Papers

For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and …

We give a uniform deterministic polynomial-time algorithm for scheduling unit-length jobs with arbitrary precedence constraints on three identical parallel machines. The algorithm constructs a schedule of minimum makespan and decides exactly whether all jobs can finish by a specified deadline. …

For arbitrary complex inputs in $L^3(\R^2)$, we prove the pointwise maximal $L^3\times L^3\to L^{3/2}$ estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and $L^{3/2}$ convergence as the lower endpoint …

We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof …

We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the …

We prove that a molecule with $M$ fixed nuclei of real charges at least one and total nuclear charge $Z$ strictly binds at most $Z+CM$ electrons, where $C$ is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with …

We prove the Unique Games Conjecture. For every fixed \(\varepsilon,\delta\in(0,1/2)\), we give a deterministic polynomial-time reduction from \(\mathrm{3SAT}\) to Unique Games over a fixed finite alphabet, with completeness at least \(1-\varepsilon\) and soundness at most \(\delta\).

We prove that weak Ma--Trudinger--Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, …

We prove global existence and uniqueness for arbitrary smooth admissible initial data in the three-dimensional, one-species relativistic Vlasov--Maxwell system. The particle density is initially compactly supported, and the electromagnetic fields have finite energy and bounded derivatives of all orders. The …

We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on $S^2\times\T^2$ for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square …

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to \(\pi\) is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.

Toeplitz matrices are among the simplest structured models of translation invariance, arising naturally in convolution, stationary covariance models, and discretizations of translation-invariant operators. For a pair of real matrices, the Böttcher--Wenzel form (BW form) is a nonnegative quartic measuring the …

We prove the sharp horizontal isoperimetric inequality in the first Heisenberg group for all sets of finite horizontal perimeter and positive finite volume, with equality precisely for left translates and dilates of the Pansu ball.

Gaussian elimination is the oldest and most popular method for solving a linear system. Its numerical stability for a given matrix is controlled by the growth factor, a measure of how large entries can become during elimination. In this work, …

We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n^{1.9992})$ time and APSP on directed $n$-vertex graphs with …

For each $m \ge 3$, an explicit connected loop-free quantum graph with cycle rank $\beta_m = 2m - 1$, no vertex of degree two and rationally independent edge lengths, under the standard Kirchhoff Laplacian, whose nodal surplus $S_m$ satisfies $\operatorname{Var}(S_m)/\beta_m …

The Kannan--Lovász--Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on $\mathbb R^n$ has a Cheeger constant bounded below by a universal positive constant. We prove that $ψ_n\le C$ for a universal constant $C>0$, resolving the KLS conjecture. We …

One fixed inclusion from $\ell_1([0,1])$ to $\ell_2([0,1])$ disproves every universal continuity-cost constant over arbitrary real Banach spaces. The construction uses nonseparable spaces and does not settle a separable variant.

The bounds are $\lfloor\log_2m\rfloor+1\le N(m)\le\lceil\log_2m\rceil+1$. This gives the sharp leading asymptotic and exact values at powers of two. It does not determine every $N(m)$, including $N(3)$.

A fixed smooth amplitude on a smooth closed curve with a straight arc gives growth of order square root of log k. This addresses the arbitrary smooth amplitudes and general parametrizations allowed in the target; it does not assert a …

The uniform estimate fails with fixed data bounds and G(p)=1-p. The proof selects gamma after the mesh; the numerical experiments are corroboration, not a uniform choice of gamma.

We study the computational complexity of static output feedback (SOF) stabilization for continuous-time linear systems. For rational plant data, we prove that this problem is complete for the existential theory of the reals: it is equivalent, under polynomial-time transformations, to …

No: ||S_n psi_0 - e^{iH} psi_0|| <= C n^{-3/8}, so n^{1/4} times the error tends to zero and no eventual lower bound c n^{-1/4} holds..

No: for n = 4, x* = (1,1,1,1) and an explicit rational D-stable matrix A, the system has a nonconstant periodic solution with all coordinates in (1/8, 4), which does not converge to x* (and so for every n >= …

We prove that the von Neumann entanglement of purification is nonadditive. For two-qubit Werner states $W(f)$ with singlet weight $1/64\le f\le1/4$, we show that canonical purification is globally optimal: its entanglement, also known as reflected entropy, gives the exact entanglement …

We give a computer-assisted counterexample to the universal absolute-continuity assertion in AIM Problem 637. The admissible parameters are two equally skilled players, logistic scale $c=1/2$, and update size $K=9/10$. The stationary law is singular continuous, has full support on the …

We construct bounded initial data for the viscous Burgers equation whose translated late-time limits contain any prescribed profile represented by a compactly supported probability measure. A uniform fixed-time locality estimate for the Cole--Hopf formula allows backward histories to be placed …

We propose a local branch of nonnegative periodic solutions of the delayed noisy leaky integrate-and-fire reset equation with inhibitory coupling and nonconstant firing rate. Translating both voltage thresholds with the coupling keeps a centered stationary reset profile fixed. An explicit …

The fixed-degree pullback matching distance of verbose Vietoris--Rips barcodes fails the triangle inequality. In degree one, three four-point ultrametric spaces, with nonzero distances only $1$ and $2$, have pairwise distances $0,0,1$. More generally, in every degree $k\ge1$ and over every …

T_*(a) = 2 for every bounded measurable a >= 0: uniform observability holds for every T > 2 and fails for every T < 2..

We give a candidate disproof of the diameter-normalized positive-Robin fundamental-gap conjecture stated as AIM problem \#021. A convex polytope with square end sections and one transverse central protrusion is made thin in two directions. The first transverse Robin energy has …

We give a proposed counterexample to the unresolved general statement in AIM Problem 506, equivalently Conjecture 4.12 of Ayanbayev, Klebanov, Lie and Sullivan. On $\ell^2$ we construct a countable product of scaled copies of the density $\rho(z)\propto e^{-z^4}$ and an …

For a centered one-dimensional Gaussian target and the Gaussian kernel, we prove the strict sign inequality governing the variance in the Gaussian moment approximation to de-regularized maximum mean discrepancy (DrMMD) flow. The result holds for every target variance $s>0$, every …

We give an explicit bounded, simply connected planar domain with smooth strictly convex boundary on which uniform exponential interior decay of boundary-normalized Steklov eigenfunctions fails. At a fixed interior point, the squared eigenfunction values have no finite positive exponential spectral …

For a fixed correlated Gaussian target, consider its optimal diagonal Gaussian approximation under $D_\alpha(p\Vert q)$, with $\alpha>1$. We give a rational three-dimensional target whose optimal covariance is the identity at $\alpha=10$, but whose second and third variances are strictly smaller …

Under the assumptions of AIM problem 457, every sufficiently fine uniform time grid gives an unweighted frame on the infinite observation window. The Bessel hypothesis on the initial sensors and the continuous lower frame bound force the normal evolution operator …

No: for d = 2, C = I and the three-quarter disc sector 0 < r < 1, 0 < theta < 3pi/2, no finite M and omega give the bound; the conormal derivative of r^{2/3} sin(2theta/3), of size r^{-1/3} …

Yes: for every d >= 2, every bounded connected C^infty domain Omega and every constant accretive C there is K(Omega, C) < infinity with ||u_phi||_inf <= K ||phi||_inf for all phi in H^{1/2} cap L^inf, from Agmon's 1960 maximum estimate …

Yes: for every nonzero f analytic in a neighbourhood of [-1,1], c_f(epsilon) <= C_f log(1/epsilon) for small epsilon, with all three global norm bounds, by an epsilon-dependent smooth cutoff and a smooth Fourier truncation..

Yes: for every d ≥ 2, χ > 0 and 1 < m < 2 + 2/d the infimum over A is attained, by a radially nonincreasing, compactly supported minimizer..

No: for n = 2, s = 1/2, λ = 1, Λ = 4 the half-plane kernel 1{tr > 0}/|x − z|^3 satisfies both conditions, while smooth v_ε have bounded K-energy and H^{1/2} energy growing like log(1/ε)..

Yes: for every such (M, g) and rho there are phi_j smooth up to the boundary with the weighted L^2 error tending to zero, by Sobolev extension of sqrt(rho) to the double of M, Erbar's subdifferential theorem there, and restriction..

No: for Z = R with the metric min{1, |s - t|}, M_infinity(Z) is disconnected, hence not path connected..

No: on the unit disk with A = I, rotating drift B = b(-x_2, x_1), a constant potential, alpha = 1/2 and T = 1, a real solution with u_0 != 0 has u(T) = 0, so no finite constant …

For every fixed order $k \ge 3$, we prove that the expected injective norm of a tensor with iid mean-zero integrable entries is comparable to its largest expected Euclidean fiber norm, uniformly over all rectangular dimensions and entry laws. Precisely, …

We show that input-to-state stability (ISS) and integral input-to-state stability (integral ISS) with respect to the space of essentially bounded functions are equivalent properties for linear systems. The proof combines recent results on Orlicz-admissibility and a duality result due to …

For every fixed real $p > 2$, the manuscript constructs full-column-rank matrices with total $\ell_p$ sensitivity $\mathcal S = \Theta_p(d)$ for which every nonnegative reweighted row subset preserving all $\|Ax\|_p^p$ within $1 \pm \varepsilon$ needs $\Omega_p((\mathcal S + d)\varepsilon^{-p})$ rows, …

Let $N = 2^d$, and sample $m$ rows of the Walsh matrix independently and uniformly, retaining repetitions. We determine, up to universal constants, the smallest $m$ for which the normalized sample has restricted isometry constant at most $1/2$ at sparsity …

In this paper we give a polynomial-time reduction from the Subset Sum Problem to unconstrained static output feedback stabilization, establishing its NP-hardness. The construction uses two scalar building blocks to impose approximate discrete choices and a weighted-sum constraint. Once the …

We give a deterministic algorithm that solves a nonsingular linear system $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ has $m$ nonzero entries and condition number $κ$, to any relative residual tolerance $0<ε\le1/2$ using $\widetilde{O(}mn\log(κ/ε))$ bit operations for inputs with logarithmically many bits per …