Papers

We establish a generalized quantum cluster algebra structure, defined by Bai–Chen–Ding–Xu, on the reduced Lê–Roger–Yang skein algebra of every quasi-triangulable punctured surface. We define Frobenius maps for generalized quantum cluster algebras at roots of unity and use these maps, together with the cluster algebra structure, to construct Frobenius maps for the (reduced) Lê–Roger–Yang skein algebras of all punctured surfaces. As applications of the Frobenius map, we determine the centers of both the Muller–Roger–Yang skein algebras and the reduced Lê–Roger–Yang skein algebras at roots of unity, and compute their ranks as modules over their centers.

Let $n=2^d$. We prove that the worst-start total-variation mixing time of the full labelled Thorp shuffle satisfies, for every fixed $0<\varepsilon<1$, \[ t_{\mathrm{mix}}(\varepsilon) \le C_* d^{5/3}+O_\varepsilon\!\left(d^{4/3}\log(d+2)\right), \qquad C_*=5.1536951454\ldots . \] Time is measured in ordinary Thorp shuffles. This improves the previously published full-deck upper bound $O(d^3)$ of Morris. The geometric starting point is elementary: label the $n=2^d$ positions by binary strings. One physical Thorp shuffle is a random swap layer on one coordinate matching of the $d$-dimensional cube, followed by a deterministic cyclic rotation of the coordinates. In a rotating frame that deterministic rotation disappears, so $d$ successive physical shuffles become one sweep: one independent random swap layer in each coordinate direction. The proof then combines five ingredients. First, a collision-graph representation and negative dependence give pointwise smoothing for ordered marked cards. Second, an entropy argument is sharpened by moving an endpoint moment up to the critical exponent $\beta_c=\log_2(3/2)$, yielding an entropy-preparation clock $s^2/(2\log(3/2))+O(s\log s)$. Third, exact multiplicities in the ordered $h$-tuple permutation representation amplify marked-card bounds into Fourier operator and energy estimates on low representation levels. Fourth, entropy truncation produces a positive mixture whose regular component retains the marked-card budget. Finally, only two long blocks need this entropy regularization: a shorter third block is used solely for additional low-level smoothing. Optimizing the three block lengths gives the constant above.

Abel's classical theorem shows that a continuously differentiable one-dimensional associative law with a two-sided unit is locally equivalent to addition under a change of coordinate. We prove an $n$-valued analogue in the algebraic and formal settings. Over a field of characteristic zero, every associative algebraic $n$-valued monoid on $\mathbb P^1$ with a strong two-sided unit is commutative. Over a reduced commutative $\mathbb Q$-algebra, any one-dimensional formal $n$-valued group is commutative. We then classify $n$-valued formal groups over arbitrary $\mathbb Q$-algebras that are integral domains: for any $n$-valued formal group there is a unique divisor $m$ of $n$ such that the law is strongly isomorphic to the $n/m$-fold diagonal of the cyclic elementary $m$-valued law $\mathbb G_m$. This proves Kholodov's classification conjecture over integral domain $\mathbb Q$-algebras.

Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets $S\subseteq\N$, the series $-\sum\mu_{\mathcal P}(\lambda)q^{|\lambda|}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\to1$. We determine when this happens. Radially, the summation kernel is a Gumbel density, and Wiener's theorem shows that for bounded $f$ the associated series tends to $L$ if and only if $f$ has mean $L$ on every interval $(x,x+cx/\log x]$; natural density does not suffice. Multiplied by $\log(1/(1-q))$, the series tends to $\delta$ if and only if $S$ has $(c\delta+o(1))x/(\log x)^2$ elements in each such interval; this gives a partition analogue of Alladi's theorem for Chebotarev sets of primes. Off the real axis the largest weights grow like $e^{\Psi_\beta/t}$ with $\Psi_\beta>0$. Along $q=e^{-t(1+i\beta)}$, the formula for residue classes modulo $m\ge3$ holds exactly when $|\beta|<\beta_m$, an explicit Clausen-function threshold with $\beta_m\to0$, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.

We show that for certain classes of matrices over octonions, a weak version of breakthrough numerical radius inequalities obtained by Kittaneh [\textit{{Stud. Math.}, 2005}] hold.

We modify a construction of Bradač \cite{Bradac26} using a vertex ordering to give, for every prime power $q$, an explicit $K_5$ free graph with more than $q^7$ vertices and independence number below $600q^4$. This shows $$r(5,t)=\Omega(t^{7/4}).$$ Additionally, pruning a graph defined by the Coulter and Matthews polynomial $X^{14}$ \cite{CM}, we give a deterministic polynomial time construction proving $$r(5,t)=\Omega\bigl(t^{20/11-\varepsilon}\bigr)$$ for every fixed $\varepsilon>0$. These results improve the constructive lower bound $\Omega(t^{5/3})$ of Kostochka, Pudlák and Rödl \cite{KPR}.

We propose a stochastic optimization framework in which football performance is modeled as the result of an efficiency-driven production process under uncertainty. The model incorporates economic inputs, a latent efficiency variable evolving according to a stochastic differential equation, and a control variable representing investment in sports development. We derive the associated discounted Hamilton–Jacobi–Bellman (HJB) equation, characterize the optimal investment policy, prove a verification theorem under explicit growth conditions, and show that a bounded investment budget is required for the control problem to be well posed. We relate our efficiency measure to the stochastic frontier and data envelopment analysis (SFA/DEA) traditions used elsewhere in sports economics, and clarify what a dynamic control formulation adds relative to those static approaches. The theoretical results are illustrated with a numerical solution of the HJB equation – validated by a grid-refinement convergence study – and Monte Carlo simulations. A cross-sectional analysis of the 24 AFCON 2025 national teams, based on GDP, population, and FIFA ranking data, estimates country-specific conditional log-efficiency residuals using heteroskedasticity-robust inference. We show that these residuals are mechanically and strongly correlated with FIFA points by construction, quantify this dependence, and interpret the results accordingly. Teams such as Senegal and Morocco consistently outperform their GDP-implied potential, whereas others, including South Africa and Tanzania, tend to underperform. While these patterns are broadly consistent with observed AFCON outcomes, the estimated residual also captures unobserved factors, such as institutional quality, diaspora talent, and demographic structure, that are not included in the model.

Benjamini asked whether every configuration of points on the real line that is in equilibrium under the inverse-square repulsive force must be an arithmetic progression. Georgakopoulos and Kolountzakis proved this when some gap between consecutive points has maximal or minimal length, and described the general (aperiodic) case as open. We show that the answer is yes. More generally, let $1<s\le2$, and let $X\subset\mathbb{R}$ be a locally finite set with at least two points such that, for every $x\in X$, the total force $\sum_{y\in X\setminus\{x\}}|y-x|^{-s}$ is finite and the net force $\sum_{y\in X\setminus\{x\}}\operatorname{sgn}(y-x)\,|y-x|^{-s}$ is zero. Then $X$ is an arithmetic progression. No a priori assumption on the gaps is needed. Subtracting the equilibrium equations of two consecutive points shows that the gaps $g_n$ form a positive harmonic function for an explicit reversible random walk on $\mathbb{Z}$ with long-range jumps. Equilibrium also bounds the ratio of consecutive gaps, by $1.5386\ldots$ when $s=2$. With this bound, an energy estimate shows that the Doob transform of the walk by $g$ is recurrent. Since $1/g$ is a positive harmonic function of the transformed walk, it is constant.

We consider a finite Ising chain in zero field and ask when the all-up configuration is as likely as zero magnetization (magnetization one for an odd number of spins). For both free and periodic ends, the crossing coupling of a long chain is half the logarithm of its length minus half the logarithm of that logarithm, up to a term that vanishes as the length grows. We compute further terms with explicit constants and, for free ends, bound the crossing at every length. The second term is missing from an earlier approximation and explains why an earlier numerical fit gives a slope below one half. To locate the crossing, we compare the known magnetization law with its Bessel approximation. The Bessel approximation is above the exact weight, and we bound its relative error for every length, coupling and magnetization. At the crossing this bound tends to zero. The central bin still has about as many domain walls as the logarithm of the length on average, so the crossing has a coupling width of order the inverse logarithm. In a fixed positive dimensionless field the last crossing comes from near the edge of the row, at a coupling of about a quarter of the logarithm of the length. For long chains it is the one-spin crossing when the field exceeds half the logarithm of two. Below that value another bin crosses later.

Braces play a fundamental role in matching theory, as they, together with bricks, constitute the basic building blocks of matching covered graphs in the tight cut decomposition. A brace is minimal if deleting any edge from it results in a graph that is not a brace. We prove that every perfect matching of a minimal brace of order at least six contains an edge whose ends both have degree three. Consequently, every such brace has at least three edges with this property. Combining this result with the forest structure induced by the noncubic vertices, we show that every minimal brace $G$ of order $n\ge6$ and size $m$, other than $K_{3,3}$, satisfies $n_3(G)\ge\max\left\{8,\left\lceil\frac{2n+8}{5}\right\rceil, \left\lceil\frac{m-n+4}{2}\right\rceil\right\}$, where $n_3(G)$ is the number of cubic vertices of $G$. Finally, we characterize the graphs attaining the constant lower bound: a minimal brace has exactly eight cubic vertices if and only if it is isomorphic to $B_8$, $B_{10}$, $Q_{10}^{+}$, or $Q_{12}$.

In Divisor Nim, a player removes from one heap a positive number of stones that divides the size of every other heap. We prove that the Sprague–Grundy (SG) value satisfies $g(P)\le 2h_{\min}+\lfloor\log_2h_{\min}\rfloor$, where $h_{\min}$ is the smallest nonempty heap, independently of the number and sizes of the other heaps. The proof starts with odd heaps and classifies removals by their factors of $2$. The largest heap in a starting board then supplies an SG bound valid throughout every successor chain. When one heap varies, we sharpen Morisawa's eventual-period bound: the smallest fixed heap controls the SG value range, the number of consecutive values compared, and an increase in the varying heap that repeats legal moves. The remaining dependence on the fixed board lies in the SG periods of direct successors. Using the largest fixed heap, we derive an explicit common period for all these recursive layers. Its possible prime divisors belong to a fixed finite set, independent of heap count; their exponents remain controlled by the total fixed stones. Finally, joining boards under one shared divisor rule preserves and can tighten the SG ceiling. Minimum 2-adic depth and its count parity determine every joined outcome, and a join with at least two odd heaps has exact SG value $0$ or $1$. We close with two conjectures: $g(P)\le2h_{\min}$, and that the least eventual SG period, when one heap varies, divides $2\operatorname{lcm}(1,\ldots,h_{\max})$. Here $h_{\max}$ is the largest fixed heap, excluding the varying heap. The second claim would remove heap-count dependence from the period bound as well.

Landing on small celestial bodies is challenging because the spacecraft must satisfy safety and actuator constraints despite gravitational forces that are difficult to model accurately. This article presents an optimal control for safe landing with gravitational uncertainty and strict actuator limits. The approach addresses spacecraft pose control with attitude represented on SO(3), where there are actuator constraints and state constraints on position, attitude, velocity, and angular velocity. The approach combines several key techniques. First, the gravitational force uncertainty is estimated using an extended high-gain observer, and we present a new dynamic upper bound on the estimation error. This gravitational estimate and dynamic bound are then used to compute optimal control forces and torques that satisfy state and actuator constraints while tracking a landing trajectory. Optimal forces and torques are obtained from the closed-form solution to a quadratic program that has a single control barrier function constraint constructed by composing multiple control barrier functions that are designed to enforce each state and actuator constraint. Finally, an optimal allocation maps control forces and torques to actuator commands that satisfy actuator constraints. The method is demonstrated in simulation using 2 vehicle configurations with severe gravitational uncertainty.

A Moore graph simultaneously asks for two things that normally pull in opposite directions: every vertex should have many neighbors, while every pair of vertices should remain close. For diameter two and degree $k$, the Moore bound is $k^2+1$, and equality forces a remarkably rigid graph: it is strongly regular with parameters $(k^2+1,k,0,1)$. Hoffman and Singleton showed that such a graph can exist only for $k=2,3,7$, and possibly 57. The degree-57 case on 3,250 vertices remains open. This paper is not an attempt to settle that open problem. Instead, we ask a more algorithmic question: How efficiently can the defining Moore structure itself be searched exactly? We begin from a human-developed exhaustive search based on the radius-two Moore tree. Leaves fall into $k$ groups of size $k-1$, and the edges between every pair of such groups form a perfect matching - a permutation. For efficiency, a partial construction is abandoned immediately when it creates a triangle or quadrangle. We then describe a sequence of exact algorithm-engineering improvements developed through a human-guided AI process and accepted only after mathematical checking and controlled regression tests. The main steps are reversible short-cycle forbid counters, bit-parallel domains, minimum-remaining-value branching, singleton propagation, matching-based AllDifferent filtering, balanced replayable parallel frontiers, residual-symmetry orbit pruning, and canonical deduplication of frontier states. On the principal $k=9$ benchmark, stronger symmetry cuts wall time from 47.6 to 17.9 seconds on a six-core laptop setup. Further parallelization speedup is reported separately; on a 16-core/32-thread machine, median wall time is reduced to 5.28 seconds. Degrees 10 and 57 are used as stress tests, and a five-minute degree-57 run uses only 161 MiB peak resident memory, showing that state size is modest.

We study domination of the bi-parameter Hilbert transform by $(1,1)$-type sparse forms over axis-parallel rectangles, and show that such domination fails even for smooth test functions compactly supported in $(0,1)^2$. Our proof combines an explicit construction of two finite lattices based on the arithmetic of $\mathbb Z[\sqrt2]$ with a randomization argument.

Sharp inequalities between the Riesz and logarithmic capacities of compact sets in $\mathbb{R}^n$ are established, and the equality conditions are fully characterized. These results naturally extend Pólya and Szegő's conjecture on the ratio of Newtonian to logarithmic capacities in $\mathbb{R}^2$, which was posed in the 1940s.

Given an $n\times n$ matrix $A=(a_{i,j})$ and subsets $I,J$ of $[1,n]:=\{1,\dotsc,n\}$, let $A_{I,J}=(a_{i,j})_{i\in I,j\in J}$ denote the $(I,J)$-submatrix of $A$. We prove that, for every totally nonnegative matrix $A$ and every $h=1,\dotsc,n-1$, \begin{equation} \operatorname{perm}(A_{[1,h],[1,h]}) \operatorname{perm}(A_{[h+1,n],[h+1,n]}) \;\geq\; \operatorname{perm}(A_{[1,n]\cap 2\mathbb Z,[1,n]\cap 2\mathbb Z}) \operatorname{perm}(A_{[1,n]\smallsetminus 2\mathbb Z,[1,n]\smallsetminus 2\mathbb Z}). \label{eq:absmain} \end{equation} In particular, these inequalities hold for the $0$-$1$ totally nonnegative matrices arising as antiadjacency matrices of unit interval orders. Our proof is based on explicit Bruhat-order comparisons between two distinguished families of permutations. Let $\mathfrak{A}_n$ be the Young subgroup $\mathfrak{S}_{1,\ldots,\left\lfloor\frac{n}{2}\right\rfloor} \times \mathfrak{S}_{\left\lfloor\frac{n}{2}\right\rfloor+1,\ldots,n}$ and let $\mathfrak{B}_n$ be the set of parity-alternating permutations \begin{equation*} \{w=w_1\cdots w_n\in\mathfrak{S}_n \mid i \text{ and } w_i \text{ have the same parity}\}. \end{equation*} For the balanced case $h=\left\lfloor\frac{n}{2}\right\rfloor$, we construct a bijection $f_n:\mathfrak{A}_n\rightarrow\mathfrak{B}_n$ satisfying $w\leq f_n(w)$ in Bruhat order. We then extend this construction to arbitrary $h$ by means of maps $g_{n,h}$ satisfying the corresponding Bruhat-order comparison. These maps yield the inequalities for all totally nonnegative matrices.

We introduce an algorithm for efficiently preconditioning log-concave and log-smooth distributions that scales logarithmically with the condition number of the underlying distribution. Based on Gaussian cooling, our multistage method approximately samples from a sequence of well-conditioned distributions to construct a preconditioner for each subsequent stage of the cooling schedule. This method is motivated by existing practical approaches for mass matrix estimation in Markov chain Monte Carlo (MCMC), in which an effective preconditioner for the target distribution is estimated from samples produced during a warm-up period. However, our approach is conceptually distinct from existing approaches and moreover permits theoretical analysis. Our algorithmic framework is flexible, allowing essentially any log-concave sampling algorithm to act as the subroutine within each cooling stage. For instance, using first-order rejection sampling (FORS) as the sampler, the first-order query complexity of our method is $\tilde{O}(d^{4/3} \log \kappa)$ for a large class of log-concave and log-smooth distributions with condition number $\kappa$ and $\tilde{O}(d^{1/3} \log \kappa)$ for translation-invariant distributions. In the translation-invariant setting, we provide numerical experiments for the lattice $\phi^4$ model, a simple lattice field theory.

We prove lower bounds for Fourier transforms of surface measures $$|\widehat{\psi\,d\sigma}(\xi)| \gtrsim_{\psi, K, \sigma, R_0} \sigma(B(p, |\xi|^{-1})), \text{ given }p\in K\Subset\{\psi>0\} ,\ \vec{n}_{p}\parallel \xi,\ |\xi|\ge R_0,$$ on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of $p\in K\Subset \{\psi>0\}$ and match the classical upper bound of Bruna-Nagel-Wainger. The key new analytic ingredient is a uniform lower bound for parameter-dependent oscillatory integrals on $\mathbb{R}$ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which $$\inf_{p\in K}|\widehat{\psi\,d\sigma}(R \vec{n}_p)|\lesssim_N R^{-N},\ \text{ for arbitrary fixed } \{0\}\Subset K\Subset\{\psi>0\}.$$ Though the estimate above has its own interest, our original motivation is to study Fourier frames for surface measures. In particular, we show that, if $\Gamma$ is a closed convex smooth surface in $\mathbb{R}^d, d\geq 2$, of finite type and at most one principal curvature vanishes at each point, then the surface measure on $\Gamma$ does not admit any Fourier frame. This weakens the non-vanishing curvature condition in a previous result of Iosevich, Lai, the second author, and Wyman. On centrally symmetric surfaces of non-vanishing Gaussian curvature, we obtain more domains without Fourier frames. For example, we show that $\{x\in S^{d-1}:0<x_d<\delta\}, d\geq 3$, does not admit any Fourier frame for all $\delta>0$, which generalizes the result of Chen and the second author on the hemisphere.

The adjoint method is a standard approach for computing gradients in PDE-constrained optimization and central to parameter estimation, model calibration, and optimal control. This tutorial presents a discrete, implementation-oriented introduction to the method with an emphasis on conceptual understanding. Starting from an inverse problem governed by a partial differential equation, it develops the corresponding optimization problem and derives the adjoint gradient from the perspective of iterative descent methods. The roles of the state, the parameters, the objective, and the constraint are made explicit, and the adjoint equation is obtained directly from the gradient computation rather than postulated. Particular attention is given to explicit derivations, implementation details, and the connection between the mathematical development and the resulting computational algorithm. The tutorial also examines the influence of discretization on the adjoint construction, explains the origin of the backward-in-time adjoint recursion, and presents practical procedures for gradient verification. A vibrating-string inverse problem serves as a running example, illustrating the complete workflow from model formulation to adjoint-based optimization. The presentation is intended for advanced undergraduate students, graduate students, and practitioners seeking a step-by-step understanding of adjoint gradient computation.

This paper studies distortion-aware clustering for uplink fronthaul-limited cell-free MIMO networks employing maximum ratio combining (MRC). While MRC is appealing for its low-complexity, its performance is limited by interference, fronthaul distortions, and channel imperfections. This motivates clustering strategies that compensate for these impairments through coordination of access points. Since instantaneous small-scale fading-based optimization is not scalable in large systems, and is impractical due to frequent channel variations, we instead optimize an objective depending only on large-scale fading coefficients. To this end, asymptotic analysis is used to derive deterministic equivalent expressions for the average network sum rate, with focus on quantization distortion, leading to a quadratic-over-linear objective. Although maximizing such an objective under binary constraints is non-convex, we exploit its structure to develop a polynomial time scheme that attains the global optimum. Numerical results show that the proposed clustering method provides performance gains over improved variants of existing literature, and remains competitive with the small-scale fading-based global optimum obtained via exhaustive search.

Analog computing with microwave signals can perform linear transformations directly in the analog domain, as the signals propagate through a microwave network. A fundamental question is which transformations can be computed with a given set of microwave components. In our previous work, we answered this question for networks of hybrid couplers and phase shifters by deriving a necessary and sufficient condition on the transformations these networks can compute, and we showed that the discrete Fourier transform (DFT) satisfies it. In this paper, we take a first step toward the formalization of analog computing with microwaves in Lean, a programming language and proof assistant that is increasingly adopted in mathematics. We formalize the considered components, their series and parallel connections, and the class of networks they can implement. Then, we formally prove the necessary and sufficient condition characterizing these networks, as well as the implementability of the DFT of any size power of two. All proofs are checked by the Lean kernel, and the code is openly available at: https://github.com/matteonerini/formalizing-analog-computing.

We prove two supercongruence statements proposed by Z.-W. Sun for Domb numbers. The first concerns the unweighted truncated sum $$S_p=\sum_{n=0}^{p-1}D_n$$ and is governed by the two proper ideal classes of the order of discriminant $-60$ in $\mathbb{Q}(\sqrt{-15})$. We prove Sun's conjectured congruences modulo $p^2$ and obtain stronger formulas modulo $p^3$ in the split cases. The second concerns Domb sums weighted by the Lucas sequences associated with $X^2-11X+1$. Sun's original formulation has the exceptional counterexample $p=3$; after excluding this prime, we prove the corrected statement, including its modulo-$p^3$ refinement when both relevant quadratic characters are $+1$. Both results arise from one modular mechanism. The twisted Domb series is a modular period on $\Gamma_0(6)+\langle 3\rangle$ and is related by a quadratic pullback to Zagier's sporadic $C$-sequence. At primes inert in the imaginary quadratic field underlying the relevant complex multiplication, the truncated $C$-period is the Hasse invariant and vanishes at supersingular reduction; a finite pullback then gives divisibility modulo $p^2$. At split primes, explicitly constructed determinant-$1$ or determinant-$3$ matrices identify the Beukers unit root, yielding congruences modulo $p^3$. The two applications correspond respectively to orders of complex multiplication of discriminants $-60$ and $-156$.

We prove ten conjectures of Z.-W. Sun on Ramanujan-type series for $1/\pi^4$ and $\pi^4$ with harmonic numbers. They follow by comparing Taylor coefficients in new Borwein-Bradley-type identities, whose products depend only on $x^3$, $x^4$, $x^6$ or $x^8$. These identities come from parametric extensions of the series of Cullen and Zhao. We derive the extensions from K. C. Au's Wilf-Zeilberger (WZ) seeds, and their sums are trigonometric. Our proofs use Guillera's periodicity argument and give new proofs of the original series. In the same way, Dougall's $_5F_4$ sum gives half-integer analogues of the identities of Koecher, Borwein-Bradley and Cohen, whose Taylor coefficients are Apéry-like series for products of zeta values.

We study a one-parameter three-dimensional quadratic flow, denoted ETM-3D, on the benchmark domain $\rho\geq0$. Its homogeneous quadratic triad preserves Euclidean quadratic energy, while the full vector field has constant divergence -8. A shifted quadratic Lyapunov function yields an explicit positively invariant absorbing set and therefore global ultimate boundedness. The equilibria reduce generically to a scalar quartic, with a separate singular case at $\rho=\sqrt{5}$. Continuation reveals two saddle-node bifurcations, a generic transcritical bifurcation, a supercritical Hopf bifurcation, and a fold of limit cycles. Stable and unstable period-one branches meet at the fold with their nontrivial Floquet multipliers approaching +1. Just beyond this point, a 25-point logarithmic parameter study gives a laminar-length scaling exponent of -0.551 (95% confidence interval [-0.621,-0.481]), consistent with type-I intermittency. At larger $\rho$, a chaotic attractor coexists with an asymptotically stable equilibrium and periodic windows occur. Structural invariants are used to distinguish the flow from several classical quadratic chaotic families.

Gaussian elimination with partial pivoting is the standard algorithm for solving a dense $n\times n$ system of linear equations $Ax=b$. The “growth factor” $\rho\,$ for this process may be as large as $2^{n-1}$, and when $\rho$ is large, the algorithm is unstable. Nevertheless, long experience has shown that Gaussian elimination is resoundingly stable in practice. Experiments indicate that among random matrices with independent normally distributed entries, growth factors $\rho \gg n^{1/2}$ are in a precise sense exponentially rare. Here we take a step toward confirming this observation by proving it not for the full growth factor, which depends on all the entries of the upper-triangular factor $U$, but for the corner entry $u_{nn}^{}$.

We study mixed-integer quadratic fractional optimization problems with binary and continuous box-constrained variables. The objective is the ratio of two quadratic functions, a structure that arises naturally in portfolio selection, signal processing, and data analytics. We propose an exact algorithm based on Dinkelbach's parametric reformulation. At each Dinkelbach iteration, the fractional problem is replaced by a nonconvex mixed-integer quadratic subproblem with box constraints. This subproblem is solved globally by a specialized branch-and-bound solver. To accelerate the outer scheme, we use a Variable Neighborhood Search heuristic to provide an initial feasible solution to the Dinkelbach scheme. The heuristic alternates between continuous local searches and a binary-space exploration phase in which the continuous variables of a feasible incumbent are fixed. Numerical results show that the proposed approach outperforms general-purpose exact algorithms based on Dinkelbach's parametric reformulation.

In a companion article \cite{RJnodyn} the present authors proved a form of the Rudolph–Johnson theorem which is effective at finite resolution, bounding the low-frequency Fourier coefficients of a $\times 2 \times 3$-invariant measure by its normalized entropy. Here we read that bound in the other direction, as a constraint on eigenfunction behavior in potential counterexamples to Furstenberg's $\times 2 \times 3$ conjecture. We show that a near return of the coordinate function $x \mapsto e^{2\pi ix}$ to itself under the semigroup generated by $2$ and $3$ forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of $x \mapsto 2x$ forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.

Counterexamples to the Eisenbud-Schreyer conjecture on the existence of Ulrich bundles have appeared recently (see Anghel [A1] and the third author [Lo]). As weakly Ulrich bundles are a useful generalization of Ulrich ones, in this paper, we classify smooth surfaces $S\subseteq \mathbb P^N$ with weakly Ulrich tangent bundle. They are Del-Pezzo surfaces of degree at least 5 and we classify the possible embeddings. An interesting feature is the appearance of two infinite series of polarizations, which is quite different from the Ulrich case [BMPT, LR]. We also achieve the classification of surfaces with initialized aCM tangent bundles and, as a consequence, of surfaces with almost Ulrich tangent bundles.

Safety filters for large-scale multi-agent systems can be formulated at the population level through mean-field control barrier functions (MF-CBFs). Most existing formulations are first-order, requiring the control input to appear explicitly in the first time derivative of the safety functional; higher-order extensions have only very recently been proposed for a single, directly actuated population. We consider instead the case in which the population to be kept safe is not directly actuated, but is transported by a field generated by a second, controlled population. For this setting, we develop a higher-order MF-CBF framework for density dynamics with safety constraints of relative degree larger than one. Our main result is that indirect actuation of this kind does not destroy the control-affine structure of the safety condition: the control-dependent part of the second time derivative of the safety functional is obtained in closed form and is linear in the control velocity field, with a kernel given by the actuated density times the spatial gradient of a first variation. Relative-degree-two safety constraints can therefore still be enforced through a convex quadratic program. We then specialize the framework to a leader-follower density control problem in the presence of dangerous regions, where safety is ensured by keeping the overlap of each population with those regions below a prescribed threshold. We show that the follower safety functional has relative degree two with respect to the leader control field, while the leader safety functional has relative degree one. We further characterize exactly when the two simultaneous constraints are infeasible, introduce an adaptive-gain mechanism to mitigate potential conflicts, and bound the tracking errors induced by the safety correction. Numerical simulations in cluttered environments illustrate the resulting safety-performance trade-off.

This paper develops a one-dimensional model for a mobile viral component propagating through a spatially distributed susceptible population. Random viral motion produces diffusion, whereas a response to the relative gradient of susceptible density generates directed transport. The existence of a positive entire traveling wave is proved when directed transport dominates diffusion, and the limiting case of equal transport and diffusion coefficients is constructed separately. A dimensionless transformation then converts the remaining equation into a logistic equation and supplies closed-form profiles. These formulas identify the unique viral maximum and the far-field behavior. A conservative implicit-explicit finite-difference method provides an independent numerical check. Diffusion is evaluated at the new time level, the gradient-driven flux explicitly at cell interfaces, and susceptible depletion pointwise. Exact traveling-wave values specify the initial data and time-dependent Dirichlet conditions at both endpoints. Thus, the computation tests preservation and translation of the analytical profiles. A calculation with eight hundred spatial subintervals reproduces the localized viral band and the monotone susceptible front. The largest uniform errors are approximately $1.70\cdot10^{-2}$ for viral density and $2.22\cdot10^{-3}$ for susceptible density. The numerical results corroborate the analytical construction and show that the conservative discretization captures the proposed propagation mechanism.