Jacobi Elliptic Cliffordian Functions

The well-known Jacobi elliptic functions sn(z), cn(z), dn(z) are defined in higher dimensional spaces by the following method. Consider the Clifford algebra of the antieuclidean vector space of dimension 2m + 1. Let x be the identity mapping on the space of scalars + vectors. The holomorphic Cliffordian functions may be viewed roughly as generated by the powers of x, namely xⁿ , their derivatives, their sums, their limits (cf: zⁿ for classical holomorphic functions). In that context it is possible to define the same type of functions as Jacobi's.     

A Remark on a Problem of Rolf Nevanlinna

Let T ( f ) and N ( r,c ) denote the usual Nevanlinna characteristic and the counting function for the c -points of a meromorphic function f , respectively. Using a result of Miles and Shea ( Quart. J. Math. Oxford , 24 (2), (1973), 377-383) and two simple estimates for trigonometric functions, we show in connection with a 1929 problem of Nevanlinna for meromorphic functions f of finite order 1 < u < X $$ \limsup\limits_{r\rightarrow \infty } { N(r, 0)+N(r, \infty ) \over T(r, \,f)}\ge {2\sqrt 2 \over \pi} {|\sin \pi \lambda | \over D(\lambda )}\ge (0.9)\, {{|\sin \pi \lambda | \over {D(\lambda )}, }} $$ with D ( u ) = q +|sin ~ u | for $ q\le \lambda \le q + \fraca {1}{2} $ and D ( u ) = q + 1 for $ q + {\fraca {1} {2}} \le \lambda \lt q + 1 $ , where $ q = \lfloor \lambda \rfloor $ .     

On Harmonic Diffeomorphisms of the Unit Disc onto a Convex Domain

We prove a theorem for harmonic diffeomorphisms between the unit disc and a convex Jordan domain, which is a generalization of Heinz theorem [E. Heinz (1959). On one-to-one harmonic mappings. Pacific J . Math ., 9 , 101-105] for harmonic diffeomorphisms of the unit disc onto itself. We give a number of corollaries of the theorem we prove.     

On minkowski metric and weighted tchebyshev norm in vector optimization ∗

The typical approach in solving vector optimization problems is to scalarize the vector cost function into a single cost function by means of some utility or value function. A very large class of utility function is given by the Minkowski’s metric proposed by Charnes and Cooper in the context of goal programming. This includes the special case of linear scalarization and the weighted Tchebyshev norm. We shall furnish a rigorous justification that there is no equivalent relationship between the general vector optimization problem and scalarized optimization problems using any Minkowski’s metric utility function. Furthermore, we also show that the weighted Tchebyshev norm is, in some sense, the best amongst the class of Minkowski’s metric utility functions since it is the only scalarization method which yields an equivalence relation between the weak vector optimization problem and a set of scalar optimization problems, without any convexity assumption     

Insurability of catastrophic risks: the stochastic optimization model

Catastrophes produce losses highly correlated in space and time, which break the law of large numbers. We derive the insurability of dependent catastrophic risks by calculating conditions that would aid insurers in deliberate selection of their portfolios. This paper outlines the general structure of a basic stochastic optimization model. Connections between the probability of ruin and nonsmooth risk functions, as well as adaptive Monte Carlo optimization procedures and path dependent laws of large numbers, are discussed     

Generalized partial derivatives and optimality conditions for non-differentiable functions

In [9] a generalization of the concept of a classical gradient was presented, which exists for a broad class of non-differentiable functions. This concept differs from traditional “set-valued”" generalizations by an expansion of a directional derivative into a special orthogonal series. Investigating the coefficients of this series and interpreting them as a kind of partial derivatives formal analogies to properties of classical partial derivatives could be drawn. Moreover the new concept reveals relations between questions appearing within the theory of optimization and results in other branches of mathematics, especially to the theory of harmonic and subharmonic functions, The aim of this paper is to show that with the new tool of generalized partial derivatives necessary and sufficient conditions for optimality can be deduced in the case, that a function under consideration is not differentiable at the point of interest. In such a way criteria of optimality can be formulated using the language of these generalized partial derivatives. Numerical considerations show, that the generalized partial derivatives can (approximately) be calculated without solving integrals by hyperinterpolation on the sphere     

Joseph Kampé de Fériet et la mécanique des fluides en France durant l'entre-deux-guerres

Up until the twentieth century, fluid mechanics was characterised by a strong division between theory and practice. The formulas of mathematicians and physicists were difficult to apply experimentally and their relevance was neither well accepted nor well understood by engineers. In France, where priority was given to the theoretical aspects of this subject, the creation of several centres of fluid mechanics in the 1930s (Lille, Marseille, Paris and Toulouse) represented an attempt at a rapprochement between fluid mechanics, experimental aerodynamics, and hydrodynamics. The aim was to re-establish France's international profile, which had been lost after 1918. In Lille, the leadership of the “Institut de mécanique des fluides” (IMFL) was entrusted to Jospeh Kampé de Fériet, who was able to address questions of both theoretical and experimental nature in the context of his work at the IMFL and at the “Commission de la turbulence atmosphérique”. On the one hand, Kampé de Fériet used a probabilistic approach to give greater mathematical rigour to the statistical theory of turbulence due to Taylor–von Kármán. On the other hand, he played an active part in his group's experimental research on atmospheric turbulence. This paper aims to show in what way Kampé de Fériet's approach to fluid mechanics enabled him to contribute to the international development of the statistical theory of turbulence during the interwar period.     

均布荷载作用下悬臂磁电弹性梁的解析解

对磁电弹性平面问题进行了研究，给出了用拟调和位移函数表达的通解，进而以试凑法按平面应力问题推导出了均布荷载作用下悬臂磁电弹性粱的解析解，所得解有易于理解、便于校对、形式统一简洁的特点。本文还将计算结果与压电材料和弹性材料相应结果进行了分析、比较，为验证各种数值计算方法提供了参考依据。     

Radiation Losses of Bent Planar Waveguides

Bent planar waveguides are conformally mapped into straight figures with equivalent index profile. Airy functions of traveling-wave form are introduced to express the radiation mode in the outer cladding region. Calculation results of radiation loss are confirmed to give excellent accuracy.