On the Deepest Fallacy in the History of Mathematics: The Denial of the Postulate about the Approximation Nature of a Simple-Iteration Method and Iterative Derivation of Cramer’s Formulas

Contrary to the opinion about approximation nature of a simple-iteration method, the exact solution of a system of linear algebraic equations (SLAE) in a finite number of iterations with a stationary matrix is demonstrated. We present a theorem and its proof that confirms the possibility to obtain the finite process and imposes the requirement for the matrix of SLAE. This matrix must be unipotent, i.e. all its eigenvalues to be equal to 1. An example of transformation of SLAE given analytically to the form with a unipotent matrix is presented. It is shown that splitting the unipotent matrix into identity and nilpotent ones results in Cramer’s analytical formulas in a finite number of iterations.

Special Unipotent Representations of Simple Linear Lie Groups of Type A

Let G be a special linear group over the real,the complex or the quaternion,or a special unitary group.In this note,we determine all special unipotent representations of G in the sense of Arthur and Barbasch-Vogan,and show in particular that all of them are unitarizable.

Transfer of Highest Weight Modules and Small Unipotent Representations

We study the transfer between small special unipotent representations for all equal rank real forms of type E_(6) and E_(7). As a consequence, one can verify these modules are unitarity using the results of Wallach and Zhu. Moreover, the K-spectra of these modules can be obtained explicitly.

The recognition of equivariant bifurcation problems

The orbit of an equivariant bifurcation problem with multiparameter is characterized under the action of the group of unipotent equivalences. When the unipotent tangent space is invariant under unipotent equivalences, the recognition problem can be solved by just using linear algebra. Sufficient conditions for a subspace to be intrinsic subspace under unipotent equivalences are given.

STABILITY OF CERTAIN RINGS OF HIGHEST WEIGHT VECTORS

The authors prove the stability of the rings of highest weight vectors of the action of Om x GLn on the complex polynomial rings on Cm,n. As an application, the structure of the rings for m = 3 is determined.

Holomorphic map with unipotent Jacobian matrices in C^3

We prove that the holomorphic unipotent Jacobian conjecture is valid when n = 3.

Construction of complete generalized algebraic groups

With one exception, the holomorph of a finite dimensional abelian connectedalgebraic group is shown to be a complete generalized algebraic group. This result on algebraic group is an analogy to that on Lie algebra.