By considering a C∞ structure on the ordered non-increasing of elements of Rn, we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.

Canonical bases for (n-1)-dimensional subspaces of n-dimensional vector space are introduced and classified in the article. This result is very prospective to utilize canonical bases at all applications. For example, maximal subalgebras of Lie algebras can be found using them.

This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, Fn(G), of real Lp(G)−spaces (when G is a reductive group in the Harish-Chandra class and p=2n), the situation may be rectified, via an outlook which generalizes the fine structure of the Hilbert space, L2(G). This success opens the door for harmonic analysis of unitary representations, G→End(Fn(G)), of G on the Hilbert-substructure Fn(G), which has hitherto been considered impossible.

Canonical bases for subspaces of a vector space are introduced as a new effective method to analyze subalgebras of Lie algebras. This method generalizes well known Gauss-Jordan elimination method.

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution space supported in the chosen point. For Lie groups, it is wellknown that this leads to the universal enveloping algebra of the Lie algebra. For Lie racks, we get rack-bialgebras, for Lie digroups, we obtain cocommutative Hopf dialgebras.

Famous K. Gauss introduced reduced row echelon forms for matrices approximately 200 years ago to solve systems of linear equations but the number of them and their structure has been unknown until 2016 when it was determined at first in the previous article given up to (n−1)×n matrices. The similar method is applied to find reduced row echelon forms for (n−2)×n matrices in this article, and all canonical bases for (n−2)-dimensional subspaces of -dimensional vector space are found also.

The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elements in G.

This paper is devoted to investigating the structure theory of a class of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we completely determine the derivation algebras, the automorphism groups and the second cohomology groups of these Lie superalgebras.

This paper presents a simple and purely geometrical Grand Unification Theory. Quantum Gravity, Electrostatic and Magnetic interactions are shown in a unified framework. Newton’s Gravitational Law, Gauss’ Electrostatics Law and Biot-Savart’s Electromagnetism Law are derived from first principles. Gravitational Lensing, Mercury Perihelion Precession are replicated within the theory. Unification symmetry is defined for all the existing forces. This alternative model does not require Strong and Electroweak forces. A 4D Shock-Wave Hyperspherical topology is proposed for the Universe which together with a Quantum Lagrangian Principle and a Dilator based model for matter result in a quantized stepwise expansion for the whole Universe along a radial direction within a 4D spatial manifold. The Hypergeometrical Standard Model for matter, Universe Topology, Simple Cosmogenesis and a new Law of Gravitation are presented. Type 1A Supernova Survey HU results is provided. A New de-Broglie Force is proposed.

The reduction of higher-dimensional theories over a coset space S/R is known to yield a residual gauge symmetry related to the number of R-singlets in the decomposition of S with respect to R. It is verified that this invariance is identical to that found by requiring that there is a subgroup of the isometry group with an action on the connection form that yields a transformation rule defined only on the base space. The Lagrangian formulation of the projection of the frame of global vector fields from S7 to the Lie group submanifold S3× S3 is considered. The structure of an octonionic Chern-Simons gauge theory is described.

The aim of this paper is to prove the celebrated Riemann Hypothesis. I have already discovered a simple proof of the Riemann Hypothesis. The hypothesis states that the nontrivial zeros of the Riemann zeta function have real part equal to 0.5. I assume that any such zero is s=a+bi. I use integral calculus in the first part of the proof. In the second part I employ variational calculus. Through equations (50) to (59) I consider (a) as a fixed exponent, and verify that a=0.5. From equation (60) onward I view (a) as a parameter (a <0.5) and arrive at a contradiction. At the end of the proof (from equation (73)) and through the assumption that (a) is a parameter, I verify again that a=0.5.