Papers

The energy-space invertibility of a screen single-layer operator need not extend to the full range of shifted Sobolev trace spaces. We construct a compact Ahlfors regular one-dimensional screen consisting of intervals that accumulate geometrically at the origin. For this screen …

We prove weak existence for a stationary porous-flow model whose drag law changes at speed one and has a negative jump. At the threshold the drag is an arbitrary vector field, as in the free-transition formulation of the model. The …

The universality conjecture for the interleaving distance asks whether every distance stable under uniform perturbations of sublevel functions is bounded by the interleaving distance on realizable persistence modules. Its formulation for each homology degree is false for ordinary zero-dimensional homology. …

For a fixed positive finite interval and a fixed exponent between zero and one, we prove the Braess--Hackbusch conjecture for the leading constant in best absolute uniform rational approximation of a fractional inverse power. The approximants have numerator degree at …

We consider integration on the unit cube for smooth functions whose every partial derivative, including the function itself, is bounded by one. For every fixed absolute error tolerance $0<\varepsilon<1$, we prove that deterministic integration requires at least $(1-\varepsilon)(\tfrac12+\sin\tfrac12)^{-d}$ function evaluations …

For the normalized transposition statistic under Jack measure on partitions of $n$, we prove a Kolmogorov bound $10\max\{n^{-1/2},\sqrt\alpha/n\}$ for every $n\ge3$ and $\alpha\ge1$. This removes the logarithmic loss in the known bound, including the transition $\alpha\asymp n$, and proves the …

Let a graph consist of a root, two leaves adjacent to it, and an infinite ray starting at the root. We prove that this graph admits no color-symmetric, one-dependent proper four-coloring whose law on every ordered simple path agrees with …

For every integer $d\geq3$, we construct a proper coloring of the infinite $d$-regular tree with finitely many colors whose law is one-dependent and invariant under every tree automorphism and every permutation of the colors. The construction uses symmetric tensors in …

We prove that no algorithm decides whether a rational polynomial in the ordinary traces of powers of one real symmetric matrix is nonnegative for all finite matrix sizes. The proof encodes nonnegative integers as spectral multiplicities. Stationary polynomial interpolation controls …

Let \(A\) have \(m=\lfloor(1-\varepsilon)n^d\rfloor\) independent columns, each the tensor product of \(d\) random vectors in \(\R^n\). All scalar coordinates are independent, centered, of variance one, and have subgaussian norm at most \(K\). We prove that, for every fixed \(\varepsilon\in(0,1/2]\) and …

Let $G_n$ be uniform among the simple directed acyclic graphs on $n$ labeled vertices. We prove that its Markov equivalence class satisfies \[ (|(G_n)|>t) C[-c(\frac{ t}{ t})^2] \] for all sufficiently large $t$, uniformly in $n$. This implies the upper …

We determine the minimax integrated squared error for estimating the spectral density of a centered stationary Gaussian time series when each observation is released under local differential privacy and the release may depend on all earlier private outputs. On a …

The oracle consistency conjecture for Greedy Sparsest Poset asks whether every starting graph can reach a globally sparsest graph by legitimate mark changes and reconstruction, without increasing the edge count. We give a counterexample under its stated Markov and restricted-faithfulness …

We determine the minimax density-estimation rate under sequentially interactive local differential privacy and an anisotropic negative Sobolev loss, when the density and discriminator smoothness vectors are proportional. The result covers coordinates whose discriminator smoothness lies on opposite sides of the …

We prove uniqueness of a smooth real scalar potential from its full Dirichlet-to-Neumann map at one energy on an arbitrary known compact, connected, smooth Riemannian manifold of dimension at least three with smooth nonempty boundary. The only spectral assumption is …

For every finite $p\geq5$, a weak $p$-harmonic function on an arbitrary open subset of $\R^3$ belongs to $W^{2,2}_{\loc}$. The conclusion concerns the full unweighted distributional Hessian and includes neighborhoods of critical points without assumptions on the critical set. We prove …

We construct smooth real mean-zero functions on the two-dimensional unit torus whose gradients stay bounded in $H^{-1/2}+L^1$, while their $H^{1/2}+W^{1,1}$ norms tend to infinity. This disproves the mixed potential estimate proposed by Bourgain and Brezis. The construction uses a finite …

We prove an exact entropy chain rule for nonnegative densities satisfying a weak continuity equation with no boundary flux on any bounded Lipschitz domain. Finite kinetic action and Fisher information, together with finite initial entropy and weak continuity in time, …

A carrying simplex reduces the nonzero long-term dynamics of a competitive population map to an invariant hypersurface. We prove that its positive interior is a $C^1$ embedded manifold under the standard $C^1$ per-capita competition and spectral-radius assumptions. Equivalently, its radial …

We give necessary and sufficient conditions for a boundary map to be the geodesic scattering map of a smooth simple Riemannian metric on a fixed disk with a prescribed boundary metric. The conditions depend only on the proposed map and …

We construct a counterexample to integral Aleksandrov-Bakelman-Pucci estimates for nonlocal inequalities of the form \begin{equation*} L_au(x):=\int_{\mathbb R^n}δ_yu(x) \frac{a(x,y)}{|y|^{n+σ}}\,dy \ge-f(x)\quad\text{in }B_1, \end{equation*} with $u\le0$ in $\mathbb R^n\setminus B_1$, where $δ_yu(x)=u(x+y)+u(x-y)-2u(x)$, and the measurable coefficient satisfies \begin{equation}\tag{C} λ\le a(x,y)\leΛ,\qquad a(x,y)=a(x,-y),\qquad x\in\mathbb R^n,\quad …

Let $\Omega\subset\R^d$, $d\geq2$, be a bounded connected Lipschitz domain, and let $C$ be a real symmetric uniformly elliptic matrix with Lipschitz entries on $\overline\Omega$. We prove that the weak Dirichlet-to-Neumann semigroup on $C(\partial\Omega)$ is holomorphic of angle $\pi/2$ and bounded …

We prove the Fourier entropy--influence inequality for every linear threshold function, with a constant below $8.937$ and no restrictions on its weights or threshold. The argument applies to regular Boolean functions and gives logarithmic bounds for their Boolean combinations and …

Among triples of pairwise disjoint nonempty connected open subsets of the unit disk, the three sectors of opening $2\pi/3$ minimize the largest first Dirichlet eigenvalue. The minimum is $j_{3/2,1}^2$, where $j_{3/2,1}$ is the first positive zero of the Bessel function …

We prove uniform null controllability of an advection--diffusion equation with Neumann boundary conditions as the diffusion coefficient tends to zero. The drift is autonomous, belongs to $W^{2,\infty}$, points strictly outward on the boundary, and its backward flow carries a neighborhood …

We construct a polynomial control-affine system in five dimensions, with two controls in $[-1,1]$, that is small-time locally controllable at the origin but admits no polynomial lower bound for its reachable neighborhoods. Every point on the negative fourth axis reachable …

We give a smooth one-dimensional obstacle and a smooth admissible initial state for which no endpoint controls can steer the parabolic obstacle problem exactly to its stationary energy minimizer in finite time. The obstacle has two interior maxima of equal …

We consider a Schrödinger field on a planar annulus with a dynamic inner boundary and a Dirichlet outer boundary. When the surface dispersion coefficient is one half of the bulk coefficient, observation of the outer normal derivative fails to control …

We consider a wave equation in a planar annulus, with a Dirichlet condition on the outer circle and a dynamic wave equation on the inner circle. The bulk and surface propagation speeds are both one. For every finite observation time, …

We consider a one-dimensional tank governed by the normalized Saint--Venant equations, with the tank acceleration as its only control. We show that local exact controllability around the flat equilibrium, with uniformly small classical trajectories in $C^1$, fails at time $T=21/10$. …

We characterize the normal heat fluxes that can be produced on an observation patch by square-integrable Dirichlet data on a disjoint control patch. The domain is any bounded connected smooth Euclidean domain in dimension at least two. The characterization consists …

We consider the heat equation on an interval with two nonnegative Dirichlet boundary controls. For every pair of distinct positive constant states, we prove that the nonnegative finite Radon measure pair attaining the minimum time is unique, under the terminal …

A complete negative theorem for every real p>2, retaining the original independent Bernoulli sampling rule, expected-size budget, and simultaneous norm-power embedding assertion.

Proof of the full original target, using Prony reconstruction and a Koszul determinant certificate; draft pending authoritative Linux verification and final independent review.

We determine the supremum of the improvement in polynomial convergence rate due to randomization for approximation of bounded linear operators between real Banach spaces using nonadaptive continuous linear measurements. The supremum is one. We construct fixed infinite-dimensional operators whose gains …

We determine the minimal time for boundary null control of the Burgers equation linearized at the stationary shock $-\tanh(x/(2\eps))$ on $(-L,L)$, with a control at the left endpoint. For arbitrary $L^2$ initial data, the worst-case $L^2$ control cost remains bounded …

We prove that Euclidean kinetic energy and arbitrary smooth potentials, considered in all dimensions, satisfy no universal Lie identities beyond those of the free polynomially graded Lie algebra of classical mechanics. Polynomial potentials already suffice to separate every nonzero element. …

For a graph $G$, let $ν(G)$ denote the maximum number of pairwise edge-disjoint triangles and let $τ(G)$ denote the minimum number of edges meeting every triangle. Tuza's conjecture asserts that $τ(G)\le 2ν(G)$ for every graph $G$. We first note that …

We prove that the local energy approximation widths of scalar, symmetric, uniformly elliptic equations in dimensions two and three decay as $\exp(-c n^{1/(d-1)})$, with no regularity assumption on the coefficients beyond measurability and boundedness. The result applies to the normalized …

For a finite simple graph $G$, let $ν(G)$ be the maximum number of pairwise edge disjoint triangles and $τ(G)$ the minimum number of edges meeting every triangle. Tuza conjectured in 1981 that every such $G$ satisfies $τ(G) \le 2 ν(G).$ …

We prove that every value of the arithmetic finite quantum dilogarithms constructed by Radchenko and Wheeler (arXiv:2609.21892) is an algebraic unit, and determine the Artin action for arbitrary quadratic pseudolattices. In particular, the associated Stark units belong to the predicted …

Local bases for finite-element potential spaces provide the algebraic decompositions needed for vertex-patch corrections. On a consistently oriented Freudenthal mesh of the cube, we prove that continuous vector polynomials of degree $k+1$ with continuous curl have a basis supported on …

Li and Wu proved that their unfiltered integrator for periodic KdV converges in $L^2$ with error $O(\tau^\gamma\log(1/\tau))$ for real, mean-zero $H^\gamma$ data, $0<\gamma\leq1$. We remove the logarithmic factor for the same time-semidiscrete algorithm, with constants uniform on bounded $H^\gamma$ sets. …

We construct a continuous complex scalar symbol with neither an inner nor an outer annular holomorphic extension, whose Toeplitz sections are not canonically distributed. Along an infinite sequence of orders, each of two points outside the symbol range is an …

We study the decision problem of global asymptotic stability (GAS) of the origin for homogeneous polynomial vector fields with rational coefficients. In the first part of the paper we give, for every fixed dimension $n$ and odd degree $2k+1$, a …

We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling--Ahlfors transform is $p-1$ if $p\geq 2$, and $(p-1)^{-1}$ if $1<p\leq 2$.

The undirected multiple-unicast conjecture [LL04] asserts that network coding offers no throughput advantage over multicommodity flow. We refute this conjecture by constructing a deterministic linear network code over $\mathbb{F}_9$ on a 182-vertex bipartite subgraph of the point-line incidence graph of …

We prove the conjecture that the irrationality exponent of \(\pi\) is \(2\). As a consequence, the classical Flint--Hills series \(\sum_{n\ge1}1/(n^3\sin^2 n)\) converges, with angles in radians.

We resolve the Falconer distance conjecture in every dimension. For every integer \(d\ge2\), a compact subset of \(\R^d\) with Hausdorff dimension greater than \(d/2\) determines a set of Euclidean distances of positive Lebesgue measure.

We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to …