REFERENCES [1] Raposo, A. P. (2018). The algebraic structure of quantity calculus. Measurement Science Review 18(4), 147–157. [2] JCGM (2012). International vocabulary of metrology– basic and general concepts and associated terms (vim). 3rd edition. [3] Kitano, M. (2013). Mathematical structure of unit sys- tems. Journal of Mathematical Physics 54, 052901–1–17. [4] Drobot, S. (1953). On the foundations of dimensional analysis. Studia Mathematica 14 (1), 84–99. [5] Whitney, H. (1968). The mathematics of physical quan- tities: Part II: Quantity structures and

## Abstract

The algebraic structure underlying the quantity calculus is defined axiomatically as an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product, and quotient spaces are considered, as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation of quantity calculus which is centered on the concept of dimension, rather than on the concept of unit, which is regarded as secondary, and uses only integer exponents of the dimensions.

## Summary

In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure [2] of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to [3], [7], [5], [6] in this formalization.

## Summary

The main aim of this article is proving properties of bilinear operators on normed linear spaces formalized by means of Mizar [1]. In the first two chapters, algebraic structures [3] of bilinear operators on linear spaces are discussed. Especially, the space of bounded bilinear operators on normed linear spaces is developed here. In the third chapter, it is remarked that the algebraic structure of bounded bilinear operators to a certain Banach space also constitutes a Banach space.

In the last chapter, the correspondence between the space of bilinear operators and the space of composition of linear opearators is shown. We referred to [4], [11], [2], [7] and [8] in this formalization.

## On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems

We review the realization theory of polynomial (transfer function) matrices via "pure" generalized state space system models. The concept of an *irreducible-at-infinity* generalized state space realization of a polynomial matrix is defined and the mechanism of the "cancellations" of "decoupling zeros at infinity" is closely examined. The difference between the concepts of *irreducibility* and *minimality* of generalized state space realizations of polynomial (transfer function) matrices is pointed out and the associated concepts of *dynamic* and *non-dynamic* variables appearing in generalized state space realizations are also examined.

## Abstract

In this paper, our aim is to introduce the notion of a composition (m, n, k)-hyperring and to analyze its properties. We also consider the algebraic structure of (m, n, k) hyperrings which is a generalization of composition rings and composition hyperrings. Also, the isomorphism theorems of ring theory are derived in the context of composition (m, n, k)-hyperrings.

## Abstract

The algebraic structure of the linear system appears in solving fractional order Poisson’s equation by Haar wavelet collocation approach is considered. The fractional derivative is described in the Caputo sense. Comparison with the classical integer case as a limiting process is illustrated. Numerical comparison is made between the solution using the Haar wavelet method and the finite difference method. The results confirms the accuracy for the Haar wavelet method.

## ABSTRACT

We continue in a direction of describing an algebraic structure of linear operators on infinite-dimensional complex Hilbert space ℋ. In [Paseka, J.- -Janda, J.: More on PT-symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65-72] there is introduced the notion of a weakly ordered partial commutative group and showed that linear operators on H with restricted addition possess this structure. In our work, we are investigating the set of self-adjoint linear operators on H showing that with more restricted addition it also has the structure of a weakly ordered partial commutative group.

## Abstract

We investigate the algebras for the double-power monad on the Sierpisnki space in the category Equ of equilogical spaces, a cartesian closed extension of Top0 introduced by Scott, and the relationship of such algebras with frames. In particular, we focus our attention on interesting subcategories of Equ. We prove uniqueness of the algebraic structure for a large class of equilogical spaces, and we characterize the algebras for the double-power monad in the category of algebraic lattices and in the category of continuous lattices, seen as full subcategories of Equ. We also analyse the case of algebras in the category Top0 of T0-spaces, again seen as a full subcategoy of Equ, proving that each algebra for the double-power monad in Top0 has an underlying sober, compact, connected space.

References [GS] GRANDE, Z.-STRO´NSKA, E.: On an ideal of linear sets , Demonstratio Math. 36 (2003), 307-311. [B] BUCZOLICH, Z.: Category of density points of fat Cantor sets , Real Anal. Exchange 29 (2003/2004), 497-502. [D] DVALISHVILI, B. P.: Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures, and Applications , in: North-Holland Mathematics Studies, Vol. 199, Elsevier, Amsterdam, 2005.