The system Galactica of free access is supplemented module for the Coulomb interaction. It is based on a high-precision method for solving differential equations of motion of N charged particles. The paper presents all the theoretical and practical issues required to use this module of system Galactica so that even the beginning researcher could study the motion of particles, atoms and molecules.

It is well-known that a ring R is semiperfect if and only if RR (or RR) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (or RR) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie d...

Our objective in this paper is to prove a rather broad generalization of some classical theorems in Iwasawa theory. We begin by recalling two of those old results. The first is a theorem of Iwasawa, which we state in terms of Galois cohomology. Suppose that K is a totally real number field, that F is a totally complex quadratic extension of K, and that ψ is the nontrivial character of Gal(F/K),...

Lifting modules plays important roles in module theory. H-supplemented are a nice generalization of lifting which have been studied extensively recently. In this article, we introduce proper via images fully invariant submodules. Let F be submodule right Rmodule M. We say that M is IF -H-supplemented case for every endomorphism φ M, there direct summand D such φ(F) + X = if and only It proved d...

Given a finite dimensional Lie algebra $L$ let $I$ be the augmentation ideal in universal enveloping $U(L)$. We study conditions on under which $Ext$-groups $Ext (k,k)$ for trivial $L$-module $k$ are same when computed category of all $U(L)$-modules or $I$-torsion $U(L)$-modules. also prove that Rees $\oplus _{n\geq 0}I^n$ is Noetherian if and only nilpotent. An application to cohomology equiva...

We parametrise the space of all possible flavour non-universal $\mathfrak{u}(1)_X$ extensions Standard Model that embed inside anomaly-free semi-simple gauge theories, including up to three right-handed neutrinos. More generally, we abelian (i.e.) by any number $\mathfrak{u}(1)$'s) SM with such completions. The resulting is a collection planes dimensions $\leq 6$. Numerically, find roughly $2.5...

Based on Morse theory for the energy functional path spaces we develop a deformation mapping of spheres into orthogonal groups. This is used to show that these are weakly homotopy equivalent, in stable range, associated Clifford representations. Given an oriented Euclidean bundle $V \to X$ rank divisible by four over finite complex $X$ derive decomposition result vector bundles sphere $\mathord...

The aim of this paper is to investigate generalizations locally artinian supplemented modules in module theory, namely radical and strongly modules. We have obtained elementary features for them. Also, we characterized by left perfect rings. Finally, proved that the reduced part a $R$-module has same property over Dedekind domain $R$.