nonzero drift. The members of this filtration are the Wiener sausages of increasing radius, and the degree-one persistence diagram records the birth and death of holes in the thickened trace as the radius varies. Our first result is a sampling theorem: for any continuous path in R d observed on a time grid $π$n the bottleneck distance between the persistence diagram of the continuous offset filtration and that of the sampled point cloud is bounded by the pathwise modulus of continuity $ω$X (|$π$n|). For Brownian motion this yields the almost-sure rate O |$π$n| log(1/|$π$n|) . Our second and main result is a law of large numbers for the drifted planar case. For every bounded Borel weight $ψ$ supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional $Φ$ $ψ$ (T ), where $β$ T 1 (r) counts the holes in the radius-r sausage at time T , satisfies $Φ$ $ψ$ (T )/T $\rightarrow$ $ρ$ $ψ$ almost surely and in L 1 for a deterministic constant $ρ$ $ψ$ . This yields a finite positive intensity measure on the radius axis that governs the linear growth of topological complexity. The proof introduces a regeneration scheme along the drift direction: projecting the planar path onto the drift axis produces a one-dimensional Brownian motion with positive drift, whose ladder hits and bounded-backtracking events generate i.i.d. path blocks. The non-additivity of topology under concatenation is controlled by a Boundary Lemma, which combines a deterministic Mayer-Vietoris estimate with a geometric bound relating integrated Betti numbers to sausage area via the coarea formula. A Betti-curve representation converts the two-parameter persistence problem into a one-parameter family of fixed-radius hole counts, making the regeneration argument possible.

We study exit times from time-dependent domains under joint perturbations of the trajectory and the domain. Representing a moving domain by a continuous barrier $Φ$ on space-time, we reduce the exit problem to a one-dimensional first-passage problem for the scalarised path $y(t) := Φ(t,x(t))$. Our first main result is a deterministic continuity theorem: the exit-time functional is continuous, under local Skorokhod $J_1$ convergence of the path and local uniform convergence of the barrier, at every configuration satisfying an explicit non-tangency condition (NT). We show that NT is sharp in the sense that it characterises the continuity set of the functional. As a direct consequence, weak convergence of exit times follows from joint weak convergence of paths and barriers whenever the limiting pair satisfies NT almost surely; no independence or structural restrictions between trajectory and domain are required. Our second main result is a functional limit theorem: the exit-time profile $u\mapstoτ(u)$, viewed as a càdlàg function of the barrier level, converges in the Skorokhod $M_1$ topology under the same hypotheses, with a concrete example showing that $J_1$ convergence can fail. Concrete verification routes for NT are provided, including a non-characteristic/Itô criterion for diffusions, and the full framework is illustrated through a worked Donsker-type example.

We derive a concrete closed string dual to any interacting Hermitian one-matrix model, away from the double-scaling limit. Matrix and string correlators manifestly agree, to all orders in the genus expansion and all orders in the 't Hooft coupling(s). The worldsheet theory consists of a supersymmetric B-twisted Landau-Ginzburg model coupled to 2d topological gravity. We provide a precise dictionary between traces of the matrix and vertex operators on the worldsheet. Matrix model correlators are explicitly mapped to computable integrals over the moduli space of Riemann surfaces. We perform several direct cross-checks on both sides of the duality. This work furnishes a detailed instantiation of gauge/string duality, in the standard 't Hooft regime, and hopefully a useful worldsheet toy model for the AdS/CFT correspondence, away from the free field limit.

Let t be the first-passage time of a continuous barrier by a c{à}dl{à}g adapted process. We show that t admits a canonical fourfold pathwise decomposition into continuous contact, contact from the left followed by an upward jump, exact hit by jump, and strict overshoot by jump from below. This refinement is more informative than the classical contact-versus-overshoot dichotomy for random-time purposes, because it separates modes with different predictability properties. In particular, the left-contact component always defines an accessible stopping time and becomes predictable under a no-premature-left-contact condition, which we prove to be both sufficient and necessary for the canonical running-supremum announcing sequence to work. On the gap side, under a structural exclusion of predictable gap-crossings, the corresponding restricted time is totally inaccessible. In the semimartingale setting, we obtain a sharp compensator criterion for the predictable-side condition, explicit compensator formulas for the jump-driven crossing modes, and a decomposition of the compensator of the default indicator into its predictable jump part and continuous part. As an application, for a mean-reverting affine jump-diffusion with upward exponential jumps, we derive the boundary-value problem governing the overshoot mode, prove that the differentiated third-order ODE is equivalent to the original problem only when a boundary compatibility condition is retained, and establish verification and uniqueness for the discounted problem. This yields an explicit Green-Volterra representation, a first-order small-q expansion expansion, and, in the undiscounted case, closed formulas for the overshoot and creeping probabilities