A Simple Method for Generating Electromagnetic Oscillations

We propose a novel approach to the generation of electromagnetic oscillations by means of a low-frequency pumping of two coupled linear oscillators. A theory of such generation mechanism is proposed, and its feasibility is demonstrated by using coupled RLC oscillators. A comparison of the theoretical results and the experimental data is presented.Comment: 5 pages, 7 figure

The proximal point method revisited

In this short survey, I revisit the role of the proximal point method in large scale optimization. I focus on three recent examples: a proximally guided subgradient method for weakly convex stochastic approximation, the prox-linear algorithm for minimizing compositions of convex functions and smooth maps, and Catalyst generic acceleration for regularized Empirical Risk Minimization.Comment: 11 pages, submitted to SIAG/OPT Views and New

BI action for gravitational and electroweak fields

This note suggests a generalization of the Born--Infeld action (1932) on case of electroweak and gravitational fields in four-dimensional spacetime. The action is constructed from Dirac matrices, $\gamma_a$, and dimensionless covariant derivatives, $\pi_{a} = - i\ell \nabla_{a}$, where $\ell$ is of order of magnitude of Planck's length. By a postulate, the action possesses additional symmetry with respect to global transformations of the Lorentz group imposed on pairs ($\gamma_{a}$, $\pi_{a}$). It's shown, that parameter of the Lorentz group is associated with a constant value of the electroweak potential at spatial infinity. It follows, that in linear and quadratic in $\ell^2$ approximation, action for gravitational field coincides with Einstein-Hilbert (EH) and Gauss-Bonnet (GB).Comment: 11 pages, 0 figure

The Size of Generating Sets of Powers

In the paper we prove for every finite algebra A that either it has the polynomially generated powers (PGP) property, or it has the exponentially generated powers (EGP) property. For idempotent algebras we give a simple criteria for the algebra to satisfy EGP property.Comment: 5 page

Strong subalgebras and the Constraint Satisfaction Problem

In 2007 it was conjectured that the Constraint Satisfaction Problem (CSP) over a constraint language $\Gamma$ is tractable if and only if $\Gamma$ is preserved by a weak near-unanimity (WNU) operation. After many efforts and partial results, this conjecture was independently proved by Andrei Bulatov and the author in 2017. In this paper we consider one of two main ingredients of my proof, that is, strong subalgebras that allow us to reduce domains of the variables iteratively. To explain how this idea works we show the algebraic properties of strong subalgebras and provide self-contained proof of two important facts about the complexity of the CSP. First, we prove that if a constraint language is not preserved by a WNU operation then the corresponding CSP is NP-hard. Second, we characterize all constraint languages that can be solved by local consistency checking. Additionally, we characterize all idempotent algebras not having a WNU term of a concrete arity $n$, not having a WNU term, having WNU terms of all arities greater than 2. Most of the results presented in the paper are not new, but I believe this paper can help to understand my approach to CSP and the new self-contained proof of known facts will be also useful

Energy of 4-Dimensional Black Hole, etc

In this letter I suggest possible redefinition of mass density, not depending on speed of the mass element, which leads to a more simple stress-energy for an object. I calculate energy of black hole.Comment: 5 pages, 0 figure

A Proof of the CSP Dichotomy Conjecture

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete. In the paper we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.Comment: the final versio

Hopf-Galois extensions with central invariants

We study Hopf-Galois extensions with central invariants for a finite dimensional Hopf algebra. We collect general facts about them and discuss some examples arising in the study of restricted Lie algebras and quantum groups at roots of unity. Our focus is on representation theory and its special feature in this situation, restriction of the central character to the subalgebra of invariants.Comment: 18 pages, LATEX, revised versio

On toric varieties and algebraic semigroups

The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a proof of the fact that every separated toric variety may be constructed from a certain fan in a Euclidean space. To our best knowledge, this proof differs essentially from the ones which can be found in the literature.Comment: LaTeX, 13 page

Born--Infeld theory of electroweak and gravitational fields: Possible correction to Newton and Coulomb laws

In this note one suggests a possibility of direct observation of the $\theta$-parameter, introduced in the Born--Infeld theory of electroweak and gravitational fields, developed in quant-ph/0202024. Namely, one may treat $\theta$ as a universal constant, responsible for correction to the Coulomb and Newton laws, allowing direct interaction between electrical charges and masses.Comment: 4p, ReVTeX 3.0, no figures, minor correction