Papers

Nonlinear filtering problem is one of the core subjects in modern control theory. In this paper, we will study the well-posedness of the three fundamental evolution equations arising in continuous-time nonlinear filtering–the robust Duncan-Mortensen-Zakai (DMZ) equation, the stochastic DMZ equation, and the Kushner-Stratonovich equation–within a unified buffered weighted formulation. An exponential-type weight function and the corresponding weighted Sobolev spaces are introduced to enable a variational treatment of the filtering equations in a more general setting, in which the coefficients of the filtering system may be unbounded with polynomial growth. Under mild and easily verifiable assumptions, we first establish the well-posedness of the weak solution to the robust DMZ equation in these weighted spaces. Using the gauge (exponential) transformation and its inverse, these results are then transferred to the stochastic DMZ equation and the Kushner-Stratonovich equation, whose solutions are shown to exist and be unique in buffered weighted Sobolev spaces, yielding a unified treatment of all three filtering equations. Sufficient conditions for the well-posedness are also summarized, which illustrate the wide applicability of the proposed framework to general nonlinear filtering systems.

The nonlinear evolution and mutual interaction of two-dimensional electron acoustic (EA) nonlinear structures in superthermal plasma environment are studied. The plasma model consists of inertial cold electrons, superthermal hot electrons described by kappa ($\kappa$) distribution, and stationary ions providing overall charge neutrality. Using the extended Poincaré-Lighthill-Kuo (PLK) reductive perturbation technique, a pair of two-sided Kadomtsev-Petviashvili (KP) equations governing right- and left-propagating EA solitary waves (EASWs) is derived. Exact analytical solutions of the KP equations, including single soliton, multisoliton, breather, and lump structures, are obtained via the Hirota bilinear method. The effects of key plasma parameters such as hot electron concentration, temperature ratio, and superthermality index on the characteristics of these nonlinear excitations are examined. Particular attention is devoted to the head-on collision dynamics between solitons, breather-soliton, and breather-breather interactions. The results reveal quasi-elastic collisions accompanied by phase shifts, transient amplitude modulation, and localized energy concentration, with clear distinctions between oscillatory and non-oscillatory mode interactions. The present study provides new insights into multidimensional electron acoustic wave (EAW) dynamics and energy redistribution mechanisms in superthermal space plasmas, with direct relevance to planetary magnetospheric environments such as Saturn's ring region.

The dynamic acoustic scattering system is governed by the wave equation $u_{tt}-\Delta u+qu=0$ in $\Bbb R^3,\,\,\,-\infty<t<\infty$, with a compactly supported potential $q$ and infinitely distant sources (controls) $f$, which initiate incoming spherical waves $u=u^f(x,t)$ provided $u^f\big|_{|x|<-t,\,\,\,t<0}=0$. These waves are focused at $x=0$ and fill up the whole space at the moment $t=0$. The system is controllable if the set of waves $u^f(\cdot,0)$ produced by all finite energy controls $f$, covers the space $L_2(\Bbb R^3)$. As we show, if the Hamiltonian $H=-\Delta+q$ has the bound states, then in the space the points $a$ appear such that the system, being refocused at $x=a$, loses controllability. The latter leads to a physical effect: the waves $u^f$ of finite energy appear, which vanish simultaneously in the past and future cones $|x|<\pm\, t$ and leave the region of inhomogeneity of $q$ without reverberation. This effect has some similarities with the wavefront reversal (Time Reversing Mirror), but is more meaningful from a mathematical point of view.

For every $\beta>0$, we establish multi-time soft-edge moment convergence for a class of Jack processes, including the Jack–Plancherel process, a discrete $\beta$-analogue of Dyson Brownian motion. The limiting moments admit an absolutely convergent expansion in terms of nonnegative Brownian bridges decorated by pairs of equal and opposite jumps. At equal times, we identify this limit with the joint Laplace statistics of the Airy$_\beta$ process for $\beta\geq1$. The same Brownian expansion yields an asymptotic $\beta$-topological expansion of the marginal $b$-conjecture and, at $\beta=2$, an explicit nonnegative formula for the Witten–Kontsevich intersection numbers.

Thyristor rectifiers (TRs) are cost-effective electrolysis power supplies for renewable power-to-hydrogen (ReP2H) systems, but their harmonics may violate grid-code limits. In contrast to conventional solutions that rely on higher-pulse (such as 24-pulse) rectifiers, this paper proposes an operational harmonic coordination scheme that enables low-cost 12-pulse TRs to meet harmonic requirements through coordinated operation. First, a harmonic model quantifies the effects of rectifier transformer (RCT) tap positions and electrolytic currents, enabling harmonic cancellation among multiple electrolyzers (ELZs). A two-layer framework then coordinates hydrogen production and harmonic mitigation. Hourly scheduling determines ELZ commitment within the harmonic feasible region under renewable uncertainty using stochastic programming and a modified progressive hedging algorithm, while minute-level dispatch tracks renewable power and mitigates harmonics. A decomposition algorithm separates production dispatch from harmonic mitigation to improve computational efficiency. Case studies based on real-life projects show that the proposed method increases profit by 31% over current-only regulation. Annual simulations further show that coordinated 12-pulse TRs can replace 24-pulse rectifiers for harmonic compliance by exchanging additional RCT tap actions for lower transformer investment, reducing rectification-stage cost by 37.5%.

A Hovey triple $(\mathcal{C}, \mathcal{F}, \mathcal{W})$ is exceptional, if $(\mathcal{C} \cap \mathcal{F}, \ \mathcal{C} \cap \mathcal{F} \cap \mathcal{W})$ is not a Frobenius pair. One has a disjoint union $\{\text{Hovey triple}\} = \{\text{Hereditary Hovey triple}\} \ \dot\bigcup \ \{\text{Exceptional Hovey triple}\}$ \ $\dot\bigcup \ \{\text{non-hereditary and non-exceptional Hovey triple}\}.$ Exceptional Hovey triples appear widely. For a selfinjective Nakayama algebra $A= kC_n/J^t$, $A\mbox{-}{\rm mod}$ admits exceptional Hovey triples if and only if $\gcd (n, t) \ge 2$ and $t \geq 3$. Their homotopy categories reveal new phenomena. Nakaoka-Palu's Theorem implies that there is a triangulation on $\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}.$ It is proved that a Hovey triple in a weakly idempotent complete extriangulated category $(\mathcal A, \mathbb E, \mathfrak s)$ is exceptional if and only if the induced extriangulated structure $\bigl(\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}, \ \overline{\mathbb E}, \ \overline{\mathfrak s}\bigr)$ is not canonically triangulated. Between this extriangulated structure and the one arising from the triangulation, the intermediate extriangulated structures are in one-to-one correspondence with Serre subcategories of $\mathrm{fp}((\frac{\mathcal P(\mathcal C\cap\mathcal F)}{\mathcal C\cap\mathcal F\cap\mathcal W})^{\mathrm{op}}, \mathrm{Ab})$.

Let k be a field of characteristic p>0. We prove that for every p-decomposition group P, the tensor-triangulated category \operatorname{DPerm}(P;k) is stratified and its Balmer spectrum is generically noetherian. As an arithmetic application, we show that for a nonarchimedean local field F with residue characteristic \ell, the category \(\operatorname{DAM}(F;k)\) of derived Artin motives is stratified if and only if \ell\neq p. In the stratified case its Balmer spectrum is generically noetherian; consequently, \operatorname{DAM}(F;k) satisfies the telescope conjecture.

For a graph $G$ without isolated vertices, $\gamma(G)\le\gamma_t(G)\le 2\gamma(G)$. While graphs attaining $\gamma(G)=\gamma_t(G)$ have been studied extensively, a complete structural description in the smallest nontrivial case $\gamma(G)=2$ has remained open. We resolve this case according to girth. When $g(G)\ne 3$, we show $\gamma_t(G)=2$ forces $G$ bipartite, give an exact degree-sum criterion for this equality, and show $\gamma_t(G)\in\{2,4\}$ under the additional hypothesis $\delta(G)\ge 2$. When $g(G)=3$, we use Golumbic's vertex-multiplication operation together with known classifications of graphs of rank $2$ through $5$ to completely list the families satisfying $\gamma(G)=\gamma_t(G)=2$. As an application, we show that every graph in the extremal family of diameter-two, dominating-vertex-free graphs identified by Erdős and Rényi and classified by Henning and Southey satisfies $\gamma_t(G)\in\{3,6\}$, so $\gamma=\gamma_t=2$ never occurs there. Together these results give a full structural dictionary translating $\gamma_t(G)=2$ into concrete, checkable graph-theoretic properties.

The standard semidefinite programming (SDP) relaxation of Max-Cut is exact when its optimum equals the maximum cut value. Delorme and Poljak resolved NP-completeness of recognizing exactness for weighted graphs and left the unweighted case open. We show that recognition is NP-complete even for connected simple unweighted graphs, and hence strongly NP-complete for nonnegative integer edge weights. The reduction provides an explicit SDP optimum and makes the additive integrality gap equal to the minimum number of unsatisfied clauses in the source formula. Recognition remains NP-complete even when an exact rational optimal primal–dual pair is supplied. We also establish strong NP-hardness of recognizing exactness of the Frieze–Jerrum Max-$k$-Cut relaxation for every fixed $k\ge3$, even for connected graphs with nonnegative integer edge weights. An independent bounded-weight construction gives a second proof for Max-Cut. Finally, reductions preserving the additive gap up to explicit factors establish strong NP-completeness of exactness recognition for a basic Max-DiCut SDP and NP-hardness for a Max-Bisection SDP.

On every infinite compact Hausdorff abelian group there is a complex-valued continuous function whose Fourier sums, taken over coefficients above a decreasing magnitude threshold, are unbounded almost everywhere. In fact, such functions form a dense \(G_\delta\) in the space of continuous functions with its uniform norm. Moreover, a divergent example may be chosen with Fourier support contained in any prescribed infinite subgroup of the dual, while avoiding any prescribed finite set of frequencies. No metrizability or zero-dimensionality is assumed. The proof combines Körner's finite circle and Walsh lemmas with a finite construction on products of odd cyclic groups. A trichotomy for the dual group, exact quotient transfer, and cyclic sampling give finite examples on every group under consideration.

Mixture-of-experts (MoE) architectures increase model capacity by combining a collection of expert predictors through input-dependent routing, while often activating only a small subset of experts for each input. Despite their growing importance in modern large-scale models, the statistical roles of their design choices, especially routing, sparse activation, and shared experts, remain only partially understood, as existing theory has largely focused on parametric or correctly specified MoE models. In this paper, we view MoE as a form of localized aggregation and show how this localization reshapes the approximation-estimation-computation tradeoff. We derive oracle risk bounds for learning dense and sparse routing with evolving experts, separating approximation, expert-learning, and router-estimation errors, and characterize how sparse Top-K routing can retain the benefits of localized aggregation while controlling per-input computation. We also interpret gating through the geometry of input space, relating routing performance to regions of local expert advantage, and show how shared experts, as adopted in architectures such as DeepSeekMoE, can extract common predictive structure so that routed experts focus on residual local variation. Together, these results provide a unified statistical framework for understanding MoE through input-dependent expert aggregation, in which expert specialization and computational tradeoffs are governed by local predictive structure.

The macroscopic collective motion of active continuum on curved manifolds is conventionally addressed through perturbative dynamic renormalization or finite-element simulations, often obscuring the underlying geometric mechanisms. Here, an exact algebraic framework is established to reformulate the active phase transition on hyperbolic spaces $\mathbb{H}^2$. By rigorously expanding the covariant Navier-Stokes-like equations and applying the Weitzenböck identity, we derive the exact critical threshold $\alpha_c = \frac{5}{4}D\kappa^2$ for macroscopic polarization, which is dictated by the geometric mass gap of the Hodge-de Rham Laplacian. We strictly define the parameter subspace $\alpha = 2D\kappa^2$ where the topological free energy reaches the Bogomolny-Prasad-Sommerfield (BPS) limit. This enables the reduction of the complex velocity field to Blaschke products via Möbius gauge symmetry. The flat-space limit ($\kappa \to 0$) exactly degenerates to the topological phase of the classical O(2) model, demonstrating that the constant negative curvature acts as an un-perturbative infrared regularization for non-linear amplitude saturation. Furthermore, mapping the non-variational active convective modes onto the defect translational zero-modes yields an intrinsically non-reciprocal interaction matrix. By analytically extending the singular integral operator of the dynamically condensed defect ring, we identify a macroscopic second-order exceptional point (EP2) characterized by a strictly algebraic dynamic scaling law $\tau \sim |\Delta \nu|^{-1/2}$. This closed-form theoretical paradigm provides exact solutions for geometric frustration and non-Hermitian topology in soft mechanics.

We introduce a conjecture for Bernoulli bond percolation on finite graphs. Roughly speaking, it asserts that if each boundary vertex is associated with a highly probable event that is increasing with respect to the percolation configuration, then, conditionally on the origin being connected to the boundary, the origin is likely to be connected to a boundary vertex whose associated event occurs. The conjecture implies a high-probability connectivity statement which is known to imply $\theta(p_c)=0$ for Bernoulli bond percolation on $\mathbb Z^d$, for every $d\geq2$. We prove the conjecture for finite planar graphs when the origin and the boundary vertices lie on the outer-face boundary, with the explicit bound $1-2\sqrt{\varepsilon}$. The proof combines a left-first depth-first exploration with a partial FKG inequality adapted to monotonicity up to a stopping time. A counterexample shows that the connectivity structure is essential: the analogous statement fails when the connectivity events are replaced by arbitrary increasing events.

In [B2] Balmer introduced the homological spectrum of a rigid tensor-triangulated category, and had the “nerves of steel” not to conjecture that it was in bijection with the prime spectrum, despite an “avalanche of examples”. Naturally, the statement came to be known as Balmer's Nerves of Steel Conjecture. More recently, in [NVY] the conjecture was stated explicitly and extended to a more general non-symmetric context. In this paper we provide a counterexample to the original suggestion in [B2], so also to the conjecture from [NVY]. A first counterexample was recently constructed in [BHR].

We report on our ongoing project to develop a computational pipeline, AutoGraphForge, for an automated graph-theoretic conjecturing-refuting-formalizing-proving system. Conjecture generation is counterexample-guided and runs in rounds: a Graffiti3 generator proposes conjectures over a small, evolving snapshot table $T$ (initially a few hundred graphs with their computed invariants) that grows only by counterexamples to its own conjectures. A novelty filter of $559$ classical and folklore relations, closed under transitive composition and linear identity substitution, decides via a linear program whether a candidate is already implied by known results. Surviving candidates are tested against a dataset of about $348,000$ graphs, unioning the complete House of Graphs invariant export, the exhaustive census of all connected graphs on at most nine vertices, several extremal families (strongly regular, minimal Ramsey, Cayley, cages, barbells, lollipops, spiders), and random models. Counterexample-search algorithms then attack the remainder. Run for several rounds on an HPC cluster, the loop yields $6,522$ conjectures that survived the refutation dataset, the novelty filter and every active-search run – among them nontrivial relations between the annihilation number and the edge-cover number for bipartite and regular graphs, which we prove by hand. A subsequent formalization and proving stage deterministically translates each surviving conjecture into a Lean 4 statement skeleton; every candidate proof is kernel-verified against a pinned mathlib4 and our custom invariant preamble. This stage integrates two neural provers – DeepSeek-Prover-V2-671B (served with vLLM) and the Lean-specialised OProver-32B – behind the independent kernel check. It is implemented end-to-end and passes initial sanity checks, with the full pipeline currently running on the cluster.

We study the override and update operators on partial functions from the perspective of restriction categories. We propose a definition of override restriction categories, in which both of the operators in question exist. We prove a number of results concerning these operators and their relationship to the structure of the ambient restriction category, as well as relating them to the existing literature on the override and update operators. We provide various examples of override restriction categories and in particular show that every classical restriction category is an override restriction category.

We study the Almost-Invariant Subspace Problem for operators defined on non-normable Fréchet spaces and especially on $\omega$. This problem is stated as follows: "Given a bounded operator $T$ on an infinite-dimensional complex Fréchet space $X$, can we always find a closed almost-invariant half-space?". In this paper we solve the problem for the space $\omega$ by showing that an operator $T$ on $\omega$ possesses a closed almost-invariant half-space if and only if $T$ is conjugate to $F + R + \lambda \operatorname{Id}$ or to $B + R + \lambda \operatorname{Id}$ where $F$ is the Forward shift and $B$ is the Backward shift and where $R$ is a finite rank operator and $\lambda \in \mathbb{C}$. We also investigate the case of operators defined on $X \oplus \omega$ where $X$ is an infinite-dimensional Fréchet space with a continuous norm and we show that in this case, every operator defined on $X \oplus \omega$ possesses a closed almost-invariant half-space.

Consider $k$ independent Bernoulli populations, each sampled $n$ times, and select the $t$ populations with the largest success counts, breaking ties uniformly. Classical monotonicity reduces the worst case over the preference zone with separation $\delta$ to the slippage family with levels $p$ and $p+\delta$, leaving only its absolute location $p\in[0,1-\delta]$ undetermined. A Gaussian approximation suggests the symmetric center $p_{\mathrm c}=(1-\delta)/2$, and the exact two-population problem is uniquely centered there for every $n\ge2$. For fixed $k,t$ and $\delta\in(0,1)$, we prove that exact eventual centering holds precisely when $k=2t$. When $k\ne2t$, the least-favorable location $p_{n,k,t}^*$ satisfies \[ p_{n,k,t}^*-p_{\mathrm c} =(k-2t)C_\delta n^{-1/2}e^{-n\Gamma_\delta}\{1+o(1)\}, \] where $C_\delta$ and $\Gamma_\delta$ are explicit and positive. In either case, the least-favorable location is eventually unique. The proof writes incorrect selection as a union of pairwise misrankings and applies inclusion–exclusion, yielding a bipartite graph expansion. A single misranking has its exact maximum at the symmetric center and determines the central curvature; two-edge intersections sharing one population determine the central slope through their multiplicity imbalance; all remaining graphs have higher large-deviation rates.

A common approach to representing disk-like and sphere-like geometries in isogeometric analysis is to use collapsed-edge singularities in the underlying parameterization. The resulting tensor-product B-spline spaces on such singular parameterizations lack the required smoothness to be used in isogeometric analysis. Polar splines (Toshniwal et al., CMAME 2017) rectify this issue by extracting a smoother subspace, and have been successfully used to discretize high-order partial differential equations, as well as to perform structure-preserving discretizations of the de Rham complex.The latter is particularly relevant for mixed formulations that appear in applications such as electromagnetism and fluid dynamics. The main contributions of this paper are twofold. We combine the polar spline construction with hierarchical splines to obtain adaptively-refinable polar splines. Moreover, we do so in the context of structure-preserving methods. That is, we show how to construct hierarchical polar spline $k$-form spaces, prove that the corresponding basis functions are linearly independent, and prove that they form a cohomologically-correct discrete de Rham complex. These contributions provide a mathematically sound foundation for adaptive structure-preserving simulations on polar geometries. We also provide numerical experiments that validate the theory and illustrate the practical behavior of the construction with problems where preserving the structure is mandatory. These include a study of spurious harmonics that highlights how cohomology-breaking refinements differ between the polar and tensor-product settings, a discretization of a Maxwell eigenvalue problem on a locally-refined polar mesh, and an adaptive convergence study for a Hodge–Laplace problem.

We study the 2-division of the Klein-Vélu elliptic normal septic. Its three non-zero 2-torsion points give rise to an explicit cubic equation over the function field of $X(7)$. We identify the three roots of this cubic with modular functions expressed in terms of septic theta functions at $\tau/2,\tau$, and $2\tau$, and show that its splitting field is precisely the function field of $X(14)$. This gives an explicit link between the 2-division geometry of the Klein-Vélu septic and the passage from full level 7 to full level $14$. Several intermediate modular function fields in the resulting $S_3$-extension are also described.

Let $(F|K,w)$ be an immediate valued function field of transcendence degree one over a rank-one perfect valued field $(K,v)$ of characteristic $p>0$. It is henselian rational if $F^h=K(Y)^h$ for some $Y\in F^h$. Kuhlmann proved henselian rationality over tame fields; we investigate how far his method extends to perfect fields. Relative approximation degrees are a central ingredient in Kuhlmann's approach. We first complete their theory over henselian fields by proving the existence of the relative approximation degree and constant of every polynomial, including for pseudo-convergent sequences of algebraic type. Using the $j$-invariants of associated monomial valuations, we describe these invariants directly through Taylor expansions and extend the henselian degree bound of Kuhlmann and Vlahu. We next study the Artin–Schreier reduction underlying the henselian rationality argument. Over perfect fields, every polynomial is Artin–Schreier equivalent to one whose relative approximation degree lies in ${1,p}$. We construct an explicit rank-one example showing that $p$ cannot always be reduced to one modulo the Artin–Schreier image of $K[X]$. Nevertheless, reduction to degree one becomes possible in this example after passing to equivalence modulo the Artin–Schreier image of $K(X)^h$, and the resulting Artin–Schreier function field is henselian rational. Finally, assume that $K$ equals its absolute ramification field, and let $L=IC(F|K,w)$ be the relative algebraic closure of $K$ in $F^h$. We prove that $F^h$ is henselian rational over $L$, and that henselian rationality descends to $K$ whenever $L|K$ is finite. This finiteness condition holds whenever some separating transcendental element induces an extension of Type II, yielding henselian rationality in this case.

Classic Lie theory establishes a correspondence between Lie groups, which are internal group objects in the category of smooth manifolds, and Lie algebras. An analogous correspondence also exists for group objects in affine schemes. Both smooth manifolds and affine schemes form tangent categories, which provide a categorical context for differential geometry. Therefore, it is natural to ask whether the Lie correspondence can be constructed entirely from the tangent structure. Building on work of Cockett and Schwarz, we develop an internal Lie group-Lie algebra correspondence in tangent categories. We introduce Lie group objects as group objects in a tangent category which admit a tangent space at the unit. For any group object, we construct a Lie functor and show it is tangentially representable exactly when the group object is a Lie group. Representability then yields an internal Lie algebra whose underlying object is the tangent space at the unit. We prove this Lie algebra is a differential Lie algebra, with a bilinear Lie bracket induced by the adjoint representation, and we extend the construction to a functor from Lie groups to differential Lie algebras. Finally, we recover the usual Lie correspondences in both differential and algebraic geometry.

A positive $D(n)$-$m$-tuple is a set $A=\{a_1,\ldots,a_m\}$ of distinct positive integers such that $a_i a_j+n$ is a square for every $i\ne j$. In 2005, Dujella and Luca obtained an absolute bound for the cardinality of a $D(p)$- or $D(-p)$-tuple of positive integers, uniformly in the prime $p$. We extend the underlying gap argument to prime powers. An explicit toric elimination certificate, proved by elementary linear algebra, replaces the unsaturated homogeneous elimination step and is valid modulo every prime power. Explicit degree and height bounds for this eliminant yield a uniform gap principle. Combined with the general bound for bounded $|n|$, this shows that every positive reduced $D(\pm p^r)$-tuple (i.e. tuple with elements not divisible by $p$) has less than $2^{121}$ elements, independently of $p$ and $r$. Thus, every positive $D(\pm p)$-tuple has at most $2^{121}$ elements, positive $D(\pm p^2)$-tuples have less than $2^{122}$ elements, and the maximal cardinality of a positive $D(\pm p^r)$-tuple is $O(r)$ uniformly in $p$.

For distinct positive integers v_1, ..., v_n let ML(v_1, ..., v_n) be the largest number L such that at some time t every t vi is at distance at least L from the nearest integer; the Lonely Runner Conjecture asserts that ML >= 1/(n+1). Write ML = p/q in lowest terms. Kravitz conjectured that whenever ML < 1/n one has q = np + 1; Fan and Sun found counterexamples for n = 4, conjectured that q = np + k with 1 <= k <= n always holds, and observed that in their data for n = 6 only k = 1 and k = 3 occur. We explain this observation. For six speeds we show that all but finitely many tuples with ML < 1/6 satisfy ML = (P-1)/(6P) for an integer P congruent to 1 or 5 modulo 6; in particular k is 1 or 3 and the denominator q is odd. The proof determines the three infinite two-parameter families of tuples on which such values concentrate, computes ML exactly on each family, and describes exactly where k = 3 occurs. An exhaustive search over the 2 x 10^9 sextuples with speeds at most 110 finds no exception. For five speeds the same method, together with Chen's classification of the tuples attaining ML = 1/5, shows that all but finitely many tuples with ML < 1/5 satisfy Kravitz's original conjecture. The computations are exact and the code is provided.

Characterizing graphs uniquely determined by their spectra (DS) is a core open problem in spectral graph theory. While this problem has been extensively investigated for simple undirected graphs, it remains relatively underexplored for oriented graphs. For a simple undirected graph $G$ equipped with an orientation $\sigma$, the corresponding oriented graph $\Sigma=(G,\sigma)$ is the digraph obtained by orienting each edge of $G$ according to $\sigma$. An oriented graph $\Sigma$ is said to be determined by its generalized skew spectrum (DGSS) if every oriented graph sharing the same generalized skew spectrum is isomorphic to $\Sigma$. This paper develops a new sufficient criterion for recognizing DGSS controllable oriented graphs, which applies to a much broader family of graphs than previously known results. Let $S$ be the skew-adjacency matrix of $\Sigma$, $W(\Sigma)=[e,Se,\ldots,S^{n-1}e]$, and $d_n$ the last invariant factor of $W(\Sigma)$. For each odd prime $p$, we define the polynomial $\Phi_p(\Sigma;x)=\gcd(\chi(S;x),\chi(S+J;x))$ over the finite field $\mathbb{F}_p$, which is invariant under generalized skew cospectrality. By analyzing the square-free part of $\Phi_p(\Sigma;x)$ and the associated $p$-main polynomial, we establish a DGSS sufficient condition under the square-free assumption on $d_n$. The proposed criterion allows higher $p$-nullity and recovers the square-free determinant criterion of Qiu, Wang and Wang~(2019) as a special case. We further provide illustrative examples to verify the wider applicability of our new condition and to highlight the role of the compatibility constraints on the irreducible factors of $\Phi_p(\Sigma;x)$.

We show that for an embedded two-sided minimal hypersurface in $\mathbb{R}^N$, there is a lower bound for the index in terms of the first Betti number and the number of ends, using ideas from the recent work of Chodosh–Gianocca. This estimate is sharp for the higher-dimensional catenoid. We also obtain a $\operatorname{Spin}(7)$ analogue of Theorem 10.1 of Chodosh–Gianocca: a complete two-sided minimal immersion $M^7\to\mathbb{R}^8$ of index one and finite total curvature is a higher-dimensional catenoid.

In this paper, we establish sampling and density results for the fractional Opdam-Cherednik transform. Using the Sturm-Liouville structure of Jacobi functions, we construct a Riesz basis associated with the Opdam-Cherednik kernel and derive an explicit sampling formula. The sampling nodes are determined by the zeros of a shifted Jacobi function. We further develop a concentration-operator approach and obtain Landau-type necessary density conditions for sampling and interpolation in fractional Opdam-Cherednik bandlimited spaces. The non-fractional Opdam-Cherednik sampling formula is recovered when $\theta=\pi/2$.

A graph is fractionally Hamiltonian if it admits a nonnegative edge weighting in $[0,1]$ of total weight equal to its order such that every nontrivial edge cut has weight at least two. Motivated by Chvátal's Toughness Conjecture, Scheinerman and Ullman conjectured that every $2$-tough graph is fractionally Hamiltonian. In this paper, we show that every connected graph on at least three vertices that is not fractionally Hamiltonian has a non-Hamiltonian chordal spanning supergraph. Since adding edges does not decrease toughness, a theorem of Kabela and Kaiser that every $10$-tough chordal graph on at least three vertices is Hamiltonian yields that every $10$-tough graph on at least three vertices is fractionally Hamiltonian. We apply this result to resistance curvature. We prove that every fractionally Hamiltonian graph is resistance positive (RP), and consequently every $10$-tough graph is RP, confirming a conjecture of Devriendt. In the other direction, for every $\varepsilon>0$, we construct a graph that is not resistance nonnegative and has toughness greater than $3/2-\varepsilon$, extending a recent construction of Agrahari, Bibby, Boros, Garcia, Heidercheidt, and Wang.

The adjacent vertex distinguishing (AVD)-total chromatic number $\chi''_{a}(G)$ of a graph $G$ is the least integer $k$ for which $G$ has a proper total coloring $f$ with $k$ colors such that $C_G(u)\neq C_G(v)$ for every edge $uv\in E(G)$, where $C_G(u)=\{f(u)\}\cup\{f(uw):uw\in E(G)\}$. The AVD-total coloring conjecture (AVD-TCC) asserts that $\chi''_{a}(G)\leq \Delta(G)+3$ for every simple graph $G$, where $\Delta(G)$ is the maximum degree of $G$. In this paper, we prove the AVD-TCC for certain classes of graph products, including Cartesian products, lexicographic products, skew products, cover products, comb products, and Indu–Bala products.

We study the semi-discrete quadratic Wasserstein energy. The energy is nonsmooth at collisions of sites. We prove local Lipschitz continuity on the full configuration space, together with global semiconcavity, coercivity, and dissipativity; show that every global minimizer is interior and collision free; and establish $C^2$ regularity on the collision-free configuration space. The gradient is expressed through the barycenters of the balanced Laguerre cells, while the Hessian is given by an explicit facet formula and satisfies a global one-sided bound. We also solve the one-dimensional problem explicitly in each ordering chamber and give a two-site example on the unit square with non-minimizing Lloyd fixed points. For $d\ge 2$, we then formulate an entropy-regularized relaxed control of the Langevin temperature. The controlled dynamics is strongly well posed, nonexplosive, and collision free. Its value function is a classical interior solution of the exploratory Hamilton-Jacobi-Bellman equation; the Laplacian of the value function is locally $C^1$, which yields a locally Lipschitz optimal temperature feedback. Independently of this optimal-control result, for every fixed Borel temperature rule bounded away from zero, and every sufficiently small step size, the associated Gaussian Euler chain is geometrically ergodic with a full-support invariant law. The raw iterates do not converge, whereas the best-so-far energy converges almost surely to the global minimum and the running record approaches the set of global minimizers.