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Summation of Multiple Fourier Series in Matrix Weighted -Spaces
Morten Nielsen
Journal of Mathematics , 2013, DOI: 10.1155/2013/135245
Abstract: This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted -spaces. We introduce a product Muckenhoupt condition for matrix weights and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space , , if and only if the weight satisfies the product Muckenhoupt condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel. 1. Introduction Let be the family of nonnegative-definite complex-valued matrices. A (periodic) matrix weight is by definition an integrable map . For a measurable vector-valued function , let where denotes the usual norm on . We let denote the family , and becomes a Banach space when we factorize over . In this paper we are interested in convergence properties of multiple trigonometric series in and how specific convergence properties of trigonometric series can be related to properties of the weight . To be more specific, let denote the univariate Dirichlet kernel, and for we define the rectangular kernel . Then defines the rectangular partial sum operator for the trigonometric system. We define the action of on vector-valued functions by letting it act separately on each coordinate function; that is, It is well known (see, e.g., [1, Theorem ]) that , as , for . An immediate corollary is that we have convergence of the partial sums , in , for in the vector-valued case. However, it is not obvious what can be said about convergence of in for a general matrix weight . The main result of the preset paper completely characterizes the special class of weights that allow convergence; converges if and only if the weight satisfies a certain matrix Muckenhoupt product condition. Moreover, the characterization relies solely on certain localization properties of the Dirichlet kernels shared by many other summation kernels. So, in addition, we prove that the rectangular Cesàro means and approximation using the Jackson kernels converge in if and only if the weight satisfies the mentioned matrix Muckenhoupt product condition. At a first glance, the study of vector-valued operators like on may seem artificial, but let us mention one important application to finitely generated shift invariant spaces where such mappings appear naturally. For a finite set of functions in , the associated shift invariant space is given by A natural question to pose is whether forms some sort of “stable” generating system for . Here stable can mean a Schauder basis, or even some weaker notion such as a block Schauder basis. Consider
Trigonometric quasi-greedy bases for $L^p(\bT;w)$
Morten Nielsen
Mathematics , 2006,
Abstract: We give a complete characterization of $2\pi$-periodic weights $w$ for which the usual trigonometric system forms a quasi-greedy basis for $L^p(\bT;w)$, i.e., bases for which simple thresholding approximants converge in norm. The characterization implies that this can happen only for $p=2$ and whenever the system forms a quasi-greedy basis, the basis must actually be a Riesz basis.
An example of an almost greedy uniformly bounded orthonormal basis for $L_p([0,1])$
Morten Nielsen
Mathematics , 2006,
Abstract: We construct a uniformly bounded orthonormal almost greedy basis for $L_p([0,1])$, $1
On Schauder Bases Properties of Multiply Generated Gabor Systems
Morten Nielsen
Mathematics , 2015,
Abstract: Let $A$ be a finite subset of $L^2(\mathbb{R})$ and $p,q\in\mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R})$ of the Gabor system $$G(1,p/q,A)=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z}, g\in A\},$$ with a specific ordering on $\mathbb{Z}\times \mathbb{Z}\times A$. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix.
Projection operators on matrix weighted $L^p$ and a simple sufficient Muckenhoupt condition
Morten Nielsen,Morten Grud Rasmussen
Mathematics , 2015,
Abstract: Boundedness for a class of projection operators, which includes the coordinate projections, on matrix weighted $L^p$-spaces is completely characterised in terms of simple scalar conditions. Using the projection result, sufficient conditions, which are straightforward to verify, are obtained that ensure that a given matrix weight is contained in the Muckenhoupt matrix $A_p$ class. Applications to singular integral operators with product kernels are considered.
Explicit construction of constrained instantons
Morten Nielsen,N. K. Nielsen
Physics , 1999, DOI: 10.1103/PhysRevD.61.105020
Abstract: Instantons in massless theories do not carry over to massive theories due to Derrick's theorem. This theorem can, however, be circumvented, if a constraint that restricts the scale of the instanton is imposed on the theory. Constrained instantons are considered in four dimensions in phi^4 theory and SU(2) Yang-Mills-Higgs theory. In each of these theories a calculational sceme is set up and solved in the lowest few orders in the mass parameter in such a way that the need for a constraint is exhibited clearly. Constrained instantons are shown to exist as finite action solutions of the field equations with exponential fall off only for specific constraints that are unique in lowest order in the mass parameter in question.
Alternative approaches to the Casalbuoni-Brink-Schwarz Superparticle
Morten Nielsen,N. K. Nielsen
Physics , 2000, DOI: 10.1006/aphy.2000.6083
Abstract: Wigner's method of induced representations is applied to the N=1 super-Poincare group, and by using a state corresponding to the basic vector of the little group as a Clifford vacuum we show that the spin operator of a supersymmetric point particle obeys Wigner's constraints. As dynamical variables for the particle we use canonical coordinates on the symmetry group manifold. The physical phase space is then constructed using a vielbein formalism. We find that the Casalbuoni-Brink-Schwarz superparticle appears as a special case of our general construction. Finally, the theory is reformulated as a gauge theory where the gauge freedom corresponds to the choice of spin constraints or, equivalently, the free choice of relativistic center of mass. In a special case the gauge symmetry reduces to the well known kappa-symmetry.
On anisotropic Triebel-Lizorkin type spaces, with applications to the study of pseudo-differential operators
Lasse Borup,Morten Nielsen
Journal of Function Spaces and Applications , 2008, DOI: 10.1155/2008/510584
Abstract: A construction of Triebel-Lizorkin type spaces associated with flexible decompositions of the frequency space ℝd is considered. The class of admissible frequency decompositions is generated by a one parameter group of (anisotropic) dilations on ℝd and a suitable decomposition function. The decomposition function governs the structure of the decomposition of the frequency space, and for a very particular choice of decomposition function the spaces are reduced to classical (anisotropic) Triebel-Lizorkin spaces. An explicit atomic decomposition of the Triebel-Lizorkin type spaces is provided, and their interpolation properties are studied. As the main application, we consider Hörmander type classes of pseudo-differential operators adapted to the anisotropy and boundedness of such operators between corresponding Triebel-Lizorkin type spaces is proved.
NN-align. An artificial neural network-based alignment algorithm for MHC class II peptide binding prediction
Morten Nielsen, Ole Lund
BMC Bioinformatics , 2009, DOI: 10.1186/1471-2105-10-296
Abstract: Here, we present a novel artificial neural network-based method, NN-align that allows for simultaneous identification of the MHC class II binding core and binding affinity. NN-align is trained using a novel training algorithm that allows for correction of bias in the training data due to redundant binding core representation. Incorporation of information about the residues flanking the peptide-binding core is shown to significantly improve the prediction accuracy. The method is evaluated on a large-scale benchmark consisting of six independent data sets covering 14 human MHC class II alleles, and is demonstrated to outperform other state-of-the-art MHC class II prediction methods.The NN-align method is competitive with the state-of-the-art MHC class II peptide binding prediction algorithms. The method is publicly available at http://www.cbs.dtu.dk/services/NetMHCII-2.0 webcite.Major histocompatibility complex (MHC) molecules play an essential role in host-pathogen interactions determining the onset and outcome of many host immune responses. Only a small fraction of the possible peptides that can be generated from proteins of pathogenic organisms actually generate an immune response. MHC class II molecules present peptides derived from proteins taken up from the extracellular environment. They stimulate cellular and humoral immunity against pathogenic microorganisms through the actions of helper T lymphocytes. In order for a peptide to stimulate a helper T lymphocyte response, it must bind MHC II in the endocytic organelles [1].The MHC class I molecule is highly specific and binds a limited set of peptides of a narrow length distribution [2]. In contrast to this, the MHC class II molecule is highly promiscuous both with respect to composition and length of the peptide ligands [3,4]. During the last decade, large efforts have been invested in developing methods to allow for in silico screening of pathogenic organisms with the purpose of identifying peptides that will
Multicultural Multilegalism – Definition and Challenges
Morten Ebbe Juul Nielsen
Les Ateliers de l’éthique , 2011,
Abstract: Multilegalism is a species of legal pluralism denoting the existence of quasi-autonomous “minority jurisdictions” for at least some legal matters within a “normal” state jurisdiction. Multiculturalism in the advocatory sense might provide the justification for establishing such minority jurisdictions. This paper aims to provide 1) a detailed idea about what such a multicultural multilegal arrangement would amount to and how it differs from certain related concepts and legal frameworks, 2) in what sense some standard multicultural arguments could provide a starting point for seriously considering multicultural multilegalism in practice, 3) how the idea fares against some standard liberal criticisms, and finally 4), to point out three salient problems for multilegalism, concerning a) choice of law problems, b) a dilemma facing us as to whether state supremacy should be upheld or not, and c) clashes with basic human rights.Le multi-légalisme est une espèce de pluralisme légal qui dénote l’existence de juridictions minoritaires quasi-autonomes, au moins en ce qui concerne certains domaines légaux au sein d’une juridiction d’état normale . Dans son acception juridique, le multiculturalisme peut justifier la mise en place de telles juridictions minoritaires. Cet article vise à 1) fournir une idée détaillée de ce à quoi un arrangement multilégal pourrait ressembler et de quelle manière celui-ci diffère de certains concepts et modèles juridiques afférents, 2) évaluer la pertinence d’arguments multiculturels standards comme point de départ pour sérieusement considérer la pratique multilégale, 3) voir comment cette idée répond à certaines critiques libérales classiques, et 4) souligner trois problèmes importants pour le multilégalisme concernant a) les problèmes de choix de la loi, b) le dilemme de savoir si la suprématie de l’état doit être maintenue ou non, et c) les tensions avec les droits humains fondamentaux.
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