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Revista Matemática Complutense Nº 18

Nombre de la Revista: Revista Matemática Complutense
Número de Sumario: 18
Fecha de Publicación: 2005/2

Revista Matemática Complutense

Servicio de Publicaciones de la Universidad Complutense de Madrid

- Volumen 18, nº 2  (2005) - 

Más información/Texto completo en  http://revistas.ucm.es/portal/ ....


3.   Sumario  

On the Amount of Information Resulting from Empirical and Theoretical Knowledge     
VAJDA, Igor | ZVAROVA, Jana | VESELY, Arnoˇst
Palabras Clave: probability space, ontology, predicate, knowledge area, state of the knowledge area, empirical knowledge, theoretical knowledge, information in empirical knowledge, information in theoretical knowledge.
: We present a mathematical model allowing formally define the concepts of empirical and theoretical knowledge. The model consists of a finite set P of predicatesand a probability space (, S, P) over a finite set called ontology which consists of objects ! for which the predicates _ 2 P are either valid (_(!) = 1) or not valid (_(!) = 0). Since this is a first step in this area, our approach is as simple as possible, but still nontrivial, as it is demonstrated by examples. More realistic approach would be more complicated, based on a fuzzy logic where the predicates _ 2 P are valid on the objects ! 2 to some degree (0 _ _(!) _ 1). We use the classical information divergence to introduce the amount of information in empirical and theoretical knowledge. By an example is demonstrated that information in theoretical knowledge is an extension of the “sematic information” introduced formerly by Bar Hillel and Carnap as an alternative to the information of Shannon.

Existence and Regularity of the Solution of a Time Dependent Hartree-Fock Equation Coupled with a Classical Nuclear Dynamics   
Palabras Clave: Hartree-Fock equation, classical dynamics, regularity, existence. 2000 Mathematics Subject Classification: 35Q40, 35Q55,
: We study an Helium atom (composed of one nucleus and two electrons) submitted to a general time dependent electric field, mod led by the Hartree-Fock equation, whose solution is the wave function of the electrons, coupled with the classical Newtonian dynamics, for the position of the nucleus. We prove a result of existence and regularity for the Cauchy problem, where the main ingredients are a preliminary study of the regularity in a nonlinear Schr¨odinger equation with semi-group techniques and a Schauder fixed point theorem

Wilks’ Factorization of the Complex Matrix Variate Dirichlet Distributions    
CUI, Xinping | K. GUPTA, Arjun | K. NAGAR, Daya
Palabras Clave: beta distribution, complex random matrix, Dirichlet distribution, Jacobian, complex multivariate gamma function, transformation, Wishart distribution.
: In this paper, it has been shown that the complex matrix variate Dirichlet type I density factors into the complex matrix variate beta type I densities. Similar result has also been derived for the complex matrix variate Dirichlet type II density. Also, by using certain matrix transformations, the complex matrix variate Dirichlet distributions have been generated from the complex matrix beta distributions. Further, several results on the product of complex Wishart and complex beta matrices with a set of complex Dirichlet type I matrices have been derived.

A Number Theoretic Approach to Sylow r-Subgroups of Classical Groups     
I. ALALI, Mashhour | HERING, Christoph | NEUMANN, Anni
Palabras Clave: Sylow r-subgroups, wreath product. 2000 Mathematics Subject Classification: 20D15.
: The purpose of this paper is to give a general and a simple approach to describe the Sylow r-subgroups of classical groups.

A Remark on Perturbed Elliptic Equations of Caffarelli-Kohn-Nirenberg Type   
ABDELLAOUI, Boumediene | FELLI, Veronica | PERAL, Ireneo
Palabras Clave: semilinear elliptic equations, critical Sobolev exponent, Caffarelli-Kohn- Nirenberg inequalities.
: Using a perturbation argument based on a finite dimensional reduction, we find positive solutions to the following class of perturbed degenerate elliptic equations with critical growt

.121221222 . . . Is Not Quadratic    
LUCA, Florian
Palabras Clave: irrationality, applications of sieve methods.
: In this note, we show that if b > 1 is an integer, f(X) 2 Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number(...) where an 2 Z and 1  |an|  K for all n  0, is neither rational nor quadratic.

Boundedness for Threefolds in P6 Containing a Smooth Ruled Surface as Hyperplane Section     
Palabras Clave: threefolds, low codimensional subvarieties, ruled surface as hyperplane section, bounded degree.
: Let X _ P6 be a smooth irreducible projective threefold, and d its degree. In this paper we prove that there exists a constant _ such that for all X containing a smooth ruled surface as hyperplane section and not contained in a fourfold of degree less than or equal to 15, d _ _. Under some more restrictive hypothesis we prove an analogous result for threefolds containing a smooth ruled surface as hyperplane section and contained in a fourfold of degree less than or equal to 15.

A Note on Non-Negative Singular Infinity-Harmonic Functions in the Half-Space  
Palabras Clave: viscosity solutions, infinity-harmonic, singular.
: In this work we study non-negative singular infinity-harmonic functions in the half-space. We assume that solutions blow-up at the origin while vanishing at infinity and on a hyperplane. We show that blow-up rate is of the order |x|−1/3.

Ribbon Knots of 1-Fusion, the Jones Polynomial, and the Casson-Walker Invariant    
Palabras Clave: Ribbon knots of 1-fusion, Jones polynomial, Casson-Walker invariant.
: We give an explicit formula for the Casson-Walker invariant of double branched covers of S3 branched along ribbon knots of 1-fusion.

Schottky Uniformizations of Z22 Actions on Riemann Surfaces    
Palabras Clave: Schottky groups, Riemann surfaces, automorphisms, uniformization.
: Given a closed Riemann surface S together a group of its conformal automorphisms H _= Z22 , it is known that there are Schottky uniformizations of S realizing H. In this note we proceed to give an explicit Schottky uniformizations for each of all different topological actions of Z22 as group of conformal automorphisms on a closed Riemann surface.

Real Hypersurfaces with Many Simple Singularities    
Palabras Clave: simple singularities, real hypersurfaces, patchworking method.
: In this paper we present constructions of real hypersurfaces with many simple singularities and deduce an asymptotical optimal existence result for hypersurfaces corresponding to T-smooth germs of the equisingular stratum. We proceed along the lines of [7] where analogous results were shown for the complex case.

On Functions of Integrable Mean Oscillation    
BLASCO, Oscar | PEREZ, M. Amparo
Palabras Clave: mean oscillation, BMO, modulus of continuity.

Entropy Numbers of General Diagonal Operators     
KUHN, Thomas
Palabras Clave: entropy numbers, diagonal operators, `p-spaces.
: We determine the asymptotic behavior of the entropy numbers of diagonal operators D : `p ! `q, (xk) 7! (_kxk), 0 < p, q _ 1, under mild regularity and decay conditions on the generating sequence (_k). Our results extend the known estimates for polynomial and logarithmic diagonals (_k). Moreover, we also consider some exotic intermediate examples like _k = exp(−plog k).

A Gradient Inequality at Infinity for Tame Functions     
D’ACUNTO, Didier | GRANDJEAN, Vincent
Palabras Clave: Lojasiewicz inequality, asymptotic critical values, bifurcation values, gradient trajectories, o-minimal structures.
: Let f be a C1 function defined over Rn and definable in a given o-minimal structure M expanding the real field. We prove here a gradient-like inequality at infinity in a neighborhood of an asymptotic critical value c. When f is C2 we use this inequality to discuss the trivialization by the gradient flow of f in a neighborhood of a regular asymptotic critical level.

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