On inductive limits of measure spaces

Assume that S is an almost excellent extension of R. Using functors HomR(S,−) and —⊗RS, we establish some connections between classes of modules Open image in new window and Open image in new window , cotorsion pairs Open image in new window and Open image in new window . If Open image in new window is a T-extension or (and) H-extension of Open image in new window , we show that Open image in new window is a (resp., monomorphic, epimorphic, special) preenveloping class if and only if so is Open image in new window . If Open image in new window is a TH-extension of Open image in new window , we obtain that Open image in new window is complete (resp., of finite type, of cofinite type, hereditary, perfect, n-tilting) if and only if so is Open image in new window .

Let Open image in new window stand for the Fubini-type product of σ-ideals Open image in new window . We consider mixed measure-category σ-ideals Open image in new window and Open image in new window (called the Mendez σ-ideals), and study some features of their structure. We show that Open image in new window and Open image in new window are not invariant with respect to nonzero rotations. Using Fremlin’s results, we describe nice Borel bases of Open image in new window and Open image in new window . The rest of the paper is devoted to uniform versions of the Nikodym theorem and the Lusin theorem for Borel functions of two variables.

In this paper, using generating functions, we study two categories Open image in new window and Open image in new window of modules for twisted affine Lie algebras Open image in new window , which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635–664]. We classify integrable irreducible Open image in new window -modules in categories Open image in new window and Open image in new window , where Open image in new window is proved to contain the well-known evaluation modules and Open image in new window to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.

Infrared spectra and normal vibrations of ions of the type Open image in new window and Open image in new window , with X and Y as S, O and N have been compared and discussed.

This chapter describes floating-point mathematics for real-time C++ using built-in floating-point types such as Open image in new window and Open image in new window . The first sections of this chapter introduce floating-point arithmetic, mathematical constants, elementary transcendental functions and higher transcendental functions. The last sections of this chapter cover more advanced topics including complex-numbered mathematics, compile-time evaluation of floating-point functions and generic numeric programming.

The heat capacity and density of solutions of cadmium and mercury diiodides in methylpyrrolidone (MP) at 298.15 K were studied by calorimetry and densimetry. The data obtained are discussed in relation to the special features of solvation and complex formation in solutions of the salts. The standard partial molar heat capacities and volumes ( Open image in new window and Open image in new window ) of the electrolytes in MP were calculated. The standard heat capacities Open image in new window and volumes Open image in new window of the Cd2+ and Hg2+ ions in solutions in MP at 298.15 K were determined.

The subject of this note is a characterization for the strong generator of a Ray resolvent. The discussion arose from the question if there exists any Hille-Yoshida type theorem in the context of Ray resolvents. More precisely the problem is the following: Given a compact space K and a linear operator L which maps a subspace of the space of continuous functions on K into , find necessary and sufficient conditions on (L, ) which ensure that L is the strong generator of a Ray resolvent on K.

We find that a generalized transition function P s,t (x, y, z,A) with two parameters s, t > 0 on a measurable space Open image in new window has a very interesting and important property: its total transition probability P s,t (x, y, z, E) is only related to the product st of the two parameters s, t > 0 and is unrelated to the states x, y, z ∈ E. To be more exact, there is a constant 0 ≤ λ ≤ + ∞ such that

In this paper, a new distribution space Open image in new window is constructed and the definition of the classical Hilbert transform is extended to it. It is shown that Open image in new window is the biggest subspace of Open image in new window on which the extended Hilbert transform is a homeomorphism and both the classical Hilbert transform for Lp functions and the circular Hilbert transform for periodic functions are special cases of the extension. Some characterizations of the space Open image in new window are given and a class of useful nonlinear phase signals is shown to be in Open image in new window . Finally, the applications of the extended Hilbert transform are discussed.

We consider the ten-dimensional vector space Open image in new window whose elements are generators of infinitesimal transformations of the tendimensional manifold Open image in new window of all the local reference frames. We indicate by Open image in new window the convex closed cone which contains the generators of the physically feasible transformations and we assume that Open image in new window is stable under a group Open image in new window of linear mappings of Open image in new window onto itself. The convex closed cones which are stable under the restricted Lorentz group form a family parametrized by a quantityl which we interpret as a new physical constant, in the simplest case a fundamental lenght. We discuss all the linear mappings which leave the cone Open image in new window stable and we show that, under reasonable assumptions, the symmetry group Open image in new window contains a group isomorphic to the restricted de Sitter groupSO2,3+ and possibly also a parity operator and the dilatations of the space Open image in new window .

1. According to a classical theorem, every semicontinuous real-valued function of a real variable can be obtained as the limit of a sequence of continuous functions. The corresponding proposition need not hold for the semicontinuous functions defined on an arbitrary topological space; indeed, it is known that the theorem holds in the general case if and only if the topology is perfectly normal [6]. For some purposes, it is important to know when a topological measure space has the property that each semicontinuous real-valued function defined thereon is almost everywhere equal to a function of the first Baire class. In the present article, we are able to resolve this question when the topology is completely regular. A most interesting by-product of our consideration of this problem is the discovery that every Lebesgue measurable function is equivalent to the limit of a sequence of approximately continuous ones. This fact enables us to answer a question, posed in an earlier work [7], concerning the existence of topological measure spaces in which every bounded measurable function is equivalent to a Baire function of the first class.

On a capacity for modular spaces

In this note we define and discuss some properties of partition of unity on *‐inductive limits of topological vector spaces. We prove that if a partition of unity exists on a *‐inductive limit space of a collection of topological vector spaces, then it is isomorphic and homeomorphic to a subspace of a *‐direct sum of topological vector spaces.

For an order-continuous Banach function space Λ and a separated inductive limit E:= indn E n, we prove that indn A {En} is a topological subspace of Λ {E}; moreover, both spaces coincide if the inductive limit is hyperstrict. As a consequence, we deduce that if E is an LF-space, then L p {E} is barrelled for 1 ≤ p ≤ ∞.

The questions of dense definiteness and boundedness of composition operators in L 2spaces are studied by means of inductive limits of operators.Methods based on projective systems of measure spaces and inductive limits of L 2 -spaces are developed.Illustrative examples are presented.