Papers

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The inclusion graph of annihilators in $R$, denoted by $\Gamma^{\prime}(R)$, is a graph with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{ann}_R(x)\subseteq \operatorname{ann}_R(y)$ or $\operatorname{ann}_R(y)\subseteq \operatorname{ann}_R(x)$. It is proved that $\Gamma^{\prime}(R)$ is not connected if and only if $R$ is reduced with $|\operatorname{Min}(R)|=2$. Also, we show that if $\Gamma^{\prime}(R)$ is a connected graph, then the diameter of $\Gamma^{\prime}(R)$ is at most $4$ and the girth of $\Gamma^{\prime}(R)$ is at most $6$, if it contains a cycle. Moreover, we study the affinity between inclusion graph of annihilators and complement of the annihilator graph (a well-known graph with the same vertices and two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{ann}_R(xy)\neq \operatorname{ann}_R(x)\cup \operatorname{ann}_R(y)$) associated with a commutative ring. Finally, we characterize all rings whose inclusion graphs of annihilators are complete.

We investigate several classic coefficient problems for the Ma–Minda starlike subclass $\mathcal{S}_{\mathcal{AP}}^{*}$ defined by the apple-like subordination function $\psi_{\mathcal{AP}}(z)=e^{z}\sqrt{1+z}$. Sharp bounds are derived for the initial inverse logarithmic coefficients $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, and the successive modulus difference $|\Gamma_2|-|\Gamma_1|$. In addition, we evaluate the second-order inverse logarithmic Hankel determinant, the generalized Fekete–Szegö functional over all real parameter domains, and the third-order Hermitian–Toeplitz determinant. The corresponding extremal functions are explicitly determined for each functional.

Steerable convolutional neural networks (Steerable-CNNs) guarantee SE(3)-equivariance by parameterizing kernels as linear combinations of steerable basis functions, but their deterministic nature precludes uncertainty quantification - limiting their use in settings where confidence estimates are essential. We propose a Bayesian Steerable-CNN that places posterior distributions over the basis coefficients, yielding stochastic kernels while preserving equivariance exactly. The loss function of the model is obtained via variational inference and minimized by Bayes-by-Backpropagation. The framework admits a decomposition of predictive uncertainty into epistemic and aleatoric components. Empirically, the model attains competitive classification accuracy alongside an expected calibration error of 0.0263 and outperforms its deterministic counterpart by up to 6.17% under distributional shift induced by additive Gaussian noise. Furthermore, we leverage the model's uncertainty estimates to enhance its performance significantly, achieving a notable gain - approximately 4% higher accuracy across 84% of the test dataset. A statistically significant negative correlation between epistemic uncertainty and prediction error confirms that the learned posterior variance is semantically meaningful. The framework unifies Bayesian uncertainty quantification with the inductive bias of equivariant CNNs.

We study labeling-sensitive Fourier complexity for finite graph kernels. After identifying the vertices of a graph with the cyclic group $\mathbb Z_N$, its adjacency matrix becomes a function on $\mathbb Z_N^2$. Minimizing the quotient of the $\ell^1$ and $\ell^2$ norms of its two-dimensional Fourier transform over all vertex labelings gives an isomorphism invariant $\operatorname{FR}_{\min}(G)$. A nuclear-norm argument gives \[\operatorname{FR}_{\min}(G) \geq \frac{\mathcal E(G)}{\sqrt{2s}},\] where $s$ is the number of edges and $\mathcal E(G)$ is the graph energy. The natural cyclic labeling attains equality for every circulant graph. We obtain exact formulas for several graph families and a labeling-sensitive complete bipartite example. We also connect the invariant with the Fourier algebra of $\mathbb Z_N^2$. The quantitative Cohen idempotent theorem implies that every Boolean kernel of bounded Fourier ratio has an exact signed coset decomposition whose length is independent of $N$. A previously established Fourier-ratio recovery theorem gives stable Frobenius approximation of a fixed labeled adjacency matrix from Bernoulli samples. We distinguish this conclusion from exact edge recovery and from the problem of finding a good labeling. For a Laplacian eigenvalue of multiplicity $m(\lambda)$, we prove \[ \operatorname{FR}_{\min}(\Pi_\lambda) \geq \sqrt{m(\lambda)}, \] with equality for circulant graphs. Strongly regular graphs and the Petersen graph show how adjacency and projector complexity can agree or differ. Direct projector sampling yields heat-kernel approximation. We conclude with a graph-signal spectral synthesis principle and asymptotic uniqueness from incomplete vertex data.

Let ${}_2F_1(1/3,1/3;1;27z)^3=\sum_{n\ge0}A_nz^n$. We prove the full prime-power supercongruence tower $A_{mp^r}\equiv A_{mp^{r-1}}\pmod{p^{4r}}$ for $p\ge5$, $m\ge1$, $r\ge1$. The level-three modular expansion used below was already recorded by Moy; the contribution here is the depth-preserving modulus $p^{4r}$, extending the previously known depth-one modulus $p^4$ case to all prime powers. The proof is modular. After replacing $A_n$ by $B_n=(-1)^nA_n$, the generating function is realized on $X_0(3)$ by $\sum_{n\ge0}B_nt^n=\eta(\tau)^9/\eta(3\tau)^3$ with $t=\eta(3\tau)^{12}/\eta(\tau)^{12}$. The logarithmic derivative $C=(\sum_{n\ge0}B_nt^n)\,(q/t)\,(dt/dq)$ is the Eisenstein series $3E_{5,\chi_0,\chi_3}$. Lagrange-Buermann gives $B_m=\mathrm{CT}_q(C(q)/t(q)^m)$. The new point is to replace the one-prime Hecke defect by the prime-power defect $T_{p^{s+1}}(C/t^{mp^{s+1}})-T_{p^s}(C/t^{mp^s})$. Its $i=0$ Hecke layer is the sparse Cartier defect controlling $A_{mp^{s+1}}-A_{mp^s}$, while all remaining Hecke layers are divisible by the required power of $p$ by induction. A Fricke involution argument on the two-dimensional space $M_5(\Gamma_0(3),\chi_3)=\mathrm{Span}\{C,tC\}$ then kills the low-order part exactly and the principal part modulo $p^{4(s+1)}$.

We prove a finite-window anti-phantom principle for scale-critical Navier–Stokes defect packages and develop a conditional localized transfer framework around it. In a fixed clean quotient, the main compactness theorem shows that if the only defect that is simultaneously invisible, exactly reproducible, and tax-free is a gauge artifact, then observation, reproduction failure, and tax control the clean quotient distance by a positive finite-window gap. We then isolate the additional localized inputs needed to use this clean gap for Navier–Stokes packages: pressure-source observability, enhanced pressure-tail geometry, chart visibility, and residual sub-budgets for localization, reproduction, and gate/tax mismatch. The localized results are conditional finite-window reductions with explicit error constants, including a comparison theorem between the enhanced-tail geometry and the original intrinsic geometry under stated projection and harmonic tail approximation assumptions. The paper should be read as a rigorous finite-window obstruction framework, not as a proof of Navier–Stokes regularity, a construction of a singular solution, or a scale-uniform regularity criterion.

Rare events in time series are critical to model but hard to learn due to data scarcity. Current generative models struggle with extreme values. We observe that rare events leave distinct topological fingerprints - transitions in Betti numbers from point-cloud embeddings - that are more stable and discriminative than statistical moments. We introduce PHINN, a flow-matching framework using dynamic Betti curves as conditioning signals and a persistence landscape loss for homology consistency. It scales to multivariate data, includes a natural-language interface to set Betti targets, supports cross-domain meta-learning and few-shot generation, and provides certified adversarial robustness. On financial, epidemiological, and multi-modal benchmarks, PHINN outperforms statistical and diffusion baselines in topological fidelity (beta-RMSE down 41-63%, transition accuracy up 84%) and matches jump-diffusion models in tail coverage while exceeding them in shape fidelity. All results have 95% confidence intervals.

Let \(k\) be a perfect field of exponential characteristic \(p\), and let \(R\) be a commutative \(\Z[1/p]\)-algebra. We prove that the first derived unramified functor \[ F\longmapsto R^1_{\mathrm{nr},R}F \] from homotopy invariant Nisnevich sheaves with transfers of \(R\)-modules to birational sheaves commutes with arbitrary small direct sums. This gives a positive answer, after inverting the exponential characteristic, to a question of Kahn and Sujatha; on smooth projective varieties no inversion is needed. We also describe an obstruction to this for the functor $R^2_{\mathrm{nr},R}$ in categorical terms, which includes the familiar Griffiths group obstruction. As applications of the motivic nature of the functors \(R^q_{\mathrm{nr}}\), we prove torsion-order bounds and a correspondence-detection statement for surfaces.

In this paper we show that the physical learning methods known as coupled learning (CL) and equilibrium propagation (EP) conserve a mass-like quantity in the trainable parameters in the continuous-time, small-nudging limit. We prove that this conservation holds in a broad range of physically relevant settings. We then show that the conservation law constrains the training dynamics in a way that makes convergence reliable in important settings for linear circuits. We conclude by discussing some practical implications of this conservation law.

Physical learning methods train physical networks to perform computational tasks using only local update rules, exploiting the physics of the system to handle the global transfer of information. We provide the first local convergence analysis of three such methods – Equilibrium Propagation (EP), Coupled Learning (CL), and a new method we call Adjoint Coupled Learning (AL) – for linear circuits, in the limit of small-nudging for both discrete and continuous time. EP and AL perform gradient descent on a natural loss function, while CL follows modified dynamics with an additional cubic correction. Assuming the existence of a solution, we identify a coercivity condition, expressed as a rank condition on a matrix built from the network's incidence structure, under which the training loss decays exponentially and the parameters converge to the solution manifold. We show that coercivity can fail by exhibiting a kite circuit in which a symmetry causes the coercivity constant to degenerate on the solution manifold, but prove using Sard's theorem that such degeneracies are non-generic: coercivity holds at every point of the solution manifold for almost every choice of desired output.

Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this paper, we make these combinatorial methods functorial. Our categorical approach not only reveals the relationship between moves, but also leads to unexpected extensions of known moves (shifts and reductions) and the characterization of out-splits as admissible epimorphisms admitting a lifting of the source map. The former allowed us to go beyond isomorphisms and Morita equivalences, and the latter yielded a factorization lemma for out-splits. Better still, the functorial understanding of maximal out-splits permitted combining them with the derived-graph construction, which resulted in computing new proper endomorphisms of Cuntz algebras. Finally, the functorial implementation of balanced in-splits required the construction of a new category of graphs, where morphisms are given as homomorphisms of graph inverse semigroups. Thus, the combinatorics of moves brought a new tool to the realm of noncommutative topology where directed graphs are viewed as quantum spaces.

Let $p>1$ be a large prime number and let $x=O((\log p)^2(\log\log p)^3$ be a real number. It is proved that the least consecutive pair of quadratic nonresidues $u\ne\pm1, v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field $\mathbb{F}_p$.

We consider two-player linear-quadratic differential games of incomplete information, in which one player has a private type initially unknown to the other. The typed player has incentive to conceal their type, while the uninformed player has the potential to infer it during play. Any ex-ante equilibrium in this setting will decompose into a deceptive, pooling phase, and a complete-information, revelatory phase. We demonstrate how to solve both phases via nested Riccati equations. Candidate equilibria are then found by maximizing the game value over a scalar revelation time, for which we provide a gradient in the case of time-homogeneous system matrices. We conclude by demonstrating our framework in a pursuit-evasion game with time-varying control advantages, finding interior optimal revelation times that confirm deception has quantifiable ex-ante value.

Inferring the strength of conditional dependence and testing conditional independence are fundamental problems in statistics. A recent breakthrough by Azadkia and Chatterjee introduced, for the first time, a conditional dependence measure that equals $0$ if and only if the variables under study are conditionally independent, and equals $1$ if and only if they are conditionally perfectly dependent. They further proposed a computationally efficient and strongly consistent estimator, $T_n$, based on an ingenious use of ranks and nearest neighbors. Despite these attractive features, the asymptotic theory of $T_n$ has remained largely undeveloped. This paper closes that gap. We prove that, under general dependence, $T_n$ is asymptotically normal and its limiting variance admits a closed form. We also construct consistent variance estimators that are computationally efficient and implementable in $O(n\log n)$ time. Taken together with existing bias-correction methods, these results provide a complete inferential theory for $T_n$.

We prove that the map sending the diffusion profile to the solution of a time-changed Parisi PDE evaluated at time-space $(0,0)$ is concave. This result strengthens the raywise concavity result proven by Auffinger and Chen (2016). As an application, for the balanced multispecies Ising spin glasses, the lower bound of Bates and Sohn (2025) matches the Hopf-type upper bound given by the Hamilton–Jacobi framework developed by Mourrat, Chen and Xia.

We prove that a homomorphism of rings of prime characteristic is b-nil formally unramified if and only if its relative Frobenius is an epimorphism. We specialize this result under different finiteness conditions such as (relative) $F$-finiteness and Noetherianity. We give an example where the absolute Frobenius is an epimorphism but is not surjective, as well as an example of a formally unramified homomorphism that is not b-nil formally unramified.

In this paper we aim to solve structured monotone inclusions using inexact warped resolvents evaluated under a relative-error criterion. The resulting algorithms admit a geometric interpretation as relaxed projection methods onto dynamically generated cuts, extending classical projection–proximal and hybrid extragradient proximal frameworks to nonlinear warped resolvent. Under mild assumptions, we establish weak convergence of the iterates. We further derive strong convergence results by incorporating projection steps onto intersections of halfspaces via Haugazeau-type scheme, as well as linear convergence under a metric subregularity assumption. The proposed algorithm provides a unified framework for incorporating inexact resolvent computations into several classical schemes arising in monotone operator theory and primal-dual optimization, such as Tseng's forward-backward-forward splitting, forward-backward-half-forward, Chambolle–Pock, and Condat–Vũ. Finally, we present applications in saddle-point and structured convex minimization problems. Numerical experiments on synthetic saddle-point instances and computed tomography reconstruction demonstrate the computational advantages of the proposed methods.

We study well-posedness of stochastic parabolic equations with gradient nonlinearities. Our analysis is based on recent maximal-regularity frameworks for nonlinear stochastic parabolic equations in critical spaces. We extend the existing results by controlling drift and noise coefficient separately. This way we can allow for less regular driving noise in case of subcritical dispersion coefficients. Our approach, based on gluings of local solutions, moreover implies new continuation criteria. We then apply our existence result and the continuation criteria to show global well-posedness of phase-field models of moving boundary problems.

We prove that two different abstract quantitative uncertainty principles are equivalent to the strict positivity of the associated abstract Schrö\-din\-ger operators. We also discuss the case of continuum Schrödinger operators, in which case our method provides an explicitly computable lower bound as well as a control theoretic application.

Linker-mediated aggregation plays an important role in modern nanoscience. We demonstrate that it departs sharply from classical Smoluchowski kinetics because cluster reactivity evolves during growth. Combining theory with DNA-linked gold-nanoparticle experiments, we establish kinetic critical point controlled by linker abundance. Below threshold, active linkers are depleted and growth arrests; above threshold, clusters accumulate reactive sites, self-accelerate, and cross over to diffusion-limited coarsening. Experiments verify the predicted arrest, accelerated growth, and scaling collapse.

We study solutions of the differential inequality $$ \Delta^{m / 2} u \ge f (x) g (u) \quad \mbox{in } B_1 \setminus \{ 0 \}, $$ where $m \ge 2$ is an even integer, $f$ and $g$ are some functions, and $B_1$ is an open unit ball in $R^n$, $n \ge 2$, centered at zero. Our aim is to obtain a necessary condition for a singularity at zero to be removable for any solution of this inequality.

We study Motzkin paths whose up-, level-, and down-step weights are affine functions of the height: alpha_k = A k + alpha_0, gamma_k = B k + gamma_0, beta_k = C k + beta_0. Let w_{n,k} be the total weight of paths of length n ending at height k, and let W_n = w_{n,0} be the return column. For affine weights the row recurrence becomes a first-order differential equation in the variable marking terminal height, and, unless beta_0 = C, that equation involves the unknown return series as boundary data. We show that the return series is nevertheless determined by the step rule alone: a characteristic ending on the floor forces a cancellation and yields an Abel-Volterra equation for the returns, while the same identity read at an interior point reconstructs the entire triangle (w_{n,k}). We call this boundary cancellation the differential kernel method; it takes the place of substituting an admissible kernel root, which is unavailable because the equation is differential rather than algebraic in the catalytic variable. When the boundary index nu = (beta_0 - C)/C is a positive integer m, the construction becomes finite and combinatorial: shifting every height by m turns the divided-difference evolution into an ordinary weighted path model on a half-line carrying m virtual levels below the visible floor, and a first-entry decomposition expresses (w_{n,k}) through two such local models. Exactly one bridge, of weight alpha_0 - A, leads from the virtual strip back to the visible region; it is closed, and the terminal-height columns factorise, precisely when alpha_0 = A. For the Dyck weights alpha_k = k+1, beta_k = k+nu+1, gamma_k = 0 this gives sum_{n>=0} w_{n,k} t^n / n! = sec^{nu+1}(t) tan^k(t).

In this paper, we study the rigidity of noncompact convex sets in hyperbolic 3-space. We prove that any intrinsic isometry between the boundaries of two closed, noncompact convex subsets of hyperbolic 3-space of dimension at least two extends to a global isometry of the ambient space, provided that their ideal boundaries are circle-type closed sets with countably many connected components. Moreover, the same conclusion holds if the ideal boundaries consist of finitely many mutually disjoint disks together with a set of one-dimensional Hausdorff measure zero. This result generalizes a recent rigidity theorem of Luo, Luo, and Rao by allowing the ideal boundaries to contain disk components. As a direct consequence, we establish a uniqueness result concerning the Weyl problem for convex surfaces in hyperbolic 3-space, as proposed by Luo and Wu. In particular, our approach provides an alternative proof of the discrete Schwarz lemma. The proof uses Pogorelov's rigidity theorem for compact convex bodies in $\mathbb{R}^3$, the Pogorelov map, and properties of locally convex surfaces in $\mathbb{R}^3$.

Jensen, King, and Su described a category $\operatorname{CM}(C)$ which categorifies the cluster structure on the homogeneous coordinate ring of a Grassmannian. In this paper we describe subcategories $\operatorname{R}(v,w) \subseteq \operatorname{CM}(C)$ which lift Leclerc's categories $\mathcal{C}_{v,w}$ in the case where $v \in \left(W^{k}\backslash W\right)^{\max}$ and $w \geq v.$ As such, these categories are Frobenius, stably 2-CY, have natural cluster characters, and induce a cluster structure in lifts of open positroid varieties.

We study $\mathbb{C}^{\times}$-actions on quiver representations by adding idempotent loops. We construct a category mod $\widehat{\operatorname{W}}(\mathrm{Q})$ and relate it to $\mathbb{C}Q$ modules with $\mathbb{C}^{\times}$-actions. We further describe a subcategory $\widehat{\operatorname{W}}_{T}(\mathrm{Q})$ equivalent to the category of equivariant modules where $\mathbb{C}^{\times}$ acts on $\mathbb{C}Q$ via a character $T \in \mathbb{Z}^{Q_1}.$ This allows us to study $\mathbb{C}^{\times}$-actions on quiver Grassmannians and we can recover known combinatorial descriptions of their Euler characteristics via the category mod $\widehat{\operatorname{W}}(\mathrm{Q}).$ We directly relate morphisms of quivers to full subcategories of mod $\widehat{\operatorname{W}}(\mathrm{Q}).$ Finally we show that the representation theory of quivers with idempotents can be applied in a useful way to preprojective algebras. We show that this gives constructions of Galois covers and cluster characters used by Geiss, Leclerc, and Schröer.

We give an exact finite-dimensional Perron–Frobenius realization of the dominant zero of the Nekrasov–Okounkov polynomials $\nop _n(z)$. For a normalized positive sequence $h=(h(n))_{n\ge 1}$ with $h(1)=1$, define $\pol _0^h(z)=1$ and, for $n\ge 1$, \[ \pol _n^h(z)=\frac{z}{h(n)}\sum_{k=1}^n \sigma(k)\pol _{n-k}^h(z),\] where $\sigma(k)$ denotes the sum of divisors of $k$. The Nekrasov–Okounkov polynomials are obtained from the specialization $h(n)=n$ by the shift $\nop _n(z)=\pol _n^h(z+1)$. We derive a Hessenberg determinant representation for $\pol _n^h(z)$. After separating the trivial zero at the origin, the remaining zeros of $\pol _n^h(-z)$ are identified with the eigenvalues of an explicit $(n-1)\times(n-1)$ nonnegative matrix $M_n^h$. We prove that $M_n^h$ is primitive and apply Perron–Frobenius theory to show that $\pol _n^h(z)$ has a unique zero of maximal modulus; this zero is real, negative, and simple. As a consequence, the same property holds for the Nekrasov–Okounkov polynomials. We also prove strict monotonicity of the associated spectral radii.

For a commutative ring $R$, we study the $R$-localization functor on the category of groups, defined as localization with respect to the homomorphism $\mathbb{Z}\to R$. Our main result is that, when $R$ is a binomial ring, the $R$-localization of a nilpotent group is again nilpotent. Taking $R=\mathbb{Z}_p$, the ring of $p$-adic integers, yields a new example of a localization functor that preserves nilpotency. To prove this, we characterize $R$-local groups in terms of $R$-groups in the sense of Myasnikov-Remeslennikov. We call an $R$-group a Hall-Petresco $R$-group if it satisfies a version of the Hall-Petresco identity, and show that these form a quasivariety closed under quotients by the center. The crucial input to our main result is that every $R$-local group carries a unique Hall-Petresco $R$-group structure.

Periodically driven dissipative systems can settle into steady orbits - fixed loops on their dynamical manifolds. In quantum mechanics, steady orbits occur in cooling engines (used to initialise quantum devices), coherent oscillators (such as lasers and masers), precision metrology devices (atomic clocks, optical and spin magnetometers), and magnetic resonance (steady state free precession, dynamic nuclear polarisation). Steady orbits and stroboscopic steady states are a promising target for quantum optimal control, but the numerical complexity is prohibitive: the infinite loop defeats gradient ascent pulse engineering (GRAPE) which relies on explicit numerical propagation in the time domain. Here we propose an efficient quantum control strategy for stroboscopic steady states and limit cycles that are approached asymptotically when a control sequence is repeated infinitely many times. The formalism is different from Floquet-Lindblad state engineering and effective Hamiltonian theories: it finds control sequences that drive a dissipative quantum system towards a steady orbit passing through user-specified waypoints. The software implementation (same numerical complexity scaling as GRAPE) is done for the Spinach library.

Covering codes for insertions and deletions arise naturally in the study of synchronization errors and differ substantially from their classical counterparts in the Hamming metric. In this paper, we study covering codes under insertion and deletion operations. We first show that, in contrast to the equivalence between insertion and deletion correction, insertion covering and deletion covering are not equivalent. We then develop bounds and constructions for insertion and deletion covering codes, with particular emphasis on the large-alphabet regime. For insertion covering codes, we extend a recent combinatorial approach for single insertions and establish a new lower bound for arbitrary fixed insertion radius. For deletion covering codes, we relate the problem to hypergraph covering and prove that the elementary counting lower bound is asymptotically tight when the alphabet size tends to infinity. We further provide a construction of asymptotically optimal non-binary single-deletion covering codes by using differential Varshamov–Tenengolts (VT) codes together with a completion argument. In addition, we study covering codes for burst deletions. We prove that binary differential VT codes are not only capable of correcting two-burst deletions but also have the corresponding covering property, and hence form binary perfect codes for two-burst deletions. Finally, we extend this construction to non-binary alphabets and obtain explicit $q$-ary two-burst-deletion covering codes.

This paper is devoted to the study of uniformly distributed (u.d.) sequences on general non-compact topological spaces. Convergence-determining classes and the equivalent characterizations of u.d. sequences on $T_4$ spaces are discussed. For Polish spaces endowed with regular probability measures, the existence of a countable convergence-determining class for u.d. sequences is proved by virtue of the tightness of Borel probability measures. It is shown that, under the condition that a countable convergence-determining class exists, both independent identically distributed (i.i.d.) randomly sampled sequences and orbit sequences of ergodic transformations are u.d.\ with full measure.