Papers

We prove that there exists a sequence of two-torsion Hermitian line bundles on closed hyperbolic three-manifolds, with volume tending to infinity, and with asymptotically optimal smallest eigenvalue of the Laplacian. The proof stems from recent advances in the program of strongly convergent unitary representations of discrete groups.

Integrating non-terrestrial networks (NTN) with terrestrial infrastructure is a key enabler of next-generation wireless systems, providing ubiquitous connectivity while meeting stringent rate and latency requirements. In particular, high altitude platform stations (HAPS) can complement terrestrial networks and jointly form vertical heterogeneous networks (vHetNets), extending coverage while delivering high-capacity, reliable, and low-latency connectivity for user equipments (UEs) including ground users and uncrewed aerial vehicles (UAVs). However, the high altitude deployment of HAPS establishes strong line-of-sight (LoS) links to UEs, creating highly correlated channels among UEs. Moreover, the wide coverage footprint of HAPS enables it to serve a large number of UEs, forcing limited radio resources to be shared among many UEs and resulting in significant intra-resource block (RB) interference. To address this challenge, we propose an interference management scheme based on UE clustering and rate-splitting multiple access (RSMA). Specifically, the network is modeled as a heterogeneous graph, and a graph neural network (GNN) is developed to efficiently allocate the common and private RSMA powers, maximizing the minimum spectral efficiency (SE) in a fast and scalable manner. Simulation results demonstrate that the proposed GNN-RSMA interference management algorithm outperforms conventional multiple access schemes while achieving fairness and worst-user performance comparable to successive convex approximation (SCA)-based optimization at only a fraction of its computational cost.

Schubitopes are a family of generalized permutahedra arising as the Newton polytopes of both Schubert polynomials and Demazure characters of $\mathrm{GL}_n$. Monical, Tokcan, and Yong conjectured that Schubitopes are Ehrhart positive. We exhibit a counterexample to this conjecture, a Schubitope whose Ehrhart polynomial contains a negative coefficient in its monomial expansion. This example was found using GPT-5.6 Sol Pro in ChatGPT.

For stochastic differential equations, we establish a relationship between the measure-theoretic entropy of the stochastic flow and the rate of volume growth of stable submanifolds under iteration. By combining the result of Kifer and Yomdin (1988), we show that under a suitable integrability condition, for systems with $C^{\infty}$ coefficients, the measure-theoretic entropy is upper semicontinuous with respect to the coefficients of SDEs.

Large systems of linear equations often contain far fewer active degrees of freedom than their ambient dimension suggests. A particularly transparent instance occurs when a large matrix factors as A = TSW, where T is m x n, S is n x n, and W is n x m, and m >> n. The action of A then passes through an n-dimensional intermediate space. Sylvester's theorem relates the nonzero eigenvalues of a product XY to those of the reversed product YX. Applied cyclically, it shows that the nonzero spectrum of the large matrix A is completely determined by either of the small matrices B = (WT)S, C = S(WT). This article develops the theorem from first principles and explains the geometry behind the factorization. The result yields exact reduced systems, both for linear algebraic equations and for first-order linear ordinary differential equations. Worked examples carry this through in detail: spectral reduction, reconstruction of full-space eigenvectors, reduced solution of shifted systems, and exact integration of a three-dimensional ODE through a two-dimensional model. A separate section takes up singular values. Those of A, B, and C do not agree in general, though they do when the bases are orthonormal, a case the same theorem settles once it is applied to A*A. The closing sections separate exact factorization from projection-based approximation, and flag what the theorem leaves undetermined: zero-eigenvalue structure, conditioning, and transient behavior. The material is classical. What this article adds is one continuous development, from first principles through to the limits of the theorem, pitched for an upper-level undergraduate course.

We give an explicit description, in terms of the quantum Bruhat graph, of the (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants $\langle \mathcal{O}(- \lambda), \mathcal{O}^{w}, \mathcal{O}_{u} \rangle_{d}$ for the (full) flag manifold $X = G/B$, where $\mathcal{O}(- \lambda)$ denotes the class in the (torus-equivariant) $K$-theory ring $K_{T}(X)$ of $X$ of the line bundle $\mathcal{O}_{X}(- \lambda) = G \times_{B} \mathbb{C}_{\lambda}$ over $X = G/B$ associated to a weight $\lambda \in W \varpi_i$ lying in the Weyl group orbit of a minuscule fundamental weight $\varpi_i$, and $\mathcal{O}_{u}$, $\mathcal{O}^{w}$ are the Schubert and opposite Schubert classes in $K_{T}(X)$ for $u, w \in W$. This result can be thought of as a partial generalization of the quantum $K$-theoretic divisor axiom, which we obtained in our previous work; our proof utilizes a generalization of the Chevalley formula in the (torus-equivariant) quantum $K$-theory ring $QK_{T}(X)$ of $X$, which computes the quantum product with the line bundle class $\mathcal{O}(- \lambda)$ associated to the weight $\lambda$ above.

We prove that the projected stochastic subgradient method with the standard anytime stepsizes $\eta_t = \gamma/\sqrt t$ can suffer expected last-iterate optimization error of order $(\log n)/\sqrt n$ already on the real line. Combined with the classical in-expectation upper bound, this establishes the sharp worst-case order of the projected stochastic subgradient method.

Strongly regular graphs, hereafter SRGs, are highly structured combinatorial objects of central interest in algebraic graph theory and quantum information, whose explicit construction becomes rapidly intractable as the number of vertices grows; Conway's still unresolved $99$-graph problem is an emblematic example. The matrix equation defining SRGs is quadratic in the binary edge variables, so the natural penalty enforcing it, a sum of squared residuals, is quartic and lies outside quadratic unconstrained binary optimization, known as QUBO. We prove a considerably stronger obstruction, namely that no quadratic function of the edge variables has the labeled SRGs as its exact set of global minimizers, except for three degenerate families given by the perfect matchings, the cocktail-party graphs and the pentagon. In particular, we formally establish that the $99$-vertex graph of Conway's problem cannot be obtained from any QUBO over the edge variables without auxiliary variables. Guided by this theorem we develop two complementary algorithms. The first operates within pure QUBO, solves the degenerate families exactly, and beyond them works as a generate–prune–verify pipeline whose output is a reduced set of candidates certified by a polynomial-time classical test. The second solves the full problem by quadratizing this quartic penalty through Rosenberg auxiliary variables, whose number grows as $\Theta(N^3)$ with the number of vertices $N$. We test the pure-QUBO route under simulated annealing on all feasible families with at most $18$ vertices and the auxiliary-variable route on representative targets up to $16$ vertices, and tabulate the algebraically feasible parameter sets for graphs with at most $49$ vertices.

Autonomous multi-agent control systems (MACSs) enable distributed coordination in emerging applications such as smart power grids and autonomous vehicle or drone networks. In many of these applications, inter-agent communication relies on wireless networks. However, conventional MACS designs neglect the interaction between the communication and control subsystems. At the same time, conventional wireless designs reserve communication resources based on worst-case MACS requirements. Consequently, control performance is degraded and wireless resources are utilized inefficiently. To address these challenges, we propose a consensus-aware co-design approach for MACSs and wireless communication systems. We introduce a system model in which agents achieve consensus by exchanging information through device-to-device (D2D) links that share spectrum with cellular users under limited bandwidth resources. Based on this model, we formulate a joint optimization problem for consensus-topology selection and wireless resource allocation to minimize bandwidth consumption while guaranteeing consensus and satisfying the quality-of-service (QoS) requirements of both MACSs and cellular users. Although the resulting problem is a mixed-integer nonlinear program (NP-hard in general), a structural property of the optimal solution enables a sequential two-stage optimization algorithm with polynomial complexity. Simulations on a smart power grid case study demonstrate that the proposed approach achieves 24-27% bandwidth savings compared with the two baselines while maintaining consensus and satisfying all QoS requirements of the control agents and the communication users.

This paper is the follow up (Part II) of arXiv:2601.21428, which proposed and analyzed time discretization schemes for the Dynamical Low-Rank Approximation (DLRA) of high-dimensional systems of Stochastic Differential Equations (SDEs). In this work, we discuss and analyze the corresponding stochastic discretization. Specifically, we consider Monte Carlo type methods to approximate the stochastic space and all related quantities of interest, leading to produce noisy interacting particle systems whose error analysis is non standard. Assuming subgaussian tails of the initial condition, in the case of elliptic diffusion, we provide convergence results for the Monte Carlo approximation of the DLR-PS-SDE integrator presented in Part I. The latter requires inversion of the Gram matrix associated to the stochastic modes. When the corresponding empirical Gramian in the Monte Carlo approximation remains full rank over the evolution, we prove a convergence rate close to the usual Monte Carlo one, i.e. $O(\frac{1}{\sqrt{M}})$, where $M$ is the sample size. If a regularization of this matrix is employed at each time-step, the rate is close to $O(\frac{1}{\sqrt[3]{M}})$. We also investigate intermediate options where the regularization is activated only when the smallest eigenvalue of the empirical covariance falls below a prescribed tolerance, showing convergence rates in between the one mentioned above.

We study the fixed-reference entropy-regularized Wasserstein barycenter problem, in which a probability measure $\eta$ on the barycenter space is chosen before optimization and the $i$th transport plan is regularized relative to $\nu_i\otimes\eta$. Earlier work established a dual formulation for this model and equality between the primal and dual optimal values. However, equality of values does not by itself guarantee that the dual supremum is achieved. Our main result shows that, on compact metric spaces with continuous transport costs, the dual problem admits maximizing potentials in the original class of continuous functions. For completeness, we also establish existence and uniqueness of the optimal tuple of transport plans and of the barycenter. The maximizing potentials recover the optimal plans through exponential primal–dual relations and give an explicit density of the barycenter with respect to $\eta$. This density is continuous and strictly positive. We further show that the potentials inherit regularity from the costs and are unique up to the natural additive gauge transformations. These results strengthen the known value-duality theory and provide a complete continuous primal–dual description of the fixed-reference regularized barycenter problem.

We read the geometry of semantic neighbourhood graphs as modal logic and give that reading a statistical form, in order to make precise the structural difference between human and machine-generated text. Texts are the worlds of a finite frame whose accessibility is the $k$-nearest-neighbour relation of a transformer embedding, and the symmetry, transitivity, Euclideanity and seriality frequencies of the two subcorpora are shown to be degrees of validation of the modal axioms $\mathsf{B}$, $\mathsf{4}$, $\mathsf{5}$, $\mathsf{D}$. Each degree is at once the proportion of instances of a rule that the subframe licenses in Negri's labelled calculus $\mathsf{G3.K}$ and a plug-in estimate of a population probability. A prompt-balanced comparison finds consistently higher artificial degrees for $\mathsf{4}$ and $\mathsf{5}$. We add a degree of groundedness and of situatedness, and recast the licensing reading in Cuconato's one-sided sequent-style tableaux, where each degree becomes a rate of set membership.

In this work, we consider the Dynamical Low Rank Approximation (DLRA) of a high-dimensional system of Stochastic Differential Equations (SDEs); DLRA is an effective reduced order model that projects the equations on a time-varying low dimensional subspace following the evolution of the solution. Well posedness results for the resulting DLRA equations have been proved in arXiv:2308.11581 under the assumption of global Lipschitz continuity and linear growth of the drift and the diffusion coefficients. In this work, we extend the existence result to the case of locally Lipschitz coefficient and we prove uniqueness of solutions under the assumption of a mild growth of the Lipschitz constant to infinity.

The booksize $\mathrm{bk}(G)$ of a graph $G$ is the largest number of triangles sharing a common edge. A classical theorem of Edwards, conjectured by Bollobás and Erdős, states that every $n$-vertex graph $G$ with $e(G)>e(T_{n,2})$ has booksize greater than $n/6$. Zhai and Lin [J. Graph Theory 102 (2023) 502–520] asked whether the same conclusion holds under the spectral condition $\lambda(G)>\lambda(T_{n,2})$, where $\lambda(G)$ is the spectral radius of the adjacency matrix. We answer this question in a strong form: every $n$-vertex graph $G\neq T_{n,2}$ with $\lambda(G)\ge\lambda(T_{n,2})$ satisfies \[ \mathrm{bk}(G)\ge\max\Big\{\frac13\lambda(G),\,\lambda(G)-\frac n3,\,2\lambda(G)-n\Big\}. \] Consequently, the condition $\lambda(G)>\lambda(T_{n,2})$ forces $\mathrm{bk}(G)\ge\lfloor n/6\rfloor+1$. The middle term is a spectral improvement of Edwards' bound $\mathrm{bk}(G)\ge\frac{2m}{n}-\frac n3$, and all three bounds are best possible. These results come from the edge-spectral setting: every graph $G$ with $m$ edges and $\lambda(G)\ge\sqrt m$ that is not a complete bipartite graph satisfies \[ \mathrm{bk}(G)\ge\max\Big\{\lambda(G)-\frac{2m}{3\lambda(G)},\,2\lambda(G)-\frac{2m}{\lambda(G)}\Big\}, \] which strengthens the bound $\mathrm{bk}(G)\ge\frac13\lambda(G)$ of Zhao, You, Zeng and Zhang. As an application of our method, we prove a triangle counting bound $t(G)\ge\frac13(\lambda(G)+1)(\lambda(G)^2-m)$, which improves the result of Bollobás and Nikiforov [J. Combin. Theory Ser B. (2007)]. Finally, we prove stability results at both thresholds: if $\lambda(G)\ge(\frac12-o(1))n$, then either $\mathrm{bk}(G)\ge(\frac16-o(1))n$ or $G$ can be made into $T_{n,2}$ by adding and deleting $o(n^2)$ edges; if $\lambda(G)\ge(1-o(1))\sqrt m$, then either $\mathrm{bk}(G)\ge(\frac13-o(1))\sqrt m$ or $G$ differs from a complete bipartite graph in $o(m)$ edges.

Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra \(\mathfrak{sp}(4,\mathbb R)\). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are $G_0=[\hbar^2(u-1)]^{-1}$ and $G_2=[\hbar^2(3u-1)]^{-1}$, where $u=(2\nu/\hbar)^2$. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every $u$; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives $G_0/G_2=1+2(\beta/\alpha)u/(u-1)$; matching Bures throughout the isotropic family would require the state-dependent relation $\beta/\alpha=1/u$. At the Bures-Fisher determinant crossing \(u=\varphi\), pointwise matching is possible only by inserting $\beta/\alpha=\varphi^{-1}$. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.

We give an elementary, coordinate-free proof that uniformly inf-sup stable nested Petrov-Galerkin methods on Hilbert spaces satisfy relaxed general quasi-orthogonality. More precisely, for any window of $N$ consecutive levels, the sum of squared Galerkin increments is bounded by the squared error at the beginning of the window times $N^\sigma$, where $\sigma<1$. The exponent $\sigma$ and the prefactor can be chosen explicitly in terms of the uniform projection bound. The proof uses Hilbert-space projection estimates, dyadic iteration, and duality, without relying on matrix factorizations. As a corollary, we obtain explicit growth bounds for finite block-LU factors. In the symmetric indefinite case, the positive and negative spectral splittings of the Galerkin detail spaces yield a characterization of full quasi-orthogonality in terms of uniform boundedness of the associated finite-window sign operators. We also construct a self-adjoint involution and a nested, uniformly inf-sup stable Galerkin sequence for which full general quasi-orthogonality fails. For every $0<\alpha<1$, the construction yields a hierarchy whose finite-window constants have growth of exact order $N^\alpha$, along with an explicit target whose Galerkin increments are not square summable.

Early-warning signals are usually computed from one or a few observables, although the underlying dynamics are high dimensional. Projection creates a structural ambiguity: the same scalar record may reflect spectral loss of stability, hidden-state memory, nonnormal amplification, a turning point of a physical balance, or a combination of these. We develop a unified operator theory that makes these alternatives explicit. For a resolved-hidden block linearization we derive exact Volterra reductions, finite and diffusive Markovian liftings, and a Schur-resolvent factorization of the stability problem, distinguishing three notions of memory: realized pathwise, structural Schur return, and finite-horizon projected. A covariance theorem gives sufficient conditions for a scalar to inherit critical slowing down from a real multiplier approaching the unit circle: spectral separation, bounded eigenvector conditioning, observability of the critical mode, stochastic excitation of that mode, and quasi-stationary forcing. We then prove failure mechanisms: a critical mode can be invisible or unexcited; an oscillatory crossing need not raise lag-one correlation; a nonnormal family can produce diverging scalar variance with a fixed stable spectrum; a physics-based turning point need not coincide with a spectral boundary; and finite-horizon data cannot identify long memory uniquely. The theory is illustrated with a 4400-year CESM AMOC hysteresis experiment. EOF1 alone detects altered susceptibility, but EOF1-EOF2 is the smallest resolved space revealing changed Schur return and projected memory after recovery; F_ovS reaches its minimum 26 years before collapse while the modal margin does not decrease toward zero: the two diagnostics probe complementary mechanisms. The application is mechanistic and single-trajectory; no validated early-warning signal is claimed.

A diagram of spaces describes a system of interacting state spaces. Motion planning in such a system is not merely an objectwise problem since the selected paths must form a natural family. This leads to a notion of topological complexity for diagrams, defined as the sectional category of their endpoint evaluation map. The role of points in this setting is played by orbits, namely diagrams whose colimit is a point. Different choices of admissible orbit shapes give rise to different levels of categorical compression. We consider the families of representable, discrete and arbitrary topological orbits, and the corresponding orbit-relative Lusternik–Schnirelmann categories. Their relationship with sectional category provides upper bounds for the topological complexity of a diagram, while its values and suitable inverse limits provide lower bounds. For diagrams indexed by a finite discrete group, the construction recovers equivariant LS-category, equivariant sectional category and equivariant topological complexity.

In this paper, we prove the mean Lindelöf hypothesis for the second moment of symmetric square $L$-functions and thereby obtain the (unconditional) error bound $O (x^{36/23+\varepsilon})$ in the prime geodesic theorem on Bianchi manifolds.

Let $G$ be a compact connected Lie group, let $\mathcal R(G)$ denote its algebra of representative functions, and let $\mathcal I_G(f)=\int_G f(g)\,\mathrm d g$ be normalized Haar integration. We prove that $\ker \mathcal I_G$ is a Mathieu–Zhao subspace of $\mathcal R(G)$ if and only if $G$ is a torus. More strongly, for every nonabelian compact connected Lie group $G$ we construct $A,P,Q\in\mathcal R(G)$, with $A\geq 0$ and $A\not\equiv 0$, such that all pure moments of $P$ vanish and the marked moments satisfy an exact Pascal-row identity $$ \mathcal I_G(Q^sP^m) =c_m\binom{m-1}{s-1}\mathcal I_G(A^{4m+s})>0 \qquad (1\leq s\leq m), $$ where $c_m=4^m(m!)^2/(2m+1)!$; the same moments vanish for $s>m$. The construction begins with a homogeneous, phase-balanced polynomial pair on $\mathbb C^2$. An exact coefficient identity on the Hopf sphere $S^3$ is transferred by orbit averaging to every compactly supported $SU(2)$-invariant measure on $\mathbb C^2$. A highest-weight representation and a visible simple-root doublet produce such a measure on every compact simple Lie group. Phase balance gives descent through every central quotient, and pullback from an adjoint simple quotient handles arbitrary nonabelian compact connected groups. The proof is direct and independent of the implication from the Mathieu conjecture to the Jacobian conjecture. The torus direction is the theorem of Duistermaat and van der Kallen.

Designing high-performance tactical wireless networks under realistic operational constraints gives rise to challenging combinatorial optimization problems, where the evaluation of candidate solutions relies on detailed physical and traffic-aware models. Although classical metaheuristics such as Tabu Search offer effective mechanisms for exploring large search spaces, their computational cost remains high because numerous candidate moves must be evaluated at every iteration. In this paper, we propose a data-driven framework that improves the efficiency of Tabu Search by learning to guide its move selection process. Rather than altering the neighborhood structure, our approach exploits the information contained in the search trajectories generated during the optimization process. At each iteration, we record both improving and non-improving edge-based transformations together with a set of descriptive features capturing the structural, geometric, and performance characteristics of the network. This information is used to train a Graph Neural Network (GNN) that predicts the impact of candidate moves on the objective function. The trained model is then integrated into the Tabu Search algorithm to rank candidate transformations according to their predicted quality, thereby reducing the number of costly objective evaluations while maintaining an effective exploration of the search space. Experimental results on synthetic benchmark instances demonstrate that the proposed learning-assisted Tabu Search notably reduces computation time while consistently producing higher-quality solutions than the standard algorithm. These findings highlight the potential of combining machine learning with metaheuristics by leveraging the implicit knowledge embedded in search trajectories, paving the way for more efficient solution methods for large-scale network design problems.

Fix an integer $L \geq 1$, and a function $\phi \colon \mathbb{Z}_{\geq 1} \rightarrow [0,L] \cap \mathbb{Z}$ with $\phi^{-1}(\{0\}) = \{ 1\}$. We define a multi-point variant of the Newton-Raphson-Simpson, which we call the max-phi method, as follows. We use $\phi$ to define the iteration number of a rooted plane tree. Then we construct formal series that are weighted generating functions of rooted plane trees with iteration number at most $N$. Finally we use these formulas to define the max-phi method applied to an arbitrary $L$-differentiable function.

We study the entire interpolation \[ \mathcal{G}(z)=\int_0^1 \binom{x}{z}\,dx \] of the Gregory coefficients. Its completion satisfies the positive Markov-transform identity \[ \frac{\pi z}{\sin(\pi z)}\mathcal{G}(z) =\sum_{n=1}^{\infty}\frac{n\left|G_n\right|}{n-z}. \] Consequently, every zero is real and simple; the negative zeros are the integers $-1,-2,\ldots$, and one zero $\rho_n$ lies in each $(n,n+1)$. We derive complete logarithmic asymptotics for $\rho_n-n$, determine the Cartwright growth and canonical products of $\mathcal{G}$, and realize $1/\rho_n$ spectrally. The resulting relative determinant yields \[ \gamma=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{\rho_n}\right). \]

Let $H^\infty(\Delta)$ be the uniform algebra of bounded analytic functions on the open unit disc $\Delta$, and let $\mathfrak{M}(H^\infty)$ be the maximal ideal space of $H^\infty(\Delta)$. Applying Wermer's embedding theorem directly, we investigate the relation between the analytic structure in $\mathfrak{M}(H^\infty)$ and certain separability conditions. Our method rests only on the corona theorem and certain properties of analytic discs. Among other things, without deep factorization theorems on Blaschke products, we derive the famous Hoffman theorem: Let $P(\xi)$ be the Gleason part of $\xi$ in $\mathfrak{M}(H^\infty)$. Then $P(\xi)$ is an analytic disc if and only if $\xi$ lies in the closure in $\mathfrak{M}(H^\infty)$ of an interpolating sequence in $\Delta$.

Motivated by the polynomial form of Kalinin's Gaussian analogue of Wolstenholme's theorem, following Kalinin, we study a two-dimensional factorial ratio over the Gaussian integers, which we call the rectangular Gaussian binomial coefficient. For a rational prime $p\equiv 3\pmod 4$, we first prove that these coefficients are $p$-integral. Our main result is a Ljunggren–Jacobsthal-type supercongruence: for $p>5$ and every $k\geq 1$, simultaneous dilation of all four parameters by $p^k$ changes the coefficient by a multiple of $p^{3k}$. In particular, this proves the inert-prime case of a conjecture of Kalinin. We also establish a rectangular Bailey-type congruence modulo $p$ for parameters consisting of a large $p$-multiple and one base-$p$ digit. Its shape parallels Bailey's prime-power refinements of Lucas's theorem, but two additional ordinary binomial factors occur, reflecting the vertical and horizontal boundary strips of a rectangular block decomposition. The proofs combine reciprocal-power-sum estimates in $\mathbb{Z}_p[i]$ with factorizations of rectangular products into complete $p^k\times p^k$ blocks.

State estimation in open client server networks is challenging because the set of active observation sources changes over time, and active sources need not be valid sensors of the latent state. We study a client-server estimation problem in which a fixed but unknown state-consistent sensing class measures the latent state when active, while nuisance sources may arrive in the active set and generate observations from a different signal class. The server observes active sources, but it does not know which agents observe valid state measurements. We propose a source-reliability-weighted observer. The state estimate is a standard fixed-prior weighted least-squares update, but the information assigned to each active observation is determined by a source-level reliability score learned from repeated innovation consistency. Numerical results show that the observer, when all active sources are considered, has a finite bias, while the proposed observer initially learns source reliability and then tracks the desired latent state after a finite learning period.

We study a graph-based cellular automaton for HIV infection dynamics in lymph-node networks, originally introduced by Mukwembi. Each vertex represents a cell site that may be healthy, infected, or dead, and the evolution is controlled by a replacement parameter $R$, which determines whether a dead cell is replaced by an infected or a healthy cell according to the number of its infected neighbors. For a graph $G$, we introduce two extinction parameters. The parameter $\mathbf{hiv}(G)$ is the smallest value of $R$ for which extinction occurs for every admissible initial configuration, whereas $\mathbf{HIV}(G)$ is the smallest threshold such that extinction occurs for every replacement parameter greater than or equal to it. We prove the general bounds $2\leq \mathbf{hiv}(G)\leq \mathbf{HIV}(G)\leq \Delta(G)+1$ and characterize the extremal case $\mathbf{HIV}(G)=\Delta(G)+1$. We also show that the gap $\mathbf{HIV}(G)-\mathbf{hiv}(G)$ is unbounded and determine both parameters for some classical families of graphs. Finally, we study the dynamics of the model using the state-transition digraph of the system and the configurations whose trajectories converge to nontrivial periodic orbits. The results show that extinction depends not only on the replacement parameter but also on the structural properties of the underlying graph.

Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $\alpha_3+\epsilon$ for every $\epsilon>0$, where $\alpha_3\approx.83600811464$ is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in $[-1/2,2/5]$ and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of $\alpha_{\rm BOV}+\epsilon$ for every $\epsilon>0$, where $\alpha_{\rm BOV}\approx 0.9563372685$ is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their $S^{k-1}$-valued Borell inequality for correlations in $[-.5843,.5843]$ for all $k\geq3$.

It is well known that, given a Steiner triple system, a quasigroup can be formed by defining an operation $\cdot$ by the identities $x \cdot x = x$ and $x \cdot y = z$ where $z$ is the third point in the block containing the pair $\{x,y\}$. The same is true for a Mendelsohn triple system where the pair $(x,y)$ is considered to be ordered. But it is not true in general for directed triple systems. However directed triple systems which form quasigroups under this operation do exist and we call these Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law $x \cdot (y \cdot x) = (x \cdot y) \cdot x$ but those associated with Latin directed triple systems need not. In a previous paper, [Discrete Mathematics 312 (2012), 597-607], we studied non-flexible Latin directed triple systems. In this paper we turn our attention to flexible Latin directed triple systems.

We study persistent-agent identification and mean tracking in an open client-server network where agents participate intermittently. The network consists of a core of highly regular persistent agents and a subset of transient/non-persistent agents that appear only sporadically. Since the server observes only binary activity indicators and scalar updates from active agents, naive averaging cannot recover the active persistent mean because transient-agent updates contaminate the aggregate, and heterogeneous participation introduces bias. To address this, we propose a two-phase estimation framework. In Phase I, we develop a two-layer window-based identification procedure: the first layer forms single-window activity decisions, while the second aggregates these decisions across windows to create a high-probability separation between persistent and transient agents. Under heterogeneous Bernoulli participation, we derive explicit finite-window bounds that specify how many windows the server must wait for this separation to occur with high probability. In Phase II, we use the resulting structural estimate to track the active persistent mean via a smoothing filter, and show that the tracking error contracts geometrically to a neighborhood determined by the target drift and the filter gain. We provide numerical simulations to illustrate the effectiveness of our theoretical results.