This article presents numerical methods in order to solve problems of tolerance analysis. A geometric specification, a contact specification and a functional requirement can be respectively characterized by a finite set of geometric constraints, a finite set of contact constraints and a finite set of functional constraints. Mathematically each constraint formalises a n-face (hyperplan of dimens...

and Applied Analysis 3 In contrast to the modified Roper-Suffridge extension operator in the unit ball, it is natural to ask if we can modify the Roper-Suffridge extension operator on the Reinhardt domains. In this paper, we will introduce the following modified operator: F z ⎛ ⎝f z1 f ′ z1 n ∑ j 2 ajz pj j , ( f ′ z1 2z2, . . . , ( f ′ z1 nzn ⎞ ⎠ ′ 1.5 on the Reinhardt domainΩn,p2,...,pn . Wew...

The Petty projection inequality for sets of finite perimeter is proved. Our approach based on Steiner symmetrization. Neither the affine Sobolev nor functional Minkowski problem used in our proof. Moreover, perimeter, we prove with respect to

In this paper, we first introduce a new concept of dual quermassintegral sum function of two star bodies and establish Minkowski's type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov– Fenchel inequality and the Brunn–Minkowski inequality for mixed intersection...

Tiling problems belong to the oldest problems in mathematics. They attracted attention of many famous mathematicians. Even one of the Hilbert's problems is devoted to the topic. The interest in tilings by unit cubes originated with a conjecture raised by Minkowski in 1907. We discuss this conjecture, its history and variations, and then we describe some problems that Minkowski's conjecture, in ...

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and −P . Then the polygon U , as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U -width, and we prove that many properties of the smooth case, which i...

The aim of the present paper is to provide a unitary exposition on possible representations for positively homogeneous functions in the dual space. Starting from the classical Minkowski–Hörmander duality result of convex analysis, we develop a scheme in order to associate to a positively homogeneous function families of sublinear functionals representing it. Such dual description allows us to c...

" Doubly-special relativity " (DSR), the idea of a Planck-scale Minkowski limit that is still a relativistic theory, but with both the Planck scale and the speed-of-light scale as nontrivial relativistic invariants, was proposed (gr-qc/0012051) as a physics intuition for several scenarios which may arise in the study of the quantum-gravity problem, but most DSR studies focused exclusively on th...

I investigate through simulations the redshift dependence of several lensing measures for two cosmological models, a flat universe with a cosmological constant (ΛCDM), and an open universe (OCDM). I argue that quintessence models can be seen as an intermediate model between these two models under the weak gravitational lensing perspective. I calculate for the convergence field the angular power...

The morphology of galaxy clusters is quantified using Minkowski functionals, especially the vector–valued ones, which contain directional information and are related to curvature centroids. The asymmetry of clusters and the amount of their substructure can be characterized in a unique way using these measures. – We briefly introduce vector–valued Minkowski functionals (also known as Quermaß vec...