Journal of Generalized Lie Theory and Applications

We investigate properties of a Type-A meander, here considered to be a certain planar graph associated to seaweed subalgebra of the special linear Lie algebra. Meanders are designed in such a way that the index of the seaweed may be computed by counting the number and type of connected components of the meander. Specifically, the simplicial homotopy types of Type-A meanders are determined in the cases where there exist linear greatest common divisor index formulas for the associate seaweed. For Type-A seaweeds, the homotopy type of the algebra, defined as the homotopy type of its associated meander, is recognized as a conjugation invariant which is more granular than the Lie algebra's index.

By using the Jiang’s function J₂ (ω) we prove that there exist infinitely many integers n such that n=2P₁, n+1=3P₂, n+k−1=(k+1) P_k are all composites for arbitrarily long k, where P₁, P₂,…, P_k are all primes. This result has no prior occurrence in the history of number theory.

Properties of fuzzy subalgebras and ideals of n-ary Lie algebras are described. Methods of construction fuzzy ideals are presented. Connections with various fuzzy quotient n-Lie algebras are proved.

Let Vect(1) be the Lie algebra of smooth vector fields on 1. In this paper, we classify aff(1) -invariant linear differential operators from Vect(1) to λ,μ;v vanishing on aff(1), where λ,μ;v≔Homdiff(λ⊗μ;v) is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential aff(1)-relative cohomology of Vect(1) with coefficients in λ,μ;v

In this paper we show a few outcomes concerning two remaining deductions on a semi-prime ring are displayed. These outcomes are identified with an outcome which is motivated by Posner's hypothesis. This outcome affirms that if R is a 2-torsion free semi-prime ring, δ and g are non-zero remaining inductions of R with the end goal that g is a surjective on R, and g(y)δ(x)=g(x)δ(y) for all x,y∈R. At that point δ g can't be a non-zero left derivation. A thought of orthogonal left derivations emerges here.

We present analytic self-similar solutions for the one, two and three dimensional Madelung hydrodynamical equation for a free particle. There is a direct connection between the zeros of the Madelung fluid density and the magnitude of the quantum potential.

The wavelets are important functions in the harmonic analysis. Up to our knowledge, apply wavelets to solve differential equations are limited to ODEs or PDEs with approximate and numerical solutions. In this paper, the novel methods based on the wavelets with two independent variables according to differential invariants are proposed. In fact, the transform groups are constructed that can be acted on the existing solutions and produced the new solution. These groups are based on the symmetry groups (obtained by the Lie symmetry method and other equivalence methods) and mother wavelets. The new method based on the wavelets are presented, new mother wavelets are produced, the corresponding wavelet type transform groups are provided and applied for solving the differential equations. Our method can be used for ODEs and PDEs at every order and give us the analytic solutions according to the existing solutions (from every method without any exception).

In this work, we apply soft theory to control theory. With the inspiration of soft theory we define soft Lie groups and construct soft control systems on Lie groups with observation. For the decision argument, we use some observability characterization and give an example.

The main feature of the flame kinematics can be described with the G-equation. We investigate the solutions of the G-equation with the well-known self-similar Ansatz. The results are discussed and the method how to get selfsimilar solutions is briefly mentioned.

Let D be a mapping from an alternative ring into itself satisfying D(a⋅ba)= D(a)⋅ba+a⋅D(b)a+a⋅bD(a) for all a, b . Under some conditions on , we show that D is additive.

A new numerical method based on the approximation of polynomials is here proposed for solving the one dimensional parabolic partial differential equation arising from unsteady state flow of heat subject to initial and boundary conditions. The method results from discretization of the parabolic partial differential equation which leads to the production of a system of algebraic equations. By solving the system of algebraic equations we obtain the problem approximate solutions.

We classify, up to isomorphism, the 2-dimensional algebras over a field . We focuse also on the case of characteristic 2, identifying the matrices of GL(2, 2) with the elements of the symmetric group Σ3. The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.

The concept of moving space-matter of New Cartesian physics, based on the identity of space and matter, offers a way to materialist explanations of the paranormal and supernatural phenomena. It moves space-matter forms not only those objects and phenomena that are realized by our consciousness, but also those that escape his attention.

Santilli’s prime chains: P_j + 1=aP_j ± b , j=1,⋯,k−1, (a,b)=1, 2|ab. If a −1 = P₁^λn P_n^λn , P₁⋯P_n|b, we have J₂(ω)→∞ as ω→∞. There exist infinitely many primes P₁ such that P₂,⋯, Pk are primes for arbitrary length k. It is the Book proof. This is a generalized Euclid-Euler proof for the existence of infinitely many primes. Therefore Euclid-Euler-Jiang theorem in the distribution of primes is advanced. It is the Book theorem.