an asymptotic solution of the first passage problem for singular diffusion in population biology the first passage problem is studied for a singular diffusion process arising in population biology with the deterministic part having a stable equilibrium point and small diffusion the laplace transform of the first passage time density satisfies an ordinary differential equation in a finite interval with two point boundary conditions a formal asymptotic solution to this problem is constructed when the singular boundary point is absorbing and the regular boundary point is either absorbing or reflecting the solution depends upon the initial value z of the process for z near the equilibrium point the density is approximated by the ornstein uhlenbeck equation for z near the singular boundary point it is approximated by the diffusion limit of the linear birth death process and elsewhere it is approximated by the wkb solution these different solutions are then connected together using these results the process is shown to exit with probability one in the limit of small diffusion the mean and varianc

singular boundary resonance with turning point resonance the resonance criteria of the singularly perturbed equation frac 1 2 varepsilon xp x varepsilon u xx x a x q x varepsilon u x su are investigated on 0 x b with 0 a b and 0 varepsilon ll 1 the matched uniform asymptotic approximations developed by the authors in a companion paper for a singular point and turning point respectively are used to calculate asymptotic approximations for the resonance criteria and the associated eigenvalue problem to exponential precision the eigenvalues can be identified as being dominated by the pure singular point problem the pure turning point problem or by combinations of the two problems an asymptotic exponential precision notation is introduced to succinctly express the resonance criteria given to leading order in varepsilon

on the canonical solution of indefinite problem with m turning points of even order abstract we consider the differential equation y 2 2 x q x y 0 on a finite interval i say i 0 1 where i contains m turning points that is here zeros of under the assumption that one of the turning points is of odd order while the others are of even order using of the asymptotic estimates provided in w eberhard g freiling a schneider connection formulae for second order differential equations with a complex parameter and having an arbitrary number of turning points math nachr 165 1994 205 229 for a special fundamental system of solutions of in i we study the infinite product representation of solutions of such representations are relevant to the inverse spectral problem

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

uniform solution of boundary layer problems exhibiting resonance a uniformly valid asymptotic solution for boundary value problems exhibiting resonance in the sense of ackerberg and o malley stud appl math 49 1970 pp 277 295 is derived in this solution the location of the right endpoint is one of the variables thus it is possible to see how the solution suddenly shifts from one with a boundary layer on the left to one with a boundary layer on the right when a certain integral changes sign during such a transition there are boundary layers at both ends

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

q functions and extreme topological measures abstract q functions provide a method for constructing topological measures we give necessary and sufficient conditions for a composition of a q function and a topological measure to be a topological measure regular and extreme step q functions are characterized by certain regions in r n then extreme q functions are used to study extreme topological measures for example we prove under some assumptions on the underlying set that given n there are different types of extreme topological measures with values 0 1 n 1 in contrast in the case of measures the only extreme points are 0 1 valued i e point masses

quasi measures with image transformations as generalized variables we introduce an image transformation as a generalization of measurable maps structure properties of the image transformation is given unique extension of image transformations from solid sets is proven the multidimensional median and sample median are presented as image transformations providing a construction suitable for probability theorists

the gelfand cetlin system and quantization of the complex flag manifolds the construction of angle action variables for collective completely integrable systems is described and the associated bohr sommerfeld sets are determined the quantization method of sniatycki applied to such systems gives formulas for multiplicities for the gelfand cetlin system on complex flag manifolds we show that these formulas give the correct answers for the multiplicities of the associated representations

construction and properties of quasi linear functionals quasi linear functionals are shown to be uniformly continuous and decomposable into a difference of two quasi integrals a predual space for the quasi linear functionals inducing the weak topology is given general constructions of quasi linear functionals by solid set functions and q functions are given

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

two turning points of second order a singular perturbation problem for a second order ordinary differential equation ode with two interior turning points of second order is considered two uniform asymptotic approximations are separately derived around the two turning points the leading and first order exponential and nonexponential terms of both approximations are matched asymptotically in an overlap domain of validity between the two turning points the resonance conditions are calculated and three sets of eigenvalues are derived to leading order related to each turning point and to the combined influence of the turning points

the quantum dynamics of the compactified trigonometric ruijsenaars schneider model we quantize a compactified version of the trigonometric ruijsenaars schneider particle model with a phase space that is symplectomorphic to the complex projective space n the quantum hamiltonian is realized as a discrete difference operator acting in a finite dimensional hilbert space of complex functions with support in a finite uniform lattice over a convex polytope viz a restricted weyl alcove with walls having a thickness proportional to the coupling parameter we solve the corresponding finite dimensional bispectral eigenvalue problem in terms of discretized macdonald polynomials with q and t on the unit circle the normalization of the wave functions is determined using a terminating version of a recent summation formula due to aomoto ito and macdonald the resulting eigenfunction transform determines a discrete fourier type involution in the hilbert space of lattice functions this is in correspondence with ruijsenaars observation that at the classical level the action angle transformation defines an anti symplectic involution of n from the perspective of algebraic combinatorics our results give rise to a novel system of bilinear summation identities for the macdonald symmetric functions

supersymmetric ruijsenaars schneider model an integrable supersymmetric generalization of the trigonometric ruijsenaars schneider model is presented whose symmetry algebra includes the super poincar e algebra moreover its hamiltonian is shown to be diagonalized by the recently introduced macdonald superpolynomials somewhat surprisingly the consistency of the scalar product forces the discreteness of the hilbert space

singularly perturbed convection diffusion problems with boundary and weak interior layers in this paper a singularly perturbed convection diffusion equation with a discontinuous source term is examined boundary and weak interior layers appear in the solution a numerical method is constructed for this problem which involves an appropriate piecewise uniform mesh the method is shown to be uniformly convergent with respect to the singular perturbation parameter

a class of exactly solvable eigenvalue problems the class of differential equation eigenvalue problems y x x2n 2y x xn ey x n 1 0 1 2 3 on the interval x can be solved in closed form for all the eigenvalues e and the corresponding eigenfunctions y x the eigenvalues are all integers and the eigenfunctions are all confluent hypergeometric functions the eigenfunctions can be rewritten as products of polynomials and functions that decay exponentially as x for odd n the polynomials that are obtained in this way are new and interesting classes of orthogonal polynomials for example when n 1 the eigenfunctions are orthogonal polynomials in x3 multiplying airy functions of x2 the properties of the polynomials for all n are described in detail

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

eight perspectives on the exponentially ill conditioned equation varepsilon y x y y 0 boundary value problems involving the linear differential equation varepsilon y x y y 0 have surprising properties as varepsilon to 0 we examine this equation from eight points of v

matched uniform approximations for a singular boundary point and an interior turning point the singular perturbation problem for a second order ode with singular boundary point and interior turning point is studied two uniform asymptotic approximations are separately constructed around singular point and turning point the restricted uniform approximation about the turning point corresponds to o malley s uniform reduction while that about the singular point is found by a similar method and is entirely new higher order matching of both dominant and recessive terms of the two approximations leads to a complete asymptotic approximation of the general solution a subsequent paper applies these results to higher order resonance criteria

continuous and numerical analysis of a multiple boundary turning point problem a singularly perturbed boundary value problem with a multiple turning point at a boundary is considered a representation of the solution is given and it is used in the construction of a uniform finite difference scheme the scheme is a first order exponentially fitted one an improved modification on a special discretization mesh is given

quasi states and quasi measures abstract it is shown that in general non linear quasi states will exist on c x where x is a compact hausdorff space we introduce the concept of a quasi measure in x and establish the connection between quasi states and quasi measures

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

on a singular perturbation problem with two second order turning points in this paper we study the singular perturbation problem ep x uxx 2 x a b x q x ux 2su 0 0 x 1 0 a b 1 where 0 el1 is a small positive parameter p x and q x are sufficiently smooth and strictly positive functions the main feature of this equation is that there are two second order turning points in the interval 0 1 based on the rigorous results on singular perturbation problems with one second order turning point in our previous work we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique

on the duality between the hyperbolic sutherland and the rational ruijsenaars schneider models we consider two families of commuting hamiltonians on the cotangent bundle of the group and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic sutherland and of the rational ruijsenaars schneider lax matrices respectively the duality symplectomorphism between these two integrable models that was constructed by ruijsenaars using direct methods can then be interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group

a noncommutative version of the bispectral problem we consider a matrix valued version of the bispectral problem involving a block tridiagonal doubly infinite matrix and a first order differential operator with matrix coefficients we give a set of necessary conditions that the coefficients need to satisfy and solve these equations under a variety of conditions the situations discussed here should make it plain that while the corresponding problem in the scalar case is relatively trivial and devoid of any interest the noncommutative version of the problem is much richer and subtle the results here should be useful for instance in the study of a noncommutative version of the nonlinear toda lattice

another look at ackerberg o malley resonance ackerberg o malley resonance in the singularly perturbed turning point problem is considered a variational approach suggested by grasman and matkowsky is exploited to explain previously overlooked resonance phenomena several examples are presented

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

asymptotic behaviour of solutions near a turning point the example of the brusselator equation abstract the brusselator equation is an example of a singularly perturbed differential equation with an additional parameter it has two turning points at x 0 and x 1 we study some properties of so called canard solutions that remain bounded in a full neighbourhood of 0 and in the largest possible domain the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions for this purpose we need a study of behaviour of the solutions near a turning point here we prove that for a large class of equations if 0 is a turning point of order p any solution y not exponentially large has in some sector centred at 0 an asymptotic behaviour when e 0 of the form y n x e e n where e p 1 e for x e x with x large enough but independent of e in the brusselator case we moreover compute a stokes constant for a particular nonlinear differential equation

the gevrey asymptotics in the case of singular perturbations abstract in this paper we present results in the gevrey asymptotics which correspond to some existing results concerning asymptotic solutions in the poincare asymptotics of singularly perturbed ordinary differential equations the main idea is based on a characterization of the gevrey flat functions and a characterizaton of the gevrey asymptotic expansions as it had been already exhibited by y sibuya 1990 lecture notes in pure and applied math vol 124 pp 393 401 dekker new york and p f hsieh and y sibuya 1999 basic theory of ordinary differential equations chap xi sect xi 2 and chap xii sect xii 5 springer verlag berlin

on an internal boundary layer problem in this paper we consider the boundary value problem ey a x y b x y 0 x x x x 0 x y x a y x b where a and b are two prescribed constants and 0 e 1 is a small positive parameter as x 0 it is assumed that a x x and b x where 0 and 1 2 3 under certain smoothness conditions on a x and b x an asymptotic solution is constructed which holds uniformly for x x x this result is proved rigorously by using the method of successive approximation

boundary layer analysis of nonlinear reaction diffusion equations in a polygonal domain abstract we propose a boundary layer analysis which fits a domain with corners in particular we consider nonlinear reaction diffusion problems posed in a polygonal domain having a small diffusive coefficient e 0 we present the full analysis of the singular behaviours at any orders with respect to the parameter e where we use a systematic nonlinear treatment initiated in jung et al 2016 the boundary layers are formed near the polygonal boundaries and two adjacent ones overlap at a corner p and the overlapping produces additional layers the so called corner layers it is noteworthy that the boundary layers are also degenerate due to the singularities of the solutions involving a negative power of the radial distance to the corner p which are present in the laplace operator on a sector sector corresponding to the part of the polygon near the corner the corner layers are then designed to absorb both the singularities and the interaction of the two boundary layers at p

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1

stokes phenomenon and matched asymptotic expansions this paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting stokes phenomenon in particular the approach highlights the way in which the local structure and the possibility of finding stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

boundary layers for the subcritical modes of the 3d primitive equations in a cube abstract in this article we study the boundary layers for the subcritical modes of the viscous linearized primitive equations lpes in a cube at small viscosity the boundary layers include the parabolic boundary layers ordinary boundary layers and their interaction corner layers the boundary layer correctors are determined by a phenomenological study reminiscent of the prandtl corrector approach and then a rigorous convergence result is proved which a posteriori justifies the phenomenological study

trigonometric sutherland systems and their ruijsenaars duals from symplectic reduction beside its usual interpretation as a system of n indistinguishable particles moving on the circle the trigonometric sutherland system can be viewed alternatively as a system of distinguishable particles on the circle or on the line and these three physically distinct systems are in duality with corresponding variants of the rational ruijsenaars schneider system we explain that the three duality relations first obtained by ruijsenaars in 1995 arise naturally from the kazhdan kostant sternberg symplectic reductions of the cotangent bundles of the group u n and its covering groups u 1 su n and r su n respectively this geometric interpretation enhances our understanding of the duality relations and simplifies ruijsenaars original direct arguments that led to their discovery

a technique for the numerical verification of asymptotic expansions a technique is presented that uses numerical solutions to verify the order of the accuracy of an asymptotic expansion for several types of problems this technique may be introduced in any beginning course on asymptotic or perturbation methods but is equally suited to verifying advanced asymptotic results

orthogonality of macdonald polynomials with unitary parameters for any admissible pair of irreducible reduced crystallographic root systems we present discrete orthogonality relations for a finite dimensional system of macdonald polynomials with parameters on the unit circle subject to a truncation relation

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

asymptotic analysis for singularly perturbed convection diffusion equations with a turning point turning points occur in many circumstances in fluid mechanics when the viscosity is small very complex phenomena can occur near turning points which are not yet well understood a model problem corresponding to a linear convection diffusion equation e g suitable linearization of the navier stokes or benard convection equations is considered our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the navier stokes or benard convection equations of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems in our case at this stage the full theoretical asymptotic analysis is provided this study is totally new to the best of our knowledge numerical treatment and more complex problems will be considered elsewhere

gevrey properties of real planar singularly perturbed systems abstract by applying geometric techniques to real analytic singularly perturbed vector fields on the plane we develop a way to give a bound on the gevrey type of the taylor development of canard manifolds at degenerate planar turning points by blowing up the phase space at the turning point we find asymptotic estimates even when such expansions w r t traditional phase space variables do not exist the asymptotic estimates are then used to give a sufficient and necessary condition on the existence of local canard solutions

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

singularly perturbed linear two point boundary value problems finding asymptotic solutions to two point boundary value problems for linear singularly perturbed second order ordinary differential equations is now quite well understood troublesome exceptions involving turning points known as problems of boundary layer resonance have been studied by many experts since the initial work of ackerberg and o malley stud appl math 49 1970 pp 277 295 this significant work is surveyed here closely related literature involves the canard phenomena new generally more geometric approaches including gevrey asymptotics and blowup promise improved understanding of such sensitive asymptotics and their generalizations

singular perturbation analysis of boundary value problems for differential difference equations iii turning point problems this paper continues a study of a class of boundary value problems for linear second order differential difference equations in which the second order derivative is multiplied by a small parameter siam j appl math 42 1982 pp 502 531 45 1985 pp 687 707 the previous papers focused on problems involving boundary and interior layer phenomena rapid oscillations and resonance behavior the problems studied here have solutions which exhibit turning point behavior i e transition regions between rapid oscillations and exponential behavior the presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval a combination of exact solutions singular perturbation methods and numerical computations are used in these studies

isotopy invariant topological measures on closed orientable surfaces of higher genus given a closed orientable surface of genus at least two we establish an affine isomorphism between the convex compact set of isotopy invariant topological measures on and the convex compact set of additive functions on the set of isotopy classes of certain subsurfaces of we then construct such additive functions and thus isotopy invariant topological measures from probability measures on together with some additional data the map associating topological measures to probability measures is affine and continuous certain dirac measures map to simple topological measures while the topological measures due to py and rosenberg arise from the normalized euler characteristic

on the numerical approximations of stiff convection diffusion equations in a circle our aim in this article is to study the numerical solutions of singularly perturbed convection diffusion problems in a circular domain and provide as well approximation schemes error estimates and numerical simulations to resolve the oscillations of classical numerical solutions due to the stiffness of our problem we construct via boundary layer analysis the so called boundary layer elements which absorb the boundary layer singularities using a p 1 p 1 classical finite element space enriched with the boundary layer elements we obtain an accurate numerical scheme in a quasi uniform mesh

boundary layers for the 3d primitive equations in a cube the supercritical modes abstract in this article we study the boundary layers for the viscous linearized primitive equations lpes when the viscosity is small the lpes are considered here in a cube besides the usual boundary layers that we analyze here too corner layers due to the interaction between the different boundary layers are also studied