Microservices Validation: Methodology and Implementation

Due to the wide spread of cloud computing, arises actual question about architecture, design and implementation of cloud applications. The microservice model describes the design and development of loosely coupled cloud applications when computing resources are provided on the basis of automated IaaS and PaaS cloud platforms. Such applications consist of hundreds and thousands of service instances, so automated validation and testing of cloud applications developed on the basis of microservice model is a pressing issue. There are constantly developing new methods of testing both individual microservices and cloud applications at a whole. This article presents our vision of a framework for the validation of the microservice cloud applications, providing an integrated approach for the implementation of various testing methods of such applications, from basic unit tests to continuous stability testing

On concentrators and related approximation constants

Pippenger ([Pippenger, 1977]) showed the existence of $(6m,4m,3m,6)$-concentrator for each positive integer $m$ using a probabilistic method. We generalize his approach and prove existence of $(6m,4m,3m,5.05)$-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from $44.5$ (established in [Kalton, Roberts, 1983]) to $39$. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in $\mathbb{R}^d$ by linear functions, and improve the estimate of this Whitney constant from $802$ (proved in [Brudnyi, Kalton, 2000]) to $73$

Unit distance graphs and algebraic integers

We answer a question of Brass about vertex degrees in unit distance graphs of finitely generated additive subgroups of $\mathbb{R}^2$

On a tropical version of the Jacobian conjecture

We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism