Papers

This paper presents an algebraic method to construct convolutional codes with guaranteed free and local minimum distances without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers $r$ and $t$ with $1 \leq r < t$, a rate-$r/t$ convolutional code $\mathcal{C}_{\text{convol}}$ can be constructed by using a chain of $r$ cyclic codes $\mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_{r-1}$ of the same length $n$ which satisfy the inclusion condition, $\mathcal{C}_0 \supset \mathcal{C}_1 \supset \ldots \supset \mathcal{C}_{r-1}$. Such a convolutional code $\mathcal{C}_{\text{convol}}$ is composed of a semi-infinite chain of identical local codes confined in a diagonal band of width $n$. Each local code $\mathcal{C}_{\text{local}}$ of $\mathcal{C}_{\text{convol}}$ is formed from the $r$ cyclic codes in the code chain and is a specially localized subcode of the mother code $\mathcal{C}_0$ in the code chain. The minimum distance $d_{\text{local}}$ of each local code of $\mathcal{C}_{\text{convol}}$ is lower bounded by the minimum distance $d_0$ of the mother code $\mathcal{C}_0$ in the code chain. The local structure of $\mathcal{C}_{\text{convol}}$ allows it to be decoded based on a designed parity-check matrix of the mother code $\mathcal{C}_0$ using a sliding window decoding scheme.

We give a computer-assisted proof that every finite union-closed family containing a nonempty set has an element in at least 0.38288525 of its members. The argument combines independent sampling with conditionally independent sampling and an explicit product lower bound for a binary-entropy kernel. We identify the limiting constant of this pointwise method through a one-variable stationary equation. We also construct a global coupling by balancing inverse union multiplicities and prove a strictly positive, quantitative entropy gain over independent sampling. Combining the two arguments gives an additional frequency bound depending on the size of the family.

We study surjectivity of the geometric Sen map for perfectoid $G$-torsors over smooth rigid analytic spaces over $\mathbb{C}_p$, where $G$ is a compact $p$-adic Lie group. We construct an affinoid perfectoid $\mathbb{Z}_p^2$-torsor over a smooth curve whose Sen map is not surjective, thereby disproving Rodríguez Camargo's conjectured implication from perfectoidness to surjectivity. We prove that the Sen map of every perfectoid $G$-torsor over a smooth curve is nevertheless surjective on a Zariski open dense subset. Finally, we show that locally analytic decompletion implies Sen surjectivity for diamondian affinoid perfectoid torsors.

We develop a least-action framework for describing how a probability distribution can evolve from an equilibrium state to a prescribed nonequilibrium state under constrained incremental changes. Taking a Gibbs distribution as the equilibrium reference, the framework gives a direct physical meaning to the geometry of the probability simplex: distance from equilibrium corresponds to nonequilibrium free energy, while changes between successive distributions carry an informational kinetic cost. The Pythagorean structure of relative entropy then provides the central insight of the work. It shows that intermediate distributions chosen via sequential information projections can reduce the kinetic cost of large transitions and establishes an energy-conservation-like relation between the kinetic expenditure along a path and the free energy accumulated in reaching the target distribution. Motivated by this geometry, we construct a greedy least-action path through successive information projections, obtain a closed-form characterization of each projection through the Lambert W function, and establish a finite-step performance guarantee. We further show that state-dependent costs can be incorporated naturally by reshaping the underlying Gibbs reference, providing a thermodynamic interpretation of path penalties as modifications of the effective energy landscape. Together, these results provide a unified view of distributional evolution through least action, information geometry, and nonequilibrium thermodynamics.

The aim of this work is to extend the results from a recent paper by Hong, Li and Liu, from bounded domains to both bounded and unbounded domains. We show the global existence and uniqueness of 3D stochastic Navier-Stokes equations with nonlinear multiplicative noise for every initial data from the Sobolev space $H^1$. We do not use any tightness argument. Instead, we firstly show the existence of a local maximal solution and then we use Lyapunov function to prove the solution is global. Our approach is motivated by a recent paper of the first named author with Ferrario, Maurelli and Zanella about a similar result for stochastic nonlinear Schrödinger Equations. The main idea is that once the local existence of strong solutions is established, a very strong noise pushing toward the origin, will make blow-up impossible.

We study zeroth-order optimization of non-convex functions with the aid of directional hints, which are cheap but potentially inaccurate approximations of the true gradient direction, given by linear subspaces at each iteration. To leverage these hints adaptively while maintaining robustness to their quality, we introduce Control-Variate Zeroth-Order Descent (CV-ZOD), a new framework that refines the classical zeroth-order gradient estimator with a control variate that can be set based on the directional hints. We first show that the oracle algorithm that optimally sets the reference vector and step size at each iteration achieves a convergence rate that interpolates between the first-order $O(1/T)$ rate and the zeroth-order $O(d/T)$ rate, depending on the quality of the hints along the trajectory. We then develop a practical variant of CV-ZOD that achieves the same oracle guarantee up to logarithmic factors, without any prior knowledge of the hint quality. We validate the method empirically on simulation-based scientific optimization tasks, demonstrating sustained progress on non-convex landscapes where zeroth-order descent is slower and existing guided methods stall as guidance deteriorates.

Classical Bayesian games assume that the joint distribution of players' types is common knowledge, an assumption that rarely holds in practical applications. We address this issue by studying a group of myopic players who repeatedly interact under the Bayesian Nash conjecture and learn a parametric joint type distribution from type profiles observed over time within a Bayesian learning framework. Players may hold heterogeneous priors and may be risk averse toward both epistemic uncertainty about the distributional parameter and aleatoric uncertainty in their rivals' types. We propose two models: a Bayesian Nash equilibrium among risk-averse Bayesian learners (BNE-RABL), in which epistemic and aleatoric risks are evaluated separately, and a BNE based on Bayesian predictive distributions (BNE-BPD), in which the two sources of uncertainty are integrated into a Bayesian predictive distribution and evaluated through a single risk measure. Under suitable conditions, we establish the existence and uniqueness of both equilibria, and derive non-asymptotic convergence rates of the equilibrium sequences toward the corresponding oracle BNE under the true type distribution as the game is repeatedly played and more type profiles are observed. These results further show that the discrepancy between the BNE-RABL and BNE-BPD strategies vanishes as the sample size grows, which illustrates that the two models can be regarded as effective approximations to each other. We apply the proposed models to a price competition problem and numerically illustrate the theoretical results.

We study iYangians, namely twisted Yangians in the Drinfeld current presentation, associated with quasi-split symmetric pairs of type $\mathsf{ADE}$ with nontrivial diagram involution. We establish minimalistic presentations in terms of degree-zero and degree-one generators. Type $\mathsf A_{2n}$ is treated separately: the isolated rank-two case requires two additional relations, whereas in higher rank these relations are forced by the neighboring noncentral orbit. As a by-product, we strengthen Theorem 3.1 in arxiv:2511.07136 by proving that the extra relation in the minimalistic presentation of split iYangian is redundant whenever $\mathfrak g$ has rank at least two. We also construct explicit injective homomorphisms from quasi-split iYangians into Yangians, identify their images as right coideal subalgebras, and obtain isomorphisms between the Drinfeld and $J$ presentations. Finally, we derive triangular estimates for the Drinfeld currents and their coproducts and apply them to the ${}^\imath\ell$-weights arising by restriction from finite-dimensional Yangian modules.

We prove Todd's relation conjecture: all $\mathbb{F}_q(\theta)$-linear relations of Thakur's multiple zeta values are generated from the fundamental binary relation by the operators $\mathcal{B}^{\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\mathrm{S}}\circ \mathcal{C}^{\mathrm{S}}$; moreover they are also generated by $\mathcal{B}^{\ast\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\ast\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$. The $\mathcal{B}^{\ast\mathrm{S}}$-part of this conjecture has been proved by Chang, Chen and Mishiba. We prove the whole conjecture for Carlitz multiple polylogarithm values, which implies the $\mathcal{B}^{\mathrm{S}}$-part for multiple zeta values. We also determine all fixed relations and binary relations. Let $\mathfrak{BR}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(\theta)$-linear space spanned by binary relations of weight $w$, and let $\operatorname{Fix}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(\theta)$-linear space spanned by fixed relations. We derive generating functions $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\operatorname{Fix}^{\mathrm{S}}_w\bigr)x^w=\frac{x^{q+1}(1-x)}{(1-2x)(1-2x+x^{q+1})}$ and $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\mathfrak{BR}^{\mathrm{S}}_w\bigr)x^w=\frac{x^q(1-x)}{(1-2x)(1-2x+x^{q+1})}.$ Our results are based on the recent work of Im-Kim-Ngo Dac on the $\mathbb{F}_q$-linear relations of Thakur's multiple zeta values, and a system of transfer theorems between multiple zeta values and multiple polylogarithm values.

We study the $p$-Laplacian quadrature surface free boundary problem on a compact Riemannian manifold $(\M,g)$ of dimension $N \geq 2$, for $1<p<\infty$. Given a nonnegative source $f \in L^p_g(\M)$ with compact support $K$, and $k>0$, one seeks a domain $\Omega \supset K$ on which the overdetermined problem $-\Delta_{p,g} u = f$ in $\Omega$, $u=0$ and $|\nabla_g u|_g = k$ on $\partial\Omega$ admits a solution. Following the geometric and variational framework of Djité–Seck, we prove that the integral condition \[ (NS)_{p,g} \qquad \int_C f\,dv(g) > k^{p-1}\,|\partial C|_g, \] where $C$ is the totally convex hull of $K$, is sufficient for the existence of a solution strictly containing $C$. We then show, by an explicit counterexample on the round sphere $S^n_R$, that $(NS)_{p,g}$ is not necessary in general under nonnegative curvature: the mechanism is that the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis—perimeter monotonicity on the admissible class inside a normal convex neighborhood of $C$—under which necessity is restored, and we discuss the critical equality case $\int_C f\,dv(g) = k^{p-1}|\partial C|_g$.

We study the asymptotic behavior of positive ground states of $N$-particle atomic Coulomb Hamiltonians in a two-cluster region. For the $N$-particle ground state $\psi$, its one-particle density $\rho$, and $(N-1)$-particle ground state $\phi$, we show that for $0<\alpha<1$, as $|x_N|\to\infty$, $\psi(x_1,\dots,x_N)/\sqrt{\rho(x_N)}= \phi(x_1,\dots,x_{N-1})+O(|x_N|^{-2+\alpha} \ln |x_N|)$ uniformly in the region $|x_i|\le |x_N|^{\alpha}$. In the region $|x_i|\le R_0\ln |x_N|$, the convergence rate is $O(|x_N|^{-2} (\ln |x_N|)^2)$. We also establish an $O(|x_N|^{-2})$ bound on the squared $L^2$ error. These results hold both below the essential spectrum and at threshold.

We study a fourth-order variational model for two-dimensional frustrated ferromagnets with competing exchange interactions and an applied magnetic field of strength $H>0$. For every $H>1/4$, we prove that the energy admits minimizers in the topological classes $Q=\pm1$ and that every minimizing sequence is precompact in $H^2$ modulo translations. The main difficulty is that spectral coercivity degenerates as $H\downarrow1/4$. Using a Helmholtz circle-mean identity, we prove that the residual energy of every nonzero-degree configuration has a uniform positive lower bound, even at the degenerate endpoint. Together with a sphere-valued $H^2$-splitting construction, this provides a threshold-stable binding inequality and yields compactness throughout the coercive regime. We also identify $H=1/4$ as the sharp spectral threshold. Below it the energy is unbounded from below, whereas at the threshold nonzero-degree configurations retain a positive energy barrier and degree-zero Weyl sequences lose compactness. Finally, in the strong-field regime, rescaled minimizers approach those minimizers of the limiting functional that maximize the Dirichlet energy, while topological-charge and normalized-energy measures concentrate on the $H^{-1/4}$-scale.

Phase noise (PN), arising from imperfect local oscillators, introduces multiplicative distortions that degrade the performance of communication systems. In uplink multiuser multiple-input multiple-output (MIMO) systems, this impairment is further compounded by the presence of independent oscillators at each transmit and receive antenna, each contributing an uncorrelated noise component. Existing PN compensation algorithms at the receiver either rely on linearization approximations that lose accuracy under severe PN conditions, or incur computational complexity that scales prohibitively with the number of antennas. To address these limitations, we propose a variational Bayes (VB) framework for joint PN estimation and data detection in uplink MIMO systems. We develop VB-based detectors that treat noise statistics as latent variables, and reformulate the inference problem by absorbing the transmitter PN into the transmitted signal, treating the resulting composite variable as the inference target. Under von Mises priors, this reformulation yields exact closed-form conjugate posterior updates, from which we derive an improved detector achieving superior performance at low complexity. Simulation results demonstrate that the proposed VB algorithm achieves lower symbol error rates than the Self-Interference Whitening (SIW) algorithm and conventional phase-noise-unaware detectors across a wide range of channel conditions, modulation orders, and PN severities, while remaining computationally scalable to large MIMO deployments.

We prove that every compact connected Riemannian isometric filling $M$ of a circle of length $2\pi$ satisfies $\operatorname{Area}(M) \geq \frac{14\zeta(3)}{\pi} \approx 5.35677$, regardless of orientability or topological types. Our new approach uses the odd Fourier coefficients of the distance functions from boundary points. For orientable fillings, we use a cubic resonant perturbation to obtain $\operatorname{Area}(M)>5.40154$.

We study the quadrature surface free boundary problem on a compact Riemannian manifold $(\M,g)$ of dimension $N \geq 2$. Given a positive source $f \in L^2_g(\M)$ with compact support $K$, and $k>0$, one seeks a domain $\Omega \supset K$ on which the overdetermined problem $-\Delta_g u = f$ in $\Omega$, $u=0$ and $|\nabla_g u|_g = k$ on $\partial\Omega$ admits a solution. Following the geometric and variational framework of Djité–Seck, we prove that the integral condition \[ (NS)_g \qquad \int_C f\,dv(g) > k\,|\partial C|_g, \] where $C$ is the totally convex hull of $K$, is sufficient for the existence of a solution strictly containing $C$. We then show, by an explicit counterexample on the round sphere $S^n_R$, that $(NS)_g$ is not necessary in general under nonnegative curvature: the phenomenon arises because the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis—perimeter monotonicity within a normal convex neighborhood of $C$—under which necessity is restored, and we discuss the critical equality case $\int_C f = k|\partial C|_g$.

We show that replication carries congruence symmetry down an explicit level tower. If a normalized replicable function $f$, holomorphic on the upper half-plane, is invariant under $\Gamma_0(N)$, then its $n$th replicate is invariant under $\Gamma_0(N/(N,n))$. Thus every replicate whose index is divisible by $N$ is the normalized modular invariant $J=j-744$. This gives a direct and classification-free proof of $J$-finality for congruence-invariant replicable functions. The argument uses only the replication identities and an elementary generation theorem for congruence subgroups; it requires neither complete replicability nor arithmetic hypotheses on the Fourier coefficients. We then apply the descent law to completely replicable functions of finite replication order. Their replication towers have a canonical terminal replicate, and the only possible terminal functions are $J$, $q^{-1}$, and $q^{-1}+q$. Moreover, a finite-order completely replicable function is $J$-final precisely when its terminal replicate is $J$, and any symmetry beyond translations forces this alternative.

We classify compact Hermitian manifolds with flat Levi-Civita connection. This is equivalent to the classification of all orthogonal complex structures on flat tori. It generalized the work of Khan, Yang, and Zheng in 2017 where they solved the case in complex dimension three.

Source-to-observation operators provide reusable environment-level descriptions for multi-query electromagnetic (EM) prediction and communication-mode analysis. However, in practical multiple-scattering models, these operators are represented with finitely many angular modes, and agreement for selected excitations or between successive truncation orders does not establish uniform accuracy of the full map or reliability of its singular channels. To close this gap, we formulate the environment-induced response as a scattering Green operator on fixed continuous source and observation spaces and derive an exact trace-space factorization that reconstructs the Maxwell scattered field. For fixed, pairwise-disjoint enclosing trace spheres and a well-posed collective problem, nested vector spherical wave function (VSWF) realizations converge in operator norm. A structural bound separates external modal tails from collective-resolvent sensitivity, and operator-norm convergence guarantees uniform convergence of the singular values. We further construct a finite metric core that preserves the nonzero singular values of each finite-order operator and reconstructs matched orthonormal source–field channels without introducing external-support discretization degrees of freedom (DoF) into the spectral problem. Full-wave benchmarks verify the finite-order implementation. A controlled near-resonant two-sphere study shows that adjacent-order agreement can precede resolution of the dominant high-order collective direction. It further shows that only part of the internal amplification appears in externally accessible gains and that resonance promotes a distinct high-order channel pair above an otherwise preserved low-order family. The resulting framework provides a convergent, metric-consistent finite-modal representation of multiple-scattering source-to-observation operators and their accessible channels.

We study the singular limit of the Camassa–Holm equation as the length scale $\ell$ tends to zero. Although the formal limit is the Burgers equation with flux $3u^2/2$, we show that, for a class of smooth initial data, convergence to entropy solutions fails after the shock formation, even along subsequences in local space-time $L^1$.

A balanced spanning tree is a spanning tree that contains an edge whose removal partitions the vertices into exactly two connected subtrees of equal size. In this paper, we establish explicit recurrence relations for the number of spanning trees in $2 \times n$ triangular strip lattices- obtained by adding a diagonal edge to each square of a $2 \times n$ grid graph- generalizing combinatorial counting techniques introduced by Raff [Raf08]. We then adapt arguments of Gallagher and Tapp [GT25] to count balanced spanning trees of arbitrary triangular strip lattices. We establish sharp asymptotic bounds for the proportion of balanced spanning trees as $n \rightarrow \infty$. Finally, we determine the probability that a spanning tree of a $2 \times n$ triangular strip lattice chosen uniformly at random is balanced as $n \rightarrow \infty$.

This work considers to numerically solve three kinds of reaction-diffusion equations with free boundaries. First, the popular front-fixing method is used to transform the considered free boundary problems into fixed boundary problems. Then, by employing the finite difference method, numerical schemes with $\mathrm{M}$-matrices as their coefficient matrices are developed for these transformed fixed-boundary problems. Next, for the developed numerical schemes, we establish numerical theory involving positivity preservation, monotonicity preservation, and stability. It is noteworthy that, unlike some existing works that impose tight restrictions on the time step-size to discuss the numerical theory, the proposed numerical schemes require only a mild restriction. Finally, numerical examples are provided to test the developed numerical schemes and to confirm the theoretical results.

In the paper J. Phys. A: Math. Theor. 53 (2020), 345303, the limiting mutual information between two intervals separated by one lattice site in a free-fermion chain was evaluated. The calculation replaces the entropy function by a regularized version depending on $\varepsilon>0$, takes the block lengths to infinity, and only then lets $\varepsilon$ tend to zero. We prove that the opposite order gives the same constant $2\log2-1$. Moreover, we show that the same limit is obtained whenever both block lengths tend to infinity and $\varepsilon$ tends to zero simultaneously, with no restriction on their relative rates. In fact, the error caused by the regularization is bounded by $(1+\varepsilon)\log(1+\varepsilon)-\varepsilon\log\varepsilon$, uniformly in both block lengths.

We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius $R$, the region where the curvature radius is smaller than $R^{-1}$ occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li–Wang.

We develop a general characteristic-free framework for studying the homological properties of a broad class of module-finite Hopf algebras. This framework makes it possible to reduce the study of the homological properties of many quantum groups at roots of unity and their multiparameter deformations to the study of the corresponding small quantum groups. For any affine Hopf algebra $H$ admitting a large central Hopf subalgebra $C$, we prove that its left homological integral space, in the sense of Lu-Wu-Zhang, is isomorphic as a bimodule to the left integral space of the identity fiber algebra, which is a finite-dimensional Hopf algebra. Consequently, $H$ is a symmetric Frobenius extension of $C$ if and only if the corresponding identity fiber algebra is unimodular and the square of the antipode of $H$ is inner, thus providing an effective criterion for the Calabi-Yau property of $H$. For a broad class of quantum groups at roots of unity, an appropriate large central Hopf subalgebra can be chosen such that the identity fiber algebra is the corresponding small quantum group. Therefore, some homological properties of these big quantum groups are governed by those of their corresponding small quantum groups. Assuming that the base field is algebraically closed, we prove that $H$ is unimodular if and only if, for some (equivalently, every) maximal ideal $\mathfrak{m}$ of $C$, the category of finite-dimensional representations of the fiber algebra at $\mathfrak{m}$ is unimodular in the sense of Yadav as a module category over the finite tensor category of finite-dimensional representations of the identity fiber algebra. As an application, we prove that all Andruskiewitsch-Angiono-Yakimov large quantum groups are affine noetherian unimodular Artin-Schelter Gorenstein Hopf algebras. We also give a necessary and sufficient condition for these large quantum groups to be Calabi-Yau.

For each level of the Clifford hierarchy from the third level onward, we construct explicit asymptotically good binary CSS codes for which a strongly transversal diagonal rotation, followed by a correction from the preceding level, implements the corresponding logical rotation. We achieve this using algebraic geometry codes over binary extension fields and performing an alphabet reduction that translates orthogonality conditions over the extension field to binary overlap conditions. In particular, we obtain the first asymptotically good family of binary triorthogonal codes. We also give a stronger construction with worse parameters for which the strongly transversal physical rotation implements the desired logical rotation, with no subsequent correction. Finally, adapting an error-correction-based distillation protocol, our binary $T$-gate families yield direct constant-overhead $T$-state block distillation from noisy $T$-state inputs under sufficiently weak input noise and ideal stabilizer operations.

Let $K/\mathbb Q$ be an imaginary quadratic extension and $p$ an odd prime. Write $\rho=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}\chi$, where $E/\mathbb Q_p$ is a finite extension and $\chi:G_K\to\mathcal O_E^\times$ is a continuous character. For a fixed $r\in\mathbb Z\setminus\{0\}$, let $\pi_{\rho,r}(X)$ denote the number of rational primes $\ell\le X$ such that $\rho$ is unramified at $\ell$ and $\operatorname{tr}\rho(\operatorname{Frob}_\ell)=r$. Let $a,b$ be the two weights of $\chi$ at $p$. We prove that if $(a,b)\notin\mathbb Q^2$, then $\pi_{\rho,r}(X)\ll_{\rho,r,\varepsilon}X^\varepsilon$ for every $\varepsilon>0$, while if $(a,b)\in\mathbb Q^2\setminus\mathbb Z^2$, then only finitely many such primes occur. These bounds are substantially sparser than the classical CM Lang–Trotter scale. The main input in the non-rational case is a rigidity theorem for algebraic curves in the \(p\)-adic analytic trace locus, combined with rigid-analytic Pila–Wilkie counting; the rational non-integral case is treated by a local ramification argument.

We investigate diffusion processes on disconnected fractal sets in one-sided Brownian environments. On the real line, it is established that the process exhibits either diffusion or trapping, each occurring with probability $1/2$. In this study, we demonstrate that for fractal sets, this behavior is governed by the geometric parameters $r$ (the reciprocal of the similitude ratio) and $N$ (the number of contraction mappings), which define the fractal structure. Two distinct regimes emerge: a diffusive regime on the environment-free side and a localization regime on the side influenced by the environment. The transition between these regimes is determined by whether the random environment first hits the threshold $\log r$ or $-\log N$. Consequently, the probability of diffusion versus trapping is explicitly characterized by the Hausdorff dimension $d_f = \log N / \log r$. This result demonstrates that fractal geometry scales the limiting distributions and dictates the stochastic regime.

For every integer $r\ge4$, and $\rho \in [0,1]$, we asymptotically determine the maximum proportion of $r$-element sets of vertices that induce either a clique or an independent set in a large graph with density $\rho$. This generalizes a result of Olpp for $r=3$. After the initial idea for the main proof was found by the authors, various AI models were used to streamline the argument and perform the calculations necessary for completion of the proof.

In this paper, we establish a Fefferman–Stein inequality in terms of area function and non-tangential maximal function associated with the Schrödinger operator $\mathcal{L} = -\Delta + V$ on stratified Lie groups $\mathcal G$, where $\Delta$ denotes the sub-Laplacian on $\mathcal G$ and $V$ is a nonnegative locally integrable function. As an application, we extend this inequality to the tensor product $\mathcal G_1 \times \mathcal G_2$ of two stratified Lie groups and develop atomic decompositions associated with the Schrödinger operator for functions in the Orlicz space $L\log^{+}L(\mathcal G_1 \times \mathcal G_2).$ Using these atomic decompositions, we further prove weak-type endpoint estimates for the area integral operator and the Riesz transforms associated with the Schrödinger operator on $L\log^{+}L(\mathcal G_1 \times \mathcal G_2),$ thereby extending the celebrated result of R.\,Fefferman and E.M.\,Stein \cite{FSt1982} to the setting of singular integrals with non-smooth kernels.

For two distinct binary words of length $n$, the separating words problem asks for a small deterministic finite automaton that accepts exactly one of them. Chase proved a $\widetilde O(n^{1/3})$ upper bound using a complex-analytic estimate for sparse polynomials. We replace that estimate by a finite-difference argument and a second-order real recurrence cutoff. The resulting elementary proof gives an explicit bound of $O(n^{1/3}(\log n)^{7/3})$ states.