Papers

To support ultra-dense connectivity in terahertz (THz) communications, this paper proposes a dynamic frequency-position-fluid antenna (D-FPFA) architecture. Frequency-tunable local oscillators (LOs) are integrated into the RF chains to access different sub-bands, thereby expanding the total bandwidth of the system and providing frequency-domain diversity. To exploit spatial diversity, the base station employs movable subarrays, and each user is equipped with a movable antenna. We first develop a two-phase beam-split-aware frequency allocation strategy. In the first phase, we divide users into disjoint sub-bands according to their channel correlation coefficients to mitigate the interference. In the second phase, we investigate the wideband near-field beam-split effect for planar arrays and reveal an astigmatism phenomenon, in which the beam at a non-central subcarrier cannot be perfectly refocused at a single spatial point. Then, we establish a beam split multiplexing strategy, where we formulate the user grouping task as a minimum dominating set problem. To maximize the sum rate, we introduce a switch network along with a distance-based antenna selection strategy to account for the near-field channel gain variations, followed by a particle swarm optimization-based algorithm that jointly optimizes the antenna positions and precoders. Numerical results show that, the proposed D-FPFA achieves approximately 2.3 times the sum rate of a conventional phase-shifter (PS)-based array-of-subarrays (AoSA) architecture. It also attains 95% of the sum rate of its TTD counterpart while providing approximately 2.8 times its energy efficiency (EE). Moreover, the fully connected variant of D-FPFA, i.e., FPFA, achieves the highest EE among all considered architectures.

In this paper, we study the asymptotic behavior of deterministic identification (DID) over binary symmetric channels (BSCs) under vanishing error constraints. By introducing a minimum error parameter, we characterize how different error-decay regimes affect the achievable DID rate. General achievability and converse bounds are derived, with explicit asymptotic characterizations in the large-deviation, moderate-deviation, and central-limit regimes. The achievability analysis combines coding-theoretic constructions with probabilistic concentration techniques, while the converse links statistical distinguishability to the minimum-distance structure of DID codes via total variation and Hamming-type bounds. Our results show that the asymptotic behavior of DID over BSCs is governed by a Hamming-shell concentration geometry of channel outputs, offering insights into the finite-blocklength behavior of deterministic identification over discrete-output channels.

We study partitions which attain the largest Littlewood-Richardson coefficient. More precisely, we prove that the largest $c^\lambda_{\mu\nu}$ is attained at partitions such that $\mu \subseteq \nu$ and $\nu/\mu$ is a disjoint union of squares. We conjecture that for $n \ge 16$, all largest $c^\lambda_{\mu\nu}$ must satisfy this property. We confirm this conjecture numerically, for $16 \le n \le 45$.

In this paper the asymptotic exponent of coupled Harish-Chandra-Itzykson-Zuber (HCIZ) integrals is studied. We consider $L$ rank-one HCIZ integrals which are coupled through additive exponent terms, with $L>1$ finite. We show that the normalized asymptotic concentration exponent in the rank-one case is, under some constraints, characterized by the operator-valued R-transform. In this way the coupled HCIZ integral provides the operator-valued free analog of the classical Laplace transform. The result can be used to recover the asymptotics of rank-one non-self-adjoint spherical integrals, if the constant matrix is R-diagonal in the sense of free probability. Furthermore, we show that $L$ coupled HCIZ integrals of rank $M(N)=O\left(N^{1/2-\epsilon}\right)$ can be factorized into $M(N)$ rank-one $L$-fold coupled spherical integrals, and characterize the operator-valued R-transform for R-cyclic matrices on the algebra of real, diagonal $L\times L$ matrices. The treatment of coupled spin glasses with correlated disorder is enabled by our asymptotic exponent.

Let $G$ be a simple nonempty graph with size $m$, chromatic number $\chi$, and least eigenvalue $\lambda$. We prove that \[ \chi(\chi-1) \le (m+1-\lambda^2)+\sqrt{(m+1-\lambda^2)^2-4(\lambda^2-1)(\lambda^2-m)} \] with equality if and only if $G$ is either a complete graph or a complete bipartite graph, with possibly isolated vertices. The inequality was conjectured recently by Tang and Elphick in [Electron. J. Combin. 33 (2026), \#P2.65].

We give a sieving algorithm for finding pairs of primes with small multiplicative orders modulo each other. This problem is a necessary condition for obtaining constructions of $2$-cycles of pairing-friendly curves, which have found use in cryptographic applications. Our database of examples suggests that, except for a well-known infinite family of such primes, instances become increasingly rare as the size of the primes increase. This leads to some interesting open questions for which we hope our database prompts further investigation.

A Heisenberg Bieberbach manifold is a compact quotient of the Heisenberg group by a discrete torsion-free subgroup of its pseudo-Hermitian automorphism group. In this paper, we study the joint spectrum of the sub-Laplacian and the Reeb vector field on three-dimensional Heisenberg Bieberbach manifolds. In particular, we explicitly compute the multiplicities of the joint spectrum using theta functions and the character theory of finite groups.

We construct a projective klt generalized pair over the complex numbers whose generalized log canonical ring is not finitely generated. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol-ultra, Fable 5, and the Danus system.

A fully implicit two-stage fourth-order two-derivative time discretization was introduced previously as a temporal method. This paper closes that sequential integrator for stiff transport–relaxation equations by pairing a conservative finite-volume residual $\mathcal L_h$ with its discrete trajectory derivative $\mathcal G_h^{\rm tr}=D\mathcal L_h\,\mathcal L_h$. An ADER/Cauchy–Kowalevski predictor provides interface states and physical time derivatives; differentiating the same numerical flux and taking shared face differences yields a conservative approximation $\widetilde{\mathcal G}_h$. For linear constant-coefficient balance laws, $\widetilde{\mathcal G}_h=\mathcal G_h^{\rm tr}=\mathcal L_h^2$ exactly, although the derivative operator is assembled independently rather than by squaring the residual matrix. For nonlinear discretizations, the fourth-order temporal theory applies to $\mathcal G_h^{\rm tr}$, while a trajectory-closure consistency estimate controls the ADER approximation. The two unknown stage vectors are solved successively through two $N$-unknown systems. The completed step is fourth order and L-stable; the parameter $C_q=5/183$ cancels the leading inverse-power term and changes the deep-stiff amplification from $O(|z|^{-1})$ to $O(|z|^{-2})$. For fixed compatible spatial spaces, a slow–fast decomposition proves a full-step asymptotic-preserving operator limit with an $O(\delta)$ estimate and gives a preparation-dependent uniform-accuracy classification. Linear finite-volume, nonlinear relaxation, one- and two-dimensional damping, diffusion-limit, and modal experiments verify the corresponding closure, accuracy, stability, and singular-limit claims within their stated scopes.

In this paper we prove that unirationality, rational connectedness and rational chain connectedness coincide for smooth projective varieties over a field $ k $ of characteristic zero. Our approach uses the MRC fibration to show that if $ X $ is a smooth projective variety, then there exists a variety $ \operatorname{MU}(X) $, together with rational maps $ \pi: X \dashrightarrow \operatorname{MU}(X) $ and $ \lambda: \operatorname{MU}(X) \dashrightarrow \operatorname{MRC}(X) $, such that i) if $ \nu: X \dashrightarrow \operatorname{MRC}(X) $, then $ \lambda \circ \pi = \nu $ on the appropriate domains; ii) the very general fibres of $ \pi $ are unirational; iii) the very general fibres of $ \lambda $ are rationally connected but not unirational. We then apply an induction argument to show that $ \operatorname{MU}(X) $ is birationally equivalent to $ \operatorname{MRC}(X) $.

Let $r>1$ with $\gcd(r,30)=1$. We construct a finite group $G_r=(C_5\times C_3\times C_r)\rtimes W$, where $|W|=32$ and $G_r'\cong C_{60r}$, together with an augmentation-preserving automorphism $\alpha_r\in\operatorname{Aut}(\mathbb{Z}G_r)$ having no Zassenhaus factorization. The image $Y_r=\alpha_r(G_r)$ is a normalized group basis which is not rationally conjugate to $G_r$, although every element of $Y_r$ is individually rationally conjugate to an element of $G_r$. Consequently, (ZC2) and (ZC3) fail for finite cyclic-by-abelian groups, resolving a problem of Margolis and del Río. The construction extends Hertweck's example uniformly: both the class-preserving obstruction and the integral gluing are independent of the order of the auxiliary factor. The smallest admissible member of this family has order $3360$ and derived subgroup $C_{420}$.

We study the oracle complexity of finding $\epsilon$-stationary points of $\rho$-weakly convex and $G$-Lipschitz functions, where stationarity is measured by the gradient of the Moreau envelope. We consider an exact first-order oracle that returns both the function value and the full subdifferential at every query point. In the principal parameter regime, we prove that every first-order algorithm, including randomized algorithms, requires $\Omega(\tfrac{\rho G^2\Delta}{\epsilon^4})$ oracle queries to succeed with any fixed positive probability, where $f(0)-\inf f\le\Delta$. This lower bound matches the best known deterministic and stochastic first-order upper bounds, up to universal constants, and establishes the optimal randomized first-order oracle complexity. The result separates smooth nonconvex and weakly convex optimization: the additional $\epsilon^{-2}$ factor is intrinsic to nonsmooth geometry rather than stochastic oracle noise.

For an odd prime $p$, we show that the homotopy type of a closed $4$-manifold with fundamental group $\mathbb{Z}/p \rtimes \mathbb{Z}$ is determined by its quadratic $2$-type together with an additional $\mathbb{Z}/p$-valued intersection form. We also give a generalization to infinite groups of a result of Hambleton–Kreck on the polarized homotopy classification of Poincaré $4$-complexes, and we prove some partial results on the homotopy classification that are applicable more generally.

We introduce and study palindromic-gap partitions: partitions whose sequence of successive differences between consecutive parts is a palindrome. We derive the generating function for this family, treating partitions with an odd and an even number of parts separately, and obtain a bivariate refinement tracking the number of parts. Specializing further, we give explicit closed formulas for the number of palindromic-gap partitions of $n$ into exactly $r$ parts. We then construct explicit bijections between palindromic-gap partitions with $2m+1$, respectively $2m$, parts and ordinary partitions with at most $m+1$ parts, in which the weight of a palindromic-gap partition is encoded by the largest part, or the two largest parts, of its image; composing these yields an explicit bijection between the odd and even families themselves. Finally, we show that a partition is palindromic-gap if and only if its Ferrers diagram, drawn with left-justified rows, is self-complementary under a $180^\circ$ rotation inside its own naturally associated rectangle, identifying palindromic-gap partitions with self-complementary partitions relative to this rectangle. This yields a partial answer to a question of Keith on the enumeration of complementable and self-complementary partitions: we give explicit generating functions for the self-complementary case and exhibit, at $n=15$, a partition that is complementable but not self-complementary, confirming that the two notions are genuinely distinct.

We study a refinement of the analytic hierarchy over second-order arithmetic under the arithmetical Church thesis, the assertion that every set of natural numbers is arithmetical. The thesis is false in the full standard model, but it is naturally satisfied in the $\omega$-model consisting of the arithmetical sets. Over $\mathsf{ACA}_0^\ast+\mathsf{ACT}$, second-order quantifiers can be replaced by quantification over codes for arithmetical sets, and this gives normal forms which distinguish first-order and second-order quantifier alternations more finely than the usual analytic hierarchy. As an application, we use these normal forms to answer a question of Frittaion on fragments of uniform reflection in second-order arithmetic: the choice assumption in his separation theorem cannot simply be omitted.

High-dimensional sensitivity analysis and uncertainty quantification for multiscale gas dynamics, spanning the continuum to rarefied regimes, require computationally efficient and mathematically consistent gradient evaluation. This paper develops a discrete adjoint method for the unified gas-kinetic scheme (UGKS) based on a dual-consistent formulation. The adjoint system is derived directly from the discrete microscopic velocity-distribution equation coupled with the macroscopic-moment compatibility conditions. To resolve the stiff cross-scale coupling, we propose an asymptotic-preserving (AP) adjoint formulation constructed via macroscopic-moment projection and microscopic lifting. Under this framework, the AP adjoint formulation eliminates the stiff collision coupling and removes the collision-time step restriction in the continuum regime. Numerically, a memory-efficient residual-evaluation algorithm that mirrors the forward UGKS cell-vertex data structure is implemented to bypass the memory bottleneck in velocity space. Furthermore, a macroscopic–microscopic predictor–corrector implicit marching scheme is designed to accelerate convergence without solving a globally coupled system. The accuracy, consistency, and robustness of the proposed AP-adjoint scheme are rigorously verified against an independent linearized UGKS solver across a wide range of Knudsen numbers, including lid-driven cavity heat conduction, microchannel thermal creep flow, and hypersonic flow past a circular cylinder.

The heart of a $t$-structure and the ideal quotient by a cluster tilting subcategory are classical constructions that produce abelian categories from triangulated categories. Their higher analogues, namely $n$-extended hearts and ideal quotient categories by $(n+1)$-cluster tilting subcategories, are generally no longer abelian, but are known to carry both pretriangulated and extriangulated structures when the underlying triangulated category is algebraic. In this article, we introduce the notion of an abelian $n$-truncated category as a common framework for such higher constructions. We extend the heart construction for cotorsion pairs to $n$-cotorsion pairs on arbitrary triangulated categories, and prove that the resulting extended heart naturally carries compatible pretriangulated and extriangulated structures forming an abelian $n$-truncated category. This construction simultaneously generalizes the $n$-extended heart of a $t$-structure and the ideal quotient by an $(n+1)$-cluster tilting subcategory. It may also be regarded as a higher-dimensional generalization of the general heart construction for cotorsion pairs on triangulated categories. Finally, we show that the heart can be realized as an extriangulated localization of a suitable relative extriangulated structure on the ambient triangulated category.

This paper investigates the minimum field size required for network maximum distance separable (MDS) codes, a critical parameter affecting computational complexity at network nodes. Focusing on generalized combination networks and Zosin Khuller networks, we develop a systematic framework for both scalar and vector network MDS codes. For scalar codes on generalized combination networks, we establish an equivalence between the minimum distance of network codes and the minimum Hamming distance of classical linear codes, converting network-level MDS constraints into coding theory conditions. This yields necessary and sufficient existence conditions linked to classical MDS codes and covering Grassmannian codes. Using refined greedy constructions and MRD code designs, we obtain improved bounds on the minimum field size, outperforming the prior universal bound. For vector network codes, we develop an analogous distance equivalence and characterize existence via covering Grassmannian codes, yielding bounds on the minimum effective field size. Notably, the gap between optimal scalar and vector MDS codes vanishes for several parameter regimes; we explicitly identify a family of such networks where vector coding offers no field size advantage over scalar coding. For Zosin Khuller networks, we derive lower bounds on the minimal effective field size of vector MDS codes using hypergraph homomorphisms and subset intersection arguments, strictly improving prior scalar bounds. We further provide a necessary and sufficient condition built upon hypergraph homomorphisms for vector MDS construction, yielding an upper bound on the minimum field size. Finally, we bound the MDS gap between optimal scalar and vector solutions.

We study basis-aligned two-orthogonal (bato) tensors, which can be written as a sum of critical rank-one approximations whose factors are singular vectors of their flattenings. As such, bato tensors admit a Tucker decomposition and a canonical polyadic decomposition that are related to each other. We prove that generic bato decompositions are identifiable and that their truncations are critical low-bato-rank approximations. We also compute the dimension of the set of bato tensors, and identify the irreducible components of its Zariski closure with isotopy classes of maximal partial Latin hyperrectangles. Many important tensors are bato, such as determinants, matrix multiplication tensors, and other structure tensors of algebras.

For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $\lambda_3$ and $\lambda_4$.

We study long-time Strichartz estimates for the one-dimensional Schrödinger equation with quasi-periodic initial data. For two-frequency data with an algebraic frequency ratio, we observe that the behavior of the linear Schrödinger evolution changes depending on the algebraic degree of the ratio. Making use of this observation, we improve the Strichartz estimates on long time intervals. We also prove an endpoint $L^4$ Strichartz estimate. Our proofs use Roth-type Diophantine inequalities and Vinogradov-type mean value estimates for the Parsell–Vinogradov systems.

We study the numerical evaluation of an energy-based vector hysteresis model and its incorporation into finite element simulations based on a vector potential formulation. The inherent non-smoothness of the hysteresis model poses challenges for both numerical analysis and implementation. Using tools from convex analysis, we characterize the forward and inverse hysteresis operators in terms of energy densities. This characterization yields well-posedness of the resulting models and leads to robust algorithms for their evaluation based on generalized semi-smooth Newton methods. It furthermore enables a seamless integration into magnetic field simulations, leading to nonlinear and non-smooth optimization problems at every load step. We discuss the finite element discretization, present a semi-smooth Newton method for the iterative solution, and establish global linear convergence with mesh-independent convergence rates. The theoretical results are illustrated by numerical experiments.

We construct a compatible sequence of resistance forms on the graphs approximating the infinite Sierpinski gasket and show that the resistance form given by the limit of the compatible sequence induces the Dirichlet form associated with the Brownian motion on the infinite Sierpinski gasket. Moreover, using this characterization of the Dirichlet form, we obtain an explicit expression of the jump kernel associated with the trace process of the Brownian motion onto the half-line $[0, \infty)$.

Multi-phonon resonance conditions underpin kinetic theories of phonon transport and lattice thermalization. We show that exact resonance matching, nonzero interaction coefficients, and network connectivity do not guarantee persistent energy diffusion. Symmetry-enforced balance relations drive exact-resonant collision currents to nonthermal zero-flux states, producing kinetic arrest from individual resonant sets to connected networks. Complete energy spreading is sustained by quasi-resonances. The thermodynamic and weak-nonlinearity limits do not commute: the leading kinetic behavior is recovered in the former, whereas at fixed finite size the thermalization time diverges through higher-order crossovers as the nonlinearity vanishes. Exact-resonance existence and connectivity are therefore kinematic, not sufficient dynamical, criteria for phonon energy diffusion.

Augmented Lagrangian (AL) methods are a classical framework for constrained optimization, but for directly verifiable approximate KKT points, known first-order complexity bounds for standard inexact AL methods are suboptimal, while the best known proximal augmented Lagrangian (PAL) bounds retain an additional logarithmic factor. We consider linearly constrained convex composite problems with a smooth convex term and a possibly nonsmooth closed proper convex term with compact domain. We develop three inexact AL schemes that preserve the standard AL subproblem structure and attain the optimal primal-dual complexity $\mathcal O(\epsilon^{-1})$ in the convex setting, improving prior AL bounds of $\mathcal O(\epsilon^{-4/3})$, $\mathcal O(\epsilon^{-7/4})$, and $\mathcal O(\epsilon^{-2})$, and removing the logarithmic factor from PAL guarantees. Two variants are parameter-free, and all three admit nonergodic guarantees, including a stronger last-iterate guarantee for one variant. These results show that proximal regularization, ergodic averaging, and prior knowledge of problem-dependent constants are not intrinsic requirements for attaining optimal verifiable primal-dual complexity within the standard AL framework. A key ingredient is a parameter-free accelerated method that computes verifiable stationarity certificates for the standard, unregularized AL subproblems with optimal complexity. In the strongly convex setting, our methods attain near-optimal complexity $\mathcal O(\epsilon^{-1/2}\log(\epsilon^{-1}))$, with two parameter-free variants. Numerical experiments on six problem classes, including elastic-net least-squares regression, group-sparse Huberized support vector machines, and a quantum semidefinite program (SDP), demonstrate substantial computational advantages over a representative PAL method, with speedups frequently ranging from $5$ to $50$ times.

Aleman, Olsen, and Saksman asked whether radial convergence may fail for bounded analytic functions on the infinite-dimensional polydisc when every radial point has non-increasing coordinates, and whether the approach may be chosen independently of the boundary point. We construct a point-dependent counterexample in which every fixed coordinate nevertheless increases to $1$. A finite discretization yields a boundary-point-independent coordinatewise decreasing approach for which convergence to the boundary function fails almost everywhere. In contrast, every boundary-point-independent approach whose fixed coordinates increase monotonically to $1$ satisfies an $L^p$ maximal inequality and the almost-everywhere Fatou theorem for $1<p<\infty$.

Lokshtanov, Ramanujan, Saurabh, and Zehavi [ICALP 2018] proved that for any CMSO formula $\phi$, testing $\phi$ on arbitrary graphs can be reduced to testing it on $(q,k)$-unbreakable graphs for appropriate parameters. Their proof is non-constructive, and they ask whether it can be made constructive. We prove that this is impossible: specifically, the parameter $q$ cannot be a computable function of $\phi$.

In this paper, we mainly investigate the overtaking collision problem between a dispersive shock wave (DSW) and a rarefaction wave (RW) propagating in the same direction in the defocusing Hirota equation framework. Based on the Whitham theory, the zero-phase and single-phase modulation systems corresponding to the defocusing Hirota equation are given, along with explicit expressions of Riemann invariants. For step-type initial conditions with left-side DSW and right-side RW propagating identically, the global initial configuration is constructed.~The modulation solution within the nonlinear collision domain is obtained via the generalized hodograph transformation and solutions of the Euler-Poisson-Darboux equation. Thus, by matching the Riemann invariants at the boundaries of DSW and RW, we can provide a detailed analytical description for the dynamical behaviors of the collision across all evolutionary stages. The accuracy of theoretical predictions is verified by comparison with direct numerical simulations.

Much causal inference research is focused on methods for optimizing dynamic treatment regimes (Murphy, 2003; Robins, 2004; Schulte et al., 2015), which are rules for deciding which treatments should be assigned and when based on evolving history. There is a certain optimism underlying this endeavor that with enough tinkering we might realize consequential improvements. Another strand of research, previously confined to the point exposure setting, considers bounds on how well any individualized treatment rule could possibly do. Here, we extend to the time-varying setting sharp bounds on the performance of an oracle strategy that selects the best treatment regime for each subject based on their unobserved potential outcomes or `response type'. For binary outcomes, the lower bound (assuming higher is better) is simply the expected outcome attained by the optimal treatment regime based on observed history. For continuous outcomes, the lower bound may strictly exceed the maximal observed covariate based value. In the continuous setting, we also consider bounds on the CDF of oracle continuous potential outcomes.

In this note, motivated by \cite{DW26}, we construct counterexamples to momentum-only Schauder estimates for linear kinetic equations with relativistic transport. The construction exploits a structural compatibility between the nonlinear momentum-to-velocity map and a single fixed pair of smooth diffusion and drift coefficients: after conjugation, the diffusion becomes the Laplacian and all induced first-order terms cancel. Thus the obstruction is realized for one fixed operator, uniformly elliptic on every bounded momentum set, rather than through a frequency-dependent family of coefficients. We further show that the failure persists with zero external forcing by constructing uniformly positive stationary solutions for which the hidden spatial oscillation is encoded in a bounded zeroth-order coefficient whose momentum Hölder seminorm tends to zero. Consequently, the full Lorentzian Hölder control used in \cite{HSTT} cannot in general be replaced by momentum-only Hölder control.