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

character generators for compact semisimple lie groups a new character formula is presented which leads automatically to character generators in the positive form the character generator for b3 is constructed as an example

two classes of ode models with switch like behavior abstract in cases where the same real world system can be modeled both by an ode system d and a boolean system b it is of interest to identify conditions under which the two systems will be consistent that is will make qualitatively equivalent predictions in this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions in contrast to the widely known class of glass networks the right hand sides of our odes are lipschitz continuous we prove that if b has certain structures consistency between d and b is implied by sufficient separation of timescales in one class of our models namely if the trajectories of b are one stepping then we prove a strong form of consistency and if b has a certain monotonicity property then there is a weaker consistency between d and b these results appear to point to more general structure properties that favor consistency between ode and boolean models

generating functions for g2 characters and subgroup branching rules the g2 character generator is given with its help generating functions are derived for branching rules for g2 irreducible representations reduced according to its maximal semisimple subgroups

boundedness theorems for the relaxation method a classical theorem by block and levin block h d s a levin 1970 on the boundedness of an iterative procedure for solving a system of linear inequalities proc amer math soc 26 229 235 states that certain variants of the relaxation method for solving systems of linear inequalities generate bounded sequences of intermediate solutions even when applied to infeasible systems using a new approach we prove a more general version of this result and answer an old open problem of quantifying the bound as a function of the input data

on fusion algebras and modular matrices we consider the fusion algebras arising in e g wess zumino witten conformal field theories affine kac moody algebras at positive integer level and quantum groups at roots of unity using properties of the modular matrix s we find small sets of primary fields equivalently sets of highest weights which can be identified with the variables of a polynomial realization of the ar fusion algebra at level k we prove that for many choices of rank r and level k the number of these variables is the minimum possible and we conjecture that it is in fact minimal for most r and k we also find new systematic sources of zeros in the modular matrix s in addition we obtain a formula relating the entries of s at fixed points to entries of s at smaller ranks and levels finally we identify the number fields generated over the rationals by the entries of s and by the fusion verlinde eigenvalues

generating functions and elementary young tableaux when constructing polynomial bases for group representations one is led naturally to consider a stretched product of states elementary multiplets are states that cannot be expressed as the stretched product of any other two states and they can be identified with terms occurring in certain generating functions in this paper the authors define an analogous stretched product of young tableaux together with elementary tableaux these may be used in a manner similar to elementary multiplets to construct generating functions they consider several examples where this method can be applied and in particular give new large n rank free branching rule generating functions for the group subgroup pairs su n su m contained in implied by su n m first four labels non zero so n contained in implied by su n first three labels non zero and sp 2n contained in implied by su 2n first five labels non zero this method of finding these generating functions uses the theory of grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux

global dynamics for switching systems and their extensions by linear differential equations abstract switching systems use piecewise constant nonlinearities to model gene regulatory networks this choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of morse graphs associated to nodes of a parameter graph the parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical morse graphs however there are many cellular processes that do not exhibit threshold like behavior and thus are not well described by a switching system we consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations we show that the parameter graphs associated to the switching system and any of its extensions are identical for each parameter graph node there is an order preserving map from the morse graph of the switching system to the morse graph of any of its extensions we provide counterexamples that show why possible stronger relationships between the morse graphs are not valid

do projections stay close together abstract we estimate the rate of convergence of products of projections on k intersecting lines in r d more generally consider the orbit of a point under any sequence of orthogonal projections on k arbitrary lines in r d assume that the sum of the squares of the distances of the consecutive iterates is less than e we show that if e tends to zero then the diameter of the orbit tends to zero uniformly for all families l of a fixed number k of lines we relate this result to questions concerning convergence of products of projections on finite families of closed subspaces of l 2

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

dynamics of actively regulated gene networks abstract a popular way of modelling gene regulatory networks is to design the models as if the genes were acting directly on each other genes are activated or inhibited by transcription factors which are direct gene products the action of a transcription factor on a gene is modelled as a binary on off dose response function or a sigmoid around a certain threshold concentration a transcription factor may act on several genes with equal or different thresholds the combined effect of several transcription factors on a gene is generally assumed to obey boolean like composition rules a mathematical challenge related to such network models is to analyse the behaviour in the narrow domains in phase space where at least one variable is close to one of its thresholds called switching domains in the present paper we analyse the motion in switching domains for models with steep sigmoidal response functions under the assumption that each transcription factor only regulates one gene at each of its thresholds this biologically reasonable assumption allows us to establish rules that determine the passage through any switching domain and the sequence of domains through which the system passes the rules can be effectively implemented in a sound automated analyser for such networks the assumption simplifies certain mathematical and computational issues but unfortunately may be at variance with one of the necessary conditions for the singular perturbation theory which our derivations are based on we give sufficient conditions ensuring that despite this the difficulties raised by our assumption can be evaded and singular perturbation approximation safely applied

branching rules for e8 so16 branching rules for e8 down arrow so16 are derived from those of e8 down arrow su2 e7 down arrow su8 and so16 down arrow u1 su8 by noting that both group chains share a common u1 su8 subgroup additional branching rules are deduced from kronecker products of e8 and so16 irreps in each case the rules distinguish unambiguously between conjugate irreps of so16 an alternative labelling scheme for the irreps of so2k based on its maximal u1 suk subgroup is outlined

bifurcation analysis of a mathematical model for genetic regulatory network with time delays in this paper we aim to investigate the dynamics of a gene regulatory network which is a time delayed version of the model proposed by elowitz and leibler nature 403 2000 335 338 based on the normal form theory and center manifold reduction hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated we also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network it shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation more precisely i the amplitude increases with small time delay while the change of amplitude is not sensitive to relatively large time delay however the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter where amplitude always increases quickly and linearly with the transcriptional rate ii the period of oscillation increases as the time delay increases but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value which implies that the robustness of period to the transcriptional rate variations occurs our numerical simulations also support the theoretical conclusions namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks

successive projections on hyperplanes abstract any sequence of points in r n obtained by successive projections of a point on elements of a finite set of hyperplanes is bounded

steered sequential projections for the inconsistent convex feasibility problem we study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step sizes this algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the bregman distances to the members of the family of convex sets the resulting algorithm is a new steered sequential bregman projection method

a generalization of the littlewood richardson rule for the general linear group gz the littlewood richardson rule see 18 193 gives a method to calculate the decomposition of a tensor product of two irreducible g representations the aim of this article is to give a generalization of this rule for all simple simply connected algebraic groups of type a b c d g2 and e we obtain also partial results for g of type f4 e and es the restrictions in the last three cases come from the fact that for the formulation of the decomposition rules we need the notion of a standard young tableau such a notion has been developed by seshadri lakshmibai musili and rajeswari in a series of articles see ll 13 14 16 but not yet for all representations of the last three exceptional groups the advantage of the notion of a young tableau developed by seshadri et al is that it is independent of the type of the group for the convenience of the reader not used to this notion we give first a seperate proof for g sz using the classical notion of a standard young tableau of course for applications the classical notion is much more appropriate in the appendix we give a translation of the notion of a standard young tableau in the sense of seshadri et al into the classical notion of a young tableau for g sp2 and spin in 3 8 we give also such a translation for g g for the other exceptional groups such a translation is not known to the author let p be a dominant weight for g and denote by i the corresponding simple g module we assume that the character of vp is given by ct eycr where the sum has to be taken over all standard young tableaux f of shape p p and v 5 is the weight of the tableau y now by the

periodic solutions of gene networks with steep sigmoidal regulatory functions abstract we address the question of existence and stability of periodic solutions in gene regulatory networks the threshold dependent network dynamics divides the phase space into domains and a qualitative description can be derived specifying which transitions between domains can occur any periodic solution must follow a cyclic sequence of domains but the problem of determining when such a cyclic sequence of domains contains a periodic solution and when it is stable has not been completely resolved though results have been obtained before for restricted classes of networks here we develop a method by which existence or non existence of such solutions can be demonstrated analytically in any given example of a general class of gene networks with steep sigmoidal interactions under the assumption that any gene product that regulates multiple other genes does so at distinct thresholds our method determines qualitative stability but we also give a procedure that where applicable allows determination of quantitative stability of a periodic solution this complements the previous development of a local analysis method for this class of systems which allows computation of trajectories through any sequence of domains together the previous and current work form the basis for rigorous computer aided assessment of qualitative dynamics of a very general class of gene network models the ability to handle periodic solutions will also increase the applicability of such a computational tool to the design of synthetic networks

analysis and generic properties of gene regulatory networks with graded response functions the mass of new genomic data has lead to a growing interest in gene regulatory models describing the regulatory aspects of gene activity in these models the rates of change of gene product concentrations are expressed as a sum of regulatory switches in terms of sums of products of sigmoid functions mimicking complex logical on off functions turning gene activity on and off the customary way to analyse such equations is to replace the sigmoid functions by step functions and disregard models with autoregulation this leads to discontinuous equations of motion but in models without autoregulation continuous solutions can easily be obtained with effective autoregulation this simple approach breaks down as real regulators have finite gain and autoregulation is ubiquitous in biological systems we propose a generalised gene regulatory model framework admitting autoregulation and graded sigmoid functions with different steepnesses using singular perturbation analysis methods we derive a set of reformulated continuous equations for the limit solution when the sigmoids approach step functions and show that the solution for steep sigmoids approaches this limit solution uniformly in a finite time interval a conspicuous feature of the limit solution is so called sliding motionduring which the solution slides along a threshold hyperplane or the intersection of such hyperplanes a particular mapping leads to a dual picture of the flow where the fast parts are magnified at the expense of the slow parts combining the two dual pictures a simple and powerful method to analyse the flow is obtained we show that for steep sigmoid functions the flow may be highly sensitive to the relative steepnesses of the sigmoids this has important consequences when stochastic effects are taken into account also it implies that the observation by glass and

analysis of continuous time switching networks abstract models of a number of biological systems including gene regulation and neural networks can be formulated as switching networks in which the interactions between the variables depend strongly on thresholds an idealized class of such networks in which the switching takes the form of heaviside step functions but variables still change continuously in time has been proposed as a useful simplification to gain analytic insight these networks called here glass networks after their originator are simple enough mathematically to allow significant analysis without restricting the range of dynamics found in analogous smooth systems a number of results have been obtained before particularly regarding existence and stability of periodic orbits in such networks but important cases were not considered here we present a coherent method of analysis that summarizes previous work and fills in some of the gaps as well as including some new results furthermore we apply this analysis to a number of examples including surprising long and complex limit cycles involving sequences of hundreds of threshold transitions finally we show how the above methods can be extended to investigate aperiodic behaviour in specific networks though a complete analysis will have to await new results in matrix theory and symbolic dynamics

on character generators for simple lie algebras we study character generating functions character generators of simple lie algebras the expression due to patera and sharp derived from the weyl character formula is first reviewed a new general formula is then found it makes clear the distinct roles of outside and inside elements of the integrity basis and helps determine their quadratic incompatibilities we review analyse and extend the results obtained by gaskell using the demazure character formulae we find that the fundamental generalized poset graphs underlying the character generators can be deduced from such calculations these graphs introduced by baclawski and towber can be simplified for the purposes of constructing the character generator the generating functions can be written easily using the simplified graphs and associated demazure expressions the rank two algebras are treated in detail but we believe our results are indicative of those for general simple lie algebras

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

standard young tableaux and character generators of classical lie groups character generators are derived for each of the classical groups su k so 2k 1 sp 2k and so 2k in order to do this use is made of various generalisations of standard young tableaux one set of these is in one to one correspondence with the contributions to the characters of each irreducible representation and the other involving shifted young diagrams provides a means of enumerating those maximal chains in the fundamental posets of each group which serve to generate via their descent subchains all possible multichains the character generators are then written down in terms of a suitable labelling of the poset elements illustrative examples are given

character generators for unitary and symplectic groups combinatorial algorithms for computing the character generators of u n su n and sp 2n c are described these algorithms produce relatively compact nested expressions for the character generators moreover the terms appearing in these expressions all have positive coefficients this feature is not shared by the expression for the character generator which uses the weyl character formula

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

on an approach for computing the generating functions of the characters of simple lie algebras we describe a general approach to obtain the generating functions of the characters of simple lie algebras which is based on the theory of the quantum trigonometric calogero sutherland model we show how the method works in practice by means of a few examples involving some low rank classical algebras