Floating-Point Mathematics

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.

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.

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 the first part of the article we give a brief review of various approaches to symbolic integration of ordinary differential equations (Liouvillian approach, power series method) from the point of view of numerical methods. We aim to show that all higher transcendental functions were considered in the past centuries as solutions of such differential equations, for which the application of the computational techniques of that time was particularly efficient. Nowadays the finite differences method is a standard method for integration of differential equations. Our main idea is that now all transcendental functions can be considered as solutions of such differential equations, for which the application of this method is particularly efficient.

* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index

In this paper using a perturbative method a series expansion of the price of a (put up and out) barrier option with time dependent parameters in the Black and Scholes world is obtained. The first three terms of this series are written explicitly as formulae involving some elementary and non elementary transcendental functions. The formula obtained has been tested on some examples taken from the financial literature and a sufficient condition for the convergence of the perturbation series expansion is given. The numerical experience shows that in the cases of practical interest considered the use of the first two or three terms of the series expansion mentioned above guarantees three or four correct significant digits in the prices computed. Similar formulae can be obtained for other types of barrier options such as step options, step barrier options and so on. The website contains some auxiliary material that helps the understanding of this paper and the computer programs needed to evaluate the formula obtained.

1 - Variables and Functions

For the statistical description of a demand process, many probability density functions (pdf) are available, each having its advantages and drawbacks. How do they compare when in application? Is it true that the shape of the pdf is not relevant? This paper deals with one application, namely a simple s,Q stock control system. Assuming for the lead time demand successively five different pdf's (Gaussian, logistic, gamma, log-normal, Weibull) and using the service level as a criterion, it is shown that for this application (1) the numerical differences between the selected functions are in fact negligible; (2) the logistic pdf produces a formula that only contains elementary transcendental functions; (3) the other pdf's lead to much more complex expressions.

Over the past 24 years, the gravity fields of homogeneous polyhedra have been studied extensively (Paul, 1974; Barnett, 1976; Okabe, 1979; Golizdra, 1981; Strakhov et al., 1986a, 1986b; Gotze and Lahmeyer, 1988; Pohanka, 1988; Ivan, 1990; Pohanka, 1990; Holstein and Ketteridge, 1996). This intensive scrutiny is well deserved because polyhedra provide a framework in which complex body shapes can be modeled parsimoniously and because the analytic expressions for the gravity fields of uniform polyhedra involve only elementary transcendental functions available as part of the run‐time library for most computational programming languages.

The relationship between the axial spontaneous emission and the laser gain is obtained by means of quantum field theory. Using the relationship, the spontaneous two-quantum Stark emission in the beam direction by a relativistic electron beam and in an electrostatic wave propagating in the antiparallel direction of the electron beam («electric wiggler») is derived from the laser gain by the stimulated free-electron two-quantum Stark emission. It is found that the spontaneous-emission power per unit solid angle in the axial direction scales as Open image in new window , whereEb is the beam energy and Open image in new window andλω are the field amplitude and wiggler wavelength of the electric wiggler, respectively.

The parabolic equation (PE) approximation—essential in solving range‐dependent propagation problem—relies on iterated fast Fourier transforms (FFTs). Preventing significant errors due to the periodicity of the finite FFT (“ringing”) requires inserting an absorber potential to confine the wave energy to the range of interest. Such potentials, or sponges, have, to date, been strictly ad hoc. The problem of creating an optimal sponge by using an optical model analogy of the reflectionless potential U = U0* [sech(a*x)]2 of 1‐D quantum mechanics is studied. This approach permits the development of closed‐form expressions for both transmission and reflection coefficients in terms of elementary transcendental and gamma functions of complex argument. Using these coefficients and an efficient technique for evaluating such gamma functions yields a quantitative study of the problem of the optimal sponge. These results demonstrate that, for relevant parameter domains, there may be several “branches” of minimal paths ...

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.