the condition of vandermonde like matrices involving orthogonal polynomials the condition number relative to the frobenius norm of the n x n matrix p pi ltxj ly j l is investigated where p p dx are orthogonal polynomials with respect to some weight distribution dx and xi are pairwise distinct real numbers if the nodes x j are the zeros of p the condition number is either expressed or estimated from below and above in terms of the christoffel numbers for dh depending on whether the p are normalized or not for arbitrary real xi and normalized p a lower bound of the condition number is obtained in terms of the christoffel function evaluated at the nodes numerical results are given for minimizing the condition number as a function of the nodes for selected classical distributions dx

a korovkin type theory for finite toeplitz operators via matrix algebras preconditioned conjugate gradients pcg are widely and successfully used methods for solving a toeplitz linear system a n vec x vec b 59 9 20 5 34 62 6 10 28 45 44 46 49 frobenius optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the frobenius distance from a n the convergence features of these pcg have been naturally studied by means of the weierstrass jackson theorem 17 36 45 owing to the profound relationship between the spectral features of the matrices a n generated by the fourier coefficients of a continuous function f and the analytical properties of the symbol f itself in this paper we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f on the other hand by modifying the korovkin theorem to study the frobenius optimal preconditioning problem we provide a new and unifying tool for analyzing all frobenius optimal preconditioners in any generic matrix algebra related to trigonometric transforms finally the multilevel case is sketched and discussed by showing that a korovkin type theory also holds in a multivariate sense

korovkin tests approximation and ergodic theory we consider sequences of s k n x t k n matrices a n f with a block structure spectrally distributed as an l 1 p variate s x t matrix valued function f and for any n we suppose that a n is a linear and positive operator for every fixed n we approximate the matrix a n f in a suitable linear space m n of s k n x t k n matrices by minimizing the frobenius norm of a n f x n when x n ranges over m n the minimizer x n is denoted by p k n a n f we show that only a simple korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts 1 the sequence p k n a n f is distributed as f 2 the sequence a n f p k n a n f is distributed as the constant function 0 i e is spectrally clustered at zero the first result is an ergodic one which can be used for solving numerical approximation theory problems the second has a natural interpretation in the theory of the preconditioning associated to cg like algorithms

how to prove that a preconditioner cannot be superlinear in the general case of multilevel toeplitz matrices we recently proved that any multilevel circulant preconditioner is not superlinear a cluster it may provide cannot be proper the proof was based on the concept of quasi equimodular matrices although this concept does not apply for example to the sine transform matrices in this paper with a new concept of partially equimodular matrices we cover all trigonometric matrix algebras widely used in the literature we propose a technique for proving the nonsuperlinearity of certain frequently used preconditioners for some representative sample multilevel matrices at the same time we show that these preconditioners are in a certain sense the best among the sublinear preconditioners with only a general cluster for multilevel toeplitz matrices

moyal quantization of 2 1 dimensional galilean systems stratonovich weyl kernels are constructed for some of the coadjoint orbits of the two dimensional extended galilean group g 2 1 as an intermediate step the unitary irreducible representations associated with a given group orbit are obtained by using the kirillov mackey theory star products are defined in the sense of moyal for functions on each of these orbits the central extension of g 2 1 with parameter k is also analyzed which results from the commutator between the generators of boosts to conclude that it originates a sort of nonrelativistic remainder of the thomas precession

on network coding for sum networks a directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources we call such a network a sum network it is shown that there exists a solvably and linear solvably equivalent sum network for any multiple unicast network and thus for any directed acyclic communication network it is also shown that there exists a linear solvably equivalent multiple unicast network for every sum network it is shown that for any set of polynomials having integer coefficients there exists a sum network which is scalar linear solvable over a finite field f if and only if the polynomials have a common root in f for any finite or cofinite set of prime numbers a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set the insufficiency of linear net work coding and unachievability of the network coding capacity are proved for sum networks by using similar known results for communication networks under fractional vector linear network coding a sum network and its reverse network are shown to be equivalent however under nonlinear coding it is shown that there exists a solvable sum network whose reverse network is not solvable

a note on the chevalley eilenberg cohomology for the galilei and poincare algebras we construct in a systematic way the complete chevalley eilenberg cohomology at form degrees 2 3 and 4 for the galilei and poincare groups the corresponding non trivial forms belong to certain representations of the spatial rotation lorentz group in the case of 2 forms they give all possible central and non central extensions of the galilei group and all non central extensions of the poincare group the procedure developed in this paper can be applied to any spacetime symmetry group

representations of the 2 1 dimensional galilean group using the mackey theory of induced representations the unitary irreducible representations of the proper galilean group in 2 1 space time dimensions are constructed all representations both true as well as the projective ones are found the concomitant representations of the lie algebra are given

lower bounds for the condition number of a real confluent vandermonde matrix lower bounds on the condition numberp vc of a real con uent vander monde matrix vc are established in terms of the dimension n or n and the largest absolute value among all nodes that deflne the con uent vander monde matrix or n and the interval that contains the nodes in particular it is proved that for any modest kmax the largest number of equal nodes p vc behaves no smaller than on 1 p 2 n and than on 1 p 2 2n if all nodes are nonnegative it is not clear whether those bounds are asymp totically sharp for modest kmax

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

reversible and irreversible information networks it is shown that there exist information networks where messages can be sent utilizing network coding more easily in one direction than in the opposite direction this is valid even though each channel is assumed to have the same capacity in both directions it is shown that irreversible information networks only have solutions that use nonlinear network coding this correspondence argues that this result is more surprising than it might appear at first sight and that it follows using ideas resembling the path integral in quantum mechanics

local representation groups group cohomology techniques are used to derive conditions for a group g to be a local splitting group for g as a first step the exactness of the inflation restriction sequence gives a characterisation of factor systems arising in locally operating representations of transitive transformation groups some examples and applications of the theory are also given

a chebychev vandermonde solver abstract let t j n j 0 be a family of chebyshev polynomials for a finite interval a b let x k n k 0 be a set of distinct nodes on a b and define the transposed chebyshev vandermonde matrix w w kj n k j 0 w kj t j x k we describe a new fast algorithm for the solution of linear systems of equations of the form w a f and compare it with a fast solution scheme recently proposed by higham the latter scheme is a modification of the bjorck pereyra algorithm and the computed solution depends on the ordering of the nodes x k our algorithm is designed so that the computed solution is essentially independent of the ordering of the x k extensive computational experience indicates that our algorithm generally yields higher accuracy in the computed solution than higham s

the projective unitary irreducible representations of the poincare group in 1 2 dimensions a complete analysis is given of the projective unitary irreducible representations of the poincare group in 1 2 dimensions applying the mackey theorem and using an explicit formula for the universal covering group of the lorentz group in 1 2 dimensions explicit formulas are given for all representations

locally operating realizations of transformation lie groups using the mackey theory of induced representations a systematic study of the locally operating multiplier realizations of a connected lie group g that acts transitively on a space time manifold is presented we obtain a mathematical characterization of the locally operating multiplier realizations and a reduction of the problem of multiplier locally operating realizations to linear ones via a splitting group g m for g in this way the locally operating multiplier realizations are obtained by induction from finite dimensional linear representations of a well determined subgroup of g some examples such as the two dimensional euclidean group the galilei group and the one dimensional newton hooke group are given

nonreversibility and equivalent constructions of multiple unicast networks we prove that for any finite directed acyclic network there exists a corresponding multiple unicast network such that for every alphabet each network is solvable if and only if the other is solvable and for every finite field alphabet each network is linearly solvable if and only if the other is linearly solvable the proof is constructive and creates an extension of the original network by adding exactly s 5m r 1 new nodes where in the original network m is the number of messages r is the average number of receiver nodes demanding each source message and s is the number of messages emitted by more than one source the construction is then used to create a solvable multiple unicast network which becomes unsolvable over every alphabet size if all of its edge directions are reversed and if the roles of source receiver pairs are reversed

time dependent quantum generators for the galilei group in 1995 doebner and mann introduced an approach to the ray representations of the galilei group in 1 1 dimensions giving rise to quantum generators with an explicit dependence on time recently 2004 wawrzycki proposed a generalization of bargmann s theory in his paper he introduce phase exponents that are explicitely dependent by 4 space point in order to find applications of such generalization we extend the approach of doebner and mann to higher dimensions as a result we determine the generators of the ray representation in 2 1 and 3 1 dimensions the differences of the outcoming formal apparatus with respect to the smaller dimension case are established

the symmetry groups of noncommutative quantum mechanics and coherent state quantization we explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two dimensional plane we show that the pertinent groups for the system are the two fold central extension of the galilei group in 2 1 space time dimensions and the two fold extension of the group of translations of r 4 this latter group is just the standard weyl heisenberg group of standard quantum mechanics with an additional central extension we also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics we build unitary irreducible representations of these various groups and construct the associated families of coherent states a coherent state quantization of the underlying phase space is then carried out which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics

relativistic field theories in three dimensions we provide a complete classification of the unitary irreducible representations of the 2 1 dimensional poincare group we show in particular that only two types of spin are available for massless field theories we also construct generalized foldy wouthuysen transformations which connect the physical uir s with covariant field theories in three dimensions

classification of ten dimensional kinematical groups with space isotropy all the abstract ten dimensional real lie algebras that contain as a subalgebra the algebra of the three dimensional rotation group generators j and decompose under the rotation group into three three vector representation spaces j itself k and p and a scalar generator h are classified in all cases the existence of a homogeneous space of dimension 4 is shown

a stable gaussian radial basis function method for solving nonlinear unsteady convection diffusion reaction equations abstract we investigate a novel method for the numerical solution of two dimensional time dependent convection diffusion reaction equations with nonhomogeneous boundary conditions we first approximate the equation in space by a stable gaussian radial basis function rbf method and obtain a matrix system of odes the advantage of our method is that by avoiding kronecker products this system can be solved using one of the standard methods for odes for the linear case we show that the matrix system of odes becomes a sylvester type equation and for the nonlinear case we solve it using predictor corrector schemes such as adams bashforth and implicit explicit imex methods this work is based on the idea proposed in our previous paper 2016 in which we enhanced the expansion approach based on hermite polynomials for evaluating gaussian radial basis function interpolants in the present paper the eigenfunction expansions are rebuilt based on chebyshev polynomials which are more suitable in numerical computations the accuracy robustness and computational efficiency of the method are presented by numerically solving several problems

fast algorithms for discrete polynomial transforms consider the vandermonde like matrix p p k cos j n j k 0 n where the polynomials p k satisfy a three term recurrence relation if p k are the chebyshev polynomials t k then p coincides with c n 1 cos jk n j k 0 n this paper presents a new fast algorithm for the computation of the matrix vector product pa in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces pa with c n 1 and a subsequent fast cosine transform the first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications numerical tests demonstrate that our fast polynomial transform realizes pa with almost the same precision as the clenshaw algorithm but is much faster for n 128

projective representations of the 1 1 dimensional poincare group the unitary irreducible representations of the central extension of the poincare group in 1 1 dimensions are constructed by an application of the kirillov theory these are then lifted to projective unitary irreducible representations of the poincare group the 1 1 galilean group is treated separately in an appendix

grafting and poisson structure in 2 1 gravity with vanishing cosmological constant we relate the geometrical construction of 2 1 spacetimes via grafting to phase space and poisson structure in the chern simons formulation of 2 1 dimensional gravity with vanishing cosmological constant on manifolds of topology mathbb r times s g where sg is an orientable two surface of genus g 1 we show how grafting along simple closed geodesics is implemented in the chern simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on sg we prove that this action is generated via the poisson bracket by a gauge invariant observable associated to the holonomy of we deduce a symmetry relation between the poisson brackets of observables associated to the lorentz and translational components of the holonomies of general closed curves on sg and discuss its physical interpretation finally we relate the action of grafting on the phase space to the action of dehn twists and show that grafting can be viewed as a dehn twist with a formal parameter satisfying 2 0

ray representations of n 2 1 dimensional galilean group for a quantum mechanical system modeled via unitary or antiunitary representations of a symmetry group g with lie algebra g the hamiltonian h is of special interest if h is an element of g or of the universal enveloping algebra u g a generic time dependence for any element in u g is given through the heisenberg picture as an example we consider a system with gal n as symmetry group with h as one of the generators for n 3 one gets from ray representations the free schrodinger equation for n 1 a peculiarity occurs the free equation has an interaction term which results from the construction and parametrization of unitary ray representations for n 2 there is another special feature there exist ray representations of the universal covering group gal 2 which induce no ray representations of gal 2 furthermore for these representations it is not possible to construct a schrodinger equation

closure solvability for network coding and secret sharing network coding is a new technique to transmit data through a network by letting the intermediate nodes combine the packets they receive given a network the network coding solvability problem decides whether all the packets requested by the destinations can be transmitted in this paper we introduce a new approach to this problem we define a closure operator on a digraph closely related to the network coding instance and we show that the constraints for network coding can all be expressed according to that closure operator thus a solution for the network coding problem is equivalent to a so called solution of the closure operator we can then define the closure solvability problem in general which surprisingly reduces to finding secret sharing matroids when the closure operator is a matroid based on this reformulation we can easily prove that any multiple unicast where each node receives at least as many arcs as there are sources solvable by linear functions we also give an alternative proof that any nontrivial multiple unicast with two source receiver pairs is always solvable over all sufficiently large alphabets based on singular properties of the closure operator we are able to generalize the way in which networks can be split into two distinct parts we also provide a new way of identifying and removing useless nodes in a network we also introduce the concept of network sharing where one solvable network can be used to accommodate another solvable network coding instance finally the guessing graph approach to network coding solvability is generalized to any closure operator which yields bounds on the amount of information that can be transmitted through a network

a minimum sobolev norm technique for the numerical discretization of pdes partial differential equations pdes are discretized into an under determined system of equations and a minimum sobolev norm solution is shown to be efficient to compute and converge under very generic conditions numerical results of a single code that can handle pdes in first order form on complicated polygonal geometries are shown for a variety of pdes variable coefficient div curl scalar elliptic pdes elasticity equation stationary linearized navier stokes scalar fourth order elliptic pdes telegrapher s equations singular pdes etc

korovkin theorems and linear positive gram matrix algebra approximations of toeplitz matrices abstract in this paper we are concerned with the approximation of toeplitz matrices generated by continuous 2 periodic functions f i r with i for this purpose we define a class of matrix algebras related to suitable choices of gram functions where we look for good preconditioners in particular we construct these preconditioners through linear operators and linear positive operators lpos approximating in some sense the function f then by making use of some matrix versions 39 42 41 of the korovkin and weierstrass theorems we analyze the convergence features of old and new preconditioners finally among the given results are adapted in order to deal with l1 generating functions and the related preconditioners are compared with the ones devised by using band toeplitz matrices 6 36

moments in quadrature problems abstract an account is given of the role played by moments and modified moments in the construction of quadrature rules specifically weighted newton cotes and gaussian rules fast and slow lagrange interpolation algorithms combined with gaussian quadrature as well as linear algebra methods based on moment equations are described for generating newton cotes formulae the weaknesses and strengths of these methods are illustrated in concrete examples involving weight functions with and without singularities new conjectures are formulated concerning the positivity of certain newton cotes formulae for jacobi weight functions and for the logistics weight with numerical evidence being provided to support them finally an inherent limitation is pointed out in the use of moment information to construct gauss type quadrature rules for the hermite weight function on bounded or half infinite intervals

graph theoretical constructions for graph entropy and network coding based communications the guessing number of a directed graph digraph equivalent to the entropy of that digraph was introduced as a direct criterion on the solvability of a network coding instance this paper makes two contributions on the guessing number first we introduce an undirected graph on all possible configurations of the digraph referred to as the guessing graph which encapsulates the essence of dependence amongst configurations we prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph therefore network coding solvability is no more a problem on the operations made by each node but is simplified into a problem on the messages that can transit through the network by studying the guessing graph of a given digraph and how to combine digraphs or alphabets we are thus able to derive bounds on the guessing number of digraphs second we construct specific digraphs with high guessing numbers yielding network coding instances where a large amount of information can transit we first propose a construction of digraphs with finite parameters based on cyclic codes with guessing number equal to the degree of the generator polynomial we then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one despite these digraphs being arbitrarily sparse these constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear although these instances are sparse and the sources are arbitrarily far from their corresponding sinks

a unifying approach to abstract matrix algebra preconditioning in earlier papers tyrtyshnikov 42 and the first author 14 considered the analysis of clustering properties of the spectra of specific toeplitz preconditioned matrices obtained by means of the best known matrix algebras here we generalize this technique to a generic banach algebra of matrices by devising general preconditioners related to convergent approximation processes 36 finally as case study we focus our attention on the tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric toeplitz systems can be constructed in this class

displacement structure approach to polynomial vandermonde and related matrices abstract we introduce a new class of what we call polynomial vandermonde like matrices this class generalizes the polynomial vandermonde matrices studied earlier by various authors who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems a displacement structure approach allows us to carry over all these results to the wider class of polynomial vandermonde like matrices

the stratonovich weyl correspondence for one dimensional kinematical groups the stratonovich weyl correspondence is a restatement of the moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively the first and most important step is to define a mapping from the manifold into the set of self adjoint operators on a hilbert space under suitable conditions this mapping is called a stratonovich weyl kernel the construction of this mapping is discussed on coadjoint orbits of the one dimensional galilei poincare and newton hooke groups as well as the two dimensional euclidean group

the galilean group in 2 1 space times and its central extension the problem of constructing the central extensions by the circle group of the group of galilean transformations in two spatial dimensions as well as that of its universal covering group is solved also solved is the problem of the central extension of the corresponding lie algebra we find that the lie algebra has a three parameter family of central extensions as does the simply connected group corresponding to the lie algebra the galilean group itself has a two parameter family of central extensions a corollary of our result is the impossibility of the appearance of non integer valued angular momentum for systems possessing galilean invariance

vandermonde matrices on chebyshev points abstract this paper deals with vandermonde matrices on chebyshev points hereafter denoted by v we present simple formulas for the determinant of v and the frobenius norm of both v and v 1 and derive an algorithm for solving the linear systems v p and vtq g numerical experiments to asses the stability of this algorithm are also reported

accurate svds of polynomial vandermonde matrices involving orthonormal polynomials abstract we present a new o n3 algorithm for computing the svd of an n n polynomial vandermonde matrix vp pi 1 xj to high relative accuracy in o n3 time the pi are orthonormal polynomials deg pi i and xj are complex nodes the small singular values of vp can be arbitrarily smaller than the largest ones so that traditional algorithms typically compute them with no relative accuracy at all we show that the singular values even the tiniest ones are usually well conditioned functions of the data xj justifying this computation we also explain how this theory can be extended to other polynomial vandermonde matrices involving polynomials that are not orthonormal or even orthogonal

central extensions of the quasi orthogonal lie algebras we determine the central extensions of a whole family of lie algebras obtained by the method of graded contractions from n arbitrary all the inhomogeneous orthogonal and pseudo orthogonal algebras are members of this family as well as a large number of other non semisimple algebras all of which have at least a semidirect structure in some cases two or more the dimensions of their second cohomology groups and the explicit expression of their central extensions are given

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

graded contractions and kinematical groups of space time g graded contractions of the complex lie algebra b2 and of its real forms are considered it is found that a particular g z2 z2 is a proper choice of the grading group preserved when graded contractions leading to lie algebras of the kinematical groups of space time are studied

vandermonde matrices with chebyshev nodes abstract for an n n vandermonde matrix v n j i 1 1 ij n with translated chebyshev zero nodes it is discovered that v n t admits an explicit qr decomposition with the r factor consisting of the coefficients of the translated chebyshev polynomials this decomposition then leads to an exact expression for the frobenius condition number of its submatrix v k n j i 1 1 i k 1 j n so called a rectangular vandermonde matrix bounds on individual singular value and more it is explained how these results can be used to establish asymptotically optimal lower bounds on condition numbers of real rectangular vandermonde matrices and nearly optimally conditioned real rectangular vandermonde matrices on a given interval extensions are also made for v n with nodes being zeros of any translated orthogonal polynomials other than chebyshev ones similar results hold for v n with translated chebyshev extreme nodes too owing to that v n t admits an explicit qr like decomposition close formulas of or tight bounds on the residuals are also presented for the conjugate gradient method the minimal residual method and the generalized minimal residual method on certain linear systems ax b with a having eigenvalues the same as the nodes mentioned above as a by product they yield positive definite linear systems for which the residuals by the conjugate gradient method are always comparable to the existing error bounds for all iteration steps

fourier legendre approximation of a probability density from discrete data we produce a positive approximation of a probability density in 0 1 when only a finite number of values possibly affected by noise is available this approximation is obtained by computing a number of legendre fourier coefficients and applying the maximum entropy method an example of application of this procedure is data smoothing in the numerical solution of an identification problem for fokker planck equation

moyal quantization with compact symmetry groups and noncommutative harmonic analysis the phase space approach to quantization of systems whose symmetry group is compact and semisimple is developed from two basic principles covariance and traciality this generalizes results and methods already implemented for spin systems j c varilly and j m gracia bondia ann phys 190 107 1989 the twisted product of phase space functions is shown to be the image of group convolution in the context of a novel fourier theory on the coadjoint orbits

the linear guessing number of undirected graphs abstract this paper investigates linear guessing number introduced by riis in connection with the network coding theory this paper establishes properties of linear guessing number on undirected graphs in particular lower and upper bounds for linear guessing number are established finally linear guessing numbers of product graphs are investigated

central extensions and realizations of one dimensional galilean systems and quantization the unitary irreducible realizations representations up to a factor of the maximal non trivial central extension of the 1 1 galilei group 1 1 are obtained via the linear unitary irreducible representations of its maximal non trivial central extension 1 1 as an application we construct the stratonovich weyl correspondence which allows moyal quantization of classical systems for two cases of great physical interest a system in a external variable force field and a variable mass system

fast inversion of chebyshev vandermonde matrices this paper contains two fast algorithms for inversion of chebyshev vandermonde matrices of the first and second kind they are based on special representations of the bezoutians of chebyshev polynomials of both kinds the paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster but also more stable than other algorithms available it is also efficient to use the above two algorithms for solving chebyshev vandermode systems of equations with preprocessing

construction of representations of poincare group using lie fields in this paper we give an explicit construction of the unitary irreducible representations of the poincare groups in 2 3 and 4 space time dimensions on hilbert spaces associated with the schrodinger representation of the weyl algebra for n 1 2 and 3 respectively our method of constructing the representations uses extension and localization of the enveloping algebras associated with these weyl algebras and the poincare algebras

rectangular vandermonde matrices on chebyshev nodes abstract we consider a rectangular vandermonde matrix v on chebyshev nodes using combinatorial identities we give an explicit factorization of the matrix v a factorization of thepseudo inverse of v and prove that the condition number 2 v does not depend onthe dimension of nodes set we also discuss the numerical properties of the proposed formulas

cayley klein algebras as graded contractions of so n 1 we study z2 x n graded contractions of the real compact simple lie algebra so n 1 and we identify within them the cayley klein algebras as a naturally distinguished subset

a korovkin based approximation of multilevel toeplitz matrices with rectangular unstructured blocks via multilevel trigonometric matrix spaces in this paper we are concerned with the approximation of multilevel toeplitz matrices generated by multivariate rectangular matrix valued continuous also l2 functions f i p rightarrow cal c s times t with i pi pi we first define a class cal a t of multilevel trigonometric matrix spaces with unstructured s times t blocks second we introduce a multi sequence of linear approximation operators cal p n x n n 1 n 2 ldots n p acting on complex matrices of dimensions n 1 cdots n p s times n 1 cdots n p t n j in cal n and giving the minimizer of the distance of x from a fixed representative of cal a t with regard to the frobenius norm then by stating and proving new versions of the korovkin theorem for f ranging in the banach space bf c i p cal c s times t cdot infty and by linking these theorems to the operators cal p n we prove the goodness of the proposed approximation in terms of the strong and weak equidistribution of the eigenvalues singular values of a n f and cal p n a n f the discussed results provide a very uniform tool for treating the preconditioning problem with applications to a wide variety of structures as analyzed in the twin paper s serra capizzano calcolo in press

graph guessing games and non shannon information inequalities guessing games for directed graphs were introduced by riis for studying multiple unicast network coding problems in a guessing game the players toss generalised dice and can see some of the other outcomes depending on the structure of an underlying digraph they later guess simultaneously the outcome of their own die their objective is to find a strategy which maximizes the probability that they all guess correctly the performance of the optimal strategy for a graph is measured by the guessing number of the digraph christofides and markstrom studied guessing numbers of undirected graphs and defined a strategy which they conjectured to be optimal one of the main results of this paper is a disproof of this conjecture the main tool so far for computing guessing numbers of graphs is information theoretic inequalities the other main result of this paper is that shannon s information inequalities which work particularly well for a wide range of graph classes are not sufficient for computing the guessing number finally we pose a few more interesting questions some of which we can answer and some which we leave as open problems

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

lie bialgebra contractions and quantum deformations of quasi orthogonal algebras lie bialgebra contractions are introduced and classified a non degenerate coboundary bialgebra structure is implemented into all pseudo orthogonal real algebras so p q starting from the one corresponding to so n 1 it allows us to introduce a set of lie bialgebra contractions which leads to lie bialgebras of quasi orthogonal algebras this construction is explicitly given for n 2 3 4 all lie bialgebra contractions studied in this paper define hopf algebra contractions for the drinfel d jimbo deformations uzso p q they are explicitly used to generate new non semisimple quantum algebras as it is the case for the euclidean poincare and galilean algebras

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given

transitive symplectic manifolds in 1 2 dimensions a complete list of all transitive symplectic manifolds of the poincare and galilei group in 1 2 dimensions is given