The Cyclical Compactness in Banach C∞(Q)-Modules

In this paper, we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion of passport Γ(X) for a faithful regular laterally complete A- modules X, which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise different cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X)= Γ(Y ). Further we study Banach A-modules in the case A = C∞(Q) or A = C∞(Q)+ i · C∞(Q). We establish the equivalence of all norms in a finite-dimensional (respectively, σ-finite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a finite-dimensionality (respectively, σ-finite-dimensionality) of a Banach A-module.

Classes of analytic functions related to Blaschke products

Basing on the well known Riesz Theorem on the canonical factorization of bounded analytic functions, we distinguished the subfamilies of the Carath?odory class of functions. The basic properties of introduced classes have been proved. Related classes of analytic functions are discussed also.

F. Riesz Theorem

Summary In this article, we formalize in the Mizar system [1, 4] the F. Riesz theorem. In the first section, we defined Mizar functor ClstoCmp, compact topological spaces as closed interval subset of real numbers. Then using the former definition and referring to the article [10] and the article [5], we defined the normed spaces of continuous functions on closed interval subset of real numbers, and defined the normed spaces of bounded functions on closed interval subset of real numbers. We also proved some related properties. In Sec.2, we proved some lemmas for the proof of F. Riesz theorem. In Sec.3, we proved F. Riesz theorem, about the dual space of the space of continuous functions on closed interval subset of real numbers, finally. We applied Hahn-Banach theorem (36) in [7], to the proof of the last theorem. For the description of theorems of this section, we also referred to the article [8] and the article [6]. These formalizations are based on [2], [3], [9], and [11].

Time-Dependent Response of Mass Continuous Solid Phase Media by Integral Form of Constitutive Equations - Mathematical Modeling

Mathematical modeling of boundary value problems in linear theory of viscoelasticity. Definitions and basic principles in the mathematical modeling theory. Constitutive functional and its transformation into a form of Stieltjes integral. Application of theory of algebraic sets and corresponding subsets. Riesz theorem of representation and its application for derivation of constitutive equations. Integral operator forms of stress-strain relationships for a solid-phase continuous media.