Papers

We study the amalgamation decision problem (ADP): given a universal first-order sentence $\Phi$, decide whether the class $\mathrm{fm}(\Phi)$ of its finite models has the amalgamation property. We call $\Phi$ semantic Horn if $\mathrm{fm}(\Phi)$ is closed under binary direct products. By McKinsey's theorem, this is equivalent to $\Phi$ being logically equivalent to a universal Horn sentence. The distinction concerns the input representation: $\Phi$ itself need not be given in Horn form, and converting it into an explicit Horn normal form can incur an exponential blow-up. We prove that the ADP is decidable under this semantic promise. To this end, we associate with $\Phi$ a finite-domain CSP template, called its completion template, and a distinguished infinite family of CSP instances, called its atlas instances. The class $\mathrm{fm}(\Phi)$ has the amalgamation property precisely when all atlas instances have a solution. For semantic Horn inputs, we prove that the completion template admits a semilattice polymorphism. Consequently, its CSP has bounded width and is decided by a fixed level of local consistency. We introduce uniform contextual strategies, finite certificates expressing this local-consistency condition simultaneously for all atlas instances, and give an effective fixed-point procedure for deciding whether such a certificate exists. The resulting algorithm runs in 2ExpTime in general and in ExpTime under any fixed bound on the arities of the input relations. More generally, the construction gives a decision procedure whenever the associated completion template has bounded width. These bounds are optimal: the semantic Horn ADP is 2ExpTime-complete in general and ExpTime-complete for every fixed arity bound of at least three. Both hardness results already hold for syntactic universal Horn sentences.

Boundary conditions for a drift-kinetic model at the collisional presheath entrance with perpendicular incidence of the magnetic field to the wall are derived and numerically implemented. The drift-kinetic model for the plasma is based on the expansion of the ion distribution function on a Hermite-Laguerre basis, and the evolution of the resulting gyromoments. A linear-plasma-device geometry is considered. Comparison with simpler simulations with previously used ad hoc boundary conditions is presented. For the new set of boundary conditions, a significant increase of the plasma outflow to the wall is observed, leading to a significantly smaller plasma density in the whole volume of the device.

Let a rational positive-definite quadratic sublevel set on a full-rank affine lattice coset be intersected with a closed slab. A deterministic two-dimensional delta=3/4 reduction selects a primitive-dual integer functional and an inner ellipse. In the branch where the corresponding Babai point lies outside that ellipse, the lattice points in the disk-slab occupy at most four integer levels of the selected functional. An explicit rational instance attains four, so the bound is sharp. A strict diameter estimate reduces any counterexample to three blocks of five consecutive levels, and a nearest-integer inequality excludes the central block. Symmetry and cell localization reduce the two side blocks to eight modes: four follow from a shared-cap inequality, and four from an exact four-variable Bernstein certificate. The certificate contains 17,640 positive coefficients, with minimum 625/2048.

Suppose that ${R}^{\alpha}$ is a rhombus with side length $1$ and with an interior angle $\alpha$, where $0<\alpha\leq \frac{\pi}{2}$. Let $\triangle$ be an equilateral triangle with a side parallel to a side of ${R}^{\alpha}$ and let $\{\triangle_{n}\}$ be a collection of homothetic copies of $\triangle$. In this paper, we show the following two results: if $0<\alpha\leq\frac{\pi}{3}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\cos\alpha+\frac{\sqrt{3}}{3}\sin\alpha)^{2}$, then these equilateral triangles can parallel cover the rhombus ${R}^{\alpha}$; if $\frac{\pi}{3}<\alpha\leq\frac{\pi}{2}$ and the sum of the areas of equilateral triangles from $\{\triangle_{n}\}$ is at least $\frac{\sqrt{3}}{4}(1+\frac{2\sqrt{3}}{3}\sin\alpha)^{2}$, then they can parallel cover the rhombus ${R}^{\alpha}$. Furthermore, these bounds are optimal on their respective intervals.

For every $n\ge 8$, we construct explicit families of generating pairs of $GL(n,\mathbb Z)$ and $SL(n,\mathbb Z)$ representing countably infinitely many Nielsen equivalence classes. These pairs arise as the homology images of explicit torsion generating pairs in $Out(F_n)$ and its index-two subgroup $SOut(F_n)$, yielding countably infinitely many Nielsen classes of generating pairs in both outer groups. Within each constructed outer family, all pairs become Nielsen equivalent after one stabilization, obtained by adjoining a trivial coordinate. We derive a relation-module obstruction to Nielsen equivalence of once-stabilized generating tuples. Evans's non-cancellation examples show that this obstruction is nontrivial and can distinguish Nielsen classes of once-stabilized generating tuples. We also record a quotient relation-module refinement for possible future applications.

The compatibility of the semiorthogonal decomposition of a cubic fourfold with Bridgeland stability beyond a generic point is the missing ingredient in the conjectured dynamical protection of the K3 atom $\AX$. We analyze it in the exactly solvable models of the noncommutative minimal model program. In the uncoupled Fano models ($\PP^1$; $\PP^1 \times \PP^1$ at resonance) the quantum cohomology path exits the geometric chamber at an explicit finite time, enters the selection region of the gluing, and never leaves; perturbing the resonance shows the $\varepsilon$-crossover is not a wall. The $A_2$ quiver provides a coupled counterpart. We examine the deformed cubic oscillator, Painlevé~I, and formulate a tau-JLO dictionary as a determinant identification and establish its perturbative layer, including computing closed-form quantum periods through $Z_4$, all-orders flatness with exact curvature the tau-divisor current, identifying it with the zeta determinant's zero divisor along a Painlevé~I trajectory. We close by conjecturing that the determinant's nonperturbative jumps are automorphisms of the underlying Joyce structure.

Under a growth condition on an increasing function $g$, we prove that integrability of a freely infinitely divisible distribution with respect to $g$ is equivalent to that of the large-jump part of its free Lévy measure. For every increasing freely submultiplicative function $g$, integrability of the distribution implies integrability of the large-jump part of its free Lévy measure. We also obtain two-sided tail comparisons between freely infinitely divisible distributions and their free Lévy measures, uniform integrability along free convolution semigroups, and integrability results and tail estimates for fractional free convolution powers.

Recovering a low-dimensional latent subspace from nonlinear observations of Gaussian covariates in high dimensions is a fundamental problem in feature learning. Here, we consider Gaussian multi-index models in which the covariates $\boldsymbol{x}_i \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0,\boldsymbol{I}_d)$ and the responses $\boldsymbol{y}_i$ depend on $\boldsymbol{x}_i$ only through its projection onto an unknown $r$-dimensional subspace. Earlier work based on approximate message passing (AMP) identified a sharp threshold for weak recovery [Troiani et al., 2025], raising the question of whether it can be attained, without side information, by a spectral method. We answer this affirmatively and develop a general random matrix theory for matrix-valued spectral estimators of the form \[\boldsymbol{D}_n=\frac{1}{n}\sum_{i=1}^n\boldsymbol{T}(\boldsymbol{y}_i)\otimes\boldsymbol{x}_i\boldsymbol{x}_i^\top,\] where $\boldsymbol{T}$ is an arbitrary bounded symmetric matrix-valued preprocessing map of fixed dimension. As $n,d \to \infty$ with $n/d\to\alpha$, we prove that the empirical spectral measure of $\boldsymbol{D}_n$ converges almost surely to a deterministic compactly supported distribution characterized by a matrix-valued self-consistent equation. We then establish a spectral phase transition for the largest eigenvalue: below threshold it sticks to the bulk edge, while above threshold an outlier emerges. We characterize the outlier location through a finite-dimensional deterministic equation and show that the associated spectral estimator achieves weak recovery of the latent subspace. Finally, we prove that the AMP-derived preprocessing of [Defilippis et al., 2025] is optimal among all bounded matrix-valued preprocessing maps of any fixed dimension. Its transition coincides with the AMP weak-recovery threshold, proving the general spectral conjecture of [Defilippis et al., 2025].

Let $\boldsymbol{X}_i=(X_{1i},X_{2i})^\top$ be independent and identically distributed observations with mean $\boldsymbol{\mu}=(\mu_1,\mu_2)^\top$ constrained by $\mu_1\leq\mu_2$. We study empirical-likelihood inference for a fixed mean vector and distinguish it from the previously known test of equality against an ordered alternative. At a fixed interior point, the constrained empirical likelihood ratio has the usual $\chi^2_2$ limit. At a fixed boundary point $(m,m)^\top$, its limit is the chi-bar-square distribution $\tfrac12\chi^2_1+\tfrac12\chi^2_2$. By contrast, profiling the unknown common mean in the equality-versus-order test yields $\tfrac12\chi^2_0+\tfrac12\chi^2_1$, the $k=2$ ordered-mean case of El Barmi (1996). We give an exact reduction of the latter statistic to the empirical likelihood of the paired differences, establish the localization step needed for the fixed-boundary expansion, and derive a local-to-boundary limit showing that interior calibration is not uniform over $n^{-1/2}$-neighborhoods of the boundary. Monte Carlo experiments under Gaussian, Student $t_5$, and shifted log-normal sampling examine fixed, boundary, and local regimes with explicit numerical-failure accounting. Illustrative paired-data analyses show the practical distinction between fixed-candidate confidence regions, directional equality tests, and ordinary scalar empirical-likelihood intervals truncated to the nonnegative parameter space.

We prove that the trigonometric system $\mathcal{E}_N = \{e_k\}_{|k| \le N}$ is not rigid in $L_p(\mathbb{T})$ for $1 \le p < 2$. That is, as $N\to\infty$, all its elements are approximated with error $o(1)$ by a subspace of dimension $o(N)$.

We prove the following “Torus Trichotomy Theorem”: every finitely generated subgroup of $\pls$ is either virtually free, a central extension of a cocompact Fuchsian group by a finite cyclic group, or a one-ended, non-acylindrically-hyperbolic group containing $\mathbb{Z}^2$. In particular, we obtain the “free or surface” alternative for word-hyperbolic groups of PL homeomorphisms of the circle.

We study two shadowing properties for flows that differ in the allowed reparametrizations of time: oriented shadowing permits arbitrary increasing reparametrizations, whereas standard shadowing requires their distortion to be uniformly close to one. We prove that these notions are distinct already for $C^\infty$ flows on every closed oriented surface. Moreover, such examples are $C^0$-dense among $C^1$ flows with a singularity and consequently, on closed oriented surfaces with non-zero Euler characteristic, they are dense among all $C^1$ flows. We then relate local standard shadowing to recurrence and entropy. A non-trivial chain-transitive set with local standard shadowing forces positive topological entropy unless it is an irreducible almost heteroclinic set. Consequently, for a zero-entropy flow with standard shadowing, every non-trivial chain-recurrent class has this form, and every non-singular one is minimal. For surface flows, we further prove that oriented shadowing together with finitely many singularities forces every chain-recurrent class to be minimal.

In this paper, we continue our research on the double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{M|N}$. Extending the Etingof-Kazhdan construction, we introduce the structure of $h$-adic quantum vertex superalgebra on the vacuum module $V^c(\mathfrak{gl}_{M|N})$ over the double Yangian of level $c\in\mathbb{C}$. Next, we construct families of central elements in the completed double Yangian for $\mathfrak{gl}_{M|N}$ at the critical level. Finally, we show that the level $c$ restricted modules over the reflection algebra of $\mathfrak{gl}_{M|N}$ are naturally equipped with the structure of quasi $V^{2c}(\mathfrak{gl}_{M|N})$-module. By using this structure, we find explicit formulas for families of central elements in the completed reflection algebra of $\mathfrak{gl}_{M|N}$ at the critical level.

The power prior of Ibrahim and Chen incorporates historical data into a Bayesian analysis by raising the historical likelihood to a power $a_0 \in [0, 1]$. The choice of the exponent has remained an open question. This paper gives a closed-form answer under the predictive log-loss. For a model with $d$ parameters, a historical sample of size $N_0$, and average Kullback–Leibler divergence $\bar{D}_0$ between the historical and current data-generating distributions, the optimal exponent is $a_0^{*} = d/(2 N_0 \bar{D}_0 + d)$. Equivalently, the optimally borrowed effective sample size obeys the harmonic law $1/E^{*} = 1/N_0 + 2\bar{D}_0/d$: compatible data are pooled in full, and any difference caps the borrowed information at $d/(2\bar{D}_0)$ observations. The result is exact for multinomial data and extends to smooth parametric families. The law benchmarks adaptive borrowing, explains the reported degeneracy of the normalized power prior, and shows that neither subsetting the data nor decaying the exponent improves on the correctly discounted constant.

This paper introduces a novel algorithm for computing eigenvalues and eigenvectors of a dense real skew-symmetric matrix $A$. Its main ingredient is the computation of a polar factor of $A$ that is both skew-symmetric and orthogonal. This polar factor is then used to transform the original problem into a Hermitian eigenvalue problem of half the size, which can be solved accurately and efficiently with standard software such as LAPACK. Numerical experiments demonstrate the stability of the method and show that its running time is competitive with existing approaches for skew-symmetric eigenvalue problems. Finally, we show that the same principle can be used to reduce an orthogonal eigenvalue problem to a unitary eigenvalue problem of half the size.

We study the normalized half moment $\alpha_{\mathbb R}^{(\lambda)}(N)$ of the size-$N$ Laguerre orthogonal ensemble for real shape $\lambda\ge0$. At integer shape this is the expected average singular value of an $N\times(N+\lambda)$ real Gaussian matrix. The square mean increases with the dimension, whereas every real shape $\lambda\ge1$ decreases. Between these two regimes the decrement is strictly increasing in $\lambda$, and hence has a unique zero in $(0,1)$ for every $N$. There is also a unique crossing of the Marchenko–Pastur limit. Both thresholds converge to $\lambda_*=1-\pi/4$, and their first corrections show that they separate on the scale $(\log N)/N$. The proof starts from an exact decomposition of the real half moment into its complex counterpart and a positive orthogonal correction. Recent unitary estimates take care of the complex term. An Abel completion, together with a Laguerre connection formula, turns the orthogonal correction into a positive diagonal series; the square case, the regime $\lambda\ge1$, and the transition can then all be read from this same series.

Let $W_{N,N+\lambda}$ have the Laguerre unitary distribution with size $N$ and real shape $\lambda\ge0$. For $s>0$, we consider the normalized moment $$C_{s,\lambda}(N)=N^{-s-1}\mathbb{E}[\operatorname{Tr}W_{N,N+\lambda}^{s}]$$ and its dimension decrement $\Gamma_{N,s,\lambda}=C_{s,\lambda}(N)-C_{s,\lambda}(N+1)$. Iterating the Laguerre moment recurrence separates this decrement into a square source and a nonnegative shape source. The square source gives the complete finite-dimensional sign diagram: $C_{s,0}(N)$ decreases for $0<s<1$ and $s>2$, increases for $1<s<2$, and is constant for $s\in\{1,2\}$. For every $s>0$ and every $N$, the decrement is strictly increasing in $\lambda$. The same decomposition determines the critical shrinking-shape scales: $N^{-2s}$ for $0<s<1/2$, $(\log N)/N$ for $s=1/2$, and $N^{-1}$ for $s>1/2$. In the convex range $1<s<2$, where the two sources have opposite signs, the transition occurs when $N\lambda_N$ is of order one, with critical constant $$\tau_s^*=\frac{s(s-1)(2-s)}{6(2s-1)}.$$ In the convex range, both crossings are unique for every finite $N$, and we determine their locations to second order. At $s=1/2$ we also obtain a bounded-shape two-term expansion, which supplies the unitary estimates used in the companion orthogonal paper.

We formulate an improved version of the monotonicity method for shape reconstruction in the inverse problem of linear elastostatics. We show that this method can reconstruct inclusions in the second Lamé parameter, i.e., the shear modulus, independently of inhomogeneities in the first Lamé parameter. This improves on earlier methods that only recover inclusions in the bulk modulus under certain assumptions on the local interplay between the Lamé parameters. Our proofs are based on linearization and divergence-free localization of energy using harmonic vector fields.

We study nonnegative submodular maximization on $n$ elements subject to a general matroid of rank $k$, when the offline algorithm is given an arbitrary controlled value oracle. Our main result is an adversarial resilience theorem for the Spiteful Greedy Swap Poisson Process (SGS-Poisson): without modifying its Poisson intensity, single-element exchange rule, or spiteful drop step, the algorithm retains limiting approximation factors $1/e$ for non-monotone objectives and $1-1/e$ for monotone objectives. More precisely, given an error bound $\xi\ge0$, under every controlled oracle $\widehat f$ satisfying $|\widehat f(S)-f(S)|\le \xi$ for every set $S$, our implementation returns a feasible set with expected value at least $(1/e-\varepsilon)OPT-O(k\xi)$ and $(1-1/e-\varepsilon)OPT-O(k\xi)$, respectively, where $OPT$ is the feasible optimum. The implementation uses a deterministically bounded budget of $O(nk^{2}\varepsilon^{-2}\log n\log^{2}(1/\varepsilon))$ oracle calls. As a consequence, an offline-to-online reduction yields full-bandit combinatorial multi-armed bandit (CMAB) algorithms for general matroid-constrained submodular rewards with exact limiting approximation-regret factors $1/e$ and $1-1/e$ and $\widetilde O(n^{1/5}k^{4/5}T^{4/5})$ regret over $T$ rounds. These online guarantees allow exploration to play sets that become independent after deleting at most one element; exploitation and the benchmark remain matroid-feasible. For unit-capacity partition matroids we obtain $\widetilde O(n^{1/5}k^{3/5}T^{4/5})$ under the same exploration relaxation. We establish a deterministic query budget by truncating the Poisson process and identify the enlarged action set needed to answer its offline queries.

We study the stability of one-dimensional planar isotropic–nematic interfaces in the Landau–de Gennes model with anisotropic elastic constant $L$. Earlier work proved the instability for $L<0$ only under an extra condition. We remove this condition and prove that any non-negative minimizer of the reduced energy within the diagonal planar class is unstable under general one-dimensional perturbations throughout $-\frac{3}{2}<L<0$. This result shows that $L=0$ is the sharp endpoint of the instability range from the negative-$L$ side: the critical operator is non-negative at $L=0$, while instability holds over the full negative-$L$ range of the reduced problem. We also define the optimal stability index and correct a factor-of-two normalization inconsistency in a previously stated interface profile.

Illingworth and Wood recently proposed the Dominating Hadwiger's Conjecture, a strengthening of Hadwiger's Conjecture which asserts that every graph with no dominating $K_t$-model is $(t-1)$-colorable. We prove two relaxations of this conjecture. First, we show that every graph with average degree $Ct (\log t)^2$ contains a dominating $K_t$-model for some absolute constant $C$. This bound improves on the $2^{t-2}$ due to Illingworth and Wood and is within an $O(\log t)$ factor from optimal. Second, we prove that the vertices of every graph with no dominating $K_t$-model can be partitioned into $t-1$ parts such that the subgraph induced by each part has bounded maximum degree.

The statistics of current fluctuations has long been a central object of study in non-equilibrium physics. Most existing work has focused on one-time statistics, while their multi-time generalisation remains comparatively less explored. We address this gap by extending the fluctuating hydrodynamics framework of Macroscopic Fluctuation Theory (MFT) to study multi-time statistics in the non-stationary state of a diffusive system on an infinite line. For the simplest cases of a non-interacting lattice gas and hard-core Brownian point particles, we present explicit solution of the MFT leading to multi-time large-deviation statistics. For generic systems, the MFT is solved perturbatively, yielding explicit results for two-time correlations. These reveal that the connection between current fluctuations and fractional Brownian motion, previously observed for flat initial conditions, does not persist for step initial conditions. We independently verify these hydrodynamic results by solving the corresponding microscopic dynamics for the non-interacting gas and for the symmetric simple exclusion process. Additional confirmation comes from numerical simulations.

We study an abstract class of bilinear infinite-dimensional systems that arises in various applications, including district heating systems and quantum control. First, we analyze the existence of mild and classical solutions for abstract bilinear systems with admissible input operators in Banach spaces and study continuous dependence on the data. When the underlying space is a Hilbert space, these results are used to show passivity for a suitably defined co-located output and for mild solutions. The results are applied to the bilinear Schrödinger equation, the Fokker–Planck equation, and a district heating or cooling pipe.

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin–Wentzell large deviation rate functional, and as the Hessian of the Benamou–Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss–Newton method yields optimal fitting of velocity fields.

Biochemical reaction networks have become a central theoretical framework for implementing molecular computation. A key challenge is finite time computational accuracy, as computation outputs are encoded in limiting steady states (LSSs) of species concentrations while practical implementations operate for only finite time. This work proposes a concise characterization of convergence rate for biochemical computational modules with multiple output species, and rigorously establishes it as being bounded by the eigenvalue with largest (least negative) real part of the Jacobian matrix. Two numerical examples illustrate how the theoretical lower bound shapes the convergence rate range and reveals its dependence on reaction rate constants. This formulation enables systematic evaluation of biochemical computation speed and provides a practical design measure for constructing high-accuracy and error-controlled biochemical computational modules.

We prove that a Hausdorff topological group is tame if and only if it is precompact.

We show that the overtwisted contact structure on $S^3$ with Hopf invariant $-1$ is non-orderable, i.e., that it admits a contractible positive loop of contactopmorphisms. This provides the first known example of a non-orderable overtwisted contact manifold. We further show the existence of a contactomorphism without translated points.

For the first time, we establish higher integrability of the gradient and local $L^\infty$ estimates of weak solutions to the $p(x)$-Laplacian without assuming $\log$-Hölder continuity of $p$. Instead, we establish these results for exponents satisfying a generalized Muckenhoupt condition. This admits discontinuous exponents acting as pointwise multipliers of the BMO class. Our framework bridges the gap between classical $p(x)$-regularity and weighted Muckenhoupt theory, providing a new foundation for the analysis of differential equations with highly irregular non-standard growth.

Cohn and Kumar conjectured in 2009 that there is a radial Schwartz function $g\colon\R^{24}\to\R$ satisfying $g(r)\leq0$ for $r\geq\sqrt6$, $\widehat g(r)\geq0$ for $r\geq0$, $g(2)>0$, and $(\widehat g(0)-g(0))/g(2)=196560$. We construct such functions from the radial Fourier interpolation basis in dimension $24$. If $a_2,b_2$ denote the basis functions dual to value and radial-derivative interpolation at radius $2$, then $g_C=a_2-Cb_2$ has exactly the nodal data needed for Poisson summation over the Leech lattice. The sphere-packing magic function identifies $b_2$ and supplies its strict signs. We prove that the removable quotients $a_2/b_2$ and $\widehat a_2/\widehat b_2$ are bounded on the required half-lines. The noncompact step follows from exact coefficient extraction in the interpolation kernel and an $S$-cusp expansion. Both quotients tend to $(43+240\log2)/15$; for all sufficiently large radii they lie on opposite sides of this limit, with an explicit first exponential correction. Consequently the admissible parameters in this affine family form a nonempty closed ray, and every member proves the conjectured identity. The same interpolation basis recovers every nontrivial Leech-shell coefficient.

Gorsky, Nekrasov, and Rubtsov introduced the moduli space of marked Higgs bundles over an elliptic curve $E$ and identified it with the Hilbert scheme of points on the cotangent bundle. Later, Groechenig constructed four analogous isomorphisms on marked rational curves, of affine Dynkin types $\widetilde D_4$, $\widetilde E_6$, $\widetilde E_7$, and $\widetilde E_8$, for the $\Gamma$-Hilbert schemes of $T^*E$ for cyclic groups $\Gamma$ with $|\Gamma|\in\{2,3,4,6\}$. We prove that the isomorphisms in all five cases are holomorphic symplectomorphisms.