Papers

We study travelling shock profiles for dissipative quasilinear hyperbolic balance laws endowed with a convex entropy principle and subject to involutive differential constraints. For systems without constraints, the classical Boillat–Ruggeri result excludes continuous shock profiles whose speed exceeds the largest characteristic velocity at the equilibrium state ahead of the shock. We show that this bound remains unchanged for homogeneous constraints and for nonhomogeneous constraints whose source Jacobian has full row rank at equilibrium; a longitudinal two-species Gaussian ten-moment plasma with Landau collisions and Gauss's law provides a physical example of the latter case. For rank-deficient constraints, travelling-wave compatibility alone leads, after restriction to the joint conservation–constraint manifold, to a quadratic correction determined by a finite-dimensional Lyapunov equation. For $2\times2$ systems with one conservation law, one genuine balance law and one scalar involutive constraint, we prove that a regular noncharacteristic rank-deficient tail cannot sustain a nonzero correction, so that the original Boillat–Ruggeri bound is recovered. The remaining degenerate case is illustrated by an exact propagated two-field model in which the constraint selects the travelling speed and the smooth profile approaches equilibrium algebraically. The same model also admits entropy-admissible Lax composite profiles containing a sub-shock, while a completely smooth connection exists between the same equilibrium states. Numerical entropy evolutions further show that, depending on the smooth compatible initial datum, the first-order model may either remain smooth or dynamically develop a sub-shock.

The large-degree phase of the Jacobi polynomial $P_n^{(\alpha,\beta)}(\cos\theta)$ contains, at each order in $\Nn^{-1}$, $\Nn=n+(\alpha+\beta+1)/2$, an additive constant $\kappa_r$ that the phase equation does not determine. These constants are antisymmetric in $(\alpha^2,\beta^2)$ and vanish for $\alpha=\beta$. We obtain three results. First, on the diagonal $\alpha=\beta$ the logarithm of the midpoint amplitude is computed in closed form: if $L_n$ denotes the coefficient of $(2\Nn)^{-n}$ in that logarithm, then $\Upsilon_A(t)=\sum_{n\ge1}L_nt^n/(n-1)!$ equals \[ \Upsilon_A(t)=\frac{e^{t}\bigl(\cosh2\alpha t-\cosh t\bigr)}{\sinh 2t}. \] The proof is algebraic: the identity is established for half-odd-integer $\alpha$, where the expansion terminates, by means of two classical recurrences for Bessel polynomials, and is extended to all $\alpha$ by a degree argument. Second, the derivative of the logarithm of the amplitude with respect to $\alpha^2$ at the Legendre point $\alpha=\beta=0$ is computed in closed form to all orders. An exact reduction of an Appell $F_3$ function to a Gauss function shows that it is a parameter derivative of a zero-balanced ${}_2F_1$ at argument $2$; the real part of its expansion is a trigamma series, and the imaginary part is the unique formal solution of an explicit central-difference equation. Third, we deduce that the coefficients $w_m$ of an explicit even power series attached to the imaginary part satisfy $\nu_2(w_m)=-2\nu_2((2m)!)$ for every $m\ge0$ ($\nu_2$ the $2$-adic valuation). This determines, for every $r$, the exact power of $2$ in the denominator of the coefficient of $\alpha^2$ in the polynomial that represents $\kappa_r$. We also give a factorisation of the amplitude into products of pairs of Gauss functions, further valuation formulas, and a numerical illustration for the zeros.

We construct and analyze structure-preserving numerical schemes for a two-stage local–nonlocal dispersal system. On general bounded connected smooth habitats, we prove fixed-$\delta$ spatial consistency. We introduce a semi-implicit scheme that is uniquely solvable for every $\Delta t>0$, preserves nonnegativity, and exactly retains the sign of the semidiscrete spectral threshold. A negative threshold yields geometric extinction of the numerical solution, while a positive threshold makes the zero state of the numerical scheme linearly unstable. On rectangular habitats, we construct a reflected Cartesian discretization that is asymptotically compatible with the Neumann local limit, with convergence estimates uniform with respect to the ratio between mesh size and interaction scale. Numerical experiments illustrate the structural, threshold, and local-limit behavior.

We introduce Envelopt, a globally convergent iterative framework for a broad class of structured optimization problems where a smooth objective is augmented by a nonsmooth convex regularizer composed with a smooth mapping, and the variables are subject to general smooth constraints. All smooth functions may be nonconvex. The method is akin to an augmented-Lagrangian method in which partial minimization with respect to a lifting variable results in smooth subproblems involving the Moreau envelope of the nonsmooth regularizer, and the original constraints are retained explicitly. Only the proximal operator of the regularizer is required. Subproblems may be solved with off-the-shelf smooth optimization solvers. We state global convergence properties, establish that feasible limit points are asymptotically stationary, and develop an infeasibility detection mechanism. We derive worst-case iteration complexity bounds when the penalty parameter is and is not bounded away from zero. The framework subsumes the classical augmented Lagrangian method and accommodates important extensions, including stabilized formulations for degenerate problems, exact penalty methods, and conic constraints. We provide a Julia implementation, Envelopt.jl, as part of the JuliaSmoothOptimizers ecosystem. Numerical experiments with low-rank matrix completion, semidefinite programming, complementarity-constrained optimization, and nonconvex regularizers demonstrate the effectiveness and versatility of Envelopt.

Let $k$ be a field of characteristic zero, let $R$ be a commutative $k$-algebra, let $I$ be a proper ideal of $R$, and assume that $A=R/I$ is Noetherian. We prove that if $a=[f]$ in $A$ is nilpotent and $d_{A/k}a=0$, then $I:f$ is not contained in $(I,f)$. If $R$ is local with maximal ideal $\mathfrak n$, then $I:f$ is not contained in $(I,f)+\mathfrak n(I:f)$. As applications, we prove the isolated hypersurface case in arbitrary dimension and extend the conclusion to non-isolated critical loci on smooth varieties. Finally, we exhibit a five-variable isolated singularity satisfying $J_f:f\subseteq\overline{J_f}$, thereby disproving the integral-closure strengthening.

Let $0<p, q<\infty$ with $p \neq q$, and suppose that $p<2$ or $q \leq 1$. We prove that the Lorentz sequence space $\ell_ {p, q}$ does not isomorphically embed into $L_ {p, q}(0,1)$. Consequently, $L_ {p, q}(0, \infty)$ does not isomorphically embed into $L_ {p, q}(0,1)$.

In measure and integration theory, the themes of extending premeasures to outer measures and extending Daniell linear functionals to Daniell integrals are governed by the foundational extension theorems of Caratheodory and Daniell. These are proven to be equivalent to Littlewood's Three Principles, which are known to be equivalent alternatives to a completeness axiom for an ordered field. The continuity theorems of measure and integration theory are consequently shown to be included in the list of equivalences.

We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least $(d/n)\log^2\{n/(4d)\}$ when $d\ge C$ and $n\ge Cd$, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius $r$ around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by $\varepsilon<1$ from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most $d^3r^2/n+d^4/n^2+d^6r^6$. Lower bounds under the same frame control yield the local minimax rate $d^3r^2/n+d^4/n^2$ when $n\ge Cd^2$ and $r$ lies in an explicit matching range. For every fixed upper bound on $\varepsilon$, approximation changes only the constants, not the powers of $d,n,r$. At the critical radius $r=n^{-1/2}$, the local risk is asymptotically negligible relative to the global risk when $n\log^2(n/d)\gg d^3$. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.

For positive integers $q$ and $m$ with $m\geq 2q+3$, consider the symmetric numerical semigroup $S_{m,q}=\langle m,m+1,qm+2q+2,qm+2q+3,\ldots,qm+m-1\rangle$. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answer this question affirmatively for every $q\geq 1$, including an independent proof of the previously known case $q=1$. Let $P_{m,q}$ be the semigroup polynomial, and set $t=m-2q-3$, $L=2(q+1)(m+1)-1$, and $D=\deg P_{m,q}=2qm+2q+2$. For $m\geq 2q+4$, let $\rho_1,\ldots,\rho_D$ be the roots of $P_{m,q}$, counted with multiplicity. An explicit computation of a single coefficient of the formal logarithm gives $\left|\sum_{j=1}^{D}\rho_j^{-L}\right|=L\left|[x^L]\log P_{m,q}(x)\right|\geq Lt-1>D$. The strict inequality forces $P_{m,q}$ to have a root off the unit circle. Hence $S_{m,q}$ is noncyclotomic whenever $m\geq 2q+4$. The boundary case $m=2q+3$ has embedding dimension $3$ and is cyclotomic. Therefore $S_{m,q}$ is cyclotomic if and only if $m=2q+3$.

Social dynamical networks significantly influence contemporary digital landscapes, affecting realms from social activism to public policy formulation. This paper investigates the use of multi-agent systems (MAS) to monitor and analyze these networks. Firstly, we propose a single-time-scale distributed inference model designed to effectively manage challenges such as latency and agent failure. Secondly, we provide sufficient conditions that ensure the stability of the proposed scheme. Notably, the observer gain design remains effective regardless of time delays. Thirdly, we develop a computationally efficient recovery mechanism for agent failures that relies on employing graph-theoretic approaches to restore network observability by replacing failed agents (due to sensing failure or unbounded delays, i.e., packet drops) by implementing computationally efficient graph-theoretic methods to assign observationally equivalent agent counterparts. Lastly, we illustrate the proposed scheme through a pedagogical example and real-world network applications.

A graph $H$ is $K_{r+1}$-Turán-good if, for every sufficiently large $n$, the Turán graph $T_r(n)$ maximizes the number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs, and it is strictly $K_{r+1}$-Turán-good when this extremal graph is unique. Morrison, Nir, Norin, Rzążewski and Wesolek proved that for every graph $H$ with at least one edge, when $r\ge 300v(H)^9$, $H$ is $K_{r+1}$-Turán-good. They asked whether the above bound could be reduced to quadratic order in $v(H)$. In this paper, we answer their question affirmatively, and give a stronger result. We prove that there is an absolute constant $C>0$ such that every graph $H$ with at least one edge is strictly $K_{r+1}$-Turán-good whenever $r\ge Cv(H)$. Our proof uses a substantially different cluster method based on polymer models. Besides proving the vertex-linear Turán-good threshold, this method appears to have further applications. As one illustration, we prove that for every graph $H$ with at least one edge, the normalized chromatic polynomial $P_H(x)/x^{v(H)}$ is strictly increasing for every real $x\ge C\Delta(H)$, which strengthens the previously best result where $x\ge C\Delta(H)^{3/2}$. This proves a conjecture of Fadnavis.

Let $F_k(G)$ be the $k$-token graph of a finite graph with nonnegative edge weights, and let $A_k$ and $D_k$ be its weighted adjacency and degree matrices. For every $-1\leq\vartheta\leq1$, we prove that $k\mapsto\lambda_{\max}(A_k+\vartheta D_k)$ is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At $\vartheta=1$ and $\vartheta=0$, this gives the signless-Laplacian and adjacency spectral-radius monotonicity conjectures of Apte, Parekh, and Sud. The spectral result follows from a finite-time theorem: for every $t\geq0$, the heat contents $\binom{n}{k}^{-1}\mathbf{1}^{*}e^{t(A_k+\vartheta D_k)}\mathbf{1}$ are log-concave in $k$. We encode all token levels in one Lorentzian polynomial. A four-variable operator symbol proves preservation by each edge heat gate, and the Lie-Trotter formula passes this preservation to the full semigroup. Large-time growth rates recover the top eigenvalues. The same construction also yields Lorentzian polynomials from top spectral projections. Finally, we show that the local symbol certifies exactly the parameter range $[-1,1]$.

Machine learning learns functions: prompt to response, image to caption. What these functions are mathematically remains hard to say. We present a method to approximate these kinds of transformations using techniques from dynamical systems that fall under the umbrella of Koopmanism. We introduce the use of composition operators, which generalize the Koopman operator and, crucially, can map between distinct spaces, motivating the perspective that LLM transformations are rectangular infinite-dimensional operators. This formalism reveals useful structure: under natural assumptions on the prompt and response distributions, the LLM operator is an isometry, and misalignment between learned representations manifests as spectral pollution of its finite sections. We then outline a method of constructing finite-dimensional approximations of an LLM operator, and demonstrate how the singular value spectrum can be used to compare tasks and models.

Let $W$ be a representation of the Weyl relations on a separable Hilbert space and write $\mathcal R_W(K)=\{W(f):f\in K\}''$. If the symplectic pairing of $K_1$ and $K_2$ is nonzero, there is a self-adjoint unitary $B\in\mathcal R_W(K_2)$ such that, for every normal state $\rho$, one can choose a self-adjoint unitary $A_\rho\in\mathcal R_W(K_1)$ with $\operatorname{Var}_\rho(-i[A_\rho,B])=4$. Hence the optimized mean-input-energy-constrained coefficient equals $4$ at every admissible energy threshold. For the fixed trigonometric Weyl witness $h$, an exact identity for $4-h^2$ reduces the deficit to a phase-fixed problem for commuting squared Weyl translations. For the one-mode quadratic energy $G_M=\frac12 R^TMR-\frac12\sqrt{\det M}$, with $M>0$ and $u^T\Omega v=\pi$, we prove $4-\gamma_{G_M,E}(h)=\frac{u^TMu+v^TMv+2\pi\sqrt{\det M}}{4E}+O(E^{-2})$. The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For $G=d\Gamma(H_1)$ and witness directions $u,v\in\operatorname{dom}H_1^{1/2}$, the fixed-witness deficit is $\Theta(E^{-1})$ whenever at least one direction lies outside $\ker H_1$; if both directions are zero modes, the constrained supremum equals the endpoint at every positive threshold.

Recently, a new record kissing arrangement of $841$ points in $\mathbb R^{12}$ was obtained numerically by optimization (Takhanov-Assylbekov-Yun, 2026). The configuration was released as a coordinate file, without a mathematical description of its structure. The purpose of this paper is to provide such a description. The key observation is that the geometry becomes transparent once we regard $\mathbb R^{12}\cong \mathbb H^3$ as the Cartesian product of three copies of the quaternion algebra. We first introduce a new $840$-point kissing arrangement with a certain quaternionic structure. It consists of three mutually orthogonal regular $24$-cells, supported on the three quaternionic coordinate factors $\mathbb H\times\{0\}\times\{0\}$, $\{0\}\times\mathbb H\times\{0\}$, $\{0\}\times\{0\}\times\mathbb H$, together with two $384$-point families obtained by lifting affine sets of the form $$\{(u,v,w)\in (\mathbb F_2^2)^3\mid u+v+w=\eta\},$$ to quaternionic triples (whose components belong to the binary octahedral group $2O$) and then applying suitable component-wise rotations and weightings. A characteristic feature of this construction is a pronounced asymmetry among the three quaternionic factors. For the $816$ vectors obtained after removing the third $24$-cell, most of the squared norm is concentrated in the first two quaternionic coordinates, while the third coordinate carries systematically less mass. Thus, the third four-dimensional factor contains more available space than the first two. We then show that this $840$-point configuration provides a natural structural model for the numerical $841$-point record. Finally, we introduce a notion of the general quaternionic construction in dimensions divisible by $4$, and check that record kissing arrangements in ${\mathbb R}^{4k}$, $k\leq 5$, admit a quaternionic construction.

Two independent attempts, separated by two decades, to encode turbulent dissipation with a fractional Laplacian both converge on the operator order as the signature of the fully developedinertial range: the additive, fixed order model, and the dynamically deforming operator of the Adaptive Fractional Navier Stokes. We show that, despite sharing this fixed point, the two constructions belongto distinct universality classes: Chen operator is a superposition of two fixed order kernels whose relative weight drifts with scale, while AFNS posits a single operator whose order itself flows continuously with the local Reynolds number. We derive a scale resolved observable, the effective spectral order, whose logarithmic anomaly term discriminates between these two mechanisms in principle, and then obtain a sharper, purely analytic verdict from the two dimensional, enstrophy conserving limit: the generative mechanism matching a stable process index to the empirical pair dispersion law has no solution once vortex stretching is kinematically switched off. Whether this dichotomy leaves a residual signature in real quasi two dimensional flows is left as an open experimental challenge.

In signaling games where a sender and a receiver have misaligned criteria, equilibrium behavior may lead to fully revealing, quantized, or randomized policies. Notably, the first arises in statistical decision theory and classical communication theoretic problems involving a fully aligned sensor and receiver, the second arises in Nash theoretic simultaneous signaling games, and the last may appear in Stackelberg type (leader-follower) Bayesian signaling games. In this paper, we investigate the Bayesian persuasion problem involving a receiver that tries to estimate the source. We show that for certain payoff structures, the equilibrium solution is such that a source observation is mapped to distinct messages with nonzero probabilities. More specifically, we completely characterize conditions under which the sender requires randomization for the Bayesian persuasion problem involving general sources with finite cardinality. In particular, regardless of whether the equilibrium solution under a deterministic policy restriction is fully revealing, quantized or noninformative, there exists a randomized sender policy that improves the sender's payoff under certain conditions characterized in the paper. Moreover, we provide an algorithmic procedure to obtain the Bayesian persuasion solution, where the algorithm compares the payoffs with finitely many posterior probability combinations. We also consider fully aligned and completely misaligned payoff structures, where the solutions respectively involve a fully revealing sender and a noninformative sender. Then, we unify these results by proving that if the sender's expected payoff with respect to posterior distributions is continuous, then the equilibrium solution involves either a fully revealing sender or a noninformative sender.

Cheng introduces the multivariate writhe polynomial, leaving a question regarding the characterization of this invariant. In this paper, we resolve this question by proving that a Laurent polynomial $f$ with integer coefficients can be realized as the multivariate writhe polynomial of some multi-virtual knot if and only if $f(1,\dots,1)=0$ and $\left. \frac{d}{dt} f(t, \dots, t) \right\vert{}_{t=1} = 0$.

The primary objective of this paper is to systematically generalize this phenomenon by replacing the standard unit disk with a family of nested, internally tangent shifted disks $\Omega_\gamma$ parameterized by $\gamma \in [0, 1)$, defined by$$\Omega_\gamma = \left\{ z \in \mathbb{C} : \left| z + \frac{\gamma}{1 - \gamma} \right| < \frac{1}{1 - \gamma},\; \gamma\in [0, 1) \right\}.$$ By exploiting the geometric characteristics of $\Omega_\gamma$ and evaluating the limiting behavior as $\gamma \to 1^-$, we establish a novel framework to determine the Bohr radius for the unbounded half-plane $\mathbb{H}_1 = \{z \in \mathbb{C} : \text{Re}(z) < 1\}$. Furthermore, we prove several sharp variations of the Bohr inequality within these domains, including refined and improved formulations for unimodular bounded analytic functions. The results obtained herein not only extend classical radius problems to unbounded regions but also illuminate the delicate interplay between domain deformation and coefficient estimates.

Mader conjectured that every $k$-strong digraph $D$ with minimum semidegree $\delta^0(D)\ge 2k+m-1$ contains a dipath $P$ of order $m$ such that $D-V(P)$ remains $k$-strong. For $k=1$, he obtained the weaker bound $\delta^0(D)\ge 2m$. We confirm the conjecture for $k=1$ by showing that the sharp bound $\delta^0(D)\ge m+1$ suffices. As a consequence, we show that for every integer $m\ge2$, every strongly connected digraph $D$ with $\delta^0(D)\ge\max\{2,m-1\}$ contains a dipath $P$ of order $m$ such that $D-A(P)$ is strongly connected.

This article arose out of notes for a CIME summer school mini-course in Cetraro. In it we develop differential-geometric constructions on the moduli space ${\cal M}_{g,n}^{\rm spin}$ of smooth spin curves, with particular emphasis on their interpretation in hyperbolic geometry. We relate the locally constant sheaf arising from a hyperbolic spin structure to the corresponding holomorphic sheaf, extending this known relationship to spin curves with Ramond marked points. We use the hyperbolic description to construct characteristic forms and spin measures on moduli spaces of hyperbolic surfaces with geodesic boundary, providing a framework for applying Teichmüller-theoretic methods to their volumes and volume recursions. Finally, we study separately the even and odd components of spin moduli space and find symmetries between their volume contributions which are not apparent from the contributions of individual boundary strata.

In 1966, Erdős, Goodman, and Pósa showed that every $n$-vertex graph $G$ admits a cover of its edges using at most $\lfloor \frac{n^2}{4}\rfloor$ cliques, with tightness witnessed by the balanced complete bipartite graph. Erdős suggested the following strengthening: every $n$-vertex graph $G$ admits an edge decomposition into cliques with total cost at most $\lfloor \frac{n^2}{4}\rfloor$, where each $i$-clique has cost $i-1$. There is another generalization conjectured by Dau, Milenkovic and Puleo: for every $t\ge4$, every $n$-vertex graph $G$ admits a cover of its $t$-cliques using at most $\prod_{j=0}^{t-1}\left\lfloor\frac{n+j}{t}\right\rfloor$ cliques. Balogh, He, Krueger, Nguyen and Wigal proved asymptotic and fractional versions of these conjectures. We solve both conjectures for large $n$ and identify the equality cases. Our proof of Erdős's conjecture combines linear programming duality and the spanning-forest polytope with a weighted stability form of Mantel's theorem, followed by explicit decomposition algorithms around an almost balanced bipartition. For the $t$-clique-cover conjecture, we combine graph removal lemma and generalized Turán stability with an exact covering construction for graphs close to $T_{n,t}$.

We provide a spectral geometric description of the KdV variational equation around cnoidal waves in terms of two interacting spectral curves.

We study the stabilization of the linearized periodic Whitham–Boussinesq system on the one-dimensional torus. We establish the well-posedness of the conservative and damped dynamics in the natural energy space and describe the spectral structure of the conservative generator, whose frequencies exhibit sublinear growth of order $|k|^{1/2}$ at high frequency. We then prove strong stability of the damped semigroup and obtain a high-frequency resolvent estimate with linear growth in the spectral parameter. Under genuinely localized damping, a family of high-frequency quasimodes provides the matching lower bound and shows that this resolvent growth is optimal. By the Borichev–Tomilov theorem [5], we deduce a $t^{-1}$ decay rate for the semigroup on the domain of the generator, corresponding to a $t^{-2}$ decay rate for the energy. Under the same localization assumption, these polynomial decay rates are optimal.

We develop a PDE-based methodology for pricing and hedging European contingent claims in general one-dimensional diffusion markets characterized solely by their scale function and speed measure, possibly without a classical SDE representation, and with constant interest rate. We derive a hedging equation whose solution generates a self-financing hedging strategy and provide sufficient conditions on scale, speed, and interest rate, under which this strategy achieves the minimal hedging capital, expressed through the no free lunch with vanishing risk (NFLVR) condition. We further prove necessary and sufficient conditions for NFLVR and characterize the class of equivalent local martingale measures through an auxiliary diffusion whose scale and speed characteristics are determined by those of the real-world diffusion and by the interest rate. When the NFLVR condition fails, the framework may produce multiple hedging equations corresponding to non-minimal strategies, whose associated prices can exceed the minimal hedging capital. We illustrate both the effectiveness and limitations of the approach through numerical experiments involving diffusion models with irregular features.

We study exact representations of $\mathrm{MAX}_N(x)=\max{x_1,\ldots,x_N}$ using two-hidden-layer ReLU neural networks. This problem has been studied in recent years in an attempt to characterize the exact number of hidden layers required to represent continuous piecewise linear functions. The best lower bound is 2, while the current upper bound is logarithmic in $N$. It remains completely open if the right answer is a constant number of hidden layers (possibly even 2!) or not. In fact, a recent breakthrough was the representation of $\mathrm{MAX}_5$ as a two-hidden-layer ReLU function obtained in [Bakaev et al., 2026], and the case of $\mathrm{MAX}_N$ was stated as open for $N\geq 6$ in that paper. Using a careful computer assisted search, we obtain two-hidden-layer ReLU representations of $\mathrm{MAX}_5, \mathrm{MAX}_6, \mathrm{MAX}_7$, and $\mathrm{MAX}_8$. We obtain these by considering rational linear combinations of terms of the form $\max\{\sum_{r=1}^{s}\max(x_{a_r},x_{b_r}),\sum_{r=1}^{s}\max(x_{c_r},x_{d_r})\}$, where $a_r,b_r,c_r,d_r\in\{1,\ldots,N\}$. Each inner maximum of two coordinates can be computed in a first hidden layer, and the outer maximum of the two side-sums can be computed in a second hidden layer. Consequently, every finite linear combination of these terms has a two-hidden-layer ReLU realization. An identity for $\mathrm{MAX}_N$ in this form therefore gives an exact two-hidden-layer ReLU representation of $\mathrm{MAX}_N$. Very recently, two-hidden-layer representations of $\mathrm{MAX}_N$ of the above form were obtained for all $N\leq 10$ in [Ruess et al., 2026]. Our representations are different and were developed independently. While our techniques share most of the high-level ideas presented in [Ruess et al., 2026], there are also some minor differences which may be of interest for future research on this problem.

Mixed-Integer Linear Programming (MILP) is a fundamental tool for combinatorial optimization with extensive real-world applications. A central challenge is designing computationally efficient MILP formulations. Large Language Models (LLMs) offer new opportunities to automate the modeling process, from deriving formulations to strengthening them. Reliable automation requires robust methods for verifying that proposed formulations preserve the underlying optimization problem. However, existing approaches evaluate formulations numerically and fail to reason about general problem instances. We resolve this limitation by introducing a constructive definition of MILP reformulation that can be formalized in Lean and machine-checked. We develop FLARE (Formulation-Level Automated Reformulation Evaluation), a method that uses an LLM-based agent and the Lean proof assistant to verify proposed reformulations against a reference formulation. To evaluate our approach, we introduce FormulationBench, a challenging dataset of 20 problems and 109 formulations. FLARE outperforms existing methods, with 100% accuracy on the NP-hard subset of FormulationBench. Furthermore, FLARE produces a machine-checkable certificate for every reformulation it accepts. For cases where formal guarantees are not necessary, we introduce FLARE-NL, a fast and cheap LLM proxy that matches FLARE's accuracy but produces no certificate. These methods enable reliable verification in automated optimization modeling.

We give an explicit polynomial deformation of the Euclidean metric on the unit three-ball which has positive Ricci curvature and strictly convex boundary but violates Escobar's conjectured lower bound for the first nonzero Steklov eigenvalue. Our proof is fully human-verifiable.

The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.

Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ are subvarieties of the flag variety parameterized by a Hessenberg function $\mathbf{h}: [n] \to [n]$ and a matrix $\mathsf{X} \in \mathfrak{gl}_n(\mathbb{C})$. In recent work, Goldin and the second author showed the existence of flat degenerations of Hessenberg varieties to nilpotent Hessenberg varieties over the minimal sheet. This implies that all Hessenberg varieties over the minimal sheet have the same dimension. Our main result generalizes this dimension result to arbitrary sheets. Specifically, we prove that for a fixed Hessenberg function $\mathbf{h}:[n] \to [n]$, all Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ defined in the type $A$ flag variety by linear operators $\mathsf{X}$ from the same sheet of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$ have the same dimension.