Cotorsion pair extensions

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.

Let ℚ denote the field of rational numbers, Open image in new window an algebraic closure of ℚ, and H : Open image in new window the absolute, multiplicative, Weil height. For each positive integer d and real number \( \mathcal{H} \geqslant 1 \), it is well known that the number Open image in new window of points α in Open image in new window having degree d over ℚ and satisfying \( H\left( \alpha \right) \leqslant \mathcal{H} \) is finite. This is the one-dimensional case of Northcott’s Theorem [8] (see also [5, page 59]). The systematic study of the counting function Open image in new window , and that of related functions in higher dimensions, was begun by Schmidt [10]. It is relatively easy to prove the existence of a positive constant C = C(d) such that Open image in new window (1) and also the existence of positive constants c = c(d) and \( \mathcal{H}_0 = \mathcal{H}_0 \left( d \right) \) such that Open image in new window (2)

We introduce a category of (topological) measure spaces in which inductive limitis exist and where the Banach spaces Open image in new window and Open image in new window (1≤p≤+∞) are isometric for arbitrary inductive systems Open image in new window of (topological) measure spaces.

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.

For an additive subgroup G of a field \( \mathbb{F} \) of characteristic zero, a Lie algebra Open image in new window (G) of Block type is defined with basis {Lα,i | α ∈ G, i ∈ ℤ+} and relations [Lα,i, Lβ,j] = (β − α)Lα+β,i+j (αj − βi)Lα+β,i+j−1. It is proved that an irreducible highest weight Open image in new window (ℤ)-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order ≻ on G and any Λ ∈ Open image in new window (G)0* (the dual space of Open image in new window (G)0 = span{L0,i | i ∈ ℤ+}), a Verma Open image in new window (G)-module M(Λ, ≻) is defined, and the irreducibility of M(Λ, ≻) is completely determined.

Let Open image in new window ⇉ M be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection M → M/ Open image in new window is a fibration if and only if Open image in new window ⇉ M is regular.

LetX be a real linear space; if Open image in new window is a linear topology onX, let Open image in new window denote the class of all bounded with respect to Open image in new window subsets ofX. In this paper it is shown that the spaceX is algebraically finite dimensional if and only if the class of all convex, absorbing and radially bounded subsets ofX is included in the intersection of all Open image in new window , where Open image in new window runs the set of all linear topologies onX.

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.

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 .

Let F be a field with |F| ≥ 3, Open image in new window be the set of all m×m (m ≥ 4) alternate matrices over F. The arithmetic distance of A,B ∈ Open image in new window is d(A,B):= rank(A−B). If d(A,B) = 2, then A and B are said to be adjacent. The diameter of Open image in new window is max{d(A,B): A, B ∈ Open image in new window }. Assume that φ: Open image in new window is a map. We prove the following are equivalent: (a) φ is a diameter preserving surjection in both directions, (b) φ is both an adjacency preserving surjection and a diameter preserving map, (c) φ is a bijective map which preserves the arithmetic distance.

The C++ standard library and the STL provide a wide selection of functions, classes and generic containers that can be used in common programming situations. There are, however, times when just the right container or function for a particular programming task is missing from the standard library and the STL. In the first part of this chapter, we will extend the C++ standard library and the STL by developing a custom Open image in new window container that has a functionality that lies between those of Open image in new window and Open image in new window Furthermore, one often encounters a good C++ compiler that lacks large parts of the C++ standard library such as the STL, C\(99\) compatibility, the time utilities in Open image in new window or the thread support library. The second half of this chapter shows how to emulate partial standard library support with certain self-written parts of the C++ standard library and the STL.

Let f be an integrable function on the unit sphere Σn−1 of Rn(n⩾3) and let σNδ be the Cesaro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that Open image in new window and Open image in new window where ω(f,t)p is the 1st-order modulus of continuity of f in Lp-metric which is defined in a way different than in the classical case of n=2.