Deformations of ternary algebras

The aim of this paper is to extend to ternary algebras the classical theory of formal deformations of algebras introduced by Gerstenhaber. The associativity of ternary algebras is available in two forms, totally associative case or partially associative case. To any partially associative algebra corresponds by anti-commutation a ternary Lie algebra. In this work, we summarize the principal definitions and properties as well as classification in dimension 2 of these algebras. Then we focuss ourselves on the partially associative ternary algebras, we construct the first groups of a cohomolgy adapted to formal deformations and then we work out a theory of formal deformation in a way similar to the binary algebras.

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On compact realifications of exceptional simple Kantor triple systems

Let A be the realification of the matrix algebra determined by Jordan algebra of hermitian matrices of order three over complex composition algebra. We define an in volutive auto morphism on A with a certain action on the triple system obtained from A which give models of simple compact Kantor triple systems. In addition, we give an explicit formula for the canonical trace form and the classification for these triples and their corresponding exceptional real simple Lie algebras. Moreover, we present all realifications of complex exceptional simple Lie algebras as Kantor algebras for a compact simple Kantor triple system defined on a structurable algebra of skew-dimension one.

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Lie-admissible co-algebras

After introducing the concept of Lie-admissible co-algebras, we study a remarkable class corresponding to co-algebras whose co-associator satisfies invariance conditions with respect to the symmetric group 3. We then study the convolution and tensor products. An interesting class of Lie-admissible co-algebras is obtained by dualizing the Gi-associative algebras. These Lie-admissible algebras has been introduced and developed. Let us point out these initially notations.

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Classification of Canonical Bases for (n−1)−dimensional Subspaces of n− Dimensional Vector Space

Canonical bases for (n-1)-dimensional sub spaces 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 sub algebras of Lie algebras can be found using them.

The canonical bases for (n-1)-dimensional sub spaces of n− dimensional vector space are introduced in the article, and all non equivalent of them are classified (Theorem 2). This result generalizes a particular result for 5-dimensional sub spaces of 6-dimensional vector space obtained in the previous article of the same author.To analyze the general case, reduced row echelon forms of matrices are utilized; about reduced row echelon forms. In addition to the principal result, all non-equivalent reduced row echelon forms for (n−1)×n matrices of the rank (n−1) are found.

How to Prove the Riemann Hypothesis

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.

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Lie Group Methods for Eigenvalue Function

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 are 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.

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Lie Group Method for Studying the Heat Generation Effect

Thermophoresis is a phenomenon which causes small particles to be driven away from a hot surface and towards a cold one. Small particles, such as dust, when suspended in a gas with a temperature gradient, experience a force in the direction opposite to the temperature gradient.

The velocity acquired by the particles is called the thermophoretic velocity and the force experienced by the suspended particles due to the temperature gradient is known as the thermophoretic force. Themagnitudes of the thermophoretic force and velocity are proportional to thetemperature gradient, thermal conductivity of aerosol particles and the carrier gas, thermophoretic coefficient and the heat capacity of the gas. This phenomenon has many practical applications in removing small particles from gas streams, in determining exhaust gas particle trajectories from combustion devices, prevention of fouling and corrosion in heat exchangers and turbines, semiconductor manufacture and ceramic powder production.

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