Papers

We study the global geometry and endpoint dynamics of positive solutions of $-\Delta u+|y|^2u+\lambda u=g(u)$, $(y,z)\in\mathbb R^d\times\mathbb R^m$, $m=N-d$. Despite the loss of coercivity at $\lambda=-d$, we construct a connected component of centered positive solutions with frequency projection $(-d,\infty)$, assuming subcritical asymptotics of $g$ at zero and infinity, a zero-mass Liouville condition, and a strict Hamiltonian sign condition. At the lower edge, $\lambda+d\rightarrow 0^+$, anisotropic rescaling selects the lowest confined harmonic mode coupled to an $m$-dimensional scalar ground state. As $\lambda\rightarrow\infty$, isotropic blow-up yields the $N$-dimensional scalar ground state, revealing a dimensional crossover. The endpoint profiles and mass laws hold for every sequence of centered positive solutions, not merely along this component. The proof combines Gaussian–exponential localization on compact frequency strips, Lyapunov–Schmidt reduction, compact spectral perturbation, and Leray–Schauder continuation. At both ends, we establish uniqueness and differentiability of the centered branch, identify the Morse indices and symmetry kernels of the real and imaginary linearizations in the full energy space, and derive sharp mass-slope asymptotics. Under standard Hamiltonian-flow hypotheses, the resulting slope criterion classifies orbital stability and instability away from the mass-critical powers. The mass map on this component, combined with endpoint control on the full centered solution set, yields existence, nonexistence, and multiplicity for normalized standing waves, revealing the roles of both critical exponents $2+4/m$ and $2+4/N$.

We study closed simply connected six-manifolds with positive sectional curvature and an effective isometric action of $SO(3)$ or $SU(2)$. Starting from Liu's corrected orbit-space analysis, we refine the stratification and recover smooth topology in several previously unresolved cases. For a boundaryless $SO(3)$ quotient, singular isotropy is connected, the weighted nonprincipal graph belongs to nine necessary families, and every action with only cyclic exceptional isotropy is classified equivariantly. The remaining branching families admit explicit rational intersection and Pontryagin formulas. For a three-ball $SO(3)$ quotient, the uniform-boundary configurations give sphere bundles over $S^4$ and a completely distinguished odd-integer family of non-spin six-manifolds. A doubled-quotient argument excludes the configuration with two boundary fixed points and an interior singular orbit, while the configuration with one fixed point and an exceptional interval has underlying manifold $\mathbb{CP}^3$. For a boundaryless $SU(2)$ quotient, an adjoint-weight matching argument reduces the exceptional three-orbit case to a unique action on the Eschenburg flag and identifies every two-orbit candidate with a weighted action on $S^2\times S^4$. These classifications imply rational ellipticity throughout the stated symmetry class. We also establish restricted curvature obstructions for the two diagonal product actions, without asserting nonexistence for arbitrary invariant metrics. Abstract graph restrictions, smooth realizability, and positive-curvature realizability are kept distinct. Some of the results in this paper are discovered by ChatGPT-6-astra and verified by the authors.

Distributed quantum computing distributes workloads that exceed a single processor's capacity across resource-constrained components. Circuit cutting enables this decomposition, but its effectiveness depends on partition choice. We present SACE, a Simulation-Aware Circuit Partitioning framework with Entanglement-Capability-Aware Screening. EP-Cost screens partitions using a parameter-dependent entanglement-capability proxy and native implementation cost. Under reconstruction constraints, finite-bond refinement selects the candidate that minimizes the worst-subcircuit discrepancy between normalized MPS simulations at bond dimensions $\chi$ and $2\chi$. Across 14 exhaustively enumerated small-scale candidate spaces, EP-Cost-guided genetic algorithm selections attain co-optimal native quasiprobability decomposition (QPD) overhead, with a median within-circuit Spearman correlation of 0.9965 between EP-Cost and QPD overhead. At matched cut counts, EP-Cost reduces QPD overhead by $25.20\times$ and $16.89\times$ for 8- and 9-qubit QFT circuits, respectively, and improves sampled reconstruction. On 12-qubit approximate QFT, deterministic reconstruction reduces Pauli-observable mean absolute error in all six low-bond conditions and total-variation distance in five. With five cuts and 7,776 QPD terms, SACE is best on all four reconstruction metrics in four of six conditions. At 24 qubits, SACE improves exact-local QFT accuracy in all 30 instance-bond-dimension conditions against each EP-Cost-only, Kernighan-Lin, and spectral baseline. The 50-qubit QFT comparisons remain favorable as reference convergence improves. Weighted QAOA circuits provide complementary, more bond-sensitive evidence. Refinement benefits persist across three search methods; larger candidate panels improve the optimized proxy without guaranteeing lower exact-local error.

Within the topological concordance class of the Hopf link, we construct infinitely many classes under the equivalence relation generated by smooth $0$-shake concordance. Each contains infinitely many smooth concordance classes represented by links that are pairwise strongly $0$-shake concordant and whose second components have unbounded smooth slice genus.

We establish sharp projection inequalities for $k$-antichains under orders defined by open convex cones. For integers $n\geq2$ and $k\ge 1$, let $C\subsetneq\mathbb R^n$ be a nonempty open convex cone. A set $A\subseteq\mathbb R^n$ is a $k$-antichain with respect to the order $x\prec_C y\iff y-x\in C$ if it contains no chain $x^1\prec_C\cdots\prec_C x^{k+1}$. For every such set $A$ and every $v\in C$, we prove \[ \mathcal H^{n-1}(A) \leq k\frac{\|v\|}{\alpha_C(v)} \mathcal H^{n-1}(P_v(A)), \] where $P_v$ is the orthogonal projection onto $v^\perp$, $\mathcal H^{n-1}$ denotes normalized Hausdorff outer measure, and $\alpha_C(v)$ is the minimum of $\langle\xi,v\rangle$ over unit vectors in the dual cone $C^*$. In particular, every weak antichain $A\subseteq\mathbb R^n$ satisfies $\mathcal H^{n-1}(A)\leq\mathcal H^{n-1}(\Delta_i(A))$ for each $i$, where $\Delta_i(x)=(x_j-x_i)_{j\ne i}$. This proves the skewed projection conjecture of Engel, Mitsis, Pelekis and Reiher. As an application, we recover the bound $\mathcal H^{n-1}(A)\leq kn$ for $k$-antichains in the unit cube under the coordinatewise order.

The Gromov-Wasserstein (GW) distance provides a principled framework for comparing two probability distributions supported on different metric spaces by matching their individual intrinsic geometric structures. In practice, the GW distance is estimated from finite samples and often incorporates regularization for statistical and computational tractability. This motivates a quantitative understanding of the stability and estimation error of regularized GW distances. In this work, we study a general family of divergence regularized GW distances, including entropy and Tsallis regularizations over the full range of powers, from a nonasymptotic perspective. Specifically, we establish Lipschitz continuity of the value of regularized GW and Hölder continuity of its optimal coupling under Wasserstein perturbations of the input probability measures. Moreover, we derive the parametric rate of convergence for the empirical regularized (2,2)-GW distance on compactly supported Euclidean spaces.

This paper considers a class of structured cardinality penalized optimization problems with nonsmooth convex loss functions, for which the proximal mapping associated with the structured $\ell_0$ penalty is generally unavailable. To overcome this difficulty, we develop a chain of exact reformulations based on variable splitting, exact penalization, and continuous relaxation, and establish rigorous equivalence relationships among these resulting models. In particular, we define a class of strong local minimizers for the original problem and show that these minimizers are equivalent to weak $d$-stationary points of the relaxation problem. Further, we propose a safeguarded accelerated proximal alternating direction method of multipliers (SAP-ADMM). We prove that the sequence generated by SAP-ADMM converges to a weak $d$-stationary point of the relaxation problem. Depending on the parameter settings and algorithmic regime, we establish $O(1/k)$ or $o(1/k)$ convergence rates for both the objective function value gap and the KKT residual. Numerical experiments on signal and image denoising demonstrate the effectiveness of both the proposed models and SAP-ADMM in terms of reconstruction quality and computational efficiency.

We study the three-dimensional primitive equations with full viscosity and horizontal thermal diffusion in bounded domains. We establish global well-posedness for admissible $H^1$ initial data and persistence of $H^2$ regularity on the flat-bottom cylinder. The higher regularity analysis combines fractional hydrostatic Stokes regularity with anisotropic estimates of the vertical transport. We also consider variable-depth domains with positive $h\in C^3(\overline M)$ and a diffusion operator adapted to the topography compatible with layers $z/h(x,y)=\mathrm{constant}$. This operator reduces to $\Delta_H$ at constant depth and generally differs from the Cartesian horizontal Laplacian over a varying bottom. Its weighted formulation yields global strong solutions for admissible $H^1$ data. Persistence of $H^2$ regularity is proved under the additional assumption $\partial_\nu h=0$ on $\partial M$, where $\nu$ is the outward horizontal unit normal.

For every $m\geq2$, we prove that any connected closed oriented smooth immersion $X:M^m\to\mathbb{R}^{m+1}$ satisfying \[ K=\left\langle X,\nu\right\rangle \] is an embedding onto the unit sphere. No convexity is assumed. At a point maximizing $|X|$, we find a strictly convex component of $\{K>0\}$. If this component does not coincide with $M$, its polar immersion bounds an unbounded convex set. A finite area lemma then extends the local surface area formula for the polar body to a global measure identity. Using the vanishing first moment of surface area measure, we show that the origin lies in the interior of the polar body, contradicting the existence of recession directions. Thus the hypersurface is strictly convex, and the classification of strictly convex shrinkers completes the proof.

We formulate a proof of Part~I of Martin's conjecture for order-preserving maps directly on the quotient of the reals by Turing equivalence. The assumptions are countable choice for reals, the perfect-set property for sets of reals, and continuous uniformization on a comeager domain. The argument applies under $\mathsf{AD}+\mathsf{DC}_{\R}$ and inside the choiceless Solovay model. We then determine the exact consistency strength of the unrestricted order-preserving dichotomy as an axiom over $\mathsf{ZF}+\mathsf{AC}_{\omega}(\R)$: it is equiconsistent with $\mathsf{ZFC}$ plus a strongly inaccessible cardinal.

We solve an open problem posed by Richard Guy in his book Unsolved Problems in Number Theory. We prove that, for every integer $n\geq10$, there exist infinitely many $n\times n$ integer matrices $A$ such that every entry is $\ge2$ and $\det A=\det A^{[3]}=1$, where $A^{[3]}$ denotes the entrywise cube of $A$. The matrices may be chosen compatibly, so they can form the leading principal blocks of a single infinite positive integer array. The order-ten case is reduced to an inhomogeneous system consisting of one linear and one diagonal cubic equation. The circle method then provides infinitely many positive integral solutions. To pass from order $n$ to order $n+1$, we introduce a bordering construction that preserves the signed cofactor vectors of both the matrix and its entrywise cube. Iterating this construction yields the result in every order $n\geq10$.

Large-scale multimodal data are a key enabler for integrating large generative artificial intelligence models into wireless networks. However, existing wireless datasets offer limited support for heterogeneous environments, particularly in terms of reusable high-fidelity scenes, scalable differentiable ray tracing, and synchronized multimodal data. These limitations are more pronounced in mobile and aerial scenarios with time-varying environments and channels. To address these limitations, we propose DeepTelecom, a material-aware, high-fidelity multimodal channel dataset and reproducible generation framework for networks involving unmanned aerial vehicles (UAVs) and reconfigurable intelligent surfaces. DeepTelecom refines coarse indoor/outdoor reconstructions and uses a large language model to map segmented surfaces to frequency-dependent electromagnetic parameters, yielding material-aware Level of Detail 3 digital twins. Then, a differentiable GPU-accelerated Sionna ray tracer simulates propagation across the sub-6 GHz/millimeter-wave/sub-terahertz bands, and specifically captures rotor-induced micro-Doppler signatures of stationary or moving UAVs at different pitch angles and translational velocities. The framework synchronously produces channel tensors, coverage maps, transmission videos, multi-view images, and weather-conditioned renderings, and supports differentiable material calibration from sparse channel anchors. The released dataset contains 10 material-aware indoor/outdoor scenes and 50,000 synchronized UAV samples; the framework supports scalable generation of additional data for user-defined scenes and system configurations. In 3D UAV localization, DeepTelecom reduces the 90th-percentile error from 31.05 m to 2.43 m compared with schemes considering only channel state information, demonstrating its significance for artificial-intelligence-enabled wireless systems.

We give new invariant-theoretic proofs of regularity for the three classical residual families DIII, balanced CII, and BDI/Spin. In DIII and balanced CII, trace contractions and a supertrace identity show that the relevant character spaces in the quadratic moment quotient vanish. These quotient-level vanishings lift to the corresponding equivariant tempered-distribution vanishings. In BDI/Spin, Gram matrices and a volume tensor give the analogous vanishing for the two single-determinant characters. We also give a direct proof of the required equivariant-distribution vanishing for EVII. These results, together with corrected distinguished-nilpotent calculations for the nice factors, establish the required componentwise regularity for the finite-component data attached to all irreducible factors and repair gaps in the previously proposed proofs. Combining these factorwise results across products and finite central quotients yields regularity for every symmetric pair of a connected complex reductive group. Generalized Harish-Chandra descent and a compatible Chevalley involution give multiplicity one for irreducible Casselman-Wallach representations in the connected-group case. A finite-component restriction argument extends this multiplicity-one statement to every possibly disconnected complex reductive $G$ whose symmetric space $G/G^\theta$ is connected. Consequently, we establish van Dijk's conjecture on the unitary Gelfand property for connected complex symmetric spaces.

Twisted equations in free groups are equations in which a finite group of symmetries acts on the free group by automorphisms and anti-automorphisms, such as inversion and reversal; moreover, the variables may be subject to rational constraints. Algorithms from the theory of equations in virtually free groups solve such equations in PSPACE, but they are intricate. Our main result is an elementary algorithm for quadratic twisted equations, which extends the algorithm of Comerford and Edmunds (1981) for quadratic equations in free groups to the twisted setting and to rational constraints. Besides deciding solvability in PSPACE and yielding EDT0L descriptions of solution sets, our algorithm computes, for every quadratic expression, finitely many parametric forms describing its values as freely reduced words. As an application, we study the palindrome length of elements of free groups, that is, the least number of palindromes whose product they are. Guided by the parametric forms that our algorithm computes for products of palindromes, we found, and then proved directly, a characterization of the freely reduced words of palindrome length at most $n$ in terms of patterns describing how palindromes are nested by twisted conjugations, thereby confirming a Catalan structure conjectured by the second author. Consequently, these words form a context-free language, and palindrome length is computable in cubic time as well as in LOGCFL, answering a question of Bardakov, Shpilrain, and Tolstykh (2005).

Let $P = \sum_{\omega} a_\omega x^\omega$ be a degree $d$ homogeneous polynomial in $x_0, \dots, x_n$ with real coefficients. Pólya showed that, if $P$ is strictly positive on the standard $n$-simplex, then $(x_0 + \cdots + x_n)^M P$ has strictly positive coefficients for all $M \gg 0$. The least such $M$ is known as the Pólya exponent of $P$. de Klerk-Laurent-Parrilo showed that the Pólya exponent is $O(d^2) P^{(0)}_{\max}/\lambda(P)$, where $P^{(0)}_{\max} := \max_\omega a_\omega/\binom{d}{\omega}$ and $\lambda(P)$ is the infimum of $P$ on the standard simplex. Pólya's Positivstellensatz is a consequence of the classical approximation of the multivariate hypergeometric distribution by the multinomial distribution, as was already implicitly known to de Klerk-Laurent-Sun. By applying known estimates of this approximation from probability theory, we obtain a different $O_n(d) L(P)/\lambda(P)$ upper bound of the Pólya exponent, where $L(P) := \max |a_\omega|/\binom{d}{\omega}$. An example shows that $O_n(d)$ is tight.

Let $T$ be the Gauss map. For each $\alpha\in [0,1]$, let $E(\alpha)$ be the survivor set consisting of all $x\in [0,1]$ whose orbit $\{T^k(x):k\ge 0\}$ never enters the interval $(\alpha,1]$. We prove that the set-valued function $\alpha\mapsto E(\alpha)$ is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function $\alpha\mapsto E(\alpha)$ remains constant. We also describe the points at which the map $\alpha\mapsto E(\alpha)$ is not locally constant. Further, we obtain the Hausdorff dimension of $E(\alpha)$ for all $\alpha\in [0,1]$. We prove that the function $\alpha\mapsto \dim_H(E(\alpha))$ forms a devil's staircase on $(0,1]$, but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical $\alpha$-continued fraction expansion and regular continued fraction expansion.

This paper investigates intrinsic critical Sobolev-curl constants on bounded domains and the structure of finite-energy solutions to the critical curl-curl equation in R^3. First, for 1< p < 3, we prove that the intrinsic critical p-curl constant on (0, a)\times D is bounded above by a^{p/3} times a Sobolev quotient on D. The proof uses an aligned test field whose optimal curl-free correction vanishes. Consequently, these constants tend to zero along thin convex boxes. For p = 2, we construct explicit convex and nonconvex domains with intrinsic constant below the sharp scalar Sobolev constant. The nonconvex example gives an affirmative answer to the domain-existence question in Problem 5 of Mederski and Szulkin(Arch. Ration. Mech. Anal. 241 (2021), 1815-1842), while the convex example disproves the lower bound stated there for convex domains. Concerning their Problem 2, we obtain an implicit Newtonian representation of finite-energy solutions and exclude two special forms of solutions. These structural results do not yet provide an explicit ground state or an unconditional ground-state decay rate as asked in Problem 2.

We study a two-sided singular stochastic control problem in which changes in consumption carry utility costs and the risk premium follows the mean-reverting Kim-Omberg model. Optimal consumption is reflected between two moving barriers given by thresholds that depend on the current risk premium. These thresholds are the stopping boundaries of a Dynkin game for a two-dimensional diffusion driven by a single Brownian motion. A sharp occupation estimate shows that they are Lipschitz in logarithmic scale, uniformly in the cost of reducing consumption, and that smooth fit holds along the diffusion direction at every contact point. Outside a bounded interval of risk premia and at most two further points, the thresholds are infinitely differentiable. As the reduction cost grows, optimal consumption converges to the ratcheting policy.

Unmanned aerial vehicle-enabled synthetic aperture radar (SAR) offers significant potential for providing communication services to ground users while sensing the surrounding environment in low-altitude integrated sensing and communication (ISAC) systems. However, its practical deployment is challenged by known non-ULM trajectories and unknown position errors, which violate conventional SAR assumptions and cause severe image defocusing. We address these non-ideal impacts by formulating the ISAC SAR imaging as an exact linear inverse problem to accurately model arbitrary trajectories with multiple-input multiple-output orthogonal frequency division multiplexing waveforms. Based on this formulation, we develop a diffusion posterior sampling (DPS)-based super-resolution SAR framework, where a pre-trained diffusion model provides a data-driven image prior and the exact physical model enforces measurement consistency during posterior inference. The inference process is further accelerated for practical implementation. For unknown position errors, we develop a joint iterative DPS and motion compensation (MoCo) optimization algorithm with discrete cosine transform reparameterization and coarse-to-fine spectral scheduling. Simulations demonstrate that our framework achieves high-accuracy reconstruction across diverse trajectories and remarkable super-resolution, maintaining greater than 21.3 dB peak signal-to-noise ratio (PSNR) even under half physical resolution truncation. Under unknown motion errors, the joint DPS and MoCo algorithm achieves 23.65 dB PSNR with a residual position-error estimation error of 0.19 cm.

Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. We prove, uniformly over all primes $p$, that \[ \log_p f(p^m)=\frac{2}{27}m^3-\frac49m^2+O(m), \] with an absolute implied constant. This improves the classical upper bounds of Sims (Proc. Lond. Math. Soc., 1965) and Newman–Seeley and matches Higman's lower bound (Proc. Lond. Math. Soc., 1960) through the quadratic term. We also prove that \emph{the proportion of $2$-groups of nilpotency class $2$ among all groups of order at most $n$, counted up to isomorphism, tends to $1$ as $n\to\infty$.} These results answer questions~22.3 and~22.19 of Blackburn–Neumann–Venkataraman (Enumeration of Finite Groups, 2007). Our proof refines Sims' enumeration framework using graded Lie brackets, constraints on deeper commutator coordinates, and incidence and Fourier estimates for alternating forms. Normalization by changes of generating bases then determines the quadratic term in the upper bound. We also prove that all but a $p^{-\Omega(m^2)}$ fraction of groups of order $p^m$ have a central elementary abelian Frattini subgroup. Combining this concentration with the sharp enumeration bound of Bläser–Li–Qiao–Rogovskyy (arXiv, 2025) for $p$-groups of Frattini class $2$ yields the linear error term. The same presentation estimates give fixed-quotient extension bounds which, together with Pyber's prescribed-Sylow theorem and the concentration result, establish domination by $2$-groups of nilpotency class $2$.

The $\mathbb{A}^1$-Milnor number of a polynomial map is a quadratic enrichment of the classical Milnor number, defined as the local $\mathbb{A}^1$-Brouwer degree of the gradient. We study which elements of the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms arise as $\mathbb{A}^1$-Milnor numbers. Over fields of characteristic not $2$, $3$, or $5$, we show that every element of the Grothendieck-Witt ring of rank at most $7$ that is a local $\mathbb{A}^1$-Brouwer degree is also an $\mathbb{A}^1$-Milnor number. Unlike for local $\mathbb{A}^1$-Brouwer degrees, however, in general not all $\mathbb{A}^1$-Milnor numbers can be obtained from two-variable polynomials. Over an arbitrary field of characteristic not $2$, $3$, $5$, or $7$, we characterize the elements of the Grothendieck-Witt ring of rank at most $8$ that are $\mathbb{A}^1$-Milnor numbers of two-variable polynomials. Over subfields of $\mathbb{R}$, this implies that there are local $\mathbb{A}^1$-Brouwer degrees of ranks $4$, $6$, and $8$ that are not the $\mathbb{A}^1$-Milnor number of a two-variable polynomial. We also prove that under a mild hypothesis one can reduce the number of variables of a polynomial while preserving the $\mathbb{A}^1$-Milnor number up to multiplication by a unit.

We prove that every finite simple triangle-free graph $G$ on $n\geq1$ vertices satisfies $q_{\min}(G)\leq7n/50$, where $q_{\min}(G)$ is the least eigenvalue of its signless Laplacian matrix $Q_G=D_G+A_G$, with $D_G$ the diagonal matrix of vertex degrees and $A_G$ the adjacency matrix. The constant is sharp, since the Higman–Sims graph and its balanced blow-ups attain equality, as shown by de Lima, Nikiforov and Oliveira (2016). This proves the bound conjectured by Balogh, Clemen, Lidický, Norin and Volec (2023). The proof uses five test column vectors associated with an ordered edge to obtain a matrix Rayleigh inequality. Our use of local Rayleigh inequalities together with flag-algebra semidefinite programming is inspired by Zhou (2026). We use semidefinite programming and rational approximation to find the required positive semidefinite matrices, and complete the proof by exact correction and verification.

We define a $d$-cube in a graph $G$ to be monophobic in $G$ if, whenever it supports a $d$-face of a discrete singular $d+1$-cube, it must also support another face. In many cases, the presence of such cubes implies the nontriviality of discrete cubical homology and homotopy groups associated with $G$. We describe several large classes of examples and characterize the property completely when $d=2$.

The Copula Active Subspaces (CAS) method of Part I estimates a non-Gaussian observation-noise density $\pi_{\boldsymbol{n}}$ from residual samples. Its KL divergence $\mathcal{D}_{\mathrm{KL}}(\pi_{\boldsymbol{n}}\,\|\,\widehat\pi_{\boldsymbol{n}})$ admits an exact decomposition into marginal, projection and reduced-density terms and bounds the posterior KL divergence averaged over the true predictive distribution. We derive a priori bounds for these terms, together with estimates or bounds from residual samples. Weighted nearest-neighbour entropy estimates are used for the marginal and reduced-density terms, with asymptotic standard errors in exact rank coordinates. For the projection error, the score is fitted on one subsample, and its mean squared magnitude outside the retained subspace is estimated on the other. Corrections for score and rank-transform errors then give a theoretical upper bound with probability at least $1-\delta$. Tests on three known noise laws in dimension $d=20$ use the estimated rank transform. In $11$ of $12$ combinations of example and sample size, the sum of the two estimated terms and the trace bound exceeds the reference divergence after averaging over the estimated models. In these tests, score-estimation error is the largest contribution to looseness. The overestimate from the Gaussian log-Sobolev inequality alone is at most a factor of $1.98$.

For a subset $A$ of a finite abelian group $G$, let $\Sigma_k(A)$ denote the set of sums of $k$ distinct elements of $A$, and let $\Sigma(A)=\bigcup_{k\ge1}\Sigma_k(A)$. The problem of determining when $\Sigma(A)=G$ goes back to Erdős and Heilbronn. Nguyen and Vu later formulated the corresponding fixed-length question: for a prescribed $k$, when does $\Sigma_k(A)=G$? For groups of odd order, we prove the sharp large-order threshold $|A|>|G|/3+1$, which forces $\Sigma_k(A)=G$ throughout the entire middle range $4\le k\le |A|-4$. This determines the limiting thresholds posed by Girard, Griffiths and Hamidoune. If $p$ is the least prime divisor of $|G|$, then, for fixed odd $p$ and sufficiently large $|G|$, the exact threshold $|A|\ge |G|/p+p-1$ forces covering throughout the middle range beginning at $2p-2$. Thus, throughout this range, prescribing the number of summands does not change the sharp covering threshold. For $3\le k\le2p-3$, the restricted and ordinary $k$-fold covering problems have the same asymptotic threshold. As an application to MDS elliptic codes on an elliptic curve $E/\F_q$, we sharpen the half-scale conjecture of Li, Wan and Zhang, obtaining the sharp $1/3$-scale bound when $|E(\F_q)|$ is odd and, more generally, the exact bound determined by the least prime divisor of $|E(\F_q)|$.

Let $X$ be a smooth projective complex surface. We obtain explicit formulas for the Chern character operators on the Hilbert schemes of points $X^{[n]}$ in terms of Nakajima operators, determining in particular the coefficients of the terms involving the canonical class $K_X$ and its square. Consequently, the cohomology ring structure of the Hilbert scheme is determined. As an application, we give an alternative proof of the action of the affine Yangian $Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1)$ on the cohomological Fock space $\mathbb{H}_X = \bigoplus_{n\geq 0} H^*(X^{[n]},\mathbb{Q})$.

An edge labeling of a connected graph $G=(V, E)$ is called a local antimagic labeling if it is a bijection $f: E\rightarrow \{1, 2, \cdots, \left | E \right | \}$ such that for any pair of adjacent vertices $u$ and $v$, $f^{+}(u)\neq f^{+}(v)$ where $f^{+}(x)=\sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels across all possible local antimagic labelings of $G$. This paper investigates two families of graphs $G(m^{[n]})$ (a graph class constructed by augmenting a bipartite structure $B(m^{[n]})$ with cyclic edges) and $K_{1,s}\times C_n$ (the direct product of a star $K_{1,s}$ and a cycle $C_n$). Under the conditions odd integer $m\ge 1$ and even integer $n\ge 4$, we establish the exact values of $\chi_{la}(G(m^{[n]}))$ and $\chi_{la}(K_{1,s}\times C_n)$ for various $s$. Furthermore, we characterize several graph families achieving $\chi_{la}(G)=2$ or $3$.

For an odd prime $p$, we prove a Smith inequality over $\mathbb{F}_p$ for exact Lagrangian Floer homology under a symplectic $\mathbb{Z}_p$-action, assuming an equivariant stable normal trivialization compatible with the Lagrangians and the required relative Pin data. As an application, we obtain a rank inequality between the symplectic Khovanov homology of a $p$-periodic link and that of its quotient. These results extend the involutive localization of Seidel and Smith. The main construction is a stabilized chain complex. At every invariant intersection point we adjoin the Morse complex of a finite-dimensional normal spectral sphere, whose generators lie in free $\mathbb{Z}_p$-orbits. The differential counts holomorphic strips coupled to sphere half-trajectories by a one-sided weighted asymptotic evaluation. We prove transversality, a uniform spectral cutoff, and the compactification and gluing statements needed for the chain identity. The stabilized complex is quasi-isomorphic to the ordinary Floer complex and contains the complex of the fixed data as a subcomplex with free quotient, so the algebraic Smith inequality applies.

We consider spin-coupled Grover walks on hypercubes within the framework of quantum Bernoulli noises (QBN). We identify parity-dependent harmonic subspaces generated by the central Hamming layers: in even dimension they give a fixed-point subspace, whereas in odd dimension the walk reduces on the harmonic subspace to an explicit eight-dimensional reduced dynamics. Using symmetric-group representation theory, we characterize the corresponding isotypic structure and determine the reduced spectrum. The harmonic reduction also yields explicit long-time probability behavior and shows that the resulting states may occupy an asymptotically vanishing fraction of the hypercube while still spanning its full Hamming diameter.

In this work, we develop a general memory-augmented phase-field (MAPF) framework based on Onsager's variational principle, in which memory effects are incorporated through a general relaxation kernel while preserving the thermodynamic consistency of the underlying system. Rather than treating fractional models as fundamental formulations, we show that they can be interpreted as reduced descriptions associated with specific power-law memory kernels. The proposed MAPF equations provide a unified modeling framework for non-Markovian phase-field dynamics. To facilitate analysis and computation, we further introduce an equivalent internal-variable representation that transforms the original nonlocal-in-time dynamics into an extended Markovian gradient system. This formulation preserves key structural properties, including energy dissipation and mass conservation, and enables the development of efficient structure-preserving numerical schemes. In particular, we construct an energy-stable scalar auxiliary variable approach combined with a fast rank-one implementation that exploits the relaxation-spectrum-induced coupling structure, resulting in a computationally efficient algorithm with reduced storage requirements. The proposed framework establishes a systematic connection between memory-dependent phase-field modeling, internal-variable thermodynamics, and structure-preserving numerical computation. Numerical experiments demonstrate the accuracy, stability, and efficiency of the proposed methods for phase-field dynamics with long-time memory effects.