convergence and superconvergence of staggered discontinuous galerkin methods for the three dimensional maxwell s equations on cartesian grids in this paper a new type of staggered discontinuous galerkin methods for the three dimensional maxwell s equations is developed and analyzed the spatial discretization is based on staggered cartesian grids so that many good properties are obtained first of all our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the gauss law moreover the mass matrices are diagonal thus time marching is explicit and is very efficient our method is high order accurate and the optimal order of convergence is rigorously proved it is also very easy to implement due to its cartesian structure and can be regarded as a generalization of the classical yee s scheme as well as the quadrilateral edge finite elements furthermore a superconvergence result that is the convergence rate is one order higher at interpolation nodes is proved numerical results are shown to confirm our theoretical statements and applications to problems in unbounded domains with the use of pml are presented a comparison of our staggered method and non staggered method is carried out and shows that our method has better accuracy and efficiency

character generators for elements of finite order in simple lie groups a1 a2 a3 b2 and g2 elements of finite order in the compact simple lie groups su 2 su 3 su 4 sp 4 or o 5 and g 2 are considered we provide the characters of the elements on irreducible representations of lie groups by assigning appro priate numerical values to the variables on which the characters of representations of the lie group depend in this way we specialize generating functions for the characters of the representations of the lie groups to the generating functions for characters of the elements of finite order particular attention is paid to rational elements all of whose characters are integers they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read gaussian elements are also studied in detail their characters are complex valued with integer real and imaginary parts

discrete and continuous exponential transforms of simple lie groups of rank 2 we develop and describe continuous and discrete transforms of class functions on compact simple lie group g as their expansions into series of uncommon special functions called here e functions in recognition of the fact that the functions generalize common exponential functions the rank of g is the number of variables in the e functions a uniform discretization of the decomposition problem is described on lattices of any density and symmetry admissible for the lie group g

fast recursion formula for weight multiplicities the purpose of this note is to describe and prove a fast recursion formula for computing multiplicities of weights of finite dimensional representations of simple lie algebras over c until now information about weight multiplicities for all but some special cases 1 2 has had to be found from the recursion formulas of freudenthal 3 or racah 4 typically these formulas become too laborious to use for hand computations for ranks 5 and dimensions 1 0 0 and for ranks 10 and dimensions 10 on a large computer 5 6 with the proposed method the multiplicities can routinely be calculated even by hand for dimensions far exceeding these as an example we present a summary of calculations 7 of all multiplicities in the first sixteen irreducible representations of es let be a semisimple lie algebra over c with root system a and weyl group w relative to a cartan subalgebra let a be the positive roots with respect to some ordering and ii at a the set of simple roots let q and p be the root and weight lattices respectively spanning the real vector space f c if x c p we denote by x the set of dominant elements of x relative to n let m be an irreducible module with highest weight a and weight system 2 an important feature of the approach is the direct determination of 2 without computing outside the dominant chamber since every w orbit is represented by one weight x 2 of the same multiplicity it suffices to compute such x s the recursion formula for computing the multiplicities is a modification proposition 4 of the freudenthal formula in which the weyl group has been exploited to collapse it as much as possible after describing the procedure we present the e example finally the necessary proofs are given

orthogonal polynomials of compact simple lie groups branching rules for polynomials polynomials in this paper are defined starting from a compact semisimple lie group a known classification of maximal semisimple subgroups of simple lie groups is used to select the cases to be considered here a general method is presented and all the cases of rank 3 are explicitly studied we derive the polynomials of simple lie groups b3 and c3 as they are not available elsewhere the results point to far reaching lie theoretical connections to the theory of multivariable orthogonal polynomials

general charge conjugation operators in simple lie groups a description of particular elements charge conjugation operators found in any compact simple lie group k is presented such elements ri transform a physical state weight vector of a basis of a representation space into others with opposite charge ith component of the weight sometime changing also the sign of the state it is demonstrated that exploitation of these elements and the finite subgroup n of k generated by them offer new powerful methods for computing with representations of the lie group their application to construction of bases in representation spaces is considered in detail it represents a completely new direction to the problem

an algorithm for computing weight multiplicities in irreducible modules for complex semisimple lie algebras abstract let g be a finite dimensional semisimple lie algebra over c having rank l and let v be an irreducible finite dimensional g module having highest weight computations of weight multiplicities in v usually based on freudenthal s formula are in general difficult to carry out in large ranks or for with large coefficients in terms of the fundamental weights in this paper we first show that in some situations these coefficients can be lowered in order to simplify the calculations we then investigate how this can be used to improve the aforementioned formula of freudenthal leading to a more efficient version of the latter in terms of complexity as well as to a way of dealing with certain computations in unbounded ranks we conclude by illustrating the last assertion with a concrete example

lie group weight multiplicities from conformal field theory dominant weight multiplicities of simple lie groups are expressed in terms of the modular matrices of wess zumino witten conformal field theories and related objects symmetries of the modular matrices give rise to new relations among multiplicities at least for some lie groups these new relations are strong enough to completely fix all multiplicities

reduction of orbits of finite coxeter groups of non crystallographic type a reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices which transform points of the orbit of any group into points of the orbits of its subgroup projection matrices and branching rules for orbits of finite coxeter groups of non crystallographic type are presented the novelty in this paper is producing the branching rules that involve non crystallographic coxeter groups moreover these branching rules are relevant to any application of non crystallographic coxeter groups including molecular crystallography and encryption

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

the signatures of finite dimensional representations of the de sitter groups so and so the signature of a finite dimensional orthogonal representation of a simple lie group is the difference between the number of positive and negative signs in the diagonal form of its symmetric bilinear invariant we derive the expressions for the signatures of all finite dimensional representations of the de sitter groups so 4 1 and so 3 2 in two ways one by means of character values of appropriate elements of adjoint order two on the representation and the other through generating functions

clebsch gordan coefficients for su 5 su 3 su 2 u 1 theories the clebsch gordan coefficients are calculated for the following tensor products of su 5 representations 5 5 5 10 5 10 15 5 15 5 10 10 5 5 10 10 5 24 and 10 24 each case is calculated twice once in a weight vector basis independent of any semisimple subgroup the second time in a basis which refers to su 3 su 2 u 1 su 5

discrete and continuous sine transform generalized to semisimple lie groups of rank two a new continuous group transform together with its discretization over a lattice of any density and admissible symmetry is defined and described for the four compact semisimple lie groups of rank 2 in the case of rank 1 the discrete version is the sine transform properties of the expansion functions of the transform called s functions are studied digital data processing is our motivating application

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

asymmetric inclusion process we introduce and explore the asymmetric inclusion process asip an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process asep in both processes random events cause particles to propagate unidirectionally along a one dimensional lattice of n sites in the asep particles are subject to exclusion interactions whereas in the asip particles are subject to inclusion interactions that coalesce them into inseparable clusters we study the dynamics of the asip derive evolution equations for the mean and probability generating function pgf of the sites occupancy vector obtain explicit results for the above mean at steady state and describe an iterative scheme for the computation of the pgf at steady state we further obtain explicit results for the load distribution in steady state with the load being the total number of particles present in all lattice sites finally we address the problem of load optimization and solve it under various criteria the asip model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with unlimited batch service and a tandem array of growth collapse processes

random neural networks with negative and positive signals and product form solution we introduce a new class of random neural networks in which signals are either negative or positive a positive signal arriving at a neuron increases its total signal count or potential by one a negative signal reduces it by one if the potential is positive and has no effect if it is zero when its potential is positive a neuron fires sending positive or negative signals at random intervals to neurons or to the outside positive signals represent excitatory signals and negative signals represent inhibition we show that this model with exponential signal emission intervals poisson external signal arrivals and markovian signal movements between neurons has a product form leading to simple analytical expressions for the system state

computation of character decompositions of class functions on compact semisimple lie groups a new algorithm is described for splitting class functions of an arbitrary semisimple compact lie group k into sums of irreducible characters the method is based on the use of elements of finite order efo in k and is applicable to a number of problems including decompositions of tensor products and various symmetry classes of tensors as well as branching rules in group subgroup reductions the main feature is the construction of a decomposition matrix d computed once and for all for a given range of problems and for a given k which then reduces any particular splitting to a simple matrix multiplication determination of d requires selection of a suitable set s of conjugacy classes of efo representing a finite subgroup of a maximal torus t of k and the evaluation of weyl group orbit sums on s in fact the evaluation of d can be coupled with the evaluation of the orbit sums in such a way as to greatly enhance the efficiency of the latter the use of the method is illustrated by some extensive examples of tensor product decompositions in e6 modular arithmetic allows all computations to be performed exactly

energy stable discontinuous galerkin methods for maxwell s equations in nonlinear optical media abstract the propagation of electromagnetic waves in general media is modeled by the time dependent maxwell s partial differential equations pdes coupled with constitutive laws that describe the response of the media in this work we focus on nonlinear optical media whose response is modeled by a system of first order nonlinear ordinary differential equations odes which include a single resonance linear lorentz dispersion and the nonlinearity comes from the instantaneous electronic kerr response and the residual raman molecular vibrational response to design efficient accurate and stable computational methods we apply high order discontinuous galerkin discretizations in space to the hybrid pde ode maxwell system with several choices of numerical fluxes and the resulting semi discrete methods are shown to be energy stable under some restrictions on the strength of the nonlinearity error estimates are also established when we turn to fully discrete methods the challenge to achieve provable stability lies in the temporal discretizations of the nonlinear terms to overcome this novel strategies are proposed to treat the nonlinearity in our model within the framework of the second order leap frog and implicit trapezoidal time integrators the performance of the overall algorithms are demonstrated through numerical simulations of kink and antikink waves and third harmonic generation in soliton propagation

properties of continuous fourier extension of the discrete cosine transform and its multidimensional generalization a versatile method is described for the practical computation of the exact discrete fourier transforms dft both the direct and the inverse ones of a continuous function g given by its values gj at the points of a uniform grid fn generated by conjugacy classes of elements of finite adjoint order n in the fundamental region f of compact semisimple lie groups the present implementation of the method is for the groups su 2 when f is reduced to a one dimensional segment and for su 2 su 2 su 2 in multidimensional cases this simplest case turns out to be a version of the discrete cosine transform dct implementations abbreviated as dgt for discrete group transform based on simple lie groups of higher ranks are to be considered separately dct is often taken to be simply a specific type of the standard dft here we show that the dct is very different from the standard dft when the properties of the continuous extensions of the two inverse discrete transforms are studied the following propertie

branching rules for the weyl group orbits of the lie algebra an the orbits of weyl groups w an of simple an type lie algebras are reduced to the union of orbits of the weyl groups of maximal reductive subalgebras of an matrices transforming points of the orbits of w an into points of subalgebra orbits are listed for all cases n 8 and for the infinite series of algebra subalgebra pairs an an k 1 ak u1 a2n bn a2n 1 cn a2n 1 dn numerous special cases and examples are shown

interior penalty dg methods for maxwell s equations in dispersive media in this paper we develop a fully discrete interior penalty discontinuous galerkin method for solving the time dependent maxwell s equations in dispersive media the model is described by a vector integral differential equation our scheme is proved to be unconditionally stable and achieve optimal error estimates in both l 2 norm and energy norm the scheme is implemented and numerical results supporting our analysis are presented

orbit orbit branching rules between simple low rank algebras and equal rank subalgebras complete orbit orbit branching rules or equivalently reduction of weyl group orbits between each simple algebra of rank 3 or less and its equal rank subalgebras and between f4 and each of its equal rank subalgebras are given the generic case an an 1 u 1 the subjoining f4 b3 u 1 and e8 e6 a2 first and seventh highest weight labels nonzero are also treated branching rules between rank 4 and 5 simple algebras and e6 and their equal rank subalgebras are available in a depository

limit laws for the asymmetric inclusion process the asymmetric inclusion process asip is a unidirectional lattice gas flow model which was recently introduced as an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process an iterative algorithm that allows the computation of the probability generating function pgf of the asip s steady state exists but practical considerations limit its applicability to small asip lattices large lattices on the other hand have been studied primarily via monte carlo simulations and were shown to display a wide spectrum of intriguing statistical phenomena in this paper we bypass the need for direct computation of the pgf and explore the asip s asymptotic statistical behavior we consider three different limiting regimes heavy traffic regime large system regime and balanced system regime in each of these regimes we obtain analytically and in closed form stochastic limit laws for five key asip observables traversal time overall load busy period first occupied site and draining time the results obtained yield a detailed limit laws perspective of the asip numerical simulations demonstrate the applicability of these laws as useful approximations

the demazure tits subgroup of a simple lie group the demazure tits subgroup of a simple lie group g is the group of invariance of clebsch gordan coefficients tables assuming an appropriate choice of basis the structure of the demazure tits subgroups of an bn cn dn and g2 is described orbits of the permutation action of the dt group in any irreducible finite dimensional representation space of a2 c2 and g2 are decomposed into the sum of irreducible representations of the dt group

orbit orbit branching rules for families of classical lie algebra subalgebra pairs complete orbit orbit branching rules are derived for each classical algebra maximal subalgebra pair cm ncm cn bm ndm bn dm ndm dn since each pair is equal rank and algebra and subalgebra weyl sectors line up the integrity basis in each case consists of the subalgebra orbits contained in the fundamental orbits of the algebra

occupation probabilities and fluctuations in the asymmetric simple inclusion process the asymmetric simple inclusion process asip a lattice gas model of unidirectional transport and aggregation was recently proposed as an inclusion counterpart of the asymmetric simple exclusion process in this paper we present an exact closed form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites our results are expressed in terms of the entries of catalan s trapezoids number arrays which generalize catalan s numbers and catalan s triangle we further prove that the asip is asymptotically governed by the following i an inverse square root law of occupation ii a square root law of fluctuation and iii a rayleigh law for the distribution of interexit times the universality of these results is discussed

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

structure of certain weyl modules for the spin groups abstract let k be an algebraically closed field of characteristic p 0 and let w be a finite dimensional k vector space of dimension greater than or equal to 5 in this paper we give the structure of certain weyl modules for g spin w in the case where p 2 as well as the dimension of the corresponding irreducible finite dimensional rational kg modules in addition we determine the composition factors of the restriction of certain irreducible finite dimensional rational k sl w modules to so w

on discretization of tori of compact simple lie groups the discrete orthogonality of special function families called c and s functions which are derived from the characters of compact simple lie groups is described in hrivnak and patera 2009 j phys a math theor 42 385208 here the results of hrivnak and patera are extended to two additional recently discovered families of special functions called ss and sl functions the main result is an explicit description of their pairwise discrete orthogonality within each family when the functions are sampled on finite fragments fsm and flm of a lattice in any dimension n 2 and of any density controlled by m and of the symmetry of the weight lattice of any compact simple lie group with two different lengths of roots

periodicity and chaos from switched flow systems contrasting examples of discretely controlled continuous systems two examples of the discrete control of a continuous variable system are analyzed these examples exhibit what may be regarded as the two extremes of complexity of the closed loop behavior one is eventually periodic the other is chaotic the examples are derived from sampled deterministic flow models these are of interest in their own right but have also been used as models for certain aspects of manufacturing systems in each case the authors give a precise characterization of the closed loop behavior

discrete and continuous cosine transform generalized to lie groups su 3 and g 2 in the paper we complete the development and description of the four variants of two dimensional generalization of the cosine transform started in patera and zaratsyan j math phys 46 053514 2005 each variant is based on a compact semisimple lie group g of rank 2 here the groups are su 3 and g 2 the cosines are generalized as the corresponding c functions of the lie group a c function is the contribution to an irreducible character from one orbit of the appropriate weyl group an explicit description is provided for expansions of functions given on the fundamental region f of the two compact simple lie groups into series of c functions the fundamental region f is an equilateral triangle for su 3 and half of such a triangle for g 2 expansion coefficients are calculated using orthogonality of c functions on f discrete expansions are set up on a grid fm f the grid is defined group theoretically for all positive integers m it consists of points in f that represent conjugacy classes of ele

time domain finite element methods various time domain finite element methods for the simulation of transient electromagnetic wave phenomena are discussed detailed descriptions of test trial spaces explicit and implicit formulations nodal and edge facet element basis functions are given along with the numerical stability properties of the different methods the advantages and disadvantages of mass lumping are examined finally the various formulations are compared on the basis of their numerical dispersion performance

finite subgroup bases for compact lie groups abstract the method is described for the derivation of generating functions for branching rules from representations of a compact lie group to those of a finite subgroup interpreted in terms of an integrity basis the generating functions define analytic polynomial basis states for the lie group which respect the finite subgroup some details are given for su 3 and its maximal finite subgroup 168

on the reflection type decomposition of the adjoint reduced phase space of a compact semisimple lie group we consider a system with symmetries whose configuration space is a compact lie group acted upon by inner automorphisms the classical reduced phase space of this system decomposes into connected components of orbit type subsets to investigate hypothetical quantum effects of this decomposition one has to construct the associated costratification of the hilbert space of the quantum system in the sense of huebschmann in the present paper instead of the decomposition by orbit types we consider the related decomposition by reflection types conjugacy classes of reflection subgroups these two decompositions turn out to coincide e g for the classical groups su n and sp n we derive defining relations for reflection type subsets in terms of irreducible characters and discuss how to obtain from that the corresponding costratification of the hilbert space of the system to illustrate the method we give explicit results for some low rank classical groups

character calculations for semisimple lie algebras a recursive method for calculation of characters of semisimple lie algebras is outlined by slight modifications algorithms are obtained for the computation of weight multiplicities kronecker products and branching rules as well as symmetrized kronecker powers

a discontinuous galerkin finite element time domain method modeling of dispersive media a method is proposed for solving the time dependent maxwell s equations via the discontinuous galerkin finite element time domain dgfetd method with dispersive media an auxiliary differential equation ade method is used to represent the constitutive relations the method is applied to drude materials as well as to multiple pole debye and lorentz materials an efficient implementation for high order runge kutta time integration schemes is presented the method is validated and is shown to exhibit high order convergence

dispersion analysis of finite difference and discontinuous galerkin schemes for maxwell s equations in linear lorentz media abstract in this paper we consider maxwell s equations in linear dispersive media described by a single pole lorentz model for electronic polarization we study two classes of commonly used spatial discretizations finite difference methods fd with arbitrary even order accuracy in space and high spatial order discontinuous galerkin dg finite element methods both types of spatial discretizations are coupled with second order semi implicit leap frog and implicit trapezoidal temporal schemes by performing detailed dispersion analysis for the semi discrete and fully discrete schemes we obtain rigorous quantification of the dispersion error for lorentz dispersive dielectrics in particular comparisons of dispersion error can be made taking into account the model parameters and mesh sizes in the design of the two types of schemes this work is a continuation of our previous research on energy stable numerical schemes for nonlinear dispersive optical media 6 7 the results for the numerical dispersion analysis of the reduced linear model considered in the present paper can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models the numerical dispersion analysis of the fully discrete fd and dg schemes for the dispersive maxwell model considered in this paper clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations their order of accuracy mesh discretization parameters and model parameters the results obtained here cannot be arrived at by considering discretizations of maxwell s equations in free space in particular our results contrast the advantages and disadvantages of using high order fd or dg schemes and leap frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors finally we highlight the limitations of the second order accurate temporal discretizations considered

the weyl groups and weight multiplicities of the exceptional lie groups the weyl group wg of each exceptional simple lie group g is described in detail its structure is defined in terms of its coset decomposition with respect to the weyl group wh of a classical semi simple lie group h embedded naturally in g the concepts of g dominance and g equivalence are defined and used to determine from the character formula of weyl the branching rule associated with the restriction of group elements from g to h the weyl group wg is used further to impose constraints on both the branching multiplicities for g to h and the weight multiplicities of g these constraints are used to evaluate the weight multiplicities of f4 e6 e7 and e8 together with the branching multiplicities for e8 to so 16

discontinuous galerkin methods for dispersive and lossy maxwell s equations and pml boundary conditions in this paper we will present a unified formulation of discontinuous galerkin method dgm for maxwell s equations in linear dispersive and lossy materials of debye type and in the artificial perfectly matched layer pml regions an auxiliary differential equation ade method is used to handle the frequency dependent constitutive relations with the help of auxiliary polarization currents in the computational and pml regions the numerical flux for the dispersive lossy maxwell s equations with the auxiliary polarization current variables is derived various numerical results are provided to validate the proposed formulation

three dimensional c s and e transforms three dimensional continuous and discrete fourier like transforms based on the three simple and four semisimple compact lie groups of rank 3 are presented for each simple lie group there are three families of special functions c s and e functions on which the transforms are built pertinent properties of the functions are described in detail such as their orthogonality within each family when integrated over a finite region f of the three dimensional euclidean space continuous orthogonality as well as when summed up over a lattice grid fm f discrete orthogonality the positive integer m sets up the density of the lattice containing fm the expansion of functions given either on f or on fm is the paper s main focus

on orbit sum values of elements of finite order the irreducible characters form a basis of the ring of class functions on a compact simply connected simple lie group another basis are the orbit sums f l s m w le m which are related to the irreducible characters by the inner multiplicities there is a w invariant sesquilinear form on the class functions defined over a finite set of elements of finite order under which the orbit sums are almost orthogonal we give an exact criterion for the orthogonality given the set of elements of order dividing a fixed n and follow some remarkable properties of the values of orbit sums on elements of finite order efo

dispersive behaviour of high order finite element schemes for the one way wave equation we study the ability of high order numerical methods to propagate discrete waves at the same speed as the physical waves in the case of the one way wave equation a detailed analysis of the finite element method is presented including an explicit form for the discrete dispersion relation and a complete characterisation of the numerical bloch waves admitted by the scheme a comparison is made with the spectral element method and the discontinuous galerkin method with centred fluxes it is shown that all schemes admit a spurious mode the spectral element method is always inferior to the finite element and discontinuous galerkin schemes a somewhat surprising result in view of the fact that in the case of the second order wave equation the spectral element method propagates waves with an accuracy superior to that of the finite element scheme the comparative behaviour of the finite element and discontinuous galerkin scheme is also somewhat surprising the accuracy of the finite element method is superior to that of the discontinuous galerkin method in the case of elements of odd order by two orders of accuracy but worse again by two orders of accuracy in the case of elements of even order

steady state solution of probabilistic gene regulatory networks we introduce a probability model for gene regulatory networks based on a system of chapman kolmogorov equations that represent the dynamics of the concentration levels of each agent in the network this unifying approach includes the representation of excitatory and inhibitory interactions between agents second order interactions which allow any two agents to jointly act on other agents and boolean dependencies between agents the probability model represents the concentration or quantity of each agent and we obtain the equilibrium solution for the joint probability distribution of each of the concentrations the result is an exact solution in product form where the joint equilibrium probability distribution of the concentration for each gene is the product of the marginal distribution for each of the concentrations the analysis we present yields the probability distribution of the concentration or quantity of all of the agents in a network that includes both logical dependencies and excitatory inhibitory relationships between agents

stability and phase error analysis of fd td in dispersive dielectrics four fd td extensions for the modeling of pulse propagation in debye or lorentz dispersive media are analyzed through studying the stability and phase error properties of the coupled difference equations corresponding to maxwell s equations and to the equations for the dispersion for good overall accuracy the author shows that all schemes should be run at their courant stability limit and that the timestep should finely resolve the medium timescales particularly for debye schemes it should be at least spl delta t 10 sup spl minus 3 spl tau while for lorentz schemes it should be spl delta t 10 sup spl minus 2 spl tau where spl tau is a typical medium relaxation time a numerical experiment with a debye medium confirms this the author has determined that two of the discretizations for debye media are totally equivalent in the lorentz medium case the author establishes that the method that uses the polarization differential equation to model dispersion is stable for all wavenumbers and that the method using the local in time constitutive relation is weakly unstable for modes with wavenumber k such that k spl delta x spl pi 2

geometrical classification scheme for weights of lie algebras every weight of a lie algebra belongs to one particular weyl orbit the depth within an orbit has been defined as the minimum number of weyl reflections associated with simple roots that are necessary to transform the weight into the dominant weight of the orbit a simple set of classification parameters that measures the depth and distinguishes between different weights of the same depth is introduced the parameters are geometrical in that they are related directly to the orientation of the weight vector they correspond to a particular member of the weyl group that transforms the weight into the dominant weight a simple rule is given for listing all the weights of an orbit in terms of the parameters

general indices of representations and casimir invariants a modified definition of the index of degree p of a finite dimensional representation of a simple lie algebra is given the definition applies equally to even and odd p the correspondence between the earlier definitions of indices even p and anomaly numbers odd p is pointed out as well as the relation to casimir invariants of the algebra a closed formula for the fifth order index is derived

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

the weyl orbits of g2 f4 e6 and e7 simple methods are described for constructing the weyl orbits of exceptional lie algebras and for identifying the orbit of an arbitrary weight the methods make use of convenient bases in weight space they are applied to g2 f4 e6 and e7 a complete table of all weights of all orbits of f4 is given the weights of the 37 orbits of smallest weight length of e6 and e7 are given in tabular form the depth structures of the orbits are discussed

generating functions and multiplicity formulas the case of rank two simple lie algebras a procedure is described that makes use of the generating function of characters to obtain a new generating function h giving the multiplicities of each weight in all the representations of a simple lie algebra the way to extract from h explicit multiplicity formulas for particular weights is explained and the results corresponding to rank two simple lie algebras are shown

bridging genetic networks and queueing theory one of the main challenges facing biology today is the understanding of the joint action of genes proteins and rna molecules interwoven in intricate interdependencies commonly known as genetic networks to this end several mathematical approaches have been introduced to date in addition to developing the analytical tools required for this task anew one can utilize knowledge found in existing disciplines specializing in the representation and analysis of systems featuring similar aspects we suggest queueing theory as a possible source of such knowledge this discipline which focuses on the study of workloads forming in a variety of scenarios offers an assortment of tools allowing for the derivation of the statistical properties of the inspected systems we argue that a proper adaptation of modeling techniques and analytical methods used in queueing theory can contribute to the study of genetic regulatory networks this is demonstrated by presenting a queueing inspired model of a genetic network of arbitrary size and structure for which the probability distribution function is derived this model is further applied to the description of the lac operon regulation mechanism in addition we discuss the possible benefits stemming for queueing theory from the interdisciplinary dialogue with molecular biology in particular the incorporation of various dynamical behaviours into queueing networks

on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved

the rings of n dimensional polytopes points of an orbit of a finite coxeter group g generated by n reflections starting from a single seed point are considered as vertices of a polytope g polytope centered at the origin of a real n dimensional euclidean space a general efficient method is recalled for the geometric description of g polytopes their faces of all dimensions and their adjacencies products and symmetrized powers of g polytopes are introduced and their decomposition into the sums of g polytopes is described several invariants of g polytopes are found namely the analogs of dynkin indices of degrees 2 and 4 anomaly numbers and congruence classes of the polytopes the definitions apply to crystallographic and non crystallographic coxeter groups examples and applications are shown

four families of weyl group orbit functions of b 3 and c 3 the properties of the four families of special functions of three real variables called here c s ss and sl functions are studied the ss and sl functions are considered in all details required for their exploitation in fourier expansions of digital data sampled on finite fragment of lattices of any density and of the 3d symmetry imposed by the weight lattices of b3 and c3 simple lie algebras groups the continuous interpolations which are induced by the discrete expansions are exemplified and compared for some model functions

on the generating function of weight multiplicities for the representations of the lie algebra c2 we use the generating function of the characters of c2 to obtain a generating function for the multiplicities of the weights entering in the irreducible representations of that simple lie algebra from this generating function we derive some recurrence relations among the multiplicities and a simple graphical recipe to compute them

branching rules for representations of simple lie algebras through weyl group orbit reduction two independent algorithms are presented which together allow the determination of branching rules from an irreducible representation of a compact lie algebra to those of a subalgebra or subjoined algebra the first gives the subalgebra weyl orbits contained in an algebra orbit the second gives the irreducible representations of an algebra contained in an orbit and by inversion of a triangular matrix the orbits contained in an irreducible representation