discrete laguerre gaussian transforms and their applications laguerre gaussian functions serve as a complete and orthonormal basis for a variety of physical problems such as 2d isotropic quantum harmonic oscillators and circularly symmetric laser modes in this paper we propose discrete laguerre gaussian functions which are defined such that some elegant physical properties are preserved and a fast computation algorithm of complexity o n log n is available discrete laguerre gaussian transforms as introduced in this paper inherit nice properties from discrete laguerre gaussian functions and admit signal analysis over circularly symmetric patterns it is demonstrated through examples that discrete laguerre gaussian transforms find applications in circular pattern keypoints selection object detection image compression rotational invariance feature for pattern recognition and rotational angle estimation

additive group actions on affine t varieties of complexity one in arbitrary characteristic abstract let x be a normal affine t variety of complexity at most one over a perfect field k where t g m n stands for the split algebraic torus our main result is a classification of additive group actions on x that are normalized by the t action this generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero with the assumption that the characteristic of k is positive we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine t varieties normalized up to a frobenius map as a preliminary result we provide a complete description of these g a actions in the toric situation

the hermite transform applications it is demonstrated how the hermite transform can be used for image coding and analysis hierarchical coding structures based on increasingly specified basic patterns i e general 2 d patterns general 1 d patterns and specific 1 d patterns such as edges and corners are presented in the image coding application the relation with existing pyramid coders is described a new coding scheme based on local one dimensional image approximations is introduced in the image analysis application the relation between the hermite transform and existing line edge detection schemes is described it is shown that by concentrating on more specific patterns the coding efficiency can be increased since fewer coefficients have to be coded meanwhile sufficient descriptive power can be maintained for approximating the most interesting features in natural images

finite two dimensional oscillator ii the radial model a finite two dimensional radial oscillator of n 1 2 points is proposed with the dynamical lie algebra so 4 su 2 x su 2 y examined in part i of this work but reduced by a subalgebra chain so 4 so 3 so 2 as before there are a finite number of energies and angular momenta the casimir spectrum of the new chain provides the integer radii 0 n and the 2 1 discrete angles on each circle are obtained from the finite fourier transform of angular momenta the wavefunctions of the finite radial oscillator are so 3 clebsch gordan coefficients we define here the hankel hahn transforms with dual hahn polynomials as finite n unitary approximations to hankel integral transforms with bessel functions obtained in the contraction limit n

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

rotational invariance based on fourier analysis in polar and spherical coordinates in this paper polar and spherical fourier analysis are defined as the decomposition of a function in terms of eigenfunctions of the laplacian with the eigenfunctions being separable in the corresponding coordinates the proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures the theory is compactly presented with an emphasis on the analogy to the normal fourier transform the relation between the polar or spherical fourier transform and the normal fourier transform is explored as examples of applications rotation invariant descriptors based on polar and spherical fourier coefficients are tested on pattern classification problems

some results on non linear optimal control problems and hamilton jacobi equations in infinite dimensions abstract the optimal control of a distributed parameter system is connected to the solution of the corresponding hamilton jacobi equation this is a first order equation in infinite dimensions with discontinuous coefficients we study the hamilton jacobi equation of a system governed either by a semilinear or by a monotone dynamics replacing the unbounded terms of this equation by their yosida approximations we prove convergence of the approximate solutions to the value function of the original problem by using the uniqueness result for viscosity solutions examples and applications are included

a counterexample to the fourteenth problem of hilbert in dimension four abstract this paper presents a family of new counterexamples to hilbert s fourteenth problem they are realized as subfields of the rational function field of four variables over a field of characteristic zero of transcendence degree three it was previously not known whether any counterexample could be found as a subfield of a rational function field of four variables or whether counterexamples with minimal transcendence degree three could exist

linear counterexamples to the fourteenth problem of hilbert abstract following ideas of a campo neuen indag math n s 5 1994 253 257 we construct linear counterexamples to the fourteenth problem of hilbert from non linear counterexamples d daigle g freudenburg j algebra 221 1999 528 535 g freudenburg transform groups 5 2000 61 71

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

the discrete harmonic oscillator harper s equation and the discrete fractional fourier transform certain solutions to harper s equation are discrete analogues of and approximations to the hermite gaussian functions they are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator and they lead to a definition of a discrete fractional fourier transform ft the discrete fractional ft is essentially the time evolution operator of the discrete harmonic oscillator

finite two dimensional oscillator i the cartesian model a finite two dimensional oscillator is built as the direct product of two finite one dimensional oscillators using the dynamical lie algebra su 2 x su 2 y the position space in this model is a square grid of points while the ordinary continuous two dimensional quantum oscillator has a symmetry algebra u 2 the symmetry algebra of the finite model is only u 1 x u 1 y because it lacks rotations in the position and momentum plane we show how to import an so 2 group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid we thus propose a finite analogue for fractional u 2 fourier transforms

generalized commuting matrices and their eigenvectors for dfts offset dfts and other periodic operations it is well known that some matrices such as dickinson steiglitz s matrix can commute with the discrete fourier transform dft and that one can use them to derive the complete and orthogonal dft eigenvector set recently candan found the general form of the dft commuting matrix in this paper we further extend the previous work and find the general form of the commuting matrix for any periodic quasi periodic and offset quasi periodic operations using the general commuting matrix we can derive the complete and orthogonal eigenvector sets for offset dfts dcts of types 1 4 5 and 8 dsts of types 1 4 5 and 8 discrete hartley transforms of types 1 and 4 the walsh transform and the projection operation the operation that maps a whole vector space into a subspace successfully moreover several novel ways of finding dft eigenfunctions are also proposed furthermore we also extend our theories to the continuous case i e if a continuous transform is periodic quasi periodic or offset quasi periodic such as the ft and some cyclic operations in optics we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform

factorial algebraic group actions and categorical quotients abstract given an action of an affine algebraic group with only trivial characters on a factorial variety we ask for categorical quotients we characterize existence in the category of algebraic varieties moreover allowing constructible sets as quotients we obtain a more general existence result which for example settles the case of a finitely generated algebra of invariants as an application we provide a combinatorial git type construction of categorical quotients for actions of not necessarily reductive groups on e g complete varieties with finitely generated cox ring via lifting to the characteristic space

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

triangular derivations of k x1 x2 x3 x4 abstract it is shown that if k is a field of characteristic zero then the kernel of any triangular k derivation of k x 1 x 2 x 3 x 4 is finitely generated as a k algebra this is obtained as a corollary of a more general result concerning triangular r derivations of r x y z for certain rings r

general hermite and laguerre two dimensional polynomials general hermite and laguerre two dimensional 2d polynomials which form a complex three parameter unification of the special hermite and laguerre 2d polynomials are defined and investigated the general hermite 2d polynomials are related to the two variable hermite polynomials but are not the same the advantage of the newly introduced hermite and laguerre 2d polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the two variable hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other the matrix which plays a role in the new definition of hermite and laguerre 2d polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two variable hermite polynomials two essentially different explicit representations of the hermite and laguerre 2d polynomials are derived where the first involves jacobi polynomials as coefficients in superpositions of special hermite or laguerre 2d polynomials and the second is a superposition of products of two hermite polynomials with decreasing indices and with coefficients related to the special laguerre 2d polynomials generating functions are derived for the hermite and laguerre 2d polynomials

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

efficient 3d shape matching and retrieval using a concrete radialized spherical projection representation we present a 3d shape retrieval methodology based on the theory of spherical harmonics using properties of spherical harmonics scaling and axial flipping invariance is achieved rotation normalization is performed by employing the continuous principal component analysis along with a novel approach which applies pca on the face normals of the model the 3d model is decomposed into a set of spherical functions which represents not only the intersections of the corresponding surface with rays emanating from the origin but also points in the direction of each ray which are closer to the origin than the furthest intersection point the superior performance of the proposed methodology is demonstrated through a comparison against state of the art approaches on standard databases

fields defined by locally nilpotent derivations and monomials abstract we prove a necessary and sufficient condition for certain fields defined by locally nilpotent derivations and monomials to be algebraically closed in a rational function field this implies that a counterexample to the fourteenth problem of hilbert in dimension four which was recently given by the author is obtained as the kernel of a derivation it was previously unknown only in dimension four whether there exists such a counterexample

discrete spherical harmonic oscillator transforms on the cartesian grids using transformation coefficients the analog harmonic oscillators are well studied in quantum physics including their energy states wavefunctions orthogonal properties and eigenfunctions of the fourier transform in addition the continuous solutions in different dimension and coordinate systems are known some discrete equivalents of the 1d wavefunctions were also studied however in the 3d spherical coordinate system the discrete equivalents of the 3d wavefunctions are not established in this paper we focus on the spherical harmonic oscillator wavefunctions shows the spherical harmonic oscillator transforms shots and their discrete implementation the shows can be synthesized by linear combinations of the hermite gaussian functions with proper transformation coefficients we find that computing the coefficients can be speeded up using the fast fourier transforms or some recursive relations these coefficients relate the hermite transforms with the shots some applications of the discrete shows and the discrete shots are introduced first the shows are exactly the eigenfunctions of the 3d dft also the shots can be used to derive the spherical harmonic oscillator descriptor shod which is a rotational invariant descriptor we find that the shod is not only compatible with the existing rotational descriptors for the spherically sampled data but also outperforms the existing rotational descriptors for 3d cartesian sampled bandlimited input data besides the shots can be used to decompose 3d signals into spherical components hence 3d signal reconstruction is done using partially chosen spherical component and 3d data compression for mri data is demonstrated using shots for medical applications

the discrete fractional fourier transform we propose and consolidate a definition of the discrete fractional fourier transform that generalizes the discrete fourier transform dft in the same sense that the continuous fractional fourier transform generalizes the continuous ordinary fourier transform this definition is based on a particular set of eigenvectors of the dft matrix which constitutes the discrete counterpart of the set of hermite gaussian functions the definition is exactly unitary index additive and reduces to the dft for unit order the fact that this definition satisfies all the desirable properties expected of the discrete fractional fourier transform supports our confidence that it will be accepted as the definitive definition of this transform

dynamical shape control of the heat equation abstract we study a shape optimization problem in one space dimension by using dynamic programming we obtain the existence and uniqueness of the generalized viscosity solution to the hamilton jacobi equation corresponding to our problem as well as the feedback formula

a condition for finite generation of the kernel of a derivation abstract we consider a homogeneous derivation on a finitely generated graded normal domain over a field and give a sufficient condition for finite generation of its kernel as its consequence we show some sufficient conditions for finite generation of the kernel of a derivation on a polynomial ring in particular we prove that the kernel of a locally nilpotent monomial derivation on a polynomial ring in four variables is finitely generated

a counterexample to hilbert s fourteenth problem in dimension 5 abstract the kernel of a certain triangular derivation of the polynomial ring k x 1 x 2 x 3 x 4 x 5 is shown to be non finitely generated over k a field of characteristic zero thus giving a new counterexample to hilbert s fourteenth problem in the lowest dimension to date

hermite and laguerre 2d polynomials abstract we define hermite 2d polynomials hm n u x y and laguerre 2d polynomials l m n u z z as functions of two variables with an arbitrary 2d matrix u as parameter and discuss their properties and their explicit representation recursion relations and generating functions for these polynomials are derived the advantage of the introduced hermite and laguerre 2d polynomials in comparison to the related usual two variable hermite polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the last one has to introduce two different kinds of such polynomials which are biorthogonal to each other