a new matrix method for the casimir operators of the lie algebras w mathfrak s mathfrak p n hbox mathbb r and i mathfrak s mathfrak p 2n hbox mathbb r a method is given to determine the casimir operators of the perfect lie algebras and the inhomogeneous lie algebras in terms of polynomials associated with a parametrized 2n 1 2n 1 matrix for the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation we further analyse the invariants for the extended schrodinger algebra in n 1 dimensions which arises naturally as a subalgebra of the method is extended to other classes of lie algebras and some applications to the missing label problem are given

generalizing capacity new definitions and capacity theorems for composite channels we consider three capacity definitions for composite channels with channel side information at the receiver a composite channel consists of a collection of different channels with a distribution characterizing the probability that each channel is in operation the shannon capacity of a channel is the highest rate asymptotically achievable with arbitrarily small error probability under this definition the transmission strategy used to achieve the capacity must achieve arbitrarily small error probability for all channels in the collection comprising the composite channel the resulting capacity is dominated by the worst channel in its collection no matter how unlikely that channel is we therefore broaden the definition of capacity to allow for some outage the capacity versus outage is the highest rate asymptotically achievable with a given probability of decoder recognized outage the expected capacity is the highest average rate asymptotically achievable with a single encoder and multiple decoders where channel side information determines the channel in use the expected capacity is a generalization of capacity versus outage since codes designed for capacity versus outage decode at one of two rates rate zero when the channel is in outage and the target rate otherwise while codes designed for expected capacity can decode at many rates expected capacity equals shannon capacity for channels governed by a stationary ergodic random process but is typically greater for general channels the capacity versus outage and expected capacity definitions relax the constraint that all transmitted information must be decoded at the receiver we derive channel coding theorems for these capacity definitions through information density and provide numerical examples to highlight their connections and differences we also discuss the implications of these alternative capacity definitions for end to end distortion source channel coding and separation

an algorithm to calculate the invariants of any lie algebra an analytical method to calculate invariants is developed based on the theory of complete systems of simultaneous linear first order partial differential equations the application of this algorithm to the general case in which the invariants are not polynomials is included for the case of lie algebras with one invariant the procedure of finding the invariant can be computerized as an example of this procedure the algorithm is applied in the calculation of the invariants of the group sa n r the explicit construction of its invariants for n 2 3 4 is given a theorem determining the order of a polynomial invariant of a lie group is proven allowing us to show that the invariant of sa n r is a homogeneous polynomial of order 1 2n n 1 in the generators from this fact an immediate consequence is that the order in the generators of translations is n the method is also applied in the calculation of invariants of the dimension six solvable lie algebras and dimension seven nilpotent

a continuous interior penalty method for viscoelastic flows in this paper we consider a finite element discretization of the oldroyd b model of viscoelastic flows the method uses standard continuous polynomial finite element spaces for velocities pressures and stresses inf sup stability and stability for convection dominated flows are obtained by adding a term penalizing the jump of the solution gradient over element faces to increase robustness when the deborah number is high we add a nonlinear artificial viscosity of shock capturing type the method is analyzed on a linear model problem and optimal a priori error estimates are proven that are independent of the solvent viscosity eta s finally we demonstrate the performance of the method on some known benchmark cases

low dimensional real lie algebras all real lie algebras of dimension up to 8 that admit a nontrivial levi decomposition are found

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

casimir operators of the exceptional group f4 the chain b4 contained in implied by f4 contained in implied by d13 expressions are given for the casimir operators of the exceptional group f4 in a concise form similar to that used for the classical groups the chain b4 contained in implied by f4 contained in implied by d13 is used to label the generators of f4 in terms of the adjoint and spinor representations of b4 and to express the 26 dimensional representation of f4 in terms of the defining representation of d13 casimir operators of any degree are obtained and it is shown that a basis consists of the operators of degree 2 6 8 and 12

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

contractions of low dimensional lie algebras theoretical background of continuous contractions of finite dimensional lie algebras is rigorously formulated and developed in particular known necessary criteria of contractions are collected and new criteria are proposed a number of requisite invariant and semi invariant quantities are calculated for wide classes of lie algebras including all low dimensional lie algebras an algorithm that allows one to handle one parametric contractions is presented and applied to low dimensional lie algebras as a result all one parametric continuous contractions for both the complex and real lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases levels and colevels of low dimensional lie algebras are discussed in detail properties of multiparametric and repeated contractions are also investigated

a computational strategy for the regularized 13 moment equations with enhanced wall boundary conditions the challenge of modeling low speed rarefied gas flow in the transition regime is well known in this paper we propose a numerical solution procedure for the regularized 13 moment equations within a finite volume framework the stress and heat flux equations arising in the method of moments are transformed into the governing equations for the stress and heat flux deviators based on their first order approximation to model confined flows a complete set of wall boundary conditions for the 13 moment equations are derived based on the maxwell wall boundary model this has been achieved by expanding the molecular distribution function to fourth order accuracy in hermite polynomials empirical correction factors are introduced into the boundary conditions and calibrated against direct simulation monte carlo data the numerical predictions obtained from the regularized 13 moment equations and the navier stokes fourier equations are compared with data generated using the direct simulation monte carlo method for planar couette flow for a range of wall velocities and knudsen numbers 0 012 1 0 the results indicate that the regularized 13 moment equations are in good qualitative agreement with the direct simulation monte carlo data the results also highlight limitations that are caused by the use of a first order expansion of the third moment

a differential operator realisation approach for constructing casimir operators of non semisimple lie algebras we introduce a search algorithm that utilises differential operator realisations to find polynomial casimir operators of lie algebras to demonstrate the algorithm we look at two classes of examples 1 the model filiform lie algebras and 2 the schrodinger lie algebras we find that an abstract form of dimensional analysis assists us in our algorithm and greatly reduces the complexity of the problem

the structure of the invariants of perfect lie algebras the structure of the invariants of perfect lie algebras with nontrivial centre is analysed it is shown that if the radical of a semidirect sum of a semisimple and a nilpotent lie algebra has a one dimensional centre then the defining representation r contains a copy of the trivial representation d0 of using this fact a criterion can be deduced to eliminate variables and to characterize the casimir operators of an algebra by means of certain subrepresentations r of r for rank 1 levi subalgebras all representations leading to perfect lie algebras with a radical isomorphic to a heisenberg lie algebra are determined we prove that the number of casimir operators of such an algebra is fixed for any dimension and moreover that they can be explicitly computed by means of determinantal formulae obtained from the brackets of the algebra for rank n 2 levi part a stabilization result of this nature is also given

successive refinement of information characterization of the achievable rates let r spl middot be the rate distortion function assume that we want to describe a source with distortion no larger than spl delta sub 1 from the rate distortion theory we know that we need to do so at a rate r sub 1 no smaller than r spl delta sub 1 spl lsqb bits symbol spl rsqb if it turns out that a more accurate description at distortion spl delta sub 2 spl delta sub 2

finite element methods for the linear regularized 13 moment equations describing slow rarefied gas flows abstract the paper is concerned with the numerical solution of the linear steady state regularized 13 moment equations in two space dimensions to facilitate the understanding of the specific challenges the equations are first divided into two subsystems before the full system is approached the arising problems mainly stem from the complicated saddle point structure as well as the non standard nature of the boundary conditions a continuous interior penalty method is presented and the pronounced advantages of utilizing high order basis functions in this setting are illustrated to render the presented approach more efficient a hybridization technique is presented that originates in the discontinuous galerkin method

wigner analysis and casimir operators of sa 4 r in theories involving gravity including qcd generated gravitylike effects in hadrons sa 4 r plays a role its single casimir invariant and that of its sa 2 r and sa 3 r subgroups are evaluated the group orbits are studied and the unitary irreducible representations are classified

structure of real lie algebras abstract restrictions are presented on the radical structure of an indecomposable lie algebra these restrictions are consequences of both the levi theorem and the representation theory of semisimple lie algebras as an application all real lie algebras of dimension 9 that admit a nontrivial levi decomposition are found these algebras are expressed in terms of known solvable and simple algebras of smaller dimensions and in terms of representations of simple algebras

on stable wall boundary conditions for the hermite discretization of the linearised boltzmann equation we define certain criteria using the characteristic decomposition of the boundary conditions and energy estimates which a set of stable boundary conditions for a linear initial boundary value problem involving a symmetric hyperbolic system must satisfy we first use these stability criteria to show the instability of the maxwell boundary conditions proposed by grad commun pure appl math 2 4 331 407 1949 we then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the hermite polynomials the block structure will help us in formulating stable boundary conditions for an arbitrary order hermite discretization of the boltzmann equation the formulation of stable boundary conditions relies upon an onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the maxwell boundary conditions at least in the lower order moments

determinantal formulae for the casimir operators of inhomogeneous lie algebras the author would like to correct some inaccuracy in formula 6 please see pdf for details

graded contractions of the gell mann graded sl 3 c the gell mann grading one of the four gradings of sl 3 c that cannot be further refined is considered as the initial grading for the graded contraction procedure using the symmetries of the gell mann grading the system of contraction equations is reduced and solved each non trivial solution of this system determines a lie algebra which is not isomorphic to the original algebra sl 3 c the resulting 53 contracted algebras are divided into two classes the first is represented by the algebras which are also continuous inonu wigner contractions the second is formed by the discrete graded contractions

graded contractions of lie algebras and central extensions we use the concept of grading of lie algebras to investigate the appearance of central charges during the contraction process as for the usual graded contractions one finds simultaneously the central extensions of classes of algebras rather than specific lie algebras to illustrate the method we consider in detail two physical applications the kinematical algebras of spacetime and the u n bosons limits of some lie algebras

entropy stable hermite approximation of the linearised boltzmann equation for inflow and outflow boundaries abstract to obtain a symmetric hyperbolic moment system from the linearised boltzmann equation we approximate the entropy variable derivative of the entropy functional with the help of multi variate polynomials in the velocity space choosing the entropy functional to be quadratic we retrieve the grad s approximation for the linearised boltzmann equation we develop a necessary and sufficient condition for the entropy stability of the grad s approximation on bounded position domain with inflow and outflow boundaries these conditions show the importance of using the onsager boundary conditions obcs rana and struchtrup 2016 28 for obtaining entropy stability and we use them to prove that a broad class of grad s approximations equipped with boundary conditions obtained through continuity of odd fluxes grad 1949 17 are entropy unstable entropy stability for the grad s approximation is obtained through entropy stabilization of the boundary conditions obtained through the continuity of odd fluxes since many practical implementations require the prescription of an inflow velocity the entropy bounds for two possible methods to achieve the same is discussed in detail both for the linearised boltzmann equation and its hermite approximation we use the discontinuous galerkin dg discretization in the physical space to study several benchmark problems to ascertain the physical accuracy of the proposed entropy stable grad s approximation

casimir operators of the exceptional group g2 degree 2 and 6 casimir operators of g2 are calculated in explicit form with the generators of g2 written in terms of the subalgebra a2

minimum expected distortion in gaussian source coding with fading side information an encoder subject to a rate constraint wishes to describe a gaussian source under squared error distortion the decoder besides receiving the encoder s description also observes side information consisting of uncompressed source symbol subject to slow fading and noise the decoder knows the fading realization but the encoder knows only its distribution the rate distortion function that simultaneously satisfies the distortion constraints for all fading states was derived by heegard and berger a layered encoding strategy is considered in which each codeword layer targets a given fading state when the side information channel has two discrete fading states the expected distortion is minimized by optimally allocating the encoding rate between the two codeword layers for multiple fading states the minimum expected distortion is formulated as the solution of a convex optimization problem with linearly many variables and constraints through a limiting process on the primal and dual solutions it is shown that single layer rate allocation is optimal when the fading probability density function is continuous and quasiconcave e g rayleigh rician nakagami and log normal in particular under rayleigh fading the optimal single codeword layer targets the least favorable state as if the side information was absent

distortion minimization in gaussian layered broadcast coding with successive refinement a transmitter without channel state information wishes to send a delay limited gaussian source over a slowly fading channel the source is coded in superimposed layers with each layer successively refining the description in the previous one the receiver decodes the layers that are supported by the channel realization and reconstructs the source up to a distortion the expected distortion is minimized by optimally allocating the transmit power among the source layers for two source layers the allocation is optimal when power is first assigned to the higher layer up to a power ceiling that depends only on the channel fading distribution all remaining power if any is allocated to the lower layer for convex distortion cost functions with convex constraints the minimization is formulated as a convex optimization problem in the limit of a continuum of infinite layers the minimum expected distortion is given by the solution to a set of linear differential equations in terms of the density of the fading distribution as the number of channel uses per source symbol tends to zero the power distribution that minimizes expected distortion converges to the one that maximizes expected capacity

the group theoretical classification of the 11 dimensional classical homogeneous kaluza klein cosmologies in the context of the classical kaluza klein cosmology the generalized bianchi models in 11 dimensions are considered these are space times whose spacelike ten dimensional sections are the hypersurfaces of transitivity for a ten dimensional isometry group of the total space time such a space time is a trivial principal fiber bundle p m g7 where m is a four dimensional physical space time with an isometry group g3 of a bianchi type and g7 is a compact isometry group of the compact internal space the isometry group of p is g10 g3 g7 hence all the generalized bianchi models are classified by enumerating the relevant groups g7 due to the compactness of g7 the result is astonishingly simple there are three distinct homogeneous internal spaces in addition to the 11 ordinary bianchi types for m

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

joint source channel codes for mimo block fading channels we consider transmission of a continuous amplitude source over an l block rayleigh fading mt x mr multiple input multiple output mimo channel when the channel state information is only available at the receiver since the channel is not ergodic shannon s source channel separation theorem becomes obsolete and the optimal performance requires a joint source channel approach our goal is to minimize the expected end to end distortion particularly in the high signal to noise ratio snr regime the figure of merit is the distortion exponent defined as the exponential decay rate of the expected distortion with increasing snr we provide an upper bound and lower bounds for the distortion exponent with respect to the bandwidth ratio among the channel and source bandwidths for the lower bounds we analyze three different strategies based on layered source coding concatenated with progressive superposition or hybrid digital analog transmission in each case by adjusting the system parameters we optimize the distortion exponent as a function of the bandwidth ratio we prove that the distortion exponent upper bound can be achieved when the channel has only one degree of freedom that is l 1 and min mt mr 1 when we have more degrees of freedom our achievable distortion exponents meet the upper bound for only certain ranges of the bandwidth ratio we demonstrate that our results which were derived for a complex gaussian source can be extended to more general source distributions as well

hybrid digital analog hda joint source channel codes for broadcasting and robust communications we consider the problem of transmitting a band limited gaussian source on an additive band limited gaussian noise channel the well known threshold effect dictates that the more powerful a code is the more sensitive it is to the exact knowledge of the channel noise a code is said to be robust if it is asymptotically optimal for a wide range of channel noise thus robust codes have a graceful degradation characteristic and are free of the threshold effect it is demonstrated that robust codes exist whenever the source and channel bandwidths are equal in the unequal bandwidth case a collection of nearly robust joint source channel codes is constructed using a hybrid digital analog hda coding technique for designing nearly robust codes a matched tandem code whose channel encoder s output is partially fully matched to its input is proposed and the existence of an asymptotically optimal matched tandem code is shown the nearly robust codes achieve the shannon limit theoretically optimum distortion and have a less severe threshold effect finally for the case of two different noise conditions the distortion regions of these codes are determined

contraction of lie algebra representations motivated by an example of inonu and wigner a general procedure for the contraction of irreducible unitary representations of semisimple compact lie algebras is given in this situation contractions mean generalized inonu wigner contractions in particular this means that by a simple limiting process one obtains representations of nonsemisimple noncompact lie algebras as examples a general treatment of so 3 is given as well as some results for su 3

intrinsic formulae for the casimir operators of semidirect products of the exceptional lie algebra g2 and a heisenberg lie algebra we show that the casimir operators of the semidirect products of the exceptional lie algebra g2 and a heisenberg algebra can be constructed explicitly from the casimir operators of g2

invariants of a semi direct sum of lie algebras we show that any semi direct sum l of lie algebras with levi factor s must be perfect if the representation associated with it does not possess a copy of the trivial representation as a consequence all invariant functions of l must be casimir operators when the number of invariants is given for all possible dimensions of l replacing the traditional method of solving the system of determining pdes by the equivalent problem of solving a system of total differential equations the invariants are found for all dimensions of the radical up to 5 an analysis of the results obtained is made and this leads to a theorem on invariants of lie algebras depending only on the elements of certain subalgebras

on the identification of a lie algebra given by its structure constants i direct decompositions levi decompositions and nilradicals abstract methods are given for identifying a lie algebra l given by its structure constants the identification involves a transformation to a canonical basis in which the structure becomes obvious thus decomposable lie algebras are already decomposed into direct sums of indecomposable ones an indecomposable lie algebra that is not simple or solvable has its radical exhibited a solvable algebra has its nilradical displayed the methods all lead to simple algorithms that have been implemented as computer programs involving some symbolic manipulations

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

virtual copies of semisimple lie algebras in enveloping algebras of semidirect products and casimir operators given a semidirect product g s r of semisimple lie algebras s and solvable algebras r we construct polynomial operators in the enveloping algebra u g of g that commute with r and transform like the generators of s up to a functional factor that turns out to be a casimir operator of r such operators are said to generate a virtual copy of s in u g and allow to compute the casimir operators of g in closed form using the classical formulae for the invariants of s the behavior of virtual copies with respect to contractions of lie algebras is analyzed applications to the class of hamilton algebras and their inhomogeneous extensions are given

continuous interior penalty finite element method for oseen s equations in this paper we present an extension of the continuous interior penalty method of douglas and dupont interior penalty procedures for elliptic and parabolic galerkin methods in computing methods in applied sciences lecture notes in phys 58 springer verlag berlin 1976 pp 207 216 to oseen s equations the method consists of a stabilized galerkin formulation using equal order interpolation for pressure and velocity to counter instabilities due to the pressure velocity coupling or due to a high local reynolds number we add a stabilization term giving l2 control of the jump of the gradient over element faces edges in two dimensions to the standard galerkin formulation boundary conditions are imposed in a weak sense using a consistent penalty formulation due to nitsche we prove energy type a priori error estimates independent of the local reynolds number and give some numerical examples recovering the theoretical results

non semisimple lie algebras with levi factor fraktur so 3 fraktur sfraktur l 2 bbb r and their invariants we analyse the number of functionally independent generalized casimir invariants for non semisimple lie algebras r with levi factors isomorphic to 3 and 2 in dependence of the pair r formed by a representation r of and a solvable lie algebra we show that for any dimension n 6 there exist lie algebras r with non trivial levi decomposition such that r 0

numerical regularized moment method of arbitrary order for boltzmann bgk equation we introduce a numerical method for solving grad s moment equations or regularized moment equations for an arbitrary order of moments in our algorithm we do not explicitly need the moment equations instead we directly start from the boltzmann equation and perform grad s moment method h grad commun pure appl math 2 1949 pp 331 407 and the regularization technique h struchtrup and m torrilhon phys fluids 15 2003 pp 2668 2680 numerically we define a conservative projection operator and propose a fast implementation which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions for the collision term the bgk model is adopted so that the production step can be done trivially based on the hermite expansion extensive numerical examples for one and two dimensional problems are presented convergence in moments can be validated by the numerical results for different numbers of moments

internal labelling operators and contractions of lie algebras we analyse under which conditions the missing label problem associated with a reduction chain of simple lie algebras can be completely solved by means of an inonu wigner contraction naturally related to the embedding this provides a new interpretation of the missing label operators in terms of the casimir operators of the contracted algebra and shows that the available labelling operators are not completely equivalent further the procedure is used to obtain upper bounds for the number of invariants of affine lie algebras arising as contractions of semi simple algebras

graded contractions of the lie algebra e 2 1 the graded contractions of pseudo euclidean lie algebra are studied the non equivalent gradings of of type and a are extended to the entire lie algebra using the action of on the abelian ideal the translations the graded contractions embed into a large family of six dimensional lie algebras the family includes solvable nilpotent and nonsolvable lie algebras both decomposable and indecomposable ones the distinction between graded and inonu wigner contractions is analysed the physically most interesting lie algebras obtained by the contractions are the inhomogeneous galilei and pseudo galilei

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

hierarchical boltzmann simulations and model error estimation abstract a hierarchical simulation approach for boltzmann s equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics but a subsequent refinement allows to successively improve the result to the complete boltzmann result we use hermite discretization or moment equations for the steady linearized boltzmann equation for a proof of concept of such a framework all representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous galerkin formulation we demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains the hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations we present various model errors for a flow through a curved channel with obstacles

a robust numerical method for the r13 equations of rarefied gas dynamics application to lid driven cavity in this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment r13 equations of rarefied gas dynamics the scheme allows fast solutions for 2d and 3d boundary value problems bvps with velocity slip and temperature jump boundary conditions the scheme is applied to the lid driven cavity problem for knudsen numbers up to 0 7 the results compare well with those obtained from more costly solvers for rarefied gas dynamics such as the integro moment method imm and the direct simulation monte carlo dsmc method the r13 equations yield better results than the classical navier stokes fourier equations for this boundary value problem since they give an approximate description of knudsen boundary layers at moderate knudsen numbers the r13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate knudsen numbers

variable rate channel capacity this paper introduces the notions of variable to fixed and fixed to variable channel capacity without feedback for channels that satisfy the strong converse these notions coincide with the conventional shannon capacity for channels that do not behave ergodically the conventional fixed rate shannon capacity only depends on least favorable channel conditions while the variable rate capacity notions are able to capture the whole range of channel states and their likelihood even in the absence of any side information about channel state at the transmitter particular emphasis is placed on memoryless channels that are governed by finitely valued states we show that single user variable to fixed channel capacity is intimately connected to the capacity region of broadcast channels with degraded message sets and we give an expression for the fixed to variable capacity

systematic lossy source channel coding the fundamental limits of systematic communication are analyzed in systematic transmission the decoder has access to a noisy version of the uncoded raw data analog or digital the coded version of the data is used to reduce the average reproduced distortion d below that provided by the uncoded systematic link and or increase the rate of information transmission unlike the case of arbitrarily reliable error correction d spl rarr 0 for symmetric sources channels where systematic codes are known to do as well as nonsystematic codes we demonstrate that the systematic structure may degrade the performance for nonvanishing d we characterize the achievable average distortion and we find necessary and sufficient conditions under which systematic communication does not incur loss of optimality the wyner ziv 1976 rate distortion theorem plays a fundamental role in our setting the general result is applied to several scenarios for a gaussian bandlimited source and a gaussian channel the invariance of the bandwidth signal to noise ratio snr in decibels product is established and the optimality of systematic transmission is demonstrated bernoulli sources transmitted over binary symmetric channels and over certain gaussian channels are also analyzed it is shown that if nonnegligible bit error rate is tolerated systematic encoding is strictly suboptimal

a pair of commuting scalars for g 2 su 2 su 2 g 2 su 2 su 2 is a two missing labels problem and therefore in order to give a complete and orthogonal specification of states of irreducible representations of g 2 in an su 2 su 2 basis one needs to find a pair of commuting hermitian operators which are scalar with respect to the su 2 su 2 subalgebra a theorem due to peccia and sharp states that there are apart from the lie algebra invariants twice as many functionally independent scalars as missing labels here two commuting su 2 su 2 scalars are obtained both of sixth order in the g 2 basis elements they are in fact combinations of five scalars of different tensorial types indicating that the functionally independent ones are in general insufficient to provide the lowest order commuting scalars an expression for the sixth order invariant of g 2 is also obtained

the streaming dmt of fading channels we study sequential transmission of a stream of messages over a block fading multi input multi output mimo channel a new message arrives in each coherence block and the decoder is required to output each message after a delay of t coherence blocks we establish the optimal diversity multiplexing tradeoff dmt and show that it can be achieved using a simple interleaving of messages the converse is based on an outage amplification technique which appears to be new we also discuss another coding scheme based on a sequential tree code this coding scheme only requires the knowledge of delay at the decoder and yet realizes the optimal dmt we finally discuss some extensions when multiple messages at uniform intervals arrive within each coherence period

casimir operators induced by the maurer cartan equations it is shown that for inhomogeneous lie algebras with only one casimir operator the latter can be explicitly constructed from the maurer cartan equations by means of wedge products it is further proved that this constraint imposes sharp bounds for the dimension of the representation r defining the semidirect product the procedure is generalized to compute also the rational invariant of some lie algebras

edge stabilization for the generalized stokes problem a continuous interior penalty method in this note we introduce and analyze a stabilized finite element method for the generalized stokes equation stability is obtained by adding a least squares penalization of the gradient jumps acro

boson representations of the real symplectic group and their applications to the nuclear collective model both non hermitian dyson and hermitian holstein primakoff representations of the sp 2d r algebra are obtained when the latter is restricted to a positive discrete series irreducible representation d n 2 1 n 2 for such purposes some results for boson representations recently deduced from a study of the sp 2d r partially coherent states are combined with some standard techniques of boson expansion theories the introduction of usui operators enables the establishment of useful relations between the various boson representations two dyson representations of the sp 2d r algebra are obtained in compact form in terms of d d 1 2 pairs of boson creation and annihilation operators and of an extra u d spin characterized by the irreducible representation 1 d in contrast to what happens when 1 d it is shown that the holstein primakoff representation of the sp 2d r algebra cannot be written in such a compact form for a generic irreducible representation explicit expansions are

invariants of solvable lie algebras of dimension six all invariant functions of the coadjoint representation are determined for real solvable lie algebras of dimension six having a nilradical of dimension four many questions related to the nature of these invariants are analysed in particular we derive the general type of functions in terms of which all the invariants can be expressed and give the characterization of all solvable lie algebras with abelian nilradical that have no non trivial invariants the existence of casimir operators among these invariants is also investigated

on the structure of maximal solvable extensions and of levi extensions of nilpotent lie algebras we establish an improved upper estimate on the dimension of any solvable lie algebra with its nilradical isomorphic to a given nilpotent lie algebra next we consider levi decomposable algebras with a given nilradical and investigate restrictions on possible levi factors originating from the structure of characteristic ideals of we present a new perspective on turkowski s classification of levi decomposable algebras up to dimension 9

on the geometry of lie algebras and poisson tensors a geometric programme to analyse the structure of lie algebras is presented with special emphasis on the geometry of linear poisson tensors the notion of decomposable poisson tensors is introduced and an algorithm to construct all solvable lie algebras is presented poisson liouville structures are also introduced to discuss a new class of lie algebras which include as a subclass semi simple lie algebras a decomposition theorem for poisson tensors is proved for a class of poisson manifolds including linear ones simple lie algebras are also discussed from this viewpoint and lower dimensional real lie algebras are analysed

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

on the distortion snr exponent of hybrid digital analog space time coding we consider the transmission of a real independent and identically distributed i i d analog source over a quasi static m input n output multiple input multiple output mimo block fading channel the relevant performance criterion is end to end average quadratic distortion d versus channel signal to noise ratio snr for given spectral efficiency eta defined as the ratio of the source bandwidth over the channel bandwidth in the limit of high snr we define the distortion snr exponent a eta as the largest a such that d esdot snr a over all possible source channel coding schemes of spectral efficiency eta we find a simple upper bound on a eta an achievable lower bound asep eta achievable by separated tandem source channel coding and a tighter lower bound ahybrid eta achievable by new hybrid digital analog space time coding schemes as a corollary we have that a eta is completely determined for the scalar case m n 1 and for the bandwidth compression case eta ges 2 min m n expiicit and simple construction of hybrid space time codes achieving ahybrid eta are also given

kinematic groups and dimensional analysis it is shown that group theory may be useful in relation to dimensional analysis the group theoretical support of dimensional analysis in universes described by kinematic groups is analysed its close relation to the structure of the corresponding kinematic group is also displayed by means of a simple dimensionalisation hypothesis the scheme of contractions relating the kinematic groups enables one to discuss the dimensionalisation method the same problem is also looked at in a different way the possibility of adding dilatation like transformations is studied finally the role of mass in both relativistic and non relativistic theories is examined

graded contractions of the pauli graded sl 3 c the lie algebra sl 3 c is considered in the basis of generalized pauli matrices corresponding grading is the pauli grading here it is one of the four gradings of the algebra which cannot be further refined the set s of 48 contraction equations for 24 contraction parameters is solved our main tools are the symmetry group of the pauli grading of sl 3 c which is essentially the finite group sl 2 z3 and the induced symmetry of the system s a list of all equivalence classes of solutions of the contraction equations is provided among the solutions 175 equivalence classes are non parametric and 13 solutions depend on one or two continuous parameters providing a continuum of equivalence classes and subsequently continuum of non isomorphic lie algebras solutions of the contraction equations of pauli graded sl 3 c are identified here as specific solvable lie algebras of dimensions up to 8 earlier algorithms for identification of lie algebras given by their structure constants had to be made more efficient in order to distinguish non isomorphic lie algebras encountered here resulting lie algebras are summarized in tabular form there are 88 indecomposable solvable lie algebras of dimension 8 77 of them being nilpotent there are 11 infinite sets of parametric lie algebra which still deserve further study

a broadcast approach for a single user slowly fading mimo channel a broadcast transmission strategy for the slowly fading gaussian multiple input multiple output mimo channel is introduced this broadcast strategy is an extension of the single input single output siso broadcast approach perfect channel state information csi is assumed known at the receiver end only this strategy facilitates to adapt the reliably decoded rate to the actual channel state without having any feedback link to the transmitter transmission of layered coded information is motivated by the theory of majorization we derive the basic equations characterizing achievable rates of the strategy several ad hoc approximations to the achievable region are considered and their performance is compared with the siso setting and the ergodic capacity it has been demonstrated that a single layer outage approach is reasonably efficient in the mimo setting in terms of the average reliably decoded rate a multiple access channel mac broadcast approach is also applied for the mimo case and demonstrated to be relatively efficient

sp 3 r coadjoint orbit theory the algebraic mean field method is applied to the symplectic lie algebra sp 3 r that describes geometrical collective states in atomic nuclei the expectations of sp 3 r generators define a symplectic density matrix the mean field approximation restricts the densities to a manifold that is a coadjoint orbit of the transformation group sp 3 r and a level surface of the symplectic casimir functions compared to representation theory mean field theory is technically simpler but yields similar predictions for physical properties of collective states the critical points of the energy functional on a coadjoint orbit surface define rotational bands the deformation kinetic energy and kelvin circulation of principal axis symplectic rotors are determined as a function of the angular momentum illustrative applications of coadjoint orbit theory are reported for the yrast rotational bands of a light 20ne and a heavy 166er deformed isotope

distortion bounds for broadcasting with bandwidth expansion we consider the problem of broadcasting a single gaussian source to two listeners over a gaussian broadcast channel with rho channel uses per source sample where rho 1 a distortion pair d1 d2 is said to be achievable if one can simultaneously achieve a mean squared error mse d1 at receiver 1 and d2 at receiver 2 the main result of this correspondence is an outer bound for the set of all achievable distortion pairs that is we find necessary conditions under which d1 d2 is achievable we then apply this result to the problem of point to point transmission over a gaussian channel with unknown signal to noise ratio snr and rho 1 we show that if a system must be optimal at a certain snrmin then asymptotically the system distortion cannot decay faster than o 1 snr as for achievability we show that a previously reported scheme due to mittal and phamdo 2002 is optimal at high snr we introduce two new schemes for broadcasting with bandwidth expansion combining digital and analog transmissions we finally show how a system with a partial feedback returning from the bad receiver to the transmitter and to the good receiver achieves a distortion pair that lies on the outer bound derived here

casimir operators of semidirect sum lie algebras a new method for constructing the casimir operators of a class of non semisimple lie algebras is presented this class includes the semidirect sums of the unitary orthogonal or symplectic algebras with a heisenberg weyl algebra w n the construction procedure is illustrated with the simple example of the wu n n u n algebras its power is then demonstrated on the more difficult case of the wsp 2n r w n sp 2n r algebras detailed results are given for the eigenvalues of the wsp 6 r casimir operators relevant to a microscopic theory of nuclear collective motions

trace formulas for the casimir operators of the unextended schrodinger algebra s n using the contraction of the centrally extended schrodinger algebra s n onto the lie algebra s n r in combination with the newton identities associated with the characteristic polynomial of a matrix we derive explicit expressions for the casimir operators of the unextended schrodinger algebra s n in terms of trace operators it is shown that these operators can be defined independently of the contraction from which a direct method for the computation of the s n invariants is deduced

boundary conditions for regularized 13 moment equations for micro channel flows boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro flows of gases in this paper we present a theory how to combine the regularized 13 moment equations derived from boltzmann s equation with boundary conditions obtained from maxwell s kinetic accommodation model while for the linear case these kinetic boundary conditions suffice we need additional conditions in the non linear case these are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to boltzmann s equation after finding a suitable set of boundary conditions and equations a numerical method for generic shear flow problems is formulated several test simulations demonstrate the stable and oscillation free performance of the new approach

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

the three dimensional real lie algebras and their contractions in this paper the general contractions of the three dimensional real lie algebras are determined a short summary will be given of the history definitions properties examples and applications of contractions of lie algebras so far mostly simple or generalized inonu wigner contractions have been used the three dimensional real lie algebras are classified in such a way that their general contractions can be easily read off all these contractions can be realized by generalized inonu wigner contractions which supports the argument that these are the interesting objects to study

on the multicast capacity of the wireless broadcast channel for a multicast network in which a common source is transmitted to n users the problem of maximizing the average rate subject to a coverage constraint minimum quality of service is studied considering such a network with single antenna nodes and assuming that the channel state information is available only at the receiver side the highest expected rate achievable by a random user in the network called the expected typical rate is derived in two scenarios hard coverage constraint and soft coverage constraint in the first case the coverage is expressed in terms of the outage probability while in the second case the expected rate should satisfy a certain minimum requirement it is shown that the optimum solution achieving the highest expected typical rate for given coverage requirements in both cases is achieved by an infinite layer superposition code for which the optimal power allocation among different layers is derived the analysis is extended to a scenario with multiple transmit antennas for the miso case a suboptimal coding scheme is proposed which is shown to be asymptotically optimal when the number of transmit antennas grows at least logarithmically with the number of users in the network

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

invariants of solvable rigid lie algebras up to dimension 8 the invariants of all complex solvable rigid lie algebras up to dimension 8 are computed moreover we show for rank 1 solvable algebras some criteria to deduce the non existence of nontrivial invariants or the existence of fundamental sets of invariants formed by rational functions of the casimir invariants of the associated nilradical