Remarks on classifications and adjunctions

Jon Barwise and Jerry Seligman introduced the category of classifications and infomorphisms to model information theoretical concepts and information flow processes. This is the dual category of the category of Chu spaces introduced by Michael Barr and Peter Chu and studied extensively by Vaughan Pratt, Gordon Plotkin and others at Stanford. It is also very closely related to the theory of institutions introduced in a different context by Joseph Goguen and Rod Burstall. Here some aspects of the theory are reviewed and some adjunctions between the category of classifications and other related categories are studied from a more abstract point of view.     

Rediscovering the contributions of Rodrigues on the representation of special functions

Following the pressure of Hermite in 1865 and Heine in 1868, the name of Rodrigues was definitively associated to his famous representation given in 1815. In this paper, we precise the contribution of Rodrigues, proving that his formula not only generates the Legendre polynomials, but also special Gegenbauer polynomials and even the corresponding second kind functions, which are never mentioned. The links with Hildebrandt polynomials and with the Bell polynomials are also briefly examined.     

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.     

Separability of the space of continuous functions that are continuous in Λ-variation

The Shao-Sablin index of a Λ-sequence Λ=(λi) is defined by . The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if SΛ<2. Also, ΛBV=ΛBVc if and only if SΛ<2, where ΛBVc denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia-Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation.     

Relaxation of Certain Integral Functionals Depending onStrain and Chemical Composition

The authors provide a relaxation result in BV×Lq, 1≤q<+∞ as the first step towards the analysis of thermochemical equilibria.     

A new property of entire functions of exponential type not vanishing in a half-plane and applications

Let H be the open upper half-plane, and H its closure. If ? is a non-constant transcendental entire function of exponential type such that |?(x)| ≥ μ > 0 on the real axis, then it may happen that |?(z)| is not greater than μ anywhere in H even if ?(z) in H. We show that if ? is an entire function of exponential type not vanishing in H such that inf _x∈?|?(x)| = μ > 0, then inf _{z∈ H} |?(z)| = μ if and only if h_? (π/2) := lim sup _y →∞y ^? log|?(iy) ≥ 0, and that |?(z)| = mu for some z ∈ H only if ? is a constant. This result, which can be seen as a ‘minimum modulus principle’ for entire functions of exponential type not vanishing in a half-plane, helps us to obtain generalizations of two inequalities of Boas, and one of Turán.     

Differential Inequalities and Starlikeness

We study some differential inequalities in the unit disc which imply starlikeness: for example if ? (z) = z + Σ^∞ _n=2 a_n (?)zⁿ is analytic in D = {z | |z| < 1} and |z(?′′(z) ? 1)| ≤ 1 ? α, z?D for some α ] [0, 1), then ? is one-to-one on D and ? (D) is a starlike domain with respect to the origin.     

Tangential Form of the Trace Condition for Pluriharmonic Functions in Cn

In this paper we introduce a tangential form of the trace condition for pluriharmonic functions on a smooth bounded domain in Cn that has some convexity. The condition obtained, in the bidimensional case, is related to the complex components of quaternionic regular functions. In the case of the unit ball of Cn, we rewrite the trace condition in terms of spherical harmonics.     

Domains of Holomorphy with Edges and Lower Dimensional Boundary Singularities

It is shown that a domain in C ^N with piecewise smooth boundary (and also of some more general shape) is a domain of holomorphy, provided the Levi form at every regular point is positively semidefinite and the tangent cone is convex at every point outside a boundary subset of zero Hausdorff (2N-2)-dimensional measure.     

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.