Papers

In this paper, we consider connected signed graphs with smallest eigenvalue at least $-3-\varepsilon$ for a small positive constant $\varepsilon$. We prove that if such a signed graph has sufficiently large minimum valency, then its smallest eigenvalue is at least $-3$, and the lattice associated with it, which is generated by squared norm $3$ vectors, is a sublattice of a direct sum of the standard lattice $\mathbb{Z}^n$ and copies of the root lattice $E_8$. Moreover, there exist infinitely many connected signed graphs with smallest eigenvalue at least $-3$ containing it as a proper induced subgraph. Furthermore, we discuss signed graphs with smallest eigenvalue $-3$ arising from rootless irreducible unimodular lattices.

We produce the first known examples of non-trivial terminal local orders in the 3-dimensional case. We exhibit two different constructions. The first produces maximal orders with given toric ramification data. Since toric terminal ramification data in dimension 3 has been classified this produces toric examples. The second construction yields maximal orders ramified on surfaces with Kleinian singularities. These are automatically terminal.

High-fidelity simulations of complex physical systems are often too expensive for repeated prediction, design, and control. Reduced-order models address this computational cost by compressing high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they often lose accuracy when online dynamics drift beyond the offline training data. Adaptive methods address this limitation by updating the reduced subspace online using external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace. To exploit this channel, we continually update the reduced representation using the model's own predictions through an incremental singular-value decomposition with a forgetting factor. This produces a trajectory-informed spectral preconditioner in which the reduced subspace remains unchanged, while the basis is reweighted and realigned according to the directions visited by the evolving dynamics. This internal reorganization prepares the basis to absorb future out-of-span corrections more effectively, improving adaptation without requiring additional external information. We expose the mechanism through a three-dimensional spiral example and demonstrate its benefits on nonlinear partial differential equations, including viscous Burgers and Fisher–KPP dynamics. We also discuss how in-span learning can be interpreted as a dynamical-systems analogue of in-context learning. More broadly, in-span learning suggests a new principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.

Many image-processing problems can be formulated as composite cardinality optimization (CCOP), whose objective is the sum of two convex terms and a cardinality function composed with a linear operator. The composite cardinality term creates major algorithmic challenges: the separability of the cardinality function is lost and convergence analysis often requires surjectivity-type assumptions on the linear operators. To overcome these challenges, we consider the stationary dual formulation of CCOP, which has more favorable structure consisting of two convex terms and a separable cardinality function. In this paper, we develop an efficient proximal point algorithm (PPA) to solve the stationary dual problem. The efficiency of our PPA stems from two aspects. Firstly, the key step of its subproblem solver minimizes a smooth convex function over a low-dimensional subspace by the classic semismooth Newton algorithm (SNA), which has global convergence and local superlinear rate under suitable conditions. Secondly, implementable inexact criteria are proposed for early termination of the SNA subroutine. These components form the basic framework of our inexact PPA. Under suitable conditions, it enjoys global convergence and local linear convergence rate. In particular, we provide examples in which the convergence assumptions are automatically satisfied. Finally, the SNA subroutine is incorporated into our inexact PPA to solve jump-sparse signal recovery and computed tomography (CT) image restoration. Numerical results demonstrate the time efficiency and solution accuracy of our proposed method.

In this paper, we study random recursive trees generated by Bernoulli sequences. Starting from a graph with two vertices and one edge, each new vertex is connected to the last vertex with probability $ p $, or to the second-last vertex with probability $ q = 1-p $, this recursive construction yields a random infinite recursive tree $T$. We prove that $T$ almost surely has exactly one topological end. Furthermore, we establish that $T$ has the infinite collision property: two independent simple random walks on $T$ collide infinitely often almost surely.

This work investigates two types of inverse interface problems in scenarios where only very limited Cauchy data is available. These problems are associated with the Laplace equation featuring a Robin-type flux jump across an internal interface. The first problem focuses on reconstructing the location of cracks along a known interface using Cauchy data measured on the outer boundary. The second problem involves determining the location of an unknown interface based on Cauchy data from the outer boundary. To address these challenges, we adopt an efficient Direct Sampling Method (DSM) and introduce innovative enhancements to the boundary conditions in the reference system, thereby maximizing the utility of the available Cauchy data. Additionally, we propose a novel refinement to further improve the robustness of the DSM against noise. We provide a detailed exposition of the general principles underlying the DSM and systematically present its computational implementation steps. Through detailed Fourier analysis and computations, we illustrate the theoretical background of the DSM as well as the effectiveness of our refinement approach. A series of numerical experiments demonstrates that our method yields highly satisfactory results, even when processing incomplete and noisy Cauchy data on the outer boundary. We introduce quantitative metrics, such as Mean Localization Error (MLE) and Contrast-to-Noise Ratio (CNR), to rigorously evaluate the performance of our method. These findings underscore the exceptional effectiveness and broad applicability of the proposed approach.

We characterize solutions of the regularized $p$-Laplace equation \[ \operatorname{div}\!\left((1+|Dv|^2)^{p/2-1}Dv\right)=0, \qquad 1<p<\infty, \] in a bounded domain $\Omega\subset\mathbb{R}^n$ by a pointwise asymptotic mean value identity. For $v\in C^2(\Omega)$, solving the equation is equivalent to \[ v(x) = \frac{\widetilde{\alpha}}{2} \left( \mathcal{S}_{\varepsilon}^{+}[v](x) + \mathcal{S}_{\varepsilon}^{-}[v](x) \right) + \widetilde{\beta} \int_{B_\varepsilon(0)} v(x+h)\rho_\varepsilon(h)\,dh + o(\varepsilon^2), \] where \[ \widetilde{\alpha} = \frac{p-2}{p+n+1}, \qquad \widetilde{\beta} = \frac{n+3}{p+n+1}. \] The kernel $\rho_\varepsilon$ is the semicircular marginal of normalized Lebesgue measure on the $(n+1)$-dimensional ball, and $\mathcal{S}_{\varepsilon}^{+}$ and $\mathcal{S}_{\varepsilon}^{-}$ are the tilted strategic functionals arising from the affine lift \[ w(x,s)=v(x)+s. \] The lifted gradient $(Dv,1)$ never vanishes, so the extremal second-order expansion is valid at every gradient regime. The characterization holds for the full range $1<p<\infty$. By standard interior regularity for nondegenerate regularized $p$-growth equations, weak solutions are smooth in the interior; the weak and viscosity viewpoints for related quasilinear $p$-Laplace equations are connected in \cite{JLM01}. The convergence of the associated projected dynamic programming scheme is established in the companion paper \cite{Moosavi26}.

We ask whether interaction is necessary for order-optimal 1-bit mean estimation over nonparametric finite-moment classes. Adaptive threshold-query protocols achieve the order-optimal 1-bit minimax rate, and the same rate is attainable with general 1-bit queries using only one adaptive transition (i.e., two stages of querying). In the non-adaptive setting, threshold and interval queries are known to be highly suboptimal, but the case of arbitrary non-adaptive quantizers remains unresolved. Can such quantizers match the adaptive rate, yielding an optimal one-shot protocol? Or is the known two-stage estimator stage-optimal, with a single adaptive transition being necessary and sufficient?

We study two inverse problems for shortest-path metrics of Okamura-Seymour instances: recognizing metrics realizable by outerplanar graphs, and computing minimum-edge Okamura-Seymour realizations. We introduce the notion of repelling pairs, a metric certificate that the shortest paths corresponding to two terminal pairs must be vertex-disjoint in every realization. Our central structural result is that, for an Okamura-Seymour metric with a prescribed cyclic order, a terminal path structure in an Okamura-Seymour instance can be realized by nonnegative edge lengths if and only if the paths assigned to every repelling pair are vertex-disjoint. Building on the notion of repelling pairs, we give algorithmic answers to the inverse problems. First, we design an algorithm that, given a metric, decides in polynomial time whether or it admits an outerplanar realization and constructs one when one exists. Second, given an Okamura-Seymour metric, we efficiently compute a canonical medial template whose crossing number equals the minimum number of edges in any Okamura-Seymour realization. The minimum-edge graph structures are exactly the primal graphs of arrangements of this template, and each can be assigned realizing edge lengths in polynomial time.

We develop a meshfree numerical framework for Hele–Shaw free boundary problems with surface tension and source terms based on geometric local parameterization and boundary integral methods. By decomposing the pressure into a particular solution and a harmonic component, the problem is reformulated into a boundary-only system, avoiding volumetric meshing of the evolving domain. For general source terms, we propose an eigenfunction-based approximation on a fixed domain and establish error estimates for both the truncation and coefficient approximation. Numerical experiments verify the accuracy and convergence of the proposed method, and an application to a tumor growth model demonstrates its effectiveness for coupled moving-boundary problems.

In this paper, we investigate the dual-function radar-communication (DFRC) design for massive multiple-input multiple-output (MIMO) systems equipped with 1-bit digital-to-analog converters (DACs) at the transmitter and 1-bit analog-to-digital converters (ADCs) at the receiver, motivated by the need for low-cost and power-efficient implementations of massive MIMO systems. We consider a downlink scenario where the transmit signal matrix is optimized to enhance sensing performance while satisfying communication quality of service (QoS) requirements. Specifically, the objective is to minimize the 1-bit Cramér-Rao bound (CRB) for estimating the azimuth angle of a point-like target under symbol-level constructive interference (CI) constraints. We conduct an asymptotic analysis of the 1-bit Fisher information, revealing its nonmonotonicity with the signal-to-noise ratio (SNR), and introduce amplitude constraints to exclude regions where the objective function value is clearly suboptimal and facilitate convergence to high-quality solutions. The resulting problem is a nonconvex optimization challenge with coupled binary and linear constraints. We transform the discrete problem into a continuous constrained one, characterize its global and local minima, and tackle it via the augmented Lagrangian method (ALM) and a spectral projected gradient (SPG) method combined with nonmonotone line search. The solution is further refined via local search and cutting-plane techniques. Extensive numerical experiments verify our analysis, showing that the proposed approach exhibits promising DFRC performance compared to benchmark schemes.

We study a nonlinear size-structured transport equation where the endogenous scalar output $E(t)=\int_{l_0}^{l_m}\chi(l)x(t,l)\,dl$ feeds back into velocity and mortality. This principal-coefficient feedback precludes a semilinear perturbation framework. Freezing the feedback path yields a non-autonomous linear evolution, reducing the closed-loop problem to a scalar Volterra fixed point $E=\mathcal K(E)$. Mass balance provides an intrinsic feedback interval, while a Bielecki-norm contraction ensures unique nonnegative weak solutions. Stationary equilibria satisfy a scalar closure equation $E=\Phi(E)$. We prove uniqueness below the sharp margin $1-\Phi'(E)>0$ and identify $\Phi'(E)=1$ as a nondegenerate fold threshold. Linearization yields a finite-memory renewal equation with characteristic equation $\mathcal E(\lambda)=1$, whose root set determines the feedback spectrum and stability. Finally, the stationary harvesting adjoint reduces to a rank-one perturbation formula. At zero discount, we establish the identity $\mathcal E(0)=\Phi'(E^*)=B(0)$, linking closure resonance, spectral crossing, and adjoint loop gain.

For $\alpha \in \mathbb{C}$, let $\mathbb{N}_0[\alpha]$ be the subsemiring of~$\mathbb{C}$ obtained as a homomorphic image of the $\alpha$-evaluation map $\mathbb{N}_0[x] \to \mathbb{C}$ defined as $p(x) \mapsto p(\alpha)$ for each polynomial $p(x) \in \mathbb{N}_0[x]$. Fundamental arithmetic and atomic aspects of the additive structure of $\mathbb{N}_0[\alpha]$ were first studied by the second author and Correa-Morris (2022). In this paper, we continue the investigation, now from the valuation-theoretic perspective. We show that for any algebraic number $\alpha$, the additive monoid of $\mathbb{N}_0[\alpha]$ contains no additive irreducibles if and only if it is isomorphic to the direct product of finitely many isomorphic valuation monoids (monoids whose principal ideals form a chain under inclusion). For any algebraic number $\alpha \in (0,1)$, these valuation monoids are precisely those where $\alpha^{-1}$ is a Perron number having no positive conjugates other than itself. In addition, we offer a description of the algebraic parameters $\alpha$ for which the additive structure of $\mathbb{N}_0[\alpha]$ is a valuation monoid. Finally, we argue that the subset of $(0,1)$ consisting of all algebraic parameters $\alpha$ such that the additive structure of $\mathbb{N}_0[\alpha]$ is a valuation monoid is dense in $(0,1)$.

The Ziegler degrees were introduced to characterize definability in group theory. In [JLSta], the authors show that the first order-theory of the Ziegler degrees (as a partial order) is undecidable. We improve this result, showing that the theory of the Ziegler degrees is bi-interpretable with true second-order arithmetic.

We establish a new PBW basis for $\mathcal A_q$, the alternating central extension of the $q$-Onsager algebra. Terwilliger showed that the alternating generators form a PBW basis in the block order $\mathcal G<\mathcal W^-<\mathcal W^+<\widetilde{\mathcal G}$. We prove that they also form a PBW basis in the different block order $\mathcal W^-<\mathcal G<\widetilde{\mathcal G}<\mathcal W^+$. Consequently, multiplication induces a vector-space isomorphism \[ \mathcal W^-\otimes\mathcal G\otimes\widetilde{\mathcal G}\otimes\mathcal W^+ \longrightarrow \mathcal A_q, \] thereby confirming a conjecture of Terwilliger.

This paper introduces the voltage stability kernel (VSK), a cofactor-based bus-wise representation of voltage stability in lossy power systems. The VSK is defined as the vector of principal cofactors of the voltage stability Laplacian (VSL), a reduced Jacobian that retains voltage source internal angles while eliminating the other variables. We show that the VSK constitutes the left kernel of the VSL, which is typically nonsymmetric in lossy power systems. We also define the voltage stability margin (VSM) as the sum of all VSK components and show that it is equal to the product of all eigenvalues of the VSL except the trivial zero eigenvalue due to phase-shift symmetry. Thus, the VSK provides a bus-wise decomposition of the VSM. Furthermore, the VSK offers an algebraic interpretation of CPF calculations with a fixed slack bus. The singularity of the Jacobian in CPF calculations obtained by deleting the slack-bus row and column is characterized by the vanishing of the VSK component selected by the slack bus. In contrast, the static bifurcation is characterized by the vanishing of the VSM. Since these two conditions are generally different, our theory explains why a CPF nose point does not necessarily correspond to a static bifurcation in lossy cases.

This paper studies the Capacitated $p$-Location Problem (C$p$LP) and its extensions incorporating equity considerations. In location science, the $p$-Median problem ($p$MP) is a classical model that selects $p$ facilities from a finite set of candidates to serve a set of customers while minimizing total allocation costs. The C$p$LP, which includes the Capacitated $p$-Median Problem (C$p$MP) and its relaxed variant (C$p$MP$^r$), extends the $p$MP by incorporating capacity constraints on facilities. We formalize the C$p$LP with Territorial Coverage Constraints (C$p$LP-TC), an extension that enforces equity across spatial units, and generalize it to a multi-scale variant (C$p$LP-MTC) that enforces equity simultaneously across nested spatial scales. We present a strengthened Integer Linear Programming (ILP) formulation with valid inequalities that enables the exact solution of medium-sized instances. To tackle larger problems, we adapt the Random Sampling Spatial Voting (RSSV) heuristic, originally proposed for the $p$MP, into a competitive open-source matheuristic that combines a heuristic reduction phase with the strengthened ILP formulation. The resulting method remains flexible, requires minimal parameter tuning, and is accessible to non-specialist users.Computational experiments on a new open-source benchmark instance set, built from a French regional case study, demonstrate the effectiveness of both the exact and heuristic approaches. Beyond their computational performance, the results quantify the trade-offs between efficiency and territorial equity under different equity constraints, providing practical insights for equitable location planning.

Singular interactions can reshape quantum spectra by changing admissible wavefunction domains. In this study, we investigate this mechanism in the Klein–Gordon equation for a non-central Schiöberg-type hyperbolic potential with double ring-shaped angular barriers. Equal scalar and vector couplings separate the radial and angular dynamics. We solve the angular equation exactly on each hemisphere with principal boundary conditions and treat the radial equation with the Greene–Aldrich approximation, obtaining Jacobi-polynomial wavefunctions and an implicit relativistic quantization condition. With this boundary realization, the equatorial inverse-square singularity splits configuration space into reflection-related sectors with identical angular spectra. Reducing the barrier to zero while retaining the equatorial wall selects the odd-reflection spectral branch of the regular full-domain problem; a single hemisphere has no intrinsic parity. The nonrelativistic limit recovers the corresponding Schrödinger formulation. State-resolved nonrelativistic and relativistic benchmarks quantify the centrifugal error. In a dimensionless relativistic example, the relative error in the remaining binding exceeds unity near threshold although the absolute energy error decreases. For NaH and Na$_2$, low-lying central-radial vibrational energies agree reasonably with spectroscopic reference values, whereas the global radial shape limits near-dissociation accuracy and gives an incorrect NaH bound-state count.

We give explicit counterexamples to two rearrangeability conjectures for shuffle-type networks. First, for every $N\ge2$ we construct a simple $N$-regular ordered two-stage graph $L_N$ with $F(L_N)=2$ and $R(L_N)\ge N$, refuting the graph-theoretic Beneš inequality $R(L)\le2F(L)$ and its partition-stabilizer form as stated on Open Problem Garden. We retain the sharp cut obstruction, exact mask-composition identity, exact middle criterion, first nontrivial-level result, and balanced-middle sufficient condition that explain which extra hypotheses can replace mere external connectivity. Second, for the standard directed shuffle-exchange network, we prove $d(k,3)=6$ for every $k\ge3$, while the known binary value is $d(2,3)=5$. Hence the shuffle-exchange conjecture $d(k,n)=2n-1$ fails already at $(k,n)=(3,3)$, and the remaining upper bound $d(k,n)\le3n-3$ for $k\ge3$ suggests $d(k,n)=3n-3$ as a natural replacement problem.

The Hatano Nelson paradigm establishes that extensive bulk nonreciprocity can destabilize Anderson localization via an imaginary gauge flux. Here, we demonstrate that extensive nonreciprocity is not a necessary ingredient: a single non-Hermitian boundary bond in a disordered one-dimensional ring suffices to drive the localization-delocalization transition. More generally, we construct an exactly isospectral family of non-Hermitian Hamiltonians that continuously interpolates between the uniform Hatano Nelson model and the single-bond limit. We show that the universal critical behavior encompassing spectral, eigenstate, and topological diagnostics is gauge invariant and governed solely by the total imaginary gauge flux, regardless of its spatial distribution. Remarkably, despite sharing identical spectra and critical exponents, different configurations within this isospectral family exhibit qualitatively distinct quantum dynamics, establishing a fundamental separation between static and dynamical universality in non-Hermitian systems. Specifically, the single boundary realization features rapid operator scrambling, oscillatory wavepacket acceleration, and a double re-entrant steady state entanglement transition. Finally, we propose an experimentally feasible realization based on multi-terminal topological transport, providing a realistic route toward observing boundary induced non Hermitian criticality and its unconventional dynamical signatures.

We reinterpret the classical Hermite generating function as a Gaussian density ratio: relative to the unit Gaussian reference, it is the density ratio of a Gaussian with shifted mean and unchanged covariance. Applying the heat semigroup in the mean-parameter variable to this generating function produces the corresponding temperature variation. Thus the heat-semigroup time variable is reinterpreted as the temperature variation of the Gaussian density ratio. This parameter-space formulation also gives a parabolic control principle for Hermite approximation errors. Since Hermite projections act in the velocity variable and the heat flow acts in the mean variable, Hermite block energies and truncation tails are subsolutions of the same parameter-space heat equation. This remains useful for heat-evolved non-Gaussian perturbations where no usable closed coefficient formula is available. For Gaussian density ratios with general covariance, the Hermite coefficients satisfy a weighted homogeneity in the mean and covariance-defect parameters. This yields Ornstein–Uhlenbeck covariance, an exact generating function for total-degree Hermite block energies, and the sharp geometric Hermite truncation rate, equal to the square root of the largest absolute covariance defect. We also derive precise isotropic block and tail asymptotics and interpret the estimates for near-Gaussian kinetic distributions.

A problem of D. H. Lehmer suggests to study the number of integers, each of which has different parity from its multiplicative inverse modulo $q$. We obtain an asymptotic formula for the number of such integers in very short intervals as long as $q$ is squarefree and has good factorizations, using arithmetic exponent pairs to estimate incomplete Kloosterman sums. This improves an early result of Z. Zheng in the above special cases. We also prove a prime-variable version with the aid of an estimate for incomplete Kloosterman sums in prime variables, due to Bourgain.

Given two sequences $\phi=(\phi_i)_{i\ge 1}$ and $\psi=(\psi_i)_{i\ge 1}$ and numbers $a,b,c$, we introduce the GKP sequence of polynomials $(p_n)_n$ using the following recurrence formula: $p_0 = 1$ and for $n\ge 1$ \[ p_{n}(x) = (ax^2+bx+c) p_{n-1}'(x) + (\phi_{n} + \psi_{n} x)p_{n-1}(x), \] where we assume that $ax^2+bx+c$ has two different real zeros. Tangent, Secant, Eulerian or Jacobi polynomials are examples of GKP sequences of polynomials. In this paper, under mild assumptions we prove that the zeros of the polynomials $p_n$ are real, simple and live between the zeros of $ax^2+bx+c$. Moreover, the zeros of $p_{n+1}$ interlace the zeros of $p_n$. We study in detail the cases when $\psi$ is constant, and $\phi=(\phi_i)_{i\ge 1}$ is constant for $i$ big enough, proving, among other results, asymptotics for the leftmost and rightmost zeros of $p_n$.

We give an exact, checkable rank-certificate method for realized planar unit-distance frameworks. The method is motivated by Vogel's computations for matchstick graphs and by the insertion-edge tests used in the Matchstick Graphs Calculator. Its algebraic core is independent of geometry. A singular square matrix $M$ is replaced by a sparse perturbation $B=M+UCV^T$. If $B$ is nonsingular and the inverse satisfies $V^TB^{-1}U=C^{-1}$, then the columns of $B^{-1}U$ and the rows of $V^TB^{-1}$ form bases of the right and left kernels of $M$, and the rank defect of $M$ is certified. Applied to the equilibrium matrix of a planar framework, this gives finite exact certificates for self-stresses, infinitesimal motions, redundant edges, and candidate edges whose constraints are already forced by the realized framework. The certificate data can be checked independently from the search that produced it, using exact matrix identities. We emphasize that matchstick frameworks are not generic: unit distances, triangles, rhombi, and symmetries can change the realized rank. The method therefore concerns the coordinate-dependent representation of a given drawing, not only the generic rigidity matroid of the abstract graph.

A problem of D. H. Lehmer suggests to study the number of integers, each of which has different parity from its multiplicative inverse modulo $q$. For large prime $q$, we obtain an asymptotic formula for the number of such integers up to $N$, where $N$ is a bit smaller than $q^{1/2}$. This beats the barrier $q^{1/2}$ in the prime modulus case. An estimate for the second moment of the error term on average over $q$ is also established. The main inputs are estimates for several bilinear forms with Kloosterman fractions.

This paper develops a graph fractional uncertainty principle in the graph fractional Fourier transform (GFRFT) domain. We introduce localization operators in the vertex domain and the graph fractional spectral domain, and build an operator framework to characterize the joint localization of graph signals. A sandwiched joint localization operator is first constructed, whose largest eigenvalue quantifies the attainable simultaneous concentration in the two domains. Then, rotated localization operators and the numerical range are used to provide a geometric description of the admissible uncertainty region, together with a polygonal approximation method for its computation. Numerical examples show that the fractional order reshapes the vertex-graph fractional spectral localization trade-off, and enlarge or shrink the uncertainty region relative to the graph Fourier transform based case. These results generalize classical graph uncertainty principles to the GFRFT domain and provide a flexible tool for graph fractional signal analysis and graph-adapted localized representations.

Sixth-Generation (6G) networks will require massive connectivity, ultra-low latency, and robust security, making reliable Active User Detection (AUD) essential for interference control and physical layer protection. This letter proposes a Grover-based physical layer security (PLS) framework for a code-domain non-orthogonal multiple access (CD-NOMA) network, where the base station employs artificial-noise (AN)-assisted beamforming and identifies the active set via Grover's quantum search algorithm. We consider two threat models: passive eavesdroppers formed by detected inactive users, and active eavesdroppers selected as the top f% most frequent transmitters among detected active users. By aligning beams and AN with the Grover-based AUD output, the proposed scheme enlarges the main-wiretap rate gap and significantly improves the average secrecy rate compared with compressive sensing and classical correlation receiver baselines, while approaching maximum-likelihood detection performance with a quadratic reduction in search complexity. The impact of the information/AN power split, the base station transmit power, and the fraction of highly active users treated as eavesdroppers on secrecy is characterized through numerical simulations, and design insights are extracted for 6G PLS under both passive and active eavesdropping.

The X-ADM mass is a generalisation of the ADM mass for asymptotically flat manifolds, recently introduced by Mantegazza and Oronzio [arXiv:2602.11372], who proved a positive mass theorem for this quantity in dimension $3$ under a certain topological condition. In this article, we prove the X-positive mass theorem with rigidity in all dimensions $n\geq3$ without any topological restrictions. To this end, we first prove that the scalar curvature condition appearing in the X-positive mass theorem implies the metric is of positive Yamabe type. We then show that the X-ADM mass of the given metric is equal to the ADM mass of a scalar-flat representative in its conformal class plus a term that is explicitly non-negative. From this, the X-positive mass theorem follows from the standard positive mass theorem. As a corollary, we obtain the Riemannian mass–charge inequality of general relativity in all dimensions, which was previously only known for spin manifolds. Finally, we give a version of the X-positive mass theorem for a manifold with boundary.

Let $A(k)$ be the largest possible number of moves in a north-east lattice path whose visited vertices contain no $k$ collinear points. Gerver (1979) and Gerver and Ramsey (1979) gave lower and upper bounds on $A(k)$ of the form \[ \exp\left(\Omega(\log(k)^2)\right)\le A(k)\le \exp\left(O(k^4)\right). \] Improving upon these results, we show that \[ \exp\left(\Omega(k^{1/3})\right)\le A(k)\le \exp\left(\left(\frac{2}{e}+o(1)\right)(k-1)^2\right). \]

We study the Bhatnagar-Gross-Krook (BGK) equation in a bounded domain with in-flow boundary condition. The BGK model is a simple relaxation of the Boltzmann equation through replacing the quadratic nonlinearity by the so-called local Maxwellian, which consists of the density, the velocity, and the temperature. The aim of this paper is twofold. First, we establish the Dirichlet boundary value problem for the BGK equation by showing a local existence result in the weighted $L^\infty$ norm if the initial data and boundary condition are close to the global Maxwellian. For the purpose of studying the inverse problem, we further derive an expansion of the BGK solution with respect to a small parameter. This allows us to decompose the nonlinear BGK into purely linear transport equations based on various order of the expansion. Second, we investigate an inverse problem of determining a general collision frequency, depending on both density and temperature, in the BGK equation from the boundary measurement operator, which maps from the in-flow boundary data to the out-flow data. We apply highly concentrated in-flow test data to the solution of the resulting linearized BGK equations to extract hidden information of the collision frequency.