In this paper, we give a probabilistic counterpart of  Mazur-Ulam theorem  in probabilistic normed groups. We show, under some conditions, that every surjective isometry between two probabilistic normed groups is a homomorphism.

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB∗-triple, and up to complete isometry, of one-sided ideals in C∗-algebras.

In this article we prove the problem on isometric mappings of S posed by Th. M. Rassias. We prove that any map f : S → S, p ≥ n > 1, preserving two angles θ and mθ (mθ < π, m > 1) is an isometry. With the assumption of continuity we prove that any map f : S → S preserving an irrational angle is an isometry.

In this paper we investigatet the efficiency of the Orthogonal Matching Pursuit for random dictionaries. We concentrate on dictionaries satisfying Restricted Isometry Property. We introduce a stronger Homogenous Restricted Isometry Property which is satisfied with overwhelming probability for random dictionaries used in compressed sensing. We also present and discuss some open problems about OMP.

When the boundary of the curve complex is connected any quasi-isometry is bounded distance from a simplicial automorphism. As a consequence, when the boundary is connected the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

In this paper , the completion of t^ω-approach spaces, isometric in space and equivalent sequences are defined. Every normed can be embedded Banach is proved, χ ̃ an isometry ψ from onto subspace F which dense introduced, as well as, shown uniquely up to isometry. addition, what mentioned above, some essential definitions, examples, important theorems included illustrate our work.

In this paper, it is proved that every s-sparse vector x ∈ R can be exactly recovered from the measurement vector z = Ax ∈ R via some `-minimization with 0 < q ≤ 1, as soon as each s-sparse vector x ∈ R is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the `-minimization can be so chosen to be about 0.6796× (1− δ2s(A)), where δ2s(A) is the restricted isome...