Papers

We determine the exact dimensional range in which a normalized longest signed sum, which we call a Bang vector, always satisfies the real polarization inequality: this prescription is universally valid if and only if \(n\leq 14\). For every \(n\geq 15\) we construct \(n\) unit vectors whose all-positive sum is, up to global sign, the unique longest signed sum, while its normalized direction has polarization product smaller than \(n^{-n/2}\). The counterexamples are given by a single explicit family, and for the same configurations we exhibit nonmaximal signed sums whose normalized directions do satisfy the polarization bound.

We investigate the fake stationarity properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs) with affine drift and long-memory (regular) kernels, both on finite horizons and in the long-run regime. By either deriving explicit closed-form specifications for the deterministic initial condition $\phi$ and the mean-reversion function $\mu$ appearing in the drift, or by introducing a deterministic stabilizing factor $\varsigma$ in the diffusion coefficient associated with the kernel while keeping $\mu$ fully flexible, we show that it is possible to induce a fake stationary regime, in the sense that all marginal distributions share the same mean and variance. Afterwards, using a refined asymptotic analysis, we further establish that, in both frameworks, the time-shifted solutions of these long-memory SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance structure, for suitable classes of diffusion coefficients. These results are applied to a class of exponential-fractional Stochastic Volterra Integral Equations driven by an $\alpha$-gamma fractional integration kernel, in the particular regime \(\alpha \geq 1\), which regularizes diffusion paths and invoke textit{ long-term memory}, persistence or long range dependence.

We derive state-dependent bounds on the difference between two quantum evolutions generated by unbounded Hamiltonians sharing a common form domain. The main technical tool is a second integration by parts, performed at the level of sesquilinear forms rather than at the operator level, which removes the need for a common invariant operator domain. The resulting estimate involves the norm of the time-integrated difference of the two generators, rather than the integral of its norm, and is therefore sensitive to the averaging effects produced by fast-oscillating terms. As an application we prove a quantitative bound on the rotating-wave approximation for spin-boson models with a structured boson field, described by an arbitrary massive dispersion relation on a general measure space and by a suitable class of form factors. The proof involves a careful analysis of the high-frequency scaling. The bound holds on a dense subspace of states, is fully explicit, and all the constants entering it depend only on the parameters of the model and not on the frequency scale, so that the approximation becomes exact in the limit of large frequency.

Revisiting the rationale provided by Picard in the seminal paper [13], we provide an independent, self-contained proof of the Maxwell compactness property for weak Lipschitz domains in the Euclidean setting, detouring technical complications like the differential forms setting, Gaffney's inequality for smooth domains, Calderon's extension theorem or regularity theory for PDEs, or systems thereof, on smooth domains that are used in this context. As a by-product we provide an independent proof of the classical Gaffney estimate for cubes using only $L^2$-completeness of the Fourier bases.

We show that the Hao-Ng isomorphism for full crossed products fails to hold. More precisely, for a non-degenerate $C^*$-correspondence $X$ over a $C^*$-algebra $\mathcal{A}$ and a generalized gauge action $G \curvearrowright X$ by a locally compact group $G$, we isolate three structural obstructions for the canonical commutation $\mathcal{O}_X\rtimes G \cong \mathcal{O}_{X\rtimes G}$ between full crossed product and Cuntz-Pimsner $C^*$-algebra. The first concerns hyperrigidity, the second concerns WEP of the coefficient $C^*$-algebra $\mathcal{A}$ for trivial actions, and the third concerns the coincidence of full and reduced crossed products by the action of $G$ on $\mathcal{A}$.

In this paper, we consider the Rosenthal – Halmos problem of almost commuting matrices with respect to the normalized Hilbert – Schmidt norm $\|\cdot\|_{2,d}=d^{-1/2}\|\cdot\|_2$. We show that if $X$ and $Y$ are self-adjoint matrices with $\|X\|\le 1$, $\|Y\|\le 1$, then there exist commuting self-adjoint matrices $X',Y'$ such that $\|X-X'\|_{2,d}+\|Y-Y'\|_{2,d}\le 5\|[X,Y]\|_{2,d}^{1/3}$, and $[X,X']=0$. Within these constraints, the exponent $1/3$ cannot be improved.

Like g-frames, b-frames were introduced to generalize the concept of frames, allowing for broader applications in signal processing and other fields. The advantage of b-frames resides in their simpler definition, which may lead to reduced processing times. In this paper, we define dual b-frames and b-Riesz bases which were not precisely defined in previous literature and provide several characterizations of b-Riesz bases. We prove that the tensor product of two sequences, each lying in a Hilbert space, constitutes a b-frame (or a b-Riesz basis) if and only if both components of the product are b-frames (or b-Riesz bases). Finally, we establish a correspondence between b-frames and g-frames and propose a process for constructing frames induced by b-frames.

Sharp Hardy–Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let $C_\ell(N)$ denote the optimal constant in the $\ell$th-order discrete Hardy–Rellich inequality on $\Z^N$. Recent independent work established the leading high-dimensional behavior $C_\ell(N)\sim2^\ell N^\ell$ for every fixed $\ell$. We determine the next two orders and prove \[ C_\ell(N)=2^\ell N^\ell+\gamma_\ell N^{\ell-1} +\delta_\ell N^{\ell-2}+O_\ell(N^{\ell-3}), \] with explicit coefficients $\gamma_\ell$ and $\delta_\ell$. The central difficulty is a degeneracy that grows with the dimension: after conjugation, the leading operator is scalar on the $2N$ nearest neighbours of the origin. We resolve this cluster using signed-permutation symmetry and an effective operator on four lattice orbits. Weighted torus estimates and Feshbach–Schur reduction justify the finite-dimensional expansion inside the full operator, while a residual bound and Temple's inequality give the stated remainder. In particular, $C_1(N)=2N-4-20/(3N)+O(N^{-2})$.

For $N\ge3$, let $C(N)$ be the optimal constant in the nearest-neighbour Hardy inequality on $\mathbb Z^N$ with $u(0)=0$, and set $A_N=(N-2)^2/4$. We prove that the continuum coefficient remains an upper bound, $C(N)\le A_N$, in every dimension, and determine the exact values $C(3)=A_3=1/4$ and $C(4)=A_4=1$. In higher dimensions we show $C(N)<A_N$ for $N=9,10$ and obtain the explicit bound \[ C(N)\le 3N-\sqrt{N^2+8N-8}<2N\qquad(N\ge3), \] which lies below $A_N$ from dimension eleven onward. The low-dimensional equalities follow from shifted radial supersolutions, angular convexity, and a discrete ground-state representation. We also show that this shifted-power mechanism cannot work at the continuum coefficient from dimension five onward. Dimension nine is treated by a Gaussian Rayleigh–Ritz construction combined with exact Jacobi theta-function estimates, while the higher-dimensional bounds are obtained through finite-dimensional orbit compressions. Finally, we derive positive spatial remainders in dimensions three and four and a spectral consequence for the associated discrete Schrödinger operators in the high-dimensional regime.

We study heat semigroups on bounded Lipschitz domains $\Omega \subset \mathbb{R}^{d}$ with dimension $d>2$ under non-local Robin boundary conditions. The boundary operator $B \in \mathcal{L}( \mathrm{L}^{2}(\partial\Omega))$ is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator $C \in \mathcal{L}(\mathrm{L}^{2}(\partial\Omega))$ such that $$ |Bu| \leq C|u| \quad\text{for every }u\in \mathrm{L}^{2}(\partial\Omega), \qquad C{\bf 1}\in \mathrm{L}^{\infty}(\partial\Omega). $$ Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from $\mathrm{L}^{2}(\Omega)$ to $\mathrm{L}^{\infty}(\Omega)$ has short-time order $t^{-d/4}$. The main step is the construction of a positive eigenfunction $\phi$ of a dominating elliptic operator such that $$ 0 \ < \ \delta \ \leq \ \phi \ \leq \ M $$ almost everywhere in $\Omega$. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an $\mathrm{L}^{\infty}$-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.

Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.

In a remarkable work published in Analysis and PDE 8 (2015) 1541-1563, M. Fazly, J. Wei, and X. Xu establish the following pointwise estimate \[-\Delta u \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac \sigma 2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n\] for any bounded, positive, $C^4$-solution $u$ to the the fourth-order Hénon equation \[\Delta^2 u = |x|^\sigma u^p \quad \text{in } \mathbf R^n\] with $n \geq 5$, $\sigma \geq 0 $ but implicitly near $0$, and $p>(n+4+2\sigma)/(n-4)$. Their argument relies on a sophisticated iteration argument in the fashion of the standard Moser proof. In this paper, we provide an alternative approach which relies on the Bernstein technique via the maximum principle. The heart of our argument is a suitable choice of an auxiliary function allowing us not only achieving the same pointwise estimate, which still holds even for $p = (n+4+2\sigma)/(n-4)$, but also relaxing the boundedness of solutions. In addition, the pointwise inequality actually holds for any $\sigma \geq 0$. This estimate leads to an interesting consequence of the existence of conformal metric of $\mathbf R^n$ having positive scalar and $Q$ curvatures. We also show that the coefficient $2/(n-4)$ is sharp, hence providing the first answer to the question in the op. cit. paper. Our proof appears to be simpler and constructive.

The concept of an $E$-frame, recently introduced in frame theory, is obtained by applying an infinite invertible complex matrix to a sequence of elements of a Hilbert space $\mathcal{H}$. Here, $E$ is considered as a matrix mapping on the sequence space $\bigoplus_{n=1}^{\infty}\mathcal{H}$. A natural question is to determine the conditions on a sequence $\Psi=\{\psi_k\}_{k=1}^{\infty}$ and a matrix $E$ under which $E\Psi$ becomes a frame. In this paper, we show that if $E$ acts surjectively on $\bigoplus_{n=1}^{\infty}\mathcal{H}$, then every ordinary frame is an $E$-frame.

Vector-valued multiscale representations are essential when signals or fields take values in $\mathbb{R}^m$ and component interactions carry meaningful information. Most multiwavelet and super-wavelet constructions are formulated in scalar Hilbert-space settings and typically produce channelwise scalar coefficients followed by recombination. We develop an intrinsic framework for vector-valued wavelets on $L^2(\mathbb{R}^d,\mathbb{R}^m)$ by endowing this space with a natural $\mathbb{M}_m(\mathbb{R})$-valued inner product, thereby turning it into a Hilbert $\mathbb{M}_m(\mathbb{R})$-module. This module viewpoint yields matrix-valued coefficients that encode cross-component interactions and provides canonical reconstruction through a Parseval-type identity. Within this setting, we introduce a constructive lifting procedure that builds separable multivariate vector-valued wavelet bases in $L^2(\mathbb{R}^d,\mathbb{R}^m)$ from scalar wavelet bases while preserving compact support, vanishing moments, and regularity.

In this paper we continue the investigation of generalizations of the strong law of large numbers to operator means which has been initiated in the earlier paper "Strong law of large numbers for generalized operator means", Adv. Math. 457 (2024), 109933 of the authors. In particular, we clarify an inaccuracy in the statement and proof of the strong law for means associated with uniformly exponentially contracting flows in Thompson metric spaces. We then significantly improve the variant of this strong law given in the mentioned paper for operator means by relaxing the restrictive required moment condition to $L\log^+(L)$ by proving almost sure convergence of implicit stochastic approximations for monotone operators under this condition.

We establish ground-state representations and critical Hardy inequalities in the discrete $L^p$ setting. In particular, we construct critical Hardy weights for the discrete Dirichlet Laplacian on the half-line and the discrete fractional Laplacian on the integers for all $p\in(1,\infty)$. Our approach uses Sobolev–Bregman forms, for which the ground-state representations are exact identities and the corresponding Hardy weights are expressed through linear operators acting on powers of positive superharmonic functions.

$\newcommand{\X}{\mathbb{X}}\newcommand{\SC}{\mathcal{C}}$ We establish the existence of a "maximally spread out" Borel probability measure on a totally bounded subset $\SC$ of a (quasi)-Banach space $\X$ under two mild conditions: (i) a growth condition on the covering numbers $N(\SC, \epsilon)$ of $\SC$, and (ii) a technical topological condition that is in particular satisfied whenever $\SC\subset\X$ is closed, bounded, and convex. More formally, condition (i) requires that the so-called lower power-exponential Minkowski dimension of $\SC$, i.e., \[s_\ast:=\liminf_{\epsilon\downarrow 0}\frac{\log\log N(\SC,\epsilon)}{\log(1/\epsilon)}\] satisfies $s_\ast>0$. Under these conditions, we construct a Borel probability measure $\mu$ on $\X$ that is critical for $\SC$, or maximally spread out, meaning that the associated outer measure $\mu^\ast$ satisfies $\mu^\ast(\X\setminus\SC)= 0$ and furthermore satisfies for every $0<s<s_\ast$ the small-ball condition \[\mu^\ast(B(x,r))\le\exp\bigl(-c(s)\cdot(1/r)^s\bigr)\quad\text{ for all }x\in\X\text{ and }0<r<r_0(s).\] The existence of such a critical measure in particular implies that the so-called power-exponential Hausdorff dimension of $\SC$ introduced in [J.~Topol.~Anal.~4(2):203–235, 2012] coincides with the lower power-exponential Minkowski dimension. Previous work [Found.~Comput.~Math.~23(1):329–392, 2023] shows that such a critical measure gives rise to a phase transition regarding lossy compression and approximation by quantized neural networks of elements of $\SC$, provided that the $\liminf$ in the definition of $s_\ast$ exists as an actual limit. There, critical measures were constructed for unit balls of certain Besov and Sobolev spaces considered as subsets of $L^2$. In contrast, our construction is completely general. In particular, our results apply to function spaces of dominating mixed smoothness.

Under mild assumptions, the main results of this paper characterize a (real or complex) symmetric operator space affiliated with a semi-finite atomless von Neumann algebra $(\mathcal M, \tau)$ possessing the Daugavet property as $L_1(\mathcal M,\tau)$ or $\mathcal M$ (up to equivalent norms, or even proportional norms). Our results are new even for complex symmetric function spaces, which extend results of \cite{AKM12, KMMW13, AKM15} concerning the Daugavet property for real symmetric function spaces.

Given Banach algebras with bounded approximate identities that are Morita equivalent in the sense of Paravicini, we give explicit formulas for the induced correspondence of closed ideals, characterize it via localized Morita submodules, determine when corresponding ideals are Morita equivalent via restricted bimodules, and show that corresponding quotients are always Morita equivalent. We prove preservation of geometric cores associated to compatible dense algebraic cores. For equivalent étale groupoids with paracompact unit spaces, Morita induction preserves dynamical ideals and inner supports. For the Morita equivalence between the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra, we prove that the latter has a bounded approximate identity of projections, identify its geometric ideals explicitly, and identify corresponding complete-lattice block decompositions of the full ideal lattices with matching extremal ideals. Under property A, or for $p=1$, these blocks collapse; for $p=2$, collapse of the minimal block characterizes property A.

Fix the spectrum of a bipartite density matrix and randomize its eigenbasis according to Haar measure. We study the probability distributions induced on the spectra of the two marginal states. For arbitrary subsystem dimensions $m$ and $n$, the joint characteristic function of the reduced density matrices is expressed as a Harish-Chandra-Itzykson-Zuber integral whose external eigenvalues are the pairwise sums $x_i + y_j$. Repeated external eigenvalues are handled by confluent determinant limits. In the two-qubit case, we derive an explicit alternating-spline formula for the joint density of the two marginal Bloch radii. Its support is the Bravyi-Klyachko compatibility region. We also obtain a compact truncated-power formula for the Bloch-radius density of either individual qubit marginal. In the qubit-qutrit case, we derive a truncated-power formula for the qubit Bloch-radius density and a bivariate spline formula for the joint density of the largest and smallest eigenvalues of the qutrit marginal. The latter two variables determine the full qutrit spectrum because the trace is fixed. The derivations combine confluent HCIZ integrals, distributional Fourier inversion, orbital measures, and the SU(2) and SU(3) derivative principles. The resulting densities are piece-wise polynomial on chambers determined by subset sums of the fixed global eigenvalues, in agreement with the Duistermaat-Heckman description of projected coadjoint-orbit measures.

Let $\mathcal M$ be a semifinite von Neumann algebra equipped with a semifinite faithful normal trace $\tau$. We establish Jensen's trace inequality in full generality and characterize its equality case, which answers two questions raised in [Kosaki2013] and [HaradaKosaki2008]. As an application, we derive noncommutative Lamperti-type inequalities and their equality conditions. Employing this result, we characterize linear isometries (not necessarily surjective) on a class of $F$-normed noncommutative Orlicz spaces, which provides a noncommutative Lamperti's theorem for linear isometries.

We solve Tingley's problem for Haagerup noncommutative $L_p$-spaces, where $1<p\ne 2<\infty$, which gives an affirmative answer to Mori's problem \cite[Problem~6.3]{Mori}.

We study the set $\Gamma$ of graphs of closed, densely defined operators in a Hilbert space $H$, regarded as a subset of the Grassmann manifold $P(H\times H)$ of orthogonal projections in $H\times H$. We show that the subset $\Gamma^b$ of graphs of bounded operators is the open unit ball of $P(H\times H)$ centered at the graph of the zero operator $P_0$ (which projects onto $H\times\{0\}$). This ball is diffeomorphic to $B(H)$ via the map $T\mapsto P_T$ ($=$ the projection onto the graph ${Gr(T)}$ of $T$). We show that graphs of unbounded closed operators lie at the boundary of $\Gamma$. We also study the existence and characteristics of minimal geodesics of $P(H\times H)$ joining two graphs $Gr(A)$, $Gr(B)$. If $A,B$ are selfadjoint, such a geodesic always exists, and we construct explicitly a distinguished exponent using the five-space decomposition associated to the pair of subspaces $Gr(A)$, $Gr(B)$. An explicit low-dimensional example shows that the geodesic joining two graphs need not remain inside $\Gamma$, i.e., does not consist entirely of graphs. We also relate graphs of compact operators to the restricted Grassmannian, and study the problem of common complements for pairs $Gr(S)$, $Gr(T)$, giving positive results when one operator is bounded or under lower boundedness conditions.

We construct a uniformly Kreiss bounded operator on a separable Hilbert space which is not strongly Cesàro bounded. This gives a negative answer to a question of Cohen, Cuny, Eisner and Lin.

We construct real Banach spaces $X$ and $Y$ and a bijection $U\colon X\to Y$ which preserves all pairwise distances on every closed ball of radius $1/4$ but is not a global isometry. The proof starts from an explicit injective map that preserves every distance at most $1/2$ and shortens one longer distance. We then enlarge the source and target through a transfinite recursion that inserts every missing target point while preserving both injectivity and this distance threshold.

This paper establishes an operator-theoretic version of Pego's compactness theorem within the framework of quantum harmonic analysis on general locally compact abelian phase spaces. We show that a bounded set of Hilbert-Schmidt operators is precompact if and only if it is uniformly equicontinuous under phase-space shifts and its Fourier-Weyl transform is uniformly equicontinuous on the dual phase space. We provide applications to quantum physics.

Let $A \in \mathbb{C}^{n \times n}$ have spectrum in the closed unit disk. Its maximal power growth $\operatorname{Power}(A) := \sup_{k \ge 0} \Vert{}A^k\Vert{}$ measures transient amplification, whereas the Kreiss constant $\operatorname{Kreiss}(A) := \sup_{\vert{}z\vert{}>1} (\vert{}z\vert{}-1) \Vert{}(zI-A)^{-1}\Vert{}$ measures the corresponding resolvent growth outside the disk. The classical finite-dimensional Kreiss theorem gives $\operatorname{Power}(A) \le en \operatorname{Kreiss}(A)$, and the linear dependence on $n$ is unavoidable for general matrices. We show that, for matrices selfadjoint with respect to an indefinite metric, the ambient dimension $n$ can be replaced by an effective dimension determined by the minimal polynomial and the inertia of the metric. Specifically, if $A^*J = JA$, where $J$ is a fundamental symmetry with inertia $(n-q,q)$, then $\operatorname{Power}(A) \le e \min\{d(A), 2q+1, 2(n-q)+1\} \operatorname{Kreiss}(A)$, where $d(A)$ is the degree of the minimal polynomial. Our proof requires no assumption on diagonalizability or reality on the spectrum. Instead, we associate each cyclic orbit with a finite-rank selfadjoint Hankel operator and transfer its rank and inertia to a coefficient estimate. Examples based on scaled nilpotent shifts show that the linear dependence on the smaller inertia index is asymptotically sharp, even when this index is negligible relative to the matrix size and $d(A)=n$. We also obtain scaled-disk decay estimates and weighted-norm extensions to arbitrary nonsingular Hermitian metrics.

This paper characterizes all the singular inner eigenfunctions of the composition operators $C_\phi$ that arise from discrete measures, when $\phi$ is an automorphism of unit disk. By establishing a connection between Beurling and model invariant subspaces, we classify all the inner functions so that the corresponding Beurling subspace is invariant under the composition operators induced by non-elliptic automorphisms. This classification involves solving the eigenfunction equation for the composition operator. Further, we present some applications of the above-mentioned connection.

Let $D\subset\Bbb Z$ with cardinality $q\ge 2$, and let $b\in \Bbb Z$ with $q<b$, and let $\mu:=\mu_{b,D}$ be the associated self-similar measure. It is well known that if there exists $L\subset \Bbb Z$ such that $(b,D,L)$ be a Hadamard triple, then the Beurling dimension of the spectrum of $\mu$ exhibits an intermediate structural property. In this paper, we establish a stronger result that both Beurling dimension and Beurling density of the spectra of $\mu$ can achieve full flexibility simultaneously. More precisely, for any $t\in(0, \frac{\log q}{\log b})$ and $s\in [0,\infty]$, there exists a spectrum $\Lambda:=\Lambda_{t,s}$ of $\mu$ such that $$\dim_{Be}(\Lambda)=t,\quad D_t^+ (\Lambda)=s.$$ Here, $\dim_{Be}$ and $D_t^+$ denote the Beurling dimension and the $t$-Beurling density, respectively. We further prove that the set of such spectrum whose Beurling dimension and Beurling density are equal to any fixed $t$ and $s$ has the cardinality of the continuum. \par This work generalizes a previous result of Lu \cite{Lu}, answers an open question raised by Dai, Fu and He \cite[Conjecture 5.3]{DaiFuHe}, and sheds new light on the fine structural properties of spectra for singularly continuous spectral measures.

This document reports over 15 counterexamples found using GPT Pro over the course of 12 months, for problems in combinatorics, number theory, convexity, analysis, and other areas. The counterexamples concern both (reasonably) well-known problems as well as lesser known instances. The counterexamples are accompanied by the repo https://github.com/suvrit/count-ex-machina, and we warmly welcome new contributions toward helping building a rich “open-source countex library.”