Asymptotic Expansions

The poetic dialogue in JH Prynne's "Conversance" (dated 28 August 2018) is a vivid and singularly affective poem.
Throughout there is the trace of the other. The  "curvature of space", the Levinasian "face" and "cheek" of the other, are all part of this Prynne dialogue, as are echoes from the  Prynne lectures on Olson's Maximus poems: love not in the lyric poem, but the epic poem, with a cosmogonic sweep encompassing  early roots in myth (shimmerings of the Egyptian world of the dead) and huge constellations (central tropes in Kafka's "Der Bau", namely RESCUE, TRUST and ENTRY  ("Rettung", "Vertrauen" and "Eingang"), are  present in this later text, also suggesting a reappraisal of Prynne;'s own literary oeuvre).

Literary criticism, the role of the reader, the status of poetic texts - all are under scrutinity in the translation exercise attempted here, a critical reading and a translation into Afrikaans, itself under severe scrutiny, and destabilised, hovering on a chasm...as is the "white memorandum" that Prynne, an octogenarian poet, has at the heart of his late work text.

A detailed structure of the Chapman-Enskog expansion for the linearized Grad moment equations is determined. A method of partial summing of the Chapman-Enskog series is introduced, and is used to remove short-wave instability of the Burnett approximations.

In this study, perturbation method is applied to obtain the solution of dynamical system which does not involve periodic solution. Since perturbation method presents results in series approximation form, accuracies with retention of different number of terms are studied. Likewise, different nonlinearities are also considered, with the level of agreements of their solutions are also presented in tabular form to facilitate ease of comparison. Numerical results obtained in all cases revealed that retention of one or two terms in the solution suffices for a reasonable level of accuracy for system with no periodic solution.

The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function _2F_1(a,b;c;z) in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of _2F_1(a,b;c;z) have been provided in the literature, they are all limited by a narrow domain of validity, either in the complex plane of the variable or in the parameter space. Overcoming this restriction, we provide new asymptotic expansions for the hypergeometric function with two large parameters, which are valid for the entire complex plane of z except for a few specific points. We show that these expansions work well even when we approach the possible singularity of _2F_1(a,b;c;z), |z|=1, where the current...

The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function ${\hspace{0pt}}_2F_1(a, b;c;z)$ in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of ${\hspace{0pt}}_2F_1(a, b;c;z)$ have been provided in the literature, they are all limited by a narrow domain of validity, either in the complex plane of the variable or in the parameter space. Overcoming this restriction, we provide new asymptotic expansions for the hypergeometric function with two large parameters, which are valid for the entire complex plane of z except for a few specific points. We show that these expansions work well even when we approach the possible singularity of ${\hspace{0pt}}_2F_1(a, b;c;z)$ , $\vert z\vert =1$ , where the current expansions typically fail. Using our results we determine asymptotically the partition function of a lattice gas in a field of traps in the different possible physical limits of few/many particles and few/many traps, illustrating the applicability of the derived asymptotic expansions of the HGF in physics.

Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real parameters are obtained and
ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie’s unified generalization.

In this paper asymptotic error expansions for mixed finite element approximations of the integro-differential equation are derived, and Richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation post-processing technique. The results of this paper provide new asymptotic expansions. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a-posteriori error estimators for this mixed finite element method. Finally, a numerical example is provided to validate the theoretical results.

Wireless networking is a method by which homes, telecommunications networks and enterprise (business) installations avoid the costly process of introducing cables into a building, or as a connection between various equipment locations. For such a reasons this technology has become popular. Though it is familiar, its wireless channel is vulnerable to the eavesdroppers during message delivery (security is the major problem). In the previous cases this problem was solved by cryptographic methods such as RSA public key cryptosystem. But due to expensive key distribution and improvement in decoding technology, the message transmitted is said to be unsecured. The problem can be overcome by using artificial noise generation. This paper investigates and studies how to deliver the message securely in the wireless network using artificial noise generation concept.

We seek analytic formulae for the pricing of extinguishable cap-floor Libor quanto flows which pay capped and/or floored foreign currency Libor conditional on survival of a named debt issuer. We assume that the foreign currency in which the Libor is paid and the credit default risk (observed for debt denominated in the domestic currency) are potentially correlated with the exchange rate between foreign and domestic currency; and further that this exchange rate may drop by a fixed relative amount in the event of a credit default event. The stochastic credit intensity model is assumed to be Black-Karasinski and the interest rate model to be beta-blend, a family of models which encompasses within its scope both the Black-Karasinski and the Hull-White cases.

The solutions are obtained as perturbation expansions which are valid in the limit of the foreign interest rate and the credit default intensity being small. A Green's function solution is found for the governing PDE which is asymptotically valid under this assumption. This is calculated to second order and used to calculate to first order accuracy the price of extinguishable cap-floor Libor quanto flows, in particular taking account of the assumed correlations. When the interest rate model is Hull-White a composite expansion is proposed which has the advantage of greater consistency with the known exact formula for the conditional bond price in the limit of zero correlation.

It is shown that the sequence of the generalized Bell polynomials Sn(x) is convex under some restrictions of the parameters involved. A kind of recurrence relation for Sn(x) is established, and some numbers related to the generalized Bell numbers and their properties are investigated.

The r-Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for 𝑟-Bell numbers with real arguments is obtained.

In this work we review some recent results on approximation of solutions of elliptic problems with high-contrast coefficients. In particular, we detail the derivation of asymptotic expansions for the solution in terms of the high-contrast of the coefficients. We consider the case of high-contrast elliptic equations and we present the case of only one high-contrast inclusion. The case of more inclusions follows similarly. In order to simplify the presentation we consider first the one dimensional case so no further complicated requirements regarding the computation of two or higher dimensional integrals are needed. We review the case of two-dimensional problems and give some numerical examples.

We study linear elasticity problems with high contrast in the coefficients using asymptotic limits recently introduced. We derive an asymptotic expansion to solve heterogeneous elasticity problems in terms of the contrast in the coefficients. We study the convergence of the expansion in the H 1 norm.

In this work we review some recent results on approximation of solutions of elliptic problems with high-contrast coefficients. In particular, we detail the derivation of asymptotic expansions for the solution in terms of the high-contrast of the coefficients. We consider the case of high-contrast elliptic equations and we present the case of only one high-contrast inclusion. The case of more inclusions follows similarly. In order to simplify the presentation we consider first the one dimensional case so no further complicated requirements regarding the computation of two or higher dimensional integrals are needed. We review the case of two-dimensional problems and give some numerical examples.