Volume 8, Issue 2

We propose a new class of random matrices that enables the recovery of signals with sparse representation in a known basis with overwhelmingly high probability. To construct a matrix in this class, we begin with a fixed non-random matrix that satisfies two very general conditions. Then we decompose the matrix into pieces of sparse matrices. A random sum (involving Bernoulli random variables) of these pieces of sparse matrices is used to construct the final matrix. We say that the random matrix is the randomized Bernoulli transform of the original matrix. The random matrix is not created by filling all its entries with random variables, as in the case of Gaussian or Bernoulli matrices. Therefore, as a benefit, far fewer number of random variables are needed to generate this new type of random matrices. We prove that the number of samples needed to recover a random signal is proportional to the sparsity of the signal, up to a logarithmic factor, and hence this number is nearly optimal.

We consider the hypergeometric translation operators associated to the Cherednik operators and the Heckman–Opdam theory on ℝd${\mathbb{R}^{d}}$ introduced by the author in [6]. Under some conditions on the root system and the multiplicity function, we prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular, we deduce for the Opdam–Cherednik kernel and the Heckman–Opdam hypergeometric function the following main results: (i) Their product formulas are positive integral transforms. (ii) We obtain for them best estimates.

In this paper, we introduce a new class of resolvent operator, the η-proximal operator, and discuss some of its properties. We consider a new generalized variational-like inclusion problem involving relaxed monotone operators in Hilbert space and construct a new iterative algorithm for proving the existence of the solutions of our problem. Our results improve and generalize many corresponding results in the recent literature.

Let X be a non-empty ground set and 𝒫⁢(X)${\mathscr{P}(X)}$ be its power set. A set-labeling (or a set-valuation) of a graph G is an injective set-valued function f:V⁢(G)→𝒫⁢(X)${f\colon\hskip-0.853583ptV(G)\hskip-0.853583pt\to\hskip-0.853583pt\mathscr{P}(% X)}$ such that the induced function f∗:E⁢(G)→𝒫⁢(X)${f^{\ast}\colon\hskip-0.853583ptE(G)\hskip-0.853583pt\to\hskip-0.853583pt% \mathscr{P}(X)}$ is defined by f∗⁢(u⁢v)=f⁢(u)∗f⁢(v)${f^{\ast}(uv)=f(u)\ast f(v)}$, where f⁢(u)∗f⁢(v)${f(u)\ast f(v)}$ is a binary operation of the sets f⁢(u)${f(u)}$ and f⁢(v)${f(v)}$. A graph which admits a set-labeling is known to be a set-labeled graph. A set-labeling f of a graph G is said to be a set-indexer of G if the associated function f∗${f^{\ast}}$ is also injective. In this paper, we introduce a new notion, namely, product set-labeling of graphs as an injective set-valued function f:V⁢(G)→𝒫⁢(ℕ)${f\colon V(G)\to\mathscr{P}(\mathbb{N})}$ such that the induced edge-function f∗:V⁢(G)→𝒫⁢(ℕ)${f^{\ast}\colon\hskip-0.853583ptV(G)\to\mathscr{P}(\mathbb{N})}$ is defined as f∗⁢(u⁢v)=f⁢(u)∗f⁢(v)${f^{\ast}(uv)=f(u)\ast f(v)}$ for all u⁢v∈E⁢(G)${uv\in E(G)}$, where f⁢(u)∗f⁢(v)${f(u)\ast f(v)}$ is the product set of the set-labels f⁢(u)${f(u)}$ and f⁢(v)${f(v)}$, where ℕ${\mathbb{N}}$ is the set of all positive integers and discuss certain properties of the graphs which admit this type of set-labeling.

In 2005, Briani and Rampazzo [4], using the results of [10, 12, 13, 14], gave a density approach to Hamilton–Jacobi equations with t -measurable Hamiltonians. In this paper we firstly give more details and do more investigation on viscosity solutions for Hamilton–Jacobi equations. Then we show, using an important result of [4], the existence and uniqueness of viscosity solutions to the Vlasov relativistic equation in Yang–Mills charged Bianchi space times, with non zero mass. Thirdly, we clearly display, in the case of the Vlasov equation, the optimal control problem. To our knowledge, the method and techniques used here are original and thus, totally different from those used in [1, 5, 6, 8, 13, 15, 16] who have studied the same equation.

In this paper, we introduce the notion of an energy graph G of order n∈ℕ${n\in\mathbb{N}}$. Energy graphs are simple, connected and finite directed graphs. The vertices, labelled u1,u2,…,un${u_{1},u_{2},\dots,u_{n}}$, are such that (ui,uj)∉A⁢(G)${(u_{i},u_{j})\notin A(G)}$ for all arcs (ui,uj)${(u_{i},u_{j})}$ with i>j${i>j}$. Initially, equal amount of potential energy is allocated to certain vertices. Then, at a point of time, these vertices transform the potential energy into kinetic energy and initiate transmission to head vertices. Upon reaching a head vertex, perfect elastic collisions with atomic particles take place and propagate energy further. Propagation rules apply which could result in energy dissipation. The total dissipated energy throughout the graph is called the black energy of the graph. The notion of the black arc number of a graph is also introduced in this paper. Mainly Jaco-type graphs are considered for the application of the new concepts.