Approximation of Functions in Multi-criterial Synthesis of Composite Materials

A prerequisite for the synthesis of composite materials as complex systems is the principles of the control paradigm of Peace and the effectiveness of mathematics (for any reality and any given (not absolute) accuracy, there is a mathematical structure that describes this reality with this accuracy; the converse is also true (homomorphism, arbitrarily close to isomorphism between reality and mathematical structures)).The proposed methodology for managing the identification process (design of composites) includes the process of human choice: the probabilistic nature of the control; the main reason for the inadequacy of a purely analytical research procedure. Here, the optimization of the control of the properties of the composite is carried out experimentally on the model as a result of the approximation of the response function: not the generalized functional is approximated, but the particular criteria of which it consists. The development of composite materials is carried out on the basis of evaluating the parameters of the formation of operational properties. The parameters of each of the kinetic processes of the formation of the physical and mechanical characteristics of the material were taken as particular criteria. Kinetic processes are asymptotic for the composites under study and contain extremum and inflection points. A method is used to approximate multidimensional table-defined functions by generalized polynomials of a particular form. In the parametric identification of kinetic processes, their parameters are considered basic. Approximating models of the main properties are presented. Vector optimization of properties (selection of recipes, technologies and methods of material quality control) is carried out by overcoming ambiguities of goals using linear convolution, introducing benchmarks, building Pareto sets, etc. The expediency of using a systematic approach (the hierarchical structure of properties and the hierarchical structure of the composite proper) to the design of building materials as complex systems is shown. The research results are introduced as prototypes of new identification systems in the development of composite materials with adjustable structure and properties, in contrast to the replication of reference applied developments of identification theory in various industries.

Integration of a Generalized Ratio of Polynomials

This paper provides a closed-form solution to the indefinite integral of a ratio of generalized polynomials where the denominator polynomial is raised to the general order r∈Z+. Such an integral arises in physics and engineering, the solution of which allows for closed-form analysis.

Separately polynomial functions

It is known that if $$f:{{\mathbb R}}^2\rightarrow {\mathbb R}$$ f : R 2 → R is a polynomial in each variable, then f is a polynomial. We present generalizations of this fact, when $${{\mathbb R}}^2$$ R 2 is replaced by $$G\times H$$ G × H , where G and H are topological Abelian groups. We show, e.g., that the conclusion holds (with generalized polynomials in place of polynomials) if G is a connected Baire space and H has a dense subgroup of finite rank or, for continuous functions, if G and H are connected Baire spaces. The condition of continuity can be omitted if G and H are locally compact or one of them is metrizable. We present several examples showing that the results are not far from being optimal.

A NEW FAMILIES OF GAUSS k-JACOBSTHAL NUMBERS AND GAUSS k-JACOBSTHAL-LUCAS NUMBERS AND THEIR POLYNOMIALS

In this paper, we define the new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal-Lucas numbers. We obtain some exciting properties of the families. We give the relationships between the family of the Gauss k-Jacobsthal numbers and the known Gauss Jacobsthal numbers, the family of the Gauss k-Jacobsthal-Lucas numbers and the known Gauss Jacobsthal-Lucas numbers. We also define the generalized polynomials for these numbers. Further, we obtain some interesting properties of the polynomials. In addition, we give the relationships between the generalized Gauss k-Jacobsthal polynomials and the known Gauss Jacobsthal polynomials, the generalized Gauss k-Jacobsthal-Lucas polynomials and the known Gauss Jacobsthal-Lucas polynomials. Furthermore, we find the new generalizations of these families and the polynomials in matrix representation. Then we prove Cassini’s Identities for the families and their polynomials.

Vector Valued Polynomials, Exponential Polynomials and Vector Valued Harmonic Analysis

Let G be a topological Abelian semigroup with unit, and let E be a Banach space. We define, for functions mapping G into E, the classes of polynomials, generalized polynomials, local polynomials, exponential polynomials, and some other relevant classes. We establish their connections with each other and find their representations in terms of the corresponding complex valued classes. We also investigate spectral synthesis and analysis in the class C(G, E) of continuous functions $$f:G \rightarrow E$$ f : G → E . It is known that if G is a compact Abelian group and E is a Banach space, then spectral synthesis holds in C(G, E). We give a self-contained proof of this fact, independent of the theory of almost periodic functions. On the other hand, we show that if G is an infinite and discrete Abelian group and E is a Banach space of infinite dimension, then even spectral analysis fails in C(G, E). We also prove that if G is discrete, has finite torsion free rank and if E is a Banach space of finite dimension, then spectral synthesis holds in C(G, E).