Papers

In this paper, we study one-dimensional semilinear stochastic wave equations driven by multiplicative space-time $G$-white noise under sublinear expectation. We establish the existence and uniqueness of global mild solutions. By developing the quadratic variation structure of space-time $G$-white noise and establishing a Burkholder-Davis-Gundy inequality for the associated stochastic integrals, we derive higher-order moment estimates and prove the quasi-sure Hölder continuity of the solutions. We further show that the mild solution satisfies the corresponding weak formulation.

We give an independent computer-assisted proof of the k=4 case of the Chen-Raspaud conjecture. We prove that every graph G with odd-girth(G) >= 9 and mad(G) < 9/4 admits a homomorphism to the Kneser graph K(9,4). The proof combines a minimal-counterexample argument with a rooted star replacement, exact finite computations in K(9,4), and a final charging argument. After all reducible local types are removed, the unique positive local type is (3,3,4). Its unit excess is transferred through its 4-thread to a relative with sufficient negative capacity. The computer-assisted statements used in the proof are certified by exact C++ bitset verifiers, and a separate Python implementation provides an independent cross-check; the complete source code and raw certificates accompany the manuscript.

Despite purely local microscopic dynamics, nonequilibrium steady states can develop long-range correlations. We investigate this phenomenon in the boundary-driven symmetric simple exclusion process by relating its exact nonlocal large-deviation functional to finite-time thermodynamic irreversibility under weak time-dependent driving. Decomposing the long-range correlations into hydrodynamic modes, we find that the lag associated with the slowest mode satisfies an upper bound determined by the excess entropy production rate at each driving frequency. This bound is approached in the quasistatic limit, whereas progressively shorter-wavelength modes do not generally satisfy the corresponding bound. These results provide a quantitative connection between the curvature of the large-deviation functional, long-range correlations in the nonequilibrium steady state, and finite-time dissipation. Within the linear-response regime considered here, they indicate that the longest-wavelength hydrodynamic mode has a distinct relation between slow relaxation and thermodynamic irreversibility.

We study stability of semilinear exterior Dirichlet problems driven by possibly nonsymmetric Levy operators. Let $L^n$ and $L$ have Lévy-Khintchine symbols $\psi_n$ and $\psi$, respectively. Our basic assumption on the operators is that $\psi_n(\xi)\to\psi(\xi)$ for every $\xi\in\mathbb R^d$. Under mild regularity assumptions on the domain and an absolute-continuity condition for the resolvent of the limiting process, we prove pointwise and $L^p$-stability of solutions. No symmetry, uniform ellipticity, comparability of Lévy kernels, or common energy space is required; nonzero exterior data are allowed; and the monotone nonlinearity is subject to no growth restriction in the solution variable. The framework therefore covers local, nonlocal and mixed operators and, in particular, nonlocal-to-local limits, as well as fractional, relativistic, anisotropic and nonsymmetric stable-type operators. The proofs combine convergence of Lévy processes with a detailed analysis of exit times and exit positions and a stability theorem for monotone backward stochastic differential equations with varying terminal times. In the symmetric case, we also obtain Mosco convergence of the corresponding part forms on $C^0$ domains and give a counterexample showing that whole-space Mosco convergence need not pass to part forms on arbitrary open sets.

{For a matrix function $A(t)$, we study the normalisation of its Wiener–Hopf factorisation $A(t)=A_-(t)D(t)A_+(t)$ which guarantees the uniqueness of the factors $A_\pm(t)$. Previously, this problem was investigated for second-order matrix functions, the stable factorisation, and the factorisation of matrix functions with distinct partial indices. The main tools for solving the problem of normalisation are the theorem of I.C. Gohberg and M.G. Krein on a general form of the factors $A_\pm$, a Birkhoff factorisation of matrix functions, and a block PLU factorisation of the numerical matrix $A_-(\infty)$. To solve the normalisation problem, a concept of P-normalised factorisation is introduced. The Birkhoff factorisation of matrix functions plays a key role in defining this concept. The permutation matrix $P$ is determined by PLU factorisation of the numerical matrix $A_-(\infty)$. We proved that any invertible matrix function $A(t)$ admits P-normalisation. This normalisation guarantees the uniqueness of the Wiener–Hopf factorisation. The P-normalised factorisation of $A(t)$ allows us to find the Birkhoff factorisation of $A(t)$. Without normalisation, it is impossible to obtain explicit estimates of the absolute errors in the approximate computation of the factorisation factors.}

The Multilevel Markov chain Monte Carlo algorithm accelerates Bayesian inversion by exploiting correlations between discretized quantities of interest across a hierarchy of model resolutions. A key practical parameter is the subsampling rate used to generate coarse-level proposals for the finer-level chains. Existing practice sets this rate equal to an integrated autocorrelation time as a proxy for proposal decorrelation, but this choice can be overly conservative and lead to substantial computational overhead. In this work we treat finite subsampling as an error-budget question. We derive a cost-error criterion for selecting a computationally efficient subsampling rate using estimates of sample variances, integrated autocorrelation times, and per-level costs. The criterion balances multilevel sampling errors across levels rather than enforcing approximate independence of the coarse proposals. We clarify the theoretical role of independence through an ideal reset-reference kernel, for which independent coarse proposals may be replaced by proposals generated from a reversible coarse-level kernel. We derive a Poisson-equation representation of the perturbation bias resulting from finite subsampling. Under a state-averaged lower-level forgetting assumption, this yields a conditional geometric perturbation-bias bound; in computation, the corresponding projected diagnostic is used as an empirical error-budget check. Numerical experiments on a highly autocorrelated elasticity beam benchmark show substantial reductions in computational cost compared with IAT-based subsampling. A second Darcy-flow inverse problem gives supporting evidence of qualitatively similar finite-subsampling behaviour in another elliptic benchmark.

We study the diagonal Specht ideals $I_\lambda\subseteq\mathcal{R}_{m,n}:=\mathbb{C}[\mathbf{x}_1,\dots,\mathbf{x}_m]$, $\mathbf{x}_r=(x_{r,1},\dots,x_{r,n})$, generated by the $\lambda$-isotypic component of $\mathcal{R}_{m,n}$ for the diagonal action of $S_n$. We characterize the set of zeros of $I_\lambda$ as $V_\lambda=\bigcup_{\mu\not\trianglelefteq\lambda}H_\mu$ and prove that $\lambda\mapsto V_\lambda$ is an isomorphism of posets from $(\mathcal{P}_n,\trianglelefteq)$ to $(\{V_\lambda:\lambda\in \mathcal{P}_n\},\supseteq)$ for all $m$, while $\lambda\mapsto I_\lambda$ is an isomorphism of posets from $(\mathcal{P}_n,\trianglelefteq)$ to $(\{I_\lambda:\lambda\in \mathcal{P}_n\},\subseteq)$ if and only if $m=1$ or $n\leq3$. Unlike the case $m=1$, where Specht ideals are always radical, radicality in the diagonal setting depends on $\lambda$ and $m$. We prove radicality for hook partitions of length at most three by providing explicit Gröbner bases, and we develop three criteria for non-radicality via content, multidegree, and total degree, showing in particular that $I_\lambda$ is not radical for any non-hook partition $\lambda$ when $m\geq\operatorname{len}(\lambda)$. Using the $\operatorname{GL}_m(\mathbb{C})$-action on $\mathcal{R}_{m,n}$, we show that $I_\lambda$ is radical for all $m$ if and only if it is radical for $m=n$. Finally, for $m\geq2$, we prove that $\mathcal{R}_{m,n}/I_\lambda$ is Cohen-Macaulay if and only if $\lambda=(n)$ or $\lambda=(n-1,1)$, and the same is true for $\mathcal{R}_{m,n}/\operatorname{rad}(I_\lambda)$.

Navier-Stokes (NS) equations lose accuracy in the transition regime when the Knudsen number is large, while the Boltzmann equation provides an adequate kinetic description with a substantially higher computational cost. In this work, a data-driven generalized Navier-Stokes (GNS) model is proposed to extend the applicability of the NS equations to the transition regime. A new relationship between the non-equilibrium variables, such as the stress tensor and the heat flux, and the equilibrium variables, including the density, macroscopic velocity, and the temperature, together with the Knudsen number, is learned from the data formed by the numerical solution to the Boltzmann equation. The standard end-to-end learning is utilized to construct the loss function, based on which, a long-term end-to-end learning method is proposed for the loss function to reduce accumulated error and improve long-term predictive accuracy. Several numerical examples, including one-dimensional wave and Riemann problems as well as two-dimensional isentropic vortex and Taylor-Green vortex problems, are studied to validate the efficiency of this new GNS model.

We prove that the upper box dimension of an inhomogeneous attractor of a $C^1$ iterated function system in $\mathbb{R}^d$ is bounded above by the maximum of the singularity dimension of the system and the upper box dimension of the condensation set. This extends the upper bound of Burrell and Fraser (2020) for inhomogeneous self-affine sets to the $C^1$ setting. The covering argument of Burrell and Fraser does not apply directly to nonlinear compositions. We overcome this difficulty by combining their approach with an extension of the recursive covering estimate of Feng and Simon (2023). We also obtain the corresponding bound for orbital sets driven by arbitrary compact subshifts.

We introduce a unified framework of $Q\text{--}P$ spaces, designed to encompass a broad class of generalized topological structures arising from the interaction of two topological properties. We establish general permanence and transfer results, including heredity, productivity, additivity, and separation properties. Specializing to the Hausdorff setting, we investigate the intermediate classes \(C\)-Hausdorff, \(PS\)-Hausdorff, and \(CC\)-Hausdorff, and obtain the hierarchy \[ T_2 \Longrightarrow Ps\text{-}Hausdorff \Longrightarrow CC\text{-}Hausdorff \Longrightarrow C\text{-}Hausdorff \Longrightarrow T_1. \] For \(C\)-Hausdorff spaces, we derive intrinsic characterizations and establish the equivalence with the simultaneous properties of \(k\)-Hausdorffness and \(KC\). We further examine their connections with \(k\)-spaces, Fréchet spaces, local compactness, realcompactness, and \(\aleph_0\)-boundedness. Finally, carefully constructed examples and counterexamples demonstrate the strictness of the hierarchy and the necessity of the hypothesis. Thus, the \(Q\!-\!P\) framework provides a unified perspective on generalized separation axioms and reveals a rich structure between classical Hausdorffness and \(T_1\)-separation.

For a finite forest $F$ let $i_k(F)$ be the number of independent sets of $F$ with $k$ vertices. Zhang and Li proved that the sequence $i_0(F),i_1(F),\dots,i_{\alpha(F)}(F)$ is unimodal for every finite forest $F$, which answers Erdős Problem 993. We give a second proof. It starts from their decomposition relative to a fixed independent set and from the bounds of Zhang and Li and of Fang, Lu, Nevo, Yao and Zheng that confine a valley of the sequence to an explicit window of ranks. For a forest with at least $25$ vertices, one moment argument excludes a valley at every rank of the window: at the activity where the hard-core mean equals the rank, the size of a random independent set is a mixture of binomial laws over an independent set of maximum weight, a valley is a moment inequality for this mixture, and it is excluded by duality given three bounds that hold for every forest, on the variance of the number of free vertices and on its Laplace transforms, and on the variance ratio. The variance bound is proved by hand up to finitely many interval checks and the other two bounds are verified by computer on finite interval-arithmetic coverings; on the resulting parameter domain a valley is excluded by exact tests on finitely many rational boxes while the mean number of free vertices is below an explicit starting mean between $19$ and $50$, and above it by one inequality, with explicit constants, for the fibers of a weighted valley kernel, proved by hand up to a finite list of explicit checks and averaged over the mixture. Forests with at most $24$ vertices are treated by exact counting, by hand except for exact rational evaluations of two explicit formulas at $43$ parameter triples. No forest is enumerated. A formal proof of the theorem in Lean 4 accompanies the paper.

We prove that, for all $x > 0$, the product of modified Bessel functions satisfies \[ \frac{1}{2\sqrt{x^2+\nu^2+1/5}}<I_\nu(x)K_\nu(x)<\frac{1}{2\sqrt{x^2+\nu^2-1/4}} , \] where the lower bound holds for $\nu>-1$ and the upper bound for $\nu\ge 1/2$, and that the sum of the logarithmic derivatives satisfies \[ -\frac{x}{x^2+\nu^2-1}<\frac{I_\nu'(x)}{I_\nu(x)}+\frac{K_\nu'(x)}{K_\nu(x)}<-\frac{x}{x^2+\nu^2+7/20}, \] where the upper bound holds for $\nu>-1$ and the lower bound for $\nu\ge 1$. All four constants are best possible. The four proofs share the same elementary structure: in each case the relevant auxiliary function satisfies a second-order differential inequality, and the inequality follows from the strong maximum principle. Among other applications, we obtain sharp monotonicity properties of the product and improved bounds for the ratios $I_{\nu-1}(x)/I_\nu(x)$ and $K_{\nu+1}(x)/K_\nu(x)$.

Given a topological space $X$ and a family of subgroups $\mathcal{F}$ of $\pi_1(X)$, the $\mathcal{F}$-category of $X$ is given as one less than the cardinality of the smallest cover of $X$ by open subsets whose fundamental groups lie in $\mathcal{F}$. In this article we study $\mathcal{F}$-categories of aspherical spaces and obtain cohomological lower bounds, a maximality result and bounded cohomology classes whose vanishing properties are crucial for determining $\mathcal{F}$-categories. For this purpose, we study the bounded Adamson cohomology of a family of subgroups and discuss universality properties of bounded cohomology classes. We apply our techniques to derive some applications to the study of the monotonicity of $\mathcal{F}$-categories under degree-one maps between manifolds and to provide a counterexample to a question of Capovilla, Löh and Moraschini.

Reliable operation of plasma processes requires a power supply that can regulate and limit the arc current despite rapid changes of the electrical load. Gas-flow fluctuations, arc motion, changing attachment conditions, and electrode interactions can cause abrupt arc-voltage variations and may lead to arc interruption. The power supply must therefore provide fast current control and a suitable actuation layer for supervisory stabilization. This paper presents an adaptive average current-mode controller for a buck converter supplying a nonlinear plasma arc load. Based on an averaged model of the converter inductor dynamics, the controller combines current-error feedback with feedforward compensation of the measured arc voltage. Online adaptation of the inductance and its effective series resistance compensates for converter-parameter uncertainty without requiring an online nonlinear arc model. A Lyapunov analysis establishes bounded estimation errors and asymptotic current tracking for the averaged closed-loop system under bounded-reference assumptions. Experiments with an argon plasma arc demonstrate contact-opening ignition and current-reference tracking under varying arc- voltage conditions, supporting the controller as a fast current-control layer for plasma applications.

Let $h_n$ denote the complete homogeneous symmetric function of degree $n$, $s_\nu$ the Schur function indexed by a partition $\nu$, and $\langle\cdot,\cdot\rangle$ the Hall inner product. The plethysm coefficient $\langle h_n[h_m], s_\nu\rangle$ is the multiplicity of the Weyl module $W_\nu(\mathbb{C}^N)$ in $\mathrm{Sym}^n(\mathrm{Sym}^m(\mathbb{C}^N))$ whenever $\nu$ has at most $N$ parts. For a partition $\lambda$, let $\lambda[mn]=(mn-|\lambda|,\lambda_1,\lambda_2,\dots)$. Bowman and Paget proved that $\langle h_n[h_m], s_{\lambda[mn]}\rangle$ is independent of $m$ and $n$ once $m,n\ge|\lambda|$. We prove that this coefficient is already constant for $m\ge\lambda_1$ and $n\ge\lfloor(|\lambda|+\lambda_1)/2\rfloor$, and that its stable value equals $\langle s_\lambda, H[H-1-h_1]\rangle$, where $H=\sum_{i\ge0}h_i$; the proof uses the skewing operators adjoint to multiplication by complete homogeneous symmetric functions. We give a short new proof of the Bowman-Paget theorem based on stability of vector-partition functions, which count the ways of writing a vector of non-negative integers as an unordered sum of non-zero such vectors (its parts). Using a Vandermonde-type determinantal identity, we deduce that the stable value is a signed sum of numbers of vector partitions all of whose parts have entries summing to at least 2. The same method gives a short proof of Brion's stability theorem. We show that the stable value equals the multiplicity of $W_\lambda(\mathbb{C}^k)$, where $k=\lfloor|\lambda|/2\rfloor$, in an explicit representation of $GL_k(\mathbb{C})$ independent of $m$ and $n$, obtained by modifying the polynomial induction of Narayanan, Paul, Prasad and Srivastava. Finally, we prove an explicit positive formula for the stable value, which counts the number of lattice points in a polytope, when $\lambda_1\le 4$.

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if, for every graph $F \in \mathcal{F}$, the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphisms from $F$ to $H$. Lovász (Acta Mathematica Academiae Scientiarum Hungarica, 1967) showed that two graphs are isomorphic if, and only if, they are homomorphism indistinguishable over all graphs. Subsequently, homomorphism indistinguishability relations of a long list of natural graph classes have been equated with natural graph isomorphism relaxations. Given the wealth of such results, Atserias, Kolaitis, & Wu (LICS 2021) asked for an axiomatic characterisation of homomorphism indistinguishability relations. By exhibiting topological and geometric structure associated with homomorphism indistinguishability, we derive such an axiomatic characterisation. Here, a central ingredient is a novel characterisation of graph parameters of the form $\hom(F, \star)$ for some graph $F$ alternative to a previous result of Lovász & Schrijver (JCTA 2010). Moreover, we investigate the topology of homomorphism indistinguishability and discuss repercussions for the Ulam–Kelly Reconstruction Conjecture.

A graph is beautiful if each of its induced subgraphs is the intersection graph of all maximal nonempty 1-rectangles of a binary matrix, with adjacency defined by a common cell. Beautiful graphs were introduced as a hereditary class of Berge graphs, but a complete forbidden-induced-subgraph characterization was not obtained. We prove that a finite graph is beautiful if and only if it has no induced $C_4$, gem, net, watch, or odd hole. More precisely, these graphs are exactly the $C_4$-free comparability graphs admitting a partial order in which every interval is a chain. The proof constructs such an order from inclusion-maximal closed neighbourhoods: their representatives induce a bipartite graph whose domination regions admit compatible orientations. One order matrix then represents the graph and, through its principal submatrices, every induced subgraph. The representation step is formulated using classical double-bound graphs and the established correspondence between maximal bicliques and interval-intersection-closed posets. Consequences include polynomial-time recognition, the exact minimal obstruction families, and a corrected characterization in the $K_4$-free case.

This work focuses on the dynamic behaviors, particularly the asymptotic periodicity, of solutions to nonlinear fractional-order dynamical systems modeled by Caputo fractional evolution equations in Banach spaces. We propose a new class of weighted pseudo $S$-asymptotically $(\xi,\tau,a)$-periodic functions, which unifies well-established function types such as standard periodic functions and $S$-asymptotically periodic functions as its special instances. We then systematically derive the fundamental properties of this new function family, and rigorously prove its completeness, corresponding convolution theorems, and nonlinear composition rules. Building on these newly established results, we demonstrate that under reasonable hypotheses, the mild solutions to a broad class of nonlinear fractional-order dynamical systems — both delay-free and with finite delay — in Banach spaces admit the (weighted pseudo) $S$-asymptotically $(\xi,\tau,a)$-periodic property, which further implies that no bifurcation or chaotic behavior emerges in these fractional-order systems. The theoretical conclusions are finally validated by illustrative examples describing memory-dependent complex dynamic processes, supplemented with supporting numerical simulations. Additionally, the presented weighted pseudo $S$-asymptotically $(\xi,\tau,a)$-periodic framework can be extended to functions with first-kind discrete discontinuities, so that systems with discontinuous nonlinear terms can be handled via the same analytical paradigm.

We prove that finite-dimensional geodesic locally compact Alexandrov spaces with curvature bounded below (CBB), and in particular complete Riemannian manifolds with lower bounded sectional curvature, satisfy a Besicovitch covering theorem by showing that they are directionally limited in the sense of Federer.

We prove Mizohata–Takeuchi-type estimates for the Fourier extension operator associated with the truncated hyperbolic paraboloid in $\R^3$. Our main result controls the weighted $L^2$ norm of the extension operator by the $L^2$ mass of the weight over a multiscale family of parallelepipeds arising from $\ell^2$ decoupling for the hyperbolic paraboloid. We deduce this result from a weighted $L^2$ estimate for functions whose Fourier transforms are supported in a small neighborhood of the hyperbolic paraboloid, which we establish using a bilinear refined decoupling estimate.

Assuming projective log resolutions, we construct semiample perturbations over arbitrary fields that preserve all log canonical places. As an application, for divisor classes with finitely generated section rings, we characterize the existence of effective divisors yielding lc or klt pairs in terms of asymptotic orders of vanishing. We also derive minimal model and semiampleness results for generalized threefolds over $F$-finite fields of characteristic $p>5$.

Let $K$ be a finite field of characteristic two with $|K| = 2^{n}$, let $\gcd(k,n) = 1$, let $d_{k} = 4^{k} - 2^{k} + 1$ be the Kasami exponent, and let $\Delta_{k} = \{(b+1)^{d_{k}} + b^{d_{k}} + 1 : b \in K\}$ be the image of the normalised derivative of the Kasami monomial in the direction $1$. We show that, for all distinct nonzero $v_{1},v_{2} \in K$, \[ \bigl|\{(x,y,z) \in \Delta_{k}^{3} : v_{1}x + v_{2}y + (v_{1}+v_{2})z = 0\}\bigr| = 2^{2n-3}. \] This establishes the cyclic-additive difference-set condition introduced by Carlet and later posed for the Kasami functions at NSUCRYPTO~2019. Starting from the known half-size property of the derivative image, we express the Fourier correction as twisted root counts and prove their required nonnegativity by an incidence argument on the Fermat cubic. An exact average over the slopes then forces equality pointwise. The argument covers every admissible pair $(n,k)$ and has been formalised and machine-checked in Lean~4 with Mathlib.

We discover a connection between the classical golden section and the online leasing problem: when the number of skis increases from one to two, the optimal competitive algorithm is to rent two pairs until the golden section point$(\sqrt{5}-1)s/2$ within [0,s], then buy one pair, and finally buy the second at time $s$, achieving the optimal competitive ratio of $(5+\sqrt{5})/4$. Building on this, we explore higher-dimensional golden sections and multiple online leasing. For the golden section, we define geometric-type and harmonic-type higher-dimensional golden sections, $\tau^{(m)}$ and $T^{(m)}$, based on two equivalent propositions dividing the unit segment into $m$ segments. We establish equations for the geometric-type ratio ${\lambda}$ and harmonic-type basis ${\omega}$, and investigate their algebraic characteristics. Extending the $k$-generalized Fibonacci sequence to a bi-infinite $m$-generalized Fibonacci sequence, we establish their relationship. We introduce novel concepts including generating sequences and the Fibonacci matrix, investigating its submatrices. Results such as a determinant formula involving the Fibonacci matrix demonstrate its effectiveness in studying generalized Fibonacci sequences. For online leasing, we consider multiple online leasing of $m$ pairs of skis. We design a control and balancing strategy and prove it is optimal via competitive analysis. We show the optimal strategy is precisely the $m$-dimensional harmonic-type golden section method. Since explicit solutions may not exist for dimensions higher than four, we propose a continuous relaxation problem. We explore using its relaxed solution to approximate the optimal solution or the largest real root. We propose the mantissa weighting method to solve the equation, achieving high accuracy through error analysis.

We study the HKT geometry of compact locally conformally hyperkähler manifolds. We prove that on these manifolds the quaternionic Monge-Ampère equation is always solvable, extending results of Alesker and of Dinew-Sroka, and we confirm the conjecture on the maximal existence time of the HKT-Ricci flow. On quaternionic Hopf surfaces we construct non-trivial HKT solitons and prove convergence of the normalized flow for symmetric initial data.

We consider the localization and removal of a stationary omnidirectional radio source using a mobile sensor with bounded bearing errors and an unknown reception radius within known bounds. The objective is to minimize expected remaining action time under a fixed continuation policy while guaranteeing reception and source removal under the sensing model. A three-disk filter ensures reception, and a minimum enclosing circle certifies coverage of all bearing-consistent source positions. Monte Carlo sample-average search ranks a finite candidate set by complete mission time, including travel, readings, terminal service, and a finite fallback scan that ensures completion under the stated assumptions. In the reference case, 5000 independent validation missions average 60.40 s (approximate 95% confidence interval: [59.96, 60.84] s), saving 19.39% relative to a prescribed lateral point and 0.79% relative to a short-travel point. All 25,000 selected-point validation missions across five initial geometries succeed, with lower mean completion times than both prescribed designs. A same-grid objective ablation shows that minimizing travel or reading count alone can increase mean completion time.

For a finite ring $R$, we investigate the probability that the product of two randomly chosen elements in $R$ is nilpotent, which we call the nilpotent product probability (NPP) of $R$ and denote by $P_{nil}(R)$. We derive bounds for $P_{nil}(R)$ that depend on the structure of $R$, and prove that among all commutative rings with identity, the class of local rings with residue field $\mathbb{Z}_{2}$ attains the maximum value of NPP, which is equal to $3/4$. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of $R$ and NPP of some subrings of the matrix ring over $R$.

We prove the Mumford-Tate conjecture in every degree for hyper-Kähler varieties over fields finitely generated over $\mathbb{Q}$. The proof establishes semisimplicity of $\ell$-adic cohomology by eliminating the unipotent radical of the algebraic monodromy group of total cohomology. We also prove the weight-monodromy conjecture for hyper-Kähler varieties over $p$-adic fields. For hyper-Kähler varieties over number fields with $b_2\geq 4$, we establish, after suitable finite extensions, strong compatibility of the associated Weil-Deligne representations valued in Mumford-Tate groups and its integral refinement away from finitely many primes.

Adaptive multiresolution methods reduce representation cost by concentrating fine-scale degrees of freedom where needed, but their tree updates are usually governed by fixed local criteria. We introduce Learned Adaptive Multiresolution Diffusion Imaging (Learned AMDI), which preserves the AMDI fixed-tree propagator and hierarchy constraints while replacing the post-propagation selector with a shared local policy trained by proximal policy optimization. Regression tests reproduce deterministic AMDI trajectories to machine precision when identical trees are used. In the Haar implementation studied here, the deterministic one-step selector accepts no refinements in 54 decisions. Across nine held-out cases, Learned AMDI executes 393 refinements and reduces the mean terminal reference discrepancy from $0.17496$ to $0.13657$, while occupancy rises from $0.13737$ to $0.26660$. Step-resolved diagnostics reveal occasional small adaptation-energy increases; fixed-tree energy stability therefore does not guarantee monotonicity of the learned outer iteration. At comparable occupancy, a validation-tuned observed-detail threshold reaches a discrepancy of $0.13792$ with slightly better RMSE and SSIM, placing both methods on essentially the same accuracy–occupancy tradeoff. A decision-1-only control reaches $0.13742$, indicating that most of the improvement on this static benchmark arises from the initial allocation. The shared actor transfers without retraining to $64\times64$ and $128\times128$ images, improving reference discrepancy, RMSE, and SSIM relative to deterministic AMDI, while the frozen threshold rule remains competitive. Learned AMDI thus provides a hierarchy-constrained, resolution-transferable mechanism for adaptive allocation and clarifies the contribution of sequential decisions.

We establish a scale-invariant spatial non-collapse estimate for hypersurface-orthogonal timelike geodesic congruences. The result extends a previous Raychaudhuri-based construction in which a comparison estimate for the expansion was used to control the volume of domains transported along the congruence. The additional ingredient here is a quantitative estimate of the spatial distance distortion induced by the expansion and shear. Assuming that the initial spatial geometry is non-collapsed up to a scale $r_0$, and that the expansion, shear, and timelike Ricci curvature satisfy suitable uniform bounds, we obtain, before the first blow-up time of the Raychaudhuri comparison solution, \begin{equation*} \operatorname{Vol}_{h_\tau}\bigl(B_{h_\tau}(p_\tau,r)\bigr) \geq \kappa_{\mathrm{L}}(\tau)\, r^3, \end{equation*} where \begin{equation*} \kappa_{\mathrm{L}}(\tau) = \kappa_0 e^{-3K\tau} \exp\!\left(\int_0^\tau \Theta(s)\,ds\right), \qquad K = \frac{\Theta_0}{3} + \Sigma_0 . \end{equation*} Here $\Theta$ denotes the solution of the scalar Raychaudhuri comparison equation and $K$ controls the spatial metric distortion. For every finite time strictly before the comparison barrier, $\kappa_{\mathrm{L}}(\tau)$ is positive. When the comparison solution has a finite blow-up time, the explicit comparison coefficient approaches zero as that time is approached from below. The comparison barrier is a boundary of the estimate and is not, by itself, a statement that the physical congruence develops a focal point or spacetime singularity. The result is structurally analogous to scale-invariant non-collapse estimates in geometric flow, but is mathematically distinct from Perelman's $\kappa$-noncollapsing theorem for Ricci flow.

We provide a very short proof of the following comparison theorem, which has recently played an important role: every centered subgaussian random vector is dominated in convex order, up to a universal constant, by a standard Gaussian random vector.