Norms on Quotient Spaces of The 2-Inner Product Space

This paper discussed about construction of some quotients spaces of the 2-inner product spaces. On those quotient spaces, we defined an inner product with respect to a linear independent set. These inner products was derived from the -inner product. We then defined a norm which induced by the inner product in these quotient spaces.

Disjoint topological transitivity for weighted translations generated by group actions

In this note, we give a sufficient and necessary condition for weighted translations, generated by group actions, to be disjoint topologically transitive in terms of the weights, the group element and the measure. The characterization of disjoint topological mixing is obtained as well. Moreover, we apply the results to the quotient spaces of locally compact groups and hypergroups.

Construction of ball spaces and the notion of continuity

Spherically complete ball spaces provide a simple framework for the encoding of completeness properties of various spaces and ordered structures. This allows to prove generic versions of theorems that work with these completeness properties, such as fixed point theorems and related results. For the purpose of applying the generic theorems, it is important to have methods for the construction of new spherically complete ball spaces from existing ones. Given various ball spaces on the same underlying set, we discuss the construction of new ball spaces through set theoretic operations on the balls. A definition of continuity for functions on ball spaces leads to the notion of quotient spaces. Further, we show the existence of products and coproducts and use this to derive a topological category associated with ball spaces.

The arity of convex spaces

The arity of convex spaces is a numerical feature which shows the ability of finite subsets spanning to the whole space via the hull operators. This paper gives it a formal and strict definition by introducing the truncation of convex spaces. The relations that between the arity of quotient spaces and the original spaces, that between the arity of subspaces and superspaces, that between the arity of product spaces and factors spaces, and that between the arity of disjoint sums and term spaces, are systematically studied. A mistake of a formula in [M. Van De Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993] is corrected. It is shown that a convex space is Alexandrov iff its arity is 1. The convex structures with arity ≤n are equivalent to structured sets with n-restricted hull operators.

Asymptotics and Quotient Spaces of Solutions of Operator-Difference Equations and Differential Equations with Small Delay

Formerly, in order to conduct the in-depth study of differential equations with delay, the author proposed the method of splitting the solution space reducing such equations to the systems of operator-difference equations. Using this method, the author assumed new conditions, i.e. the absolute domains for coefficients sufficient for the existence of special (slowly changing) solutions, and proved the presence of approximating and asymptotically approximating properties in them, as well as the asymptotic one-dimensional space of solutions of the initial problems for linear scalar differential equations with insignificantly retarded argument and the corresponding operator-difference equation systems (special solutions correspond, to the solutions with a slowly changing first component and a relatively small second component). For the purposes of the single-point representation of the obtained results and other data related to the theory of dynamic systems (the distance between the solution values tends to zero alongside the unlimited increase in argument), throughout this research paper the author uses the concept of the asymptotic equivalence of solutions for dynamic systems, as it was introduced by the author in their previous research. In order to shape the new mathematical objects, the concept of asymptotic Hausdorff equivalence of solutions for dynamic systems is introduced (the distance between solution values tends to zero with unlimited increase in argument of one solution and monotonic transformation of argument of another solution).

Vector Spaces

The first three sections of this chapter provide a thorough presentation of the concepts of basis and dimension. The approach is unified in the sense that it does not treat finite and infinite-dimensional spaces separately. Important concepts such as algebraic complements, quotient spaces, direct sums, projections, linear functionals, and invariant subspaces make their first debut in section 3.4. Section 3.5 is a brief summary of matrix representations and diagonalization. Then the chapter introduces normed linear spaces followed by an extensive study of inner product spaces. The presentation of inner product spaces in this section and in section 4.10 is not limited to finite-dimensional spaces but rather to the properties of inner products that do not require completeness. The chapter concludes with the finite-dimensional spectral theory.

The space of Keplerian orbits and a family of its quotient spaces

Distance functions on the set of Keplerian orbits play an important role in solving problems of searching for parent bodies of meteoroid streams. A special kind of such functions are distances in the quotient spaces of orbits. Three metrics of this type were developed earlier. These metrics allow to disregard the longitude of ascending node or the argument of pericenter or both. Here we introduce one more quotient space, where two orbits are considered identical if they differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (the longitude of pericenter). The function q is defined to calculate distance between two equivalence classes of orbits. The algorithm for calculation of ̺6 value is provided along with a reference to the corresponding program, written in C++ language. Unfortunately, ̺6 is not a full-fledged metric. We proved that it satisfies first two axioms of metric space, but not the third one: the triangle inequality does not hold, at least in the case of large eccentricities. However there are two important particular cases when the triangle axiom is satisfied: one of three orbits is circular, longitudes of pericenters of all three orbits coincide. Perhaps the inequality holds for all elliptic orbits, but this is a matter of future research.