Related papers: The Frame Potential, on Average
The problem of existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are…
Let $A$ be a normal operator in a Hilbert space $\mathcal{H}$, and let $\mathcal{G} \subset \mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $\mathcal{G}$ , and $L$ that makes the system of iterations…
It is now well known that sparse or compressible vectors can be stably recovered from their low-dimensional projection, provided the projection matrix satisfies a Restricted Isometry Property (RIP). We establish new implications of the RIP…
Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular…
The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions $\leq 151$ and few higher dimensions as high as $1155$. All known solutions with the exception of the Hoggar solutions…
Since its popularization in the 1970s the Fiedler vector of a graph has become a standard tool for clustering of the vertices of the graph. Recently, Mendel and Noar, Dumitriu and Radcliffe, and Radcliffe and Williamson have introduced…
Let H be a separable, infinite dimensional Hilbert space and let S be a countable subset of H. Then most positive operators on H have the property that every nonzero vector in the span of S is cyclic, in the sense that the set of operators…
Frames play an important role in various practical problems related to signal and image processing. In this paper, we define computable frames in computable Hilbert spaces and obtain computable versions of some of their characterizations.…
First, we review local concepts defined previously. A (local) reference frame $\mathrm{F}$ can be defined as an equivalence class of admissible spacetime charts (coordinate systems) having a common domain $\mathrm{U}$ and exchanging by a…
General relativity is derived from an action which is quadratic in the covariant derivative of certain spinor one-form gravitational potentials. Either a pair of 2-component spinor one-forms or a single Dirac spinor one-form can be…
We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related modular operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea…
Two $n$-dimensional vectors $A$ and $B$, $A,B \in \mathbb{R}^n$, are said to be \emph{trivially orthogonal} if in every coordinate $i \in [n]$, at least one of $A(i)$ or $B(i)$ is zero. Given the $n$-dimensional Hamming cube $\{0,1\}^n$, we…
In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular…
Photonic bound states in the continuum (BICs) are nonradiating eigenmodes of structures with open scattering channels. Most often, BICs are studied in highly symmetric structures with one open scattering channel. In this simplest case, the…
Bound states in the continuum (BICs) are generally considered unusual phenomena. In this work, we provide a method to analyze the spatial structure of particle's bound states in the presence of a minimal length, which can be used to find…
The present contribution proposes a universal framework to formulate generalized section-section interaction potentials (SSIP) within the geometrically exact beam theory. By exploiting the fundamental kinematic assumption of undeformable…
We consider the vector functions in a domain homeomorphic to a spherical layer bounded by twice continuously differentiable surfaces. Additional restrictions are imposed on the domain, which allow to conduct proofs using simple methods. On…
Binary Parseval frames share many structural properties with real and complex ones. On the other hand, there are subtle differences, for example that the Gramian of a binary Parseval frame is characterized as a symmetric idempotent whose…
In this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are…
As needed for the construction of rank $n$ continuous frames on a right quaternionic Hilbert space the so-called S-spectrum of a right quaternionic operator is studied. Using the S-spectrum, as for the case of complex Hilbert spaces, along…