Algebro-geometric proof of Christoph's Theorem

In this paper an algebraic proof of Christoph's theorem is provided. This theorem from algebraic-geometry is about the existence of a finite automaton for computing coefficient of a series for an algebraic function

Concept of A.I. Based Knowledge Generator

An important feature of the currently used artificial intelligence systems is their anthropomorphism. The tool of inductive empirical systems is a neural network that simulates the human brain and operates in the "black box" mode. Deductive analytical systems for representation of knowledge use transparent formalized models and algorithms, for example, algorithms of logical inference. They solve many intellectual problems, the solution of which can do without a "deep" anthropomorphic AI. On the other hand, the solution of these problems leads to the formation of alternative artificial intelligence systems. We propose the formation of artificial intelligence systems based on the following principles: exclusion of black box technologies; domination of data conversion systems: the use of direct mathematical modeling. The base of the system is a simulator - a module that simulates a given object. The ontological module selectively extracts structured sets of functional links from the simulator and fills them with corresponding data sets. The final (custom) representation of knowledge is carried out with the help of special interfaces. The concept of simulation-ontological artificial intelligence, based on the principles outlined above, is implemented in the form of parametric analysis in the configuration space and forms the methodological basis of the AI-platform for e-learning

The images of non-commutative polynomials evaluated on 2×22\times 2 matrices

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$. We prove the conjecture for $n=2$

Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In Section 2.1 we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin's problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. Section 2.2 is dedicated to particular cases of Plotkin's problem. Section 2.3 is devoted to Plotkin's problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last two sections deal with algorithmic problems for noncommutative and commutative algebraic geometry. Section 3.1 is devoted to the Gr\"obner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. Section 3.2 is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.Comment: In this review we partially used results of arXiv:1512.06533, arXiv:math/0512273, arXiv:1812.01883 and arXiv:1606.01566 and put them in a new contex