Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying in an arbitrary totally bounded metric space where rationals replaced with countable hierarchy “well-spread” points, which we refer to as abstract prove various Jarník–Besicovitch type dimension bounds and investigate their sharpness.

1 Hadamard matrices in Space Communications One hundred years ago, in 1893, Jacques Hadamard 21] found square matrices of orders 12 and 20, with entries 1, which had all their rows (and columns) orthogonal. These matrices, X = (x ij), satissed the equality of the following inequality jdet Xj 2 n i=1 n X j=1 jx ij j 2 and had maximal determinant. Hadamard actually asked the question of matrices ...

Let $X$ be a vector space over a field $K$ of real or complex numbers. We will prove the superstability of the following Go{l}c{a}b-Schinzel type equation$$f(x+g(x)y)=f(x)f(y), x,yin X,$$where $f,g:Xrightarrow K$ are unknown functions (satisfying some assumptions). Then we generalize the superstability result for this equation with values in the field of complex numbers to the case of an arbitr...

On page 23 of his famous monograph [2], D. V. Anosov writes Every five years or so, if not more often, someone 'discovers' the theorem of Hadamard and Perron proving it either by Hadamard's method or Perron's. I myself have been guilty of this. If (X, d X) and (Y, d Y) are metric spaces and T : X → Y is a map then the Lipschitz constant of T is the quantity lip(T) = sup d Y (T (x 1), T (x 2))

The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak -compactness theorem for integral currents in dual spaces of separable Banach spaces. Our theorem...

The xed point result in Mustafa-Sims metrical structures obtained by Karapinar and Agarwal[Fixed Point Th. Appl., 2013, 2013:154] is deductible from a corresponding one stated in terms ofanticipative contractions over the associated (standard) metric space.

The purpose of this paper is to establish fixed point results for a single mapping in a partially ordered modular metric space, and to prove a common fixed point theorem for two self-maps satisfying some weak contractive inequalities.