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Universal codes and unimodular lattices

Robin Chapman, Patrick Solé (1996)

Journal de théorie des nombres de Bordeaux

Binary quadratic residue codes of length p + 1 produce via construction B and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction A modulo 4 . We prove in a direct way the equivalence of these two constructions for p ≤ 31 . In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice B W 32 . The equivalence between construction B modulo 4 plus density doubling and construction...

Universally typical sets for ergodic sources of multidimensional data

Tyll Krüger, Guido F. Montúfar, Ruedi Seiler, Rainer Siegmund-Schultze (2013)

Kybernetika

We lift important results about universally typical sets, typically sampled sets, and empirical entropy estimation in the theory of samplings of discrete ergodic information sources from the usual one-dimensional discrete-time setting to a multidimensional lattice setting. We use techniques of packings and coverings with multidimensional windows to construct sequences of multidimensional array sets which in the limit build the generated samples of any ergodic source of entropy rate below an h 0 with...

Upper and lower set formulas: restriction and modification of the Dempster-Pawlak formalism

Ismail Türkşen (2002)

International Journal of Applied Mathematics and Computer Science

A modification of Dempster's and Pawlak's constructs forms a new foundation for the identification of upper and lower sets formulas. Also, in this modified Dempster-Pawlak construct we require that subsets of the power set be restricted to the well-known information granules of the power set. An aggregation of upper information granules amongst each other and lower information granules amongst each other determine upper and lower set formulas for both crisp and fuzzy sets. The results are equivalent...

Variantes sur un théorème de Candès, Romberg et Tao

Jean-Pierre Kahane (2013)

Annales de l’institut Fourier

Le théorème CRT dit comment reconstruire un signal à partir d’un échantillonnage de fréquences parcimonieux. L’hypothèse sur le signal, considéré comme porté par un groupe cyclique d’ordre N , est qu’il est porté par un petit nombre de points, s , et la méthode est de choisir aléatoirement C s log N fréquences et de minimiser dans l’algèbre de Wiener le prolongement à ℤ / N ℤ de la transformée de Fourier du signal réduite à ces fréquences. Quand C est grand, la probabilité de reconstruire le signal est voisine...

Variational approximation for detecting point-like target problems

Gilles Aubert, Daniele Graziani (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.

Variational approximation for detecting point-like target problems*

Gilles Aubert, Daniele Graziani (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.

Velocity and Entropy of Motion in Periodic Potentials

Andreas Knauf (1996/1997)

Séminaire Équations aux dérivées partielles

This is a report on recent joint work with J. Asch, and with T. Hudetz and F. Benatti.We consider classical, quantum and semiclassical motion in periodic potentials and prove various results on the distribution of asymptotic velocities.The Kolmogorov-Sinai entropy and its quantum generalization, the Connes-Narnhofer-Thirring entropy, of the single particle and of a gas of noninteracting particles are related.

Violations of the Ingleton inequality and revising the four-atom conjecture

Nigel Boston, Ting-Ting Nan (2020)

Kybernetika

The entropy region is a fundamental object of study in mathematics, statistics, and information theory. On the one hand, it involves pure group theory, governing inequalities satisfied by subgroup indices, whereas on the other hand, computing network coding capacities amounts to a convex optimization over this region. In the case of four random variables, the points in the region that satisfy the Ingleton inequality (corresponding to abelian groups and to linear network codes) form a well-understood...

Visual simultaneous localisation and map-building supported by structured landmarks

Robert Bączyk, Andrzej Kasiński (2010)

International Journal of Applied Mathematics and Computer Science

Visual simultaneous localisation and map-building systems which take advantage of some landmarks other than point-wise environment features are not frequently reported. In the following paper the method of using the operational map of robot surrounding, which is complemented with visible structured passive landmarks, is described. These landmarks are used to improve self-localisation accuracy of the robot camera and to reduce the size of the Kalman-filter state-vector with respect to the vector...

Wavelet transform and binary coalescence detection

Jean-Michel Innocent, Bruno Torrésani (1997)

Banach Center Publications

We give a short account of some time-frequency methods which are relevant in the context of gravity waves detection. We focus on the case of wavelet analysis which we believe is particularly appropriate. We show how wavelet transforms can lead to efficient algorithms for detection and parameter estimation of binary coalescence signals. In addition, we give in an appendix some of the ingredients needed for the construction of discrete wavelet decompositions and corresponding fast algorithms.

Weighted entropies

Bruce Ebanks (2010)

Open Mathematics

We present an axiomatic characterization of entropies with properties of branching, continuity, and weighted additivity. We deliberately do not assume that the entropies are symmetric. The resulting entropies are generalizations of the entropies of degree α, including the Shannon entropy as the case α = 1. Such “weighted” entropies have potential applications to the “utility of gambling” problem.

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