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The ☆ -product approach for linear ODEs: A numerical study of the scalar case

Pozza, Stefano, Van Buggenhout, Niel (2023)

Programs and Algorithms of Numerical Mathematics

Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be straightforwardly extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the...

The spectral test of the Boolean function linearity

Piotr Porwik (2003)

International Journal of Applied Mathematics and Computer Science

The paper discusses the problem of recognizing the Boolean function linearity. A spectral method of the analysis of Boolean functions using the Walsh transform is described. Linearity and nonlinearity play important roles in the design of digital circuits. The analysis of the distribution of spectral coefficients allows us to determine various combinatorial properties of Boolean functions, such as redundancy, monotonicity, self-duality, correcting capability, etc., which seems more difficult be...

The structure-from-motion reconstruction pipeline – a survey with focus on short image sequences

Klaus Häming, Gabriele Peters (2010)

Kybernetika

The problem addressed in this paper is the reconstruction of an object in the form of a realistically textured 3D model from images taken with an uncalibrated camera. We especially focus on reconstructions from short image sequences. By means of a description of an easy to use system, which is able to accomplish this in a fast and reliable way, we give a survey of all steps of the reconstruction pipeline. For the purpose of developing a coherent reconstruction system it is necessary to integrate...

The sum-product algorithm: algebraic independence and computational aspects

Francesco M. Malvestuto (2013)

Kybernetika

The sum-product algorithm is a well-known procedure for marginalizing an “acyclic” product function whose range is the ground set of a commutative semiring. The algorithm is general enough to include as special cases several classical algorithms developed in information theory and probability theory. We present four results. First, using the sum-product algorithm we show that the variable sets involved in an acyclic factorization satisfy a relation that is a natural generalization of probability-theoretic...

The tree of shapes of an image

Coloma Ballester, Vicent Caselles, P. Monasse (2003)

ESAIM: Control, Optimisation and Calculus of Variations

In [30], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation...

The tree of shapes of an image

Coloma Ballester, Vicent Caselles, P. Monasse (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem...

The UD RLS algorithm for training feedforward neural networks

Jarosław Bilski (2005)

International Journal of Applied Mathematics and Computer Science

A new algorithm for training feedforward multilayer neural networks is proposed. It is based on recursive least squares procedures and U-D factorization, which is a well-known technique in filter theory. It will be shown that due to the U-D factorization method, our algorithm requires fewer computations than the classical RLS applied to feedforward multilayer neural network training.

The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces

Frédéric A. B. Edoukou (2009)

Journal de Théorie des Nombres de Bordeaux

We study the functional codes C 2 ( X ) defined on a projective algebraic variety X , in the case where X ⊂ ℙ 3 ( 𝔽 q ) is a non-degenerate Hermitian surface. We first give some bounds for # X Z ( 𝒬 ) ( 𝔽 q ) , which are better than the ones known. We compute the number of codewords reaching the second weight. We also estimate the third weight, show the geometrical structure of the codewords reaching this third weight and compute their number. The paper ends with a conjecture on the fourth weight and the fifth weight of the code C 2 ( X ) .

Theoretical foundation of the weighted Laplace inpainting problem

Laurent Hoeltgen, Andreas Kleefeld, Isaac Harris, Michael Breuss (2019)

Applications of Mathematics

Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers...

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