Papers

Local quantum interactions generate dynamics in an exponentially large Hilbert space, yet locality and entanglement restrict the information that can spread and accumulate. Bounds on propagation and entanglement growth, together with tensor networks, exploit these restrictions to discard information unnecessary for describing the evolution. This leads to a sharper question: how much information must any low-rank representation retain? Here we determine these limits, up to logarithmic factors, for short-range interactions. For time-independent evolution, the optimal rank obeys $\log D=\widetilde O(t+\sqrt{\log(1/\epsilon)})$, with matching lower bounds fixing the accuracy exponent $1/2$; for arbitrary driving, matching fixed-time bounds instead give $2/3$. Correspondingly, the corresponding dynamical entanglement spectra exhibit distinct small-$\alpha$ Rényi laws, $\alpha^{-1}$ and $\alpha^{-2}$. In one dimension, the static limit is constructively attained by an explicit MPO algorithm, with an analogous extension to Liouvillian dynamics. Together, these results determine the irreducible information required to represent local quantum evolution and uncover distinct entanglement structures in static and driven dynamics.

We prove that the Turán density of the tetrahedron $K_4^{(3)}$ satisfies $\pi(K_4^{(3)}) \le 14993367693127837/26880000000000000 < 0.557789$, improving Baber's upper bound of $0.5615$ and closing about $62\%$ of the gap to the conjectured value $5/9$. The proof uses an exact seven-vertex flag-algebra certificate incorporating degree-stationarity from Razborov's differential method. To find the certificate, we combine the established techniques of cutting planes and column generation to optimize jointly over flag families whose types have at most five vertices. We give a complete formal proof of this Turán density bound in Lean 4.

We establish a quantitative sound-soft scattering limit for a finite bubbly crystal near the upper edge of the first Bloch band of the corresponding infinite crystal. The inclusions form a dense periodic array of period $\varepsilon$ in a bounded Lipschitz domain, and their density contrast is $\delta=\varepsilon^2$, with fixed positive wave speeds. Under coordinate-reflection symmetry, we prove that the normalized capacitance symbol has a unique nondegenerate maximum on the Brillouin torus. Its Hessian defines a positive Dirichlet elliptic operator governing the limiting spectral detunings. A mean-constrained variational formulation allows us to compare the actual finite-array capacitance matrix with the truncated infinite-lattice operator and to prove exponential localization of their difference near the sample boundary. We obtain an $O(\varepsilon)$ norm-resolvent approximation and connect it to the full acoustic scattering problem, retaining the second-order Bloch correction required by the frequency scaling. For rescaled detunings outside the effective Dirichlet spectrum, the exterior $L^2$ and far-field discrepancies are $O(\varepsilon)$, while the interior $L^2$ field is $O(\varepsilon^{1/2})$. Numerical experiments illustrate the far-field convergence and the effective spectral modes. Explicit one-dimensional calculations describe finer Fabry–Pérot transmission windows and show that a fixed frequency strictly inside the first band need not have a unique scattering limit.

We study ball-covering properties in Lipschitz-free spaces. We establish an extension criterion for proving that $\mathcal F(M)$ fails the ball-covering property and apply it to several classes of nonseparable metric spaces. In contrast, we construct a nonseparable uniformly discrete metric space $M$ such that $\mathcal F(M)$ has the uniform ball-covering property and is isomorphic to $\ell_1(2^\omega)$. More precisely, $\mathcal F(M)$ has the $\alpha$-ball-covering property for every $\alpha\in[-1,1)$. This example also shows that these quantitative ball-covering properties are not hereditary within the class of Lipschitz-free spaces. Finally, we prove some stability results under sufficiently small bi-Lipschitz perturbations of the metric.

Large coupled systems may be chaotic at the microscopic level while exhibiting stable collective behaviour at the macroscopic level. Distinguishing between these two forms of instability has been a problem in the physics literature, and several approaches have been proposed. In this paper, we address this problem by introducing and studying observable Lyapunov exponents, which measure the rate at which perturbations of initial conditions grow or decay when viewed through a chosen observable, rather than in the full phase space. For globally coupled maps, we consider finite-time observable Lyapunov exponents and define their mean-field counterpart by first taking the infinite-system limit and then the long-time limit. Our main result shows that, for a broad class of symmetric macroscopic observables, these exponents turn out to be determined by the linearisation of the self-consistent transfer operator, which is the nonlinear operator governing the evolution of the population distribution. We then apply this result to weakly coupled uniformly expanding maps and prove that their mean-field Lyapunov exponents are negative near a stationary mean-field state. Thus, although the microscopic dynamics is chaotic, perturbations decay at the macroscopic level. Our results provide a rigorous framework for distinguishing microscopic from macroscopic chaos in large interacting systems.

We give snakes that are longer than the previous best known in dimensions 13 through 20 and improve the general lower bound for every dimension $d \geq 21$. Our explicit snakes reach 371,711 edges in dimension 20. Twenty compatible paths in that cube allow generalisation to give snakes of length at least $(17/48)2^d$ for every $d \geq 21$. Their controlled overlaps allow copies to be joined across the layers of a larger cube without creating shortcuts. Four additional paths give the same bound for coils. We explain the construction, prove the joining rule, and then count its length. The finite paths and their required intersections are independently verifiable.

We prove that a stable noncompact hypersurface with constant anisotropic mean curvature in $\mathbb{R}^{n+1}$ has only one end for $n \le 6$, under a natural ellipticity condition on the anisotropy. This provides an anisotropic counterpart of the one-end theorem for stable constant mean curvature hypersurfaces. We remark that the result is also new in $\mathbb R^7 (n=6)$, in the case of anisotropic minimal hypersurfaces. The proof relies on the existence of a Sobolev-type inequality.

Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. A subalgebra $\mathfrak{A}$ in $\mathcal{B}(\mathcal{H})$ has the left (resp.\ right) partial factorization property if for any invertible operator $S\in\mathcal{B}(\mathcal{H})$, there exists an isometry (resp.\ a co-isometry) $U\in\mathcal{B}(\mathcal{H})$ such that $U^*S, S^{-1}U\in\mathfrak{A}$. We show that if $\mathfrak{A}$ is weak operator topology closed with the left (resp.\ right) partial factorization property, then $\mathfrak{A}$ is the nest algebra associated with its invariant subspace lattice. In particular, if $\mathfrak{A}$ is transitive, then $\mathfrak{A}=\mathcal{B}(\mathcal{H})$. This gives a positive answer to Question 6.3 raised by B.V.R. Bhat and M. Kumar in Publ. Res. Inst. Math. Sci. 60(2024), 507–537.

We study strong measurability and adapted approximations for maps taking values in possibly nonseparable metric spaces. After recalling the metric-space Pettis criterion, we give conditions under which measurability implies strong measurability. Under the continuum hypothesis (CH), every measurable map from a countably generated measurable space into a metric space is strongly measurable for every measure on the domain. For finite or $\sigma$-finite measures, the same conclusion holds when the target density is strictly smaller than every real-valued measurable cardinal. Under CH this includes targets of density at most $\mathfrak c$; if no real-valued measurable cardinal exists, it holds for all metric targets. For function-valued maps, we show that pointwise measurability and continuity of the sections need not ensure measurability in the function-space topology. We provide a verification criterion using countably many evaluations and essential separability of the range. Further, we obtain elementary $L^p$-approximations of progressive or jointly measurable adapted processes under suitable filtration assumptions, Lipschitz functionals with path and Wasserstein variables, and smooth approximations based on finitely many observations of a process with left- or right-continuous paths. Finally, under CH and suitable assumptions, we apply these results to controlled stochastic differential equations in separable Hilbert spaces, allowing for jumps, random coefficients, path dependence, and state- and control-law dependence; we establish convergence of states and costs uniformly over admissible controls, convergence of optimal values, and transfer of near-optimal controls.

The exponential decay of weak solutions towards the constant steady state of general cross-diffusion systems in bounded domains with no-flux boundary conditions is investigated. The proof of quantitative decay rates is based on the relative entropy method and involves two parameters: the exponent determining the entropy density and the exponent in the entropy production integral. The key ingredient are novel convex Sobolev inequalities, which extend existing results in the literature to a broad range of values of the two parameters. These inequalities are established using convexity arguments and the Gagliardo-Nirenberg inequality.

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

We study sequential testing of a fixed family of hypotheses when observations are costly and a sampling horizon is specified in advance. The challenge is to pool evidence for earlier decisions when the number and identities of false hypotheses are unknown, while controlling the probability of any false rejection at level $\alpha$. Existing merges attain the pooled growth rate at a single number of false hypotheses: averaging when one is false, multiplying when all are. Under an independent-stream model with common simple null and alternative distributions, we test each intersection with a prior-weighted mixture of products of marginal likelihood ratios. Closed testing combines these elementary-symmetric-polynomial mixtures to identify individual false hypotheses. Design-specific boundary calibration gives finite-horizon family-wise error control, with exact finite-state guarantees or a confidence qualification for Monte Carlo calibration. The prior-matched mixture uniquely maximizes expected log evidence at each horizon. Mixtures assigning positive weight to every nonempty subset of streams attain log-growth rate $lD$ when $l$ streams follow the alternative. Here $D$ is the mean log likelihood ratio per alternative observation, and a round supplies one observation per stream. This rate attains the first-order intersection-delay lower bound as $\alpha\downarrow0$ at fixed dimension, configuration, weights, and a long enough horizon. Power for an individual hypothesis cannot exceed the best single-stream power at a given deadline, but closure removes the multiplicity penalty when all are false. Gaussian, basket-trial, language-model and advertising studies illustrate both. The primary basket boundaries are 40-52% below $1/\alpha$. Across 41 simulated configurations, the median reduction in capped mean patient outcomes relative to prespecified interim-look Bonferroni tests is 31%.

We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound $\sup_{n\ge1}\|I-T^n\|\le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd power of $T$ a Ritt operator. The sharp unconditional threshold for $T$ itself is $\sqrt3$. If $\Phi_*(q)$ denotes the optimal universal Ritt resolvent bound below this threshold, then $\Phi_*(q)=2/(\sqrt3-q)+O(1)$ as $q\uparrow\sqrt3$. At every higher peripheral threshold the optimal finite-peripheral resolvent bound has reciprocal order of growth. We also obtain constructive bounds for general weighted Wiener symbols. For each $1<q<2$, an angular escape invariant characterises exactly the sampling sequences for which a displacement bound by $q$, together with the peripheral spectral condition, forces the Ritt property. We compute this invariant for asymptotically geometric sequences. At the endpoint $q=1$, suitable phase conditions imply the full displacement bound without assuming power boundedness; normal contractions admit an exact scalar criterion. Finally, we establish quantitative Hilbert-space and $L^p$ resolvent estimates, a characterisation of operators with odd Ritt powers on uniformly convex spaces, and strict norm gaps for operator-valued disc-algebra functions, with explicit matrix-polynomial bounds from Fejér–Riesz factorisation.

We study positive mass and related Brown–York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi–Tam type construction to obtain a Brown–York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.

Let $G$ be the line graph of a finite simple graph, with $n\geq1$ vertices and maximum degree $\Delta$. We prove that single-site Glauber dynamics for uniform proper $q$-colorings mixes in $O_\Delta(n\log(n/\varepsilon))$ steps for every integer $q\geq\Delta+5$. Our proof uses the Bochner framework of Chen and Liu (2026).

For any integers \(N \ge 2\) and \(m \ge 1\), we prove that there exist infinitely many pairs of real quadratic fields \(\QQ(\sqrt{D}), \QQ(\sqrt{D + m})\), with \(D \in \ZZ\) and \(D > 0\), such that the class numbers of both fields are divisible by \(N\). Write \(N = 2^{e}n\) with \(n\) odd. If \(n > 1\), then the class groups of both fields in each such pair contain an element of order \(n\).

Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\ldots,p_n$, and let $S=\sum_i B_i$. Hillion and Johnson proved that the Shannon entropy of $S$ is jointly concave in the parameter vector and proposed corresponding critical-order conjectures for R'enyi and Tsallis entropies, with predicted thresholds $2$ and approximately $3.65986$, respectively. We determine both thresholds exactly. For every $0<q<1$, the power sum $\sum_k \mathbb P(S=k)^q$ is jointly concave in $(p_1,\ldots,p_n)$, and strictly concave on the open parameter cube. Consequently, the R'enyi and Tsallis entropies of order $q$ are jointly concave. At $q=1$ this agrees with the Shannon theorem. For every $q>1$, joint concavity fails already for the sum of two Bernoulli variables: a transverse interpolation in which the two parameters move in opposite directions gives strict local convexity for both entropies. Hence the universal joint-concavity range for both families is exactly $0<q\leq 1$. Below order one, the proof combines the Hillion–Johnson transport inequality with an explicit nonlinear telescoping correction. The corrected local curvature reduces to a two-dimensional quadratic form. An exact Riccati identity, together with a one-sided zero-crossing argument, proves positivity of its determinant throughout the full range $0<q<1$.

We obtain the relation between non-hermite skew information $NI_{\rho,\alpha}(A)$, $NU_{\rho,\alpha}(A)$, variance $NV_{\rho}(A)$ and $NJ_{\rho,\alpha}(A)$ by using the maximal eigenvalue and the minimal eigenvalue of density $\rho$. And also we obtain their refinements.

We construct a graded Lie algebra structure on the space of cyclically skew-symmetric cochains whose Maurer-Cartan elements characterize double Lie algebra structures. Twisting by a fixed double Lie bracket yields a cochain complex that is isomorphic to the positive arity part of the complex introduced by Fairon and Valeri after reindexing and degreewise sign adjustment. Using the Koszul resolution of the double Lie properad, we identify the resulting differential graded Lie algebra with the convolution differential graded Lie algebra on the deformation complex of the fixed double Lie algebra. We also construct a graded Lie algebra governing skew-symmetric Rota-Baxter operators on a finite-dimensional symmetric Frobenius algebra. For a finite-dimensional vector space $V$ and $A=\en(V)$, we lift the known correspondence between double Lie algebra structures on $V$ and skew-symmetric Rota-Baxter operators on $A$ to an isomorphism of their governing graded Lie algebras. After twisting by corresponding Maurer-Cartan elements, this isomorphism becomes an isomorphism of differential graded Lie algebras and hence induces an isomorphism between the associated cohomology groups. We further prove that the cohomology of a skew-symmetric Rota-Baxter operator on a symmetric Frobenius algebra can be seen as the cyclic cohomology of its descendent associative algebra. Thus the cohomology of a double Lie algebra is identified with the cyclic cohomology of the descendent associative algebra associated with the skew-symmetric Rota-Baxter operator on the matrix algebra.

In this paper, we construct axially symmetric kernels on the sphere. We start by taking an isotropic kernel and transform it into an axially symmetric kernel via a specific deformation. For a general deformation, an explicit spherical-harmonic representation is not necessarily available. For our deformation, we can obtain such a representation of the deformed kernel in terms of the expansion coefficients of the original isotropic kernel. We derive this representation by explicitly describing the effect of the deformation on the spherical harmonic basis. From this, an explicit finite sum formula for the coupling coefficients is deduced and we show that, for each fixed input mode, the coefficients decay geometrically at a rate controlled by our deformation parameter. We further note that if the native space of the isotropic kernel is norm equivalent to a Sobolev-space, then the deformed kernel has the same native space with an equivalent norm. In our numerical experiments, we show that our deformation can improve prediction when the underlying random field has an axially symmetric covariance.

We investigate a class of differential inclusions governed by non-autonomous and autonomous maximal monotone operators, involving set-valued perturbations with a Volterra integral term and subject to a nonlocal condition. Under suitable assumptions on the governing operator, the set-valued perturbation, and the Volterra kernel, we establish existence results for solutions. In particular, the set-valued perturbation is assumed to satisfy a one-sided Lipschitz condition, while the Volterra kernel is required to be Lipschitz continuous. The existence of a trajectory is established by means of an iterative construction and an application of Zorn's lemma. Finally, several examples are presented to illustrate the applicability of the abstract results.

In this paper, we develop a bordism-theoretic approach to rigidity problems for toric and quasitoric manifolds. We prove that two toric manifolds are isomorphic as varieties if and only if they are weakly equivariantly unitary bordant. We also establish a parallel rigidity result for omnioriented quasitoric manifolds satisfying the injectivity condition, showing that their equivariant unitary bordism classes completely determine their omniorientation-preserving equivariant homeomorphism types. Thus, equivariant bordism provides a topological framework for detecting geometric and combinatorial rigidity.

In our previous results, we have proposed a new free boundary problem representing the bread baking process in a hot oven. The problem consists of conservation laws for the internal energy and water mass, the Stefan condition, the boundary and initial conditions. We note that the boundary condition for the water content contains the temperature at the boundary. Accordingly, we do not expect a strong solution for $W^{1,2}$-initial data. To address this difficulty, we established strong solvability by assuming stricter conditions for the initial functions in [3]. In contrast, this paper aims to prove the existence of a weak solution, when these assumptions for initial functions are relaxed.

Proximal-Tracking is a novel algorithm proposed in [Falsone and Prandini, Automatica, 135 (2022), 109938] for nonsmooth distributed consensus optimization with local set constraints. The global convergence and numerical behavior of Proximal-Tracking have been well studied in [Falsone and Prandini, Automatica, 135 (2022), 109938]. As a complement, in this paper we provide a theoretical analysis regarding its linear convergence rate under additional strong convexity and calmness assumptions. In particular, this analysis does not require assumptions such as differentiability or smoothness, thus preserving the original advantages of the algorithm.

Basing on new curvature conditions for the Bochner technique of Petersen-Wrink in \cite{PW21}, in this paper, we give some vanishing properties for weighted $L^Q$ $p$-harmonic $\ell$-forms on weighted manifolds, where $p>1, Q>p-1.$ These results generalize some previous theorems for $L^Q$ harmonic $\ell$-forms of Dung-Dung-Hung on the paper \cite{DDH25}.

A latent severity has a bounded lower tail with density of shape alpha and scale L. It is observed only through a fixed Markov kernel K that is biased and non-additive. The relative conditional spread of K diverges at the endpoint. Our sample is i.i.d. from the marginal Q alone, with no anchoring covariate or instrument. We prove a dichotomy. The shape index alpha is identifiable: for every admissible choice of the class constants, any two observationally equivalent members of a lean class share alpha, determined by a near-endpoint expansion of Q. The rate, namely L and the fixed-scale exceedance p_tau, does not survive. There exist admissible shared class constants and two members of a smaller regularity class whose observed laws coincide exactly. Across the pair alpha agrees, whereas L and p_tau move. A degenerate Le Cam two-point bound excludes any uniformly consistent estimator of either, and pointwise consistency fails at one member. Only the rate needs an anchor. We conjecture that a known kernel family with known edge map identifies the rate fiber by fiber if and only if the family satisfies a fixed-scale injectivity clause, and we prove the sufficiency direction. In surrogate safety, uncalibrated conflict data give the shape of near-crash risk, not its absolute rate.

This paper is concerned with the inverse source problem for the stochastic fractional Helmholtz equation driven by white noise. For every fractional order $0<\alpha<1$, we prove the existence and uniqueness of the outgoing distributional solution to the direct problem with resolvent estimates at high frequencies, and establish its stochastic representation. For the inverse problem, we demonstrate that the variance of the source can be uniquely determined by the correlated random exterior data at a single frequency. We further establish an increasing stability estimate for the inverse problem by using the multifrequency correlated exterior data. Our stability result shows that as the upper bound of the bandwidth of the utilized frequency increases, the stability will also improve. The analysis employs the construction of geometric optics solutions, which connects the correlated data to the X-ray transform of the variance.

This paper proves the Gromov–Yomdin property for smooth projective varieties with finite Albanese morphisms. Our approach is to study the orbits of stability conditions under the action of autoequivalences on Bridgeland stability manifolds. For a finite generating set of the derived category, we consider the locus on which all its objects are semistable. Using this locus, we formulate an orbit escape principle, which provides a sufficient condition for an autoequivalence to satisfy the Gromov–Yomdin equality. Together with the stability of simple semihomogeneous bundles with respect to arbitrary numerical stability conditions, this principle readily yields the result for abelian varieties. We then reduce the case of varieties with finite Albanese morphisms to the abelian case by passing to suitable finite étale covers. Finally, we construct autoequivalences with positive categorical entropy on every abelian variety and show that a smooth projective variety with a finite Albanese morphism admits such an autoequivalence if and only if it is not of general type.

Weighted sum-rate maximization in multicell multiple-input multiple-output (MIMO) networks is commonly addressed by the weighted minimum mean-square error (WMMSE) algorithm or fractional programming (FP), both of which, after fixing their auxiliary variables, solve a power-constrained quadratic transmit problem that requires costly dense operations for large arrays. Replacing the underlying curvature matrix with a scaled identity can avoid the matrix inverse operation and thereby reduce complexity, but it discards the curvature eigenvalue structure and yields a loose lower bound. We propose Spectral-NFP, which retains selected dominant curvature eigenpairs and uses a scaled identity matrix to bound the curvature on the remaining subspace from above. The retained rank thus traces a continuous path from NFP to the exact WMMSE transmit update. With the surrogate curvature fixed, Spectral-NFP can be interpreted as Euclidean projected-gradient ascent after a linear coordinate transformation, admitting Nesterov-type acceleration. We derive a lower bound on the one-step transmit-objective gain of Spectral-NFP relative to WMMSE, expressed in terms of the curvature eigenvalues. Under an idealized Wishart model, we analyze this bound in both finite dimensions and the large-system limit, obtaining an asymptotic rank-selection rule. Experimental results show that retaining at most 20% of the transmit dimension, and often less than 10%, achieves more than 99% of the WMMSE WSR. In large-array settings, the measured update time is below 20% of that required by WMMSE.

For a finite simple graph $G$, the matching complex $M(G)$ is the simplicial complex whose vertex set is the edge set of $G$ and whose simplices are all the matchings in $G$. The topology of the matching complex of the $m\times n$ grid graph $G_{m\times n}$ is known only for $m = 1,2$, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex $M(G_{3 \times n})$. We prove that for $n\ge2$, its reduced homology vanishes in dimensions $i \leq n-2$ and in top dimension, while $\tilde{H}_{n-1}(M(G_{3\times n}))\neq 0$. We also show that $M(G_{3 \times n})$ is simply connected for $n \geq 3$. Consequently, the topological connectivity of $M(G_{3\times n})$ is $n-2$.