connection between orthogonal polynomials on the unit circle and bounded interval in this paper we establish the connection between measures on a bounded interval and on the unit circle by a transformation related with the szego classical transformation we transform a measure on the interval 1 1 into a measure on 0 2 in the same form in which the orthogonality measure of the chebyshev polynomials of fourth kind becomes the lebesgue measure we relate the sequences of orthogonal polynomials with respect to both measures and we also relate the coefficients of the three term recurrence relation with the schur parameters when the measures belong to the szego class we study the asymptotic behavior of the orthogonal polynomials on the interval outside the support of the measure as well as inside we also transform the generalized polynomials and we study the orthogonality properties of the new polynomials obtaining new interesting results and finally we solve two inverse problems connected with the transformation studied

calculation of the threshold surface for nerve equations in this paper calculations are presented for the threshold surface mathcal m in mckean s caricature for nerve conduction u t u xx u h u a and for the fitzhugh nagumo equation u t u xx u 1 u u a the previous construction of this surface gave no information about data on mathcal m or in particular the value of the amplitude which makes data critical this amplitude is computed by both a standard discretization of the partial differential equation and for mckean s equation by means of an asymptotic calculation for the integral equation satisfied by the free boundary m t defined by u m t t a on this numerical evidence it is conjectured that for a large class of initial data u 0 on mathcal m the value u 0 l 1 depends only ona

existence and uniqueness of traveling wave front of a nonlinear singularly perturbed system of reaction diffusion equations with a heaviside step function abstract the main purpose of this paper is to show the existence and uniqueness of a traveling wave front u w u x v e t w x v e t of the following nonlinear singularly perturbed system of reaction diffusion equations with a heaviside step function u t 2 u x 2 2 h u u w w t 2 w x 2 e u w

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

power law subordinacy and singular spectra i half line operators we study hausdorff dimensional spectral properties of certain whole line quasiperiodic discrete schrodinger operators by using the extension of the gilbert pearson subordinacy theory that we previously developed in 19

multiple impulse solutions to mckean s caricature of the nerve equation i existence mckean s caricature of the nerve equation is considered the h in 1 is the heaviside function we prove the existence of multiple impulse solutions consisting of any finite number of pulses these solutions are referred to as n ple impulse solutions where n is an arbitrary positive integer

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

traveling waves of calcium in a pancreatic acinar cells model construction and bifurcation analysis abstract we construct and study a model for intracellular calcium wave propagation with particular attention to pancreatic acinar cells the model is based on a model of the inositol trisphosphate ip3 receptor which assumes that calcium modulates the binding affinity of ip3 to the receptor two versions of the model one simpler than the other are studied numerically in both versions solitary waves in the excitable regime arise via homoclinic bifurcations in the traveling wave equations as the background concentration of ip3 is increased the wave speed increases and for some values of the ip3 concentration the initial pulse gives rise to secondary pulses that travel in both directions this can give rise to irregular spatio temporal behavior or to trains of pulses in the simpler model these secondary waves are related to the presence of a t point a heteroclinic cycle and an associated spiral of homoclinic orbits which terminate the branch of homoclinic orbits

the 2 blocks of the covering groups of the symmetric groups abstract let n be a double cover of the finite symmetric group s n of degree n i e n has a central involution z such that n z s n an irreducible character of n is called ordinary or spin according to whether it has z in its kernel or not the purpose of this paper is to determine the distribution of the spin characters of n into 2 blocks the methods applied here are essentially different from those applied to previous questions of this type we also discuss some consequences of our main result for the decomposition numbers an analogue of james well known result for the decomposition numbers of the symmetric groups is proved providing also a generalization of a theorem of benson ben theorem 1 2 in section 1 we present the background for our results and give some preliminaries in section 2 we give an explicit formula for the number of spin characters in a 2 block we also prove a result about the weight of a block containing a given non self associate spin character which will be important for the proof of our theorem on the 2 block distribution of spin characters section 3 presents some fundamental combinatorial concepts used in sections 4 and 5 the theorem concerning the spin characters in a given 2 block is proved in section 4 and in section 5 we present our results on the decomposition numbers

traveling wave solutions to a coupled system of spatially discrete nagumo equations we consider a coupled system of discrete nagumo equations and derive traveling wave solutions to this system using mckean s caricature of the cubic a certain form of this system is used to model ephaptic coupling between pairs of nerve axons we study the difference g c a 1 a 2 between the detuning parameters a i that is required to make both waves move at the same speed c of particular interest is the effect of a coupling parameter alpha and an alignment parameter a on the function g numerical investigation indicates that for fixed a there exists a time delay value beta that results in g 0 and for large enough wave speeds multiple such beta values exist also numerical results indicate that the perturbation of alpha away from zero will yield additional solutions with positive wave speed when a 1 over 2 we employ both analytical and numerical results to demonstrate our claims

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

multiple impulse solutions to mckean s caricature of the nerve equation ii stability we study mckean s caricature of a nerve conduction equation where h is the heaviside function it is proved that an n ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n 1 and at least n unstable modes exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to the spacings the n amplitude modes always exist we prove also that for an n ple impulse solution resembling the superposition of n stable solitary impulses there are at most n 1 unstable modes and all of them are of spacing type

weak almost periodic motions minimality and stability in impulsive semidynamical systems abstract in this paper we study topological properties of semidynamical systems whose continuous dynamics are interrupted by abrupt changes of state first we establish results which relate various concepts as stability of lyapunov weakly almost periodic motions recurrence and minimality in the sequel we study the stability of zhukovskij for impulsive systems and we obtain some results about uniform attractors

existence and stability of traveling pulse solutions of the fitzhugh nagumo equation abstract the fitzhugh nagumo model is a reaction diffusion equation describing the propagation of electrical signals in nerve axons and other biological tissues one of the model parameters is the ratio of two time scales which takes values between 0 001 and 0 1 in typical simulations of nerve axons based on the existence of a singular limit at 0 it has been shown that the fitzhugh nagumo equation admits a stable traveling pulse solution for sufficiently small 0 here we prove the existence of such a solution for 0 01 both for circular axons and axons of infinite length this is in many ways a completely different mathematical problem in particular it is non perturbative and requires new types of estimates some of these estimates are verified with the aid of a computer the methods developed in this paper should apply to many other problems involving homoclinic orbits including the fitzhugh nagumo equation for a wide range of other parameter values

homoclinic orbits in a parametrized saddle focus system abstract we consider a parametrized dynamical system with a saddle focus equilibrium point and which for one value of the parameter has a homoclinic orbit conditions on the eigenvalues for the equilibrium point together with transversality conditions imply the existence of an infinite discrete set of parameter values for which the system has a homoclinic orbit such systems arise in the study of nerve axon equations where a homoclinic orbit corresponds to a finite train of nerve impulses traveling at a velocity identified by the associated parameter value

evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations abstract the main purposes of this paper are to accomplish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations u t u x x h u u w w t e u w and to establish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation u t u x x h u u under different conditions on the model constants to establish the bifurcation for the system we will study the existence and instability of a standing pulse solution if 0 2 1 the existence and stability of two standing wave fronts if 2 1 and 2 e 1 the existence and instability of two standing wave fronts if 2 1 and 0 2 e 1 the existence and instability of an upside down standing pulse solution if 0 1 2 1 to establish the bifurcation for the scalar equation we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 2 the existence and stability of two standing wave fronts if 2 the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 2 by the way we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation u t u x x h u u w 0 where w 0 2 0 is a positive constant if 0 2 to achieve the main goals we will make complete use of the special structures of the model equations and we will construct evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators it turns out that a complex number 0 is an eigenvalue of the linear differential operator if and only if 0 is a zero of the evans function the stability instability and bifurcations of the nonlinear waves follow from the zeros of the evans functions a very important motivation to study the existence stability instability and bifurcations of the nonlinear waves is to study the existence and stability instability of infinitely many fast slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations the existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience

on cartan matrices and lower defect groups for covering groups of symmetric groups we determine the elementary divisors of the cartan matrices of spin p blocks of the covering groups of the symmetric groups when p is an odd prime as a consequence we also compute the determinants of these cartan matrices and in particular we confirm a conjecture by brundan and kleshchev that these determinants depend only on the weight but not on the sign of the block

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

existence and stability of multiple impulse solutions of a nerve equation a piecewise linear partial differential equation which models the propagation of a nerve impulse along an axon is studied for the equation parameters used there are two solutions which represent solitary impulses traveling with fixed form one with a slower velocity c s1 than the velocity c f1 of the other we show that near these velocities there exist multiple impulse solutions and exhibit four such solutions each of which has two pulses the local stability of these solutions is analyzed the two solutions whose velocities are near c s1 demonstrate instabilities which indicate growth or decay of the separate pulses for one of the multiple impulse solutions with velocity near c s1 and for one of those with velocity near c f1 the second pulse deomonstrates an instability which indicates a change in the spacing between the two pulses the other solution with velocity near c f1 is stable

the mckean s caricature of the fitzhugh nagumo model i the space clamped system within the context of lienard equations we present the fitzhugh nagumo model with an idealized nonlinearity we give an analytical expression i for the transient regime corresponding to the emission of a finite number of action potentials or spikes and ii for the asymptotic regime corresponding to the existence of a limit cycle we carry out a global analysis to study periodic solutions the existence of which is linked to the solutions of a system of transcendental equations the periodic solutions are obtained with the help of the harmonic balance method or as limit behavior of the transient regime we show how the appearance of periodic solutions corresponds either to a fold limit cycle bifurcation or to a hopf bifurcation at infinity the resultsobtained are in agreement with local analysis methods i e the melnikov method and the averaging method the generalization of the model leads us to formulate two conjectures concerning the number of limit cycles for the piecewise linear lienard equ

bifurcations of systems with structurally unstable homoclinic orbits and moduli of ω equivalence abstract bifurcations of both two dimensional diffeomorphisms with a homoclinic tangency and three dimensional flows with a homoclinic loop of an equilibrium state of saddle focus type are studied in one and two parameter families due to the well known impossibility of a complete study of such bifurcations the problem is restricted to the study of the bifurcations of the so called low round periodic orbits in this connection the idea of taking moduli continuous invariants of the topological conjugacy on the nonwandering set as the main control parameters together with the standard splitting parameter is proposed on this way new bifurcational effects are found which do not occur at a one parameter analysis in particular the density of cusp bifurcations is revealed

anisotropy propagation failure and wave speedup in traveling waves of discretizations of a nagumo pde this article is concerned with effect of spatial and temporal discretizations on traveling wave solutions to parabolic pdes nagumo type possessing piecewise linear bistable nonlinearities solution behavior is compared in terms of waveforms and in terms of the so called a c relationship where a is a parameter controlling the bistable nonlinearity by varying the potential energy difference of the two phases and c is the wave speed of the traveling wave uniform spatial discretizations and a stable linear multistep methods in time are considered results obtained show that although the traveling wave solutions to parabolic pdes are stationary for only one value of the parameter a a0 spatial discretization of these pdes produce traveling waves which are stationary for a nontrivial interval of a values which include a0 i e failure of the solution to propagate in the presence of a driving force this is true no matter how wide the interface is with respect to the discretization for temporal discretizations at large wave speeds the set of parameter a values for which there are traveling wave solutions is constrained an analysis of a complete discretization points out the potential for nonuniqueness in the a c relationship

half line schrodinger operators with no bound states acting in l2 0 c with the boundary condition r for convenience we require that the potential v be uniformly locally square integrable we write l l 2 for the banach space of such functions the free operators that is when v 0 can be diagonalized by the fourier transform this shows that they have spectra 2 2 and 0 co respectively and that in both cases the spectrum is purely absolutely continuous for a general discrete operator the mere fact that the spectrum is contained in 2 2 forces it to be purely absolutely continuous this is our first main result

commutation methods for jacobi operators abstract we offer two methods of inserting eigenvalues into spectral gaps of a given background jacobi operator the single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi bounded background jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps moreover we prove unitary equivalence of the commuted operators restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues with the original background operator in addition we compute the matrix valued weyl m functions of the commuted operator in terms of the background weyl m functions finally we show how to iterate the above methods and give explicit formulas for various quantities such as eigenfunctions and spectra of the iterated operators in terms of the corresponding background quantities and scattering matrix concrete applications include an explicit realization of the isospectral torus for algebro geometric finite gap jacobi operators and the n soliton solutions of the toda and kac van moerbeke lattice equations with respect to arbitrary background solutions

a strong szegő theorem for jacobi matrices we use a classical result of golinskii and ibragimov to prove an analog of the strong szeg theorem for jacobi matrices on l 2 mathbb n in particular we consider the class of jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that sum k n infty b k and sum k n infty a k 2 1 lie in l 2 1 the linearly weighted l 2 space