In group theory the chief factors allow a group to be studied by its representation theory on particularly natural irreducible modules. It is to be expected, therefore, that they will play an important role in the study of Lie algebras. In this article we survey a few of their properties.

In previous papers the structure of the jet bundle as $P$-module has been studied using different techniques. In this paper we use techniques from algebraic groups, sheaf theory, generliazed Verma modules, canonical filtrations of irreducible $\mathrm{SL}(V)$-modules and annihilator ideals of highest weight vectors to study the canonical filtration $U_l(\mathfrak{g})L^d$ of the irreducible $\mathrm{SL}(V)$-module $\mathrm{H}^0 (X,\mathcal{O}_X(d))^*$ where $X = \mathbb{G}(m, m + n)$. We study $U_l(\mathfrak{g})L^d$ using results from previous papers on the subject and recover a well known classification of the structure of the jet bundle $\mathcal{P}^l(\mathcal{O}(d))$ on projective space $\mathcal{P}^l(\mathcal{O}_X(V*))$ as $P$-module. As a consequence we prove formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the $P$-module of the first order jet bundle $\mathcal{P}_X^1(\mathcal{O}_X(d))$ for any $d ≥ 1$. We study the incidence complex for the line bundle $\mathcal{O}(d)$ on the projective line and show it is a resolution of the ideal sheaf of $I^l (\mathcal{O}_X(d))$ - the incidence scheme of $\mathcal{O}_X(d)$. The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians.

We consider a (real or complex) analytic manifold $M$. Assuming that $F$ is a ring of all analytic functions, full or truncated with respect to the local coordinates on $M$; we study the $(m ≥ 2)$-derivations of all involutive analytic distributions over $F$ and their respective normalizers.

In this paper, we present a relative version of the concept of lower marginal series and give some isomorphisms among $\mathcal{V}G$-marginal factor groups. Also, we conclude a generalized version of the Stalling’s theorem. Finally, we present a sufficient condition under which the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of its factor groups.

In this paper, we prove that equation $E ≡ u_1-u_{_x2_t}+u_xf(u)-au_xu_{}x^2-buu_{x^3}=0$ is self-adjoint and quasi self-adjoint, then we construct conservation laws for this equation using its symmetries. We investigate a symmetry classification of this nonlinear third order partial differential equation, where $f$ is smooth function on $u$ and $a$, $b$ are arbitrary constans. We find Three special cases of this equation, using the Lie group method.

In this paper, we introduce the notion of representation of Bol algebra. We prove an analogue of the classical Engel’s theorem and the extension of Ado-Iwasawa theorem for Bol Algebras. We study the category of representations of Bol algebras and show that it is a tensor category. In the case of regular representations of Bol algebras, we give a complete classification of them for all two-dimensional Bol algebras.

This work deals with an extension of the Black-Scholes model for rating options with the Heston volatility model. A Lie-algebraic analysis of this equation is applied to reduce its order and compute some of its solutions. As a result of this method, a five-parameter family of solutions is obtained. Though, these solutions do not match the terminal and boundary conditions, they can be used for the validation of numerical schemes.

It is proved that the Hilbert class field of a real quadratic field $Q(\sqrt{D})$ modulo a power $m$ of the conductor $f$ is generated by the Fourier coefficients of the Hecke eigenform for a congruence subgroup of level $fD$.

It emphasizes the mathematical aspects of the formation of sikidy. The sikidy as an art of divination is transmitted by oral tradition, the knowledge of these mathematical relationships allows for a more consistent language of sikidy. In particular, one can calculate systematically all ”into sikidy” special tables of Sikidy used in the ”ody” (kind of talismans).

Let Χ a diferentiable paracompact manifold. Under the hypothesis of a linear connection r with free torsion Τ on Χ, we are going to give more explicit the proofs done by Vey for constructing a Riemannian structure. We proposed three ways to reach our object. First, we give a sufficient and necessary condition on all of holonomy groups of the connection ∇ to obtain Riemannian structure. Next, in the analytic case of $Χ$, the existence of a quadratic positive definite form g on the tangent bundle ΤΧ such that it was invariant in the infinitesimal sense by the linear operators ∇$^k$R, where R is the curvature of ∇, shows that the connection ∇ comes from a Riemannian structure. At last, for a simply connected manifold Χ, we give some conditions on the linear envelope of the curvature R to have a Riemannian structure.

Analytic solutions for cylindrical thermal waves in solid medium are given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation.We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous.

The modern Science has now a lot of its branches and meanders, where are working the numerous specialists and outstanding scientists everywhere in the whole world. The theme of this article is devoted to mathematics in general and to such a new subsidiary science as the Cartesian infinitology (± ∞: x y and x y z) in a whole.

The young and adult modern people of our time, among them, in first turn, are such ones as the usual citizens, students or schoolchildren, have a very poor imagination about those achievements and successes that made by our scientists in the different parts and divisions of many fundamental sciences, especially in mathematics. This article is a short description of the numerous ideas of a new science that is named by its inventor as the mathematical infinitology.

The infinity as the scientific category is a very complicated conception and the difficult theme for professional discussing of its properties and features even by the academicians and the Nobelists as well. In spite of all problems, the author have found his own road to this Science and worked out independently, even not being a mathematician at all, the universal, from his point of view, and unusual theories and scientific methods, which helped him to find and name It as the mathematical infinitology, that may be now studied in rectangular system of Cartesian or other coordinates, in orthogonal ones, for example, as easy and practically as we study the organic chemistry or Chinese language at the middle school or in the University.

The mathematical infinitology, as a separate or independent science, has been never existed in the mathematics from the ancient times up to the 90-th years of the XX-th century. All outstanding mathematicians of the past times were able only approximately to image to themselves and explain to their colleagues and pupils in addition, what is an infinity indeed: the scientific abstraction or the natural mathematical science that can be not only tested by one’s tooth or touched by hands, but study and investigate it in schools or the Institutions of higher learning too.

The article author without no one imagination, what it is indeed. Very long time working hours spent by the inventor with this mathematical toy or the simplest logical entertainment helped him to penetrate into the mysteries of this usual intellectual mathematical object and see in it the fantastic perspectives and possibilities as for science as for himself in further studying and it investigating. In a result of the own purposefulness and interests to the re-invented mathematical idea of the famous American mathematician S.M.Ulam, the new science was born in the World, and after long time experiments, it was named as the mathematical or Cartesian infinitology (±∞ : x y and x y z).

We consider aspects of the relationship between nilpotent orbits in a semisimple real Lie algebra $\mathfrak{g}$ and those in its complexification $\mathfrak{g}_{\mathbb{C}}$. In particular, we prove that two distinct real nilpotent orbits lying in the same complex orbit are incomparable in the closure order. Secondly, we characterize those $\mathfrak{g}$ having non-empty intersections with all nilpotent orbits in $\mathfrak{g}_{\mathbb{C}}$. Finally, for $\mathfrak{g}$ quasi-split, we characterize those complex nilpotent orbits containing real ones.