Related papers: Cones of ball-ball separable elements

We show that the two-dimensional minimum-volume central section of the $n$-dimensional cross-polytope is attained by the regular $2n$-gon. We establish stability-type results for hyperplane sections of $\ell_p$-balls in all the cases where…

Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$-ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$-ball…

Let $M$ be a unital JB$^*$-algebra whose closed unit ball is denoted by $\mathcal{B}_M$. Let $\partial_e(\mathcal{B}_M)$ denote the set of all extreme points of $\mathcal{B}_M$. We prove that an element $u\in \partial_e(\mathcal{B}_M)$ is a…

We characterize the boundary of the convex compact set of absolutely separable states, referred as {\bf AS}, that cannot be transformed to entangled states by global unitary operators, in $2\otimes d$ Hilbert space. However, we show that…

This paper investigates the solutions to the two-phase Serrin's problem, an overdetermined boundary value problem motivated by shape optimization. Specifically, we study the torsional rigidity of composite beams, where two distinct…

We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the…

The solvability in $W^{2}_{p}(\bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the…

Let $d$ be a fixed positive integer and let $\epsilon>0$. It is shown that for every sufficiently large $n\geq n_0(d,\epsilon)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side…

We consider a Brownian particle with diffusion coefficient $D$ in a $d$-dimensional ball of radius $R$ with reflecting boundaries. We study the maximum $M_x(t)$ of the trajectory of the particle along the $x$-direction at time $t$. In the…

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the…

The problem of bounding of the distance between the two bodies of volume $\varepsilon$ located inside the $n$-dimensional body $B$ of unit volume where $n \to \infty$ is considered. In some cases such distances are bounded by function…

The top-quark, $W$ and $Z^0$ bosons have widths that are a sizable fraction of their masses and will be produced copiously at upcoming accelerators. Yet S-matrix theory cannot treat unstable particles as external states. Dealing with…

We prove that among all doubly connected domains of $\mathbb{R}^n$ of the form $B_1\backslash \overline{B_2}$, where $B_1$ and $B_2$ are open balls of fixed radii such that $\overline{B_2}\subset B_1$, the first nonzero Steklov eigenvalue…

For an $r$-tuple $(\gamma_1,\ldots,\gamma_r)$ of special orthogonal $d\times d$ matrices, we say that the Euclidean $(d-1)$-dimensional sphere $S^{d-1}$ is $(\gamma_1,\ldots,\gamma_r)$-divisible if there is a subset $A\subseteq S^{d-1}$…

We consider the unit ball $\Omega\subset \mathbb{R}^N$ ($N\ge2$) filled with two materials with different conductivities. We perform shape derivatives up to the second order to find out precise information about locally optimal…

Let $P$ and $Q$ be two convex polytopes both contained in the interior of an Euclidean ball $r\textbf{B}^{d}$. We prove that $P=Q$ provided that their sight cones from any point on the sphere $rS^{d-1}$ are congruent. We also prove an…

A one-dimensional model of coupled spin-1/2 spins and pseudospin-1/2 orbitals with nearest-neighbor interaction is rigorously shown to exhibit spin-orbital separation by means of a non-local unitary transformation. On an open chain, this…

A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G.…

Let S_k be the set of separable states on B(C^m \otimes C^n) admitting a representation as a convex combination of k pure product states, or fewer. If m>1, n> 1, and k \le max(m,n), we show that S_k admits a subset V_k such that V_k is…

We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with…