parameters for integrating periodic functions of several variables a number theoretical method for numerical integration of periodic functions of several variables was developed some years ago this paper presents lists of numerical parameters to be used in implementing that method the parameters define quadrature formulas for functions of 2 3 8 variables error bounds for those formulas are also tabulated the derivation of the parameters and error bounds is described

existence of heteroclinic orbits for systems satisfying monotonicity conditions we prove existence of heteroclinic orbits for systems of ordinary differential equations satisfying monotonicity conditions the proof is carried out by means of the implicit function theorem we apply these results to prove existence of travelling waves for the system describing a multicomponent plasma sustained by a laser beam

plasma produced by a laser in a medium with convection and free surface satisfying a hamiltonian jacobi equation a nonlinear diffusion equation is considered which models the temperature distribution in a laser sustained plasma subject to wind as certain parameters are small a singular perturbation problem arises and the method of matched asymptotic expansions is applied to approximate the solution an essential role in this problem is played by the plasma front this is a free surface separating the plasma and non plasma phases one of the main results is that we derive a 1st order nonlinear p d e of hamilton jacobi type which describes certain parts of the free surface in the stationary case this equation is analysed for various wind directions it appears that for the initial conditions for these parts of this free surface there are different possibilities depending on the wind direction we show further that the solution of the hamilton jacobi equation can contain singularities of corner type furthermore the effect of wind on the stability region of the stationary full plasma solution is analysed the method of analysis presented in this paper is not restricted to the cone like geometry or the specific form of the non linearity of the problem considered here but has potentially a much wider scope however the case under study in this paper is certainly representative for the effects that have to be taken into account

a quasirandom approach to integration in bayesian statistics on passe en revue la theorie existante et on introduit de nouveaux criteres d identification de suites quasi aleatoires on suggere des extensions des regles d integration de base

lattice rules by component scaling the authors introduce a theory of rectangular scaling of integer lattices this may be used to construct families of lattices they determine the relation between the zaremba index rho lambda of various members of the same family it appears that if one member of a family has a high index some of the other family members of higher order may have extraordinarily high indices they have applied a technique based on this theory to lists of good lattices available this has enabled the authors to construct lists of excellent previously unknown lattices of high order in three and four dimensions and of moderate order in five dimensions 10 refs 12 tabs

depth zero supercuspidal l packets and their stability in this paper we verify the local langlands correspondence for pure inner forms of unramied p adic groups and tame langlands parameters in general position for each such parameter we explicitly construct in a natural way a nite set l packet of depth zero supercuspidal representations of the appropriate p adic group and we verify some expected properties of this l packet in particular we prove with some conditions on the base eld that the appropriate sum of characters of the representations in our l packet is stable no proper subset of our l packets can form a stable combination our l packets are also consistent with the conjectures of b gross and d prasad on restriction from so2n 1 to so2n 24 these l packets are in general quite large for example sp2n has an l packet containing 2 n representations of which exactly two are generic in fact on a quasi split form eachl packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced bruhattits building when the group has connected center every depth zero generic supercuspidal representation appears in one of these l packets we emphasize that there is nothing new about the representations we construct they are induced from deligne lusztig representations on subgroups of nite

unrefined minimal k types for p adic groups via their restriction to compact open subgroups was begun by mautner shalika and tanaka for groups of type ai in contrast to real reductive groups where the representation theory of a maximal compact subgroup k is given in terms of slight modifications to cartan s theory of the highest weight the representation theory of a compact parahoric subgroup is quite complicated there is still no comprehensive theory for classifying the irreducible representa tions in the case of

intermediate rank lattice rules for multidimensional integration every lattice rule in s dimensions may be characterised by an integer m lying between 1 and s inclusive called the rank or index of the rule and a set of m positive integers called the invariants earlier work has shown that in a certain precise sense lattice rules of rank s are better than the commonly used rank 1 rules here this earlier work is extended by showing that a similar result holds for certain lattice rules of intermediate rank m s

some properties of rank 2 lattice rules a rank 2 lattice rule is a quadrature rule for the unit ital s dimensional hypercube of the form ital qf 1 ital n sub 1 ital n sub 2 summation sub ital j sub 1 1 ital j sub 2 1 sup ital n sub 1 summation sub ital j sub 2 1 sup ital n sub 2 ital bar f ital j sub 1 ital z1 ital n sub 1 ital j sub 2 ital z2 ital n sub 2 which cannot be re expressed in an analogous form with a single sum here ital bar f is a periodic extension of ital f and ital z sub 1 ital z sub 2 are integer vectors in this paper we discuss these rules in detail in particular we categorize a special subclass whose leading one and two dimensional projections contain the maximum feasible number of abscissas we show that rules of this subclass can be expressed uniquely in a simple tricycle form

the existence of efficient lattice rules for multidimensional numerical integration in this contribution to the theory of lattice rules for multidimensional numerical integration we first establish bounds for various efficiency measures which lead to the conclusion that in the search for efficient lattice rules one should concentrate on lattice rules with large first invariant then we prove an existence theorem for efficient lattice rules of rank 2 with prescribed invariants which extends an earlier result of the author for lattice rules of rank 1

stationary measures and invariant subsets of homogeneous spaces ii let g be a real lie group be a lattice in g and be a compactly generated closed subgroup of g if the zariski closure of the group ad is semisimple with no compact factor we prove that every orbit closure in g is a nite volume homogeneous space we also establish related equidistribution properties

supercuspidal l packets of positive depth and twisted coxeter elements the local langlands correspondence is a conjectural connection between representations of groups g k for connected reductive groups g over a padic field k and certain homomorphisms langlands parameters from the galois or weil deligne group of k into a complex lie group g which is dual in a certain sense to g and which encodes the splitting structure of g over k more introductory remarks on the local langlands correspondence can be found in 21 wheng gl1 this correspondence should reduce to local abelian class field theory for g gln the langlands correspondence is uniquely determined by local factors 24 and was shown to exist in 23 and 25 so far this correspondence is not completely explicit but much progress has been made in this direction see 9 10 for example for groups other than gln or pgln the theory is much less advanced new phenomena appear arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of langlands parameters this means that a single langlands parameter should determine not just one but a finite set of representations these are the l packets of the title however since local factors have not been defined in general there is no precise characterization of an l packet for general groups one can at present only hope to define finite sets of representations attached to langlands parameters and show that they have properties expected or perhaps unexpected of l packets see 14 chap 3 for some of these properties one is thereby proposing a definition of local factors for the representations in the sets cf 4 chap 3 this paper is a sequel to 14 the aim of both papers is to verify in an explicit and natural way the local langlands correspondence for the simplest kinds of non abelian extensions of k and the simplest kinds of

a lower bound for rank 2 lattice rules we give a lower bound for a quality measure of rank 2 lattice rules which shows that an existence result of niederreiter is essentially best possible

deligne s topological central extension is universal abstract we give a purely local proof of the fact that the topological central extension of g k g an absolutely almost simple algebraic group defined and isotropic over a nonarchimedean local field k by the finite group k of roots of unity in k constructed by pierre deligne is a universal topological central extension of g k

on computing the lattice rule criterion r lattice rules are integration rules for approximating integrals of periodic functions over the s dimensional unit cube one criterion for measuring the goodness of lattice rules is the quantity r this quantity is defined as a sum which contains about n sup s 1 terms where n is the number of quadrature points although various bounds involving r are known a procedure for calculating r itself does not appear to have been given previously here we show how an asymptotic series can be used to obtain an accurate approximation to r whereas an efficient direct calculation of r requires o nn sub 1 operations where n sub 1 is the largest invariant of the rule the use of this asymptotic expansion allows the operation count to be reduced to o n a complete error analysis for the asymptotic expansion is given the results of some calculations of r are also given 19 refs 2 tabs

a singularly perturbed free boundary problem describing a laser sustained plasma for a nonlinear diffusion equation with nonlocal terms which models a laser sustained lasma solutions are analyzed by the method of matched asymptotic expansions several stationary solutions are found and their stability is discussed as a function of the laser intensity at a certain critical intensity bifurcation of a stable stationary plasma into a pulsating plasma takes place the pulsation process is of an unusual type showing repeated formation of travelling plasma shells which die out later on

a useful result in searches for good vertex modified number theoretic rules number theoretic rules are particularly suited to the evaluation of multiple integrals in which the integrand is periodic for nonperiodic integrands an alternative is to use vertex modified versions of number theoretic rules good vertex modified number theoretic rules may be found by doing a computer search based on some criterion of goodness such criteria include a variant of the l2 discrepancy and the vertex variance here we present a result which may be used to speed up searches for good vertex modified number theoretic rules based on these criteria when the generator vectors are of the one parameter form of korobov

on the congruence subgroup problem determination of the metaplectic kernel let f be a global field i e either a number field or a function field in one variable over a finite field and let oo be the set of its archimedean places let g be a finite non empty set of places of f containing oo let o o denote the ring of g integers of f let a denote the ring of g adeles i e the restricted direct product of the completions f v for r eg let f be an absolutely simple simply connected subgroup of sl defined over f recall that a subgroup f of c f is an g arithmetic subgroup if fnsl n o has finite index in f as well as in o fy f c sl n o an g arithmetic subgroup f is a congruence subgroup if there exists a non zero ideal a in o o g such that

lattice methods for multiple integration theory error analysis and examples this paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension the concept of a lattice rule is introduced and error bounds are developed in terms of the dual lattice examples of lattice rules discussed in the paper are the rectangle rule the number theoretic good lattice methods of korobov and others the body centred cubic rule and a generalization of the latter to a family w nr the w nr rules appear to have interesting extrapolation properties both for fixed r and for r n

variance reduction via lattice rules this is a review article on lattice methods for multiple integration over the unit hypercube with a variance reduction viewpoint it also contains some new results and ideas the aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science these models can usually be reformulated as integration problems over the unit hypercube with a large sometimes infinite number of dimensions we examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low dimensional subspaces the criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators numerical examples illustrate the effectiveness of the approach

a computer search of rank 2 lattice rules for multidimensional quadrature for certain lattice rules of rank 2 it has been shown that a unique representation exists in a form suitable for computer evaluation the present paper describes computer searches of such rules reports results and identifies rules that appear promising for the numerical evaluation of practical multidimensional integrals

lattice rules for multiple integration and discrepancy upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established in this way the applicability of lattice rules is extended to nonperiodic integrands

the representation of lattice quadrature rules as multiple sums we provide a classification of lattice rules applying elementary group theory we assign to each s dimensional lattice rule a rank m and a set of positive integer invariants n sub 1 n sub 2 n sub s the number nu q of abscissas required by the rule is the product n sub 1 n sub 2 n sub s and the rule may be expressed in a canonical form with m independent summations under this classification an n point number theoretic rule in the sense of korobov and conroy is a rank m 1 rule having invariants n 1 1 1 and the product trapezoidal rule using n sup s points in rank m s rule having invariants n n n besides providing a canonical form we give some of the properties of copy rules and of projections into lower dimensions

an application of diophantine approximation to the construction of rank 1 lattice quadrature rules lattice quadrature rules were introduced by frolov 1977 sloan 1985 and sloan and kachoyan 1987 they are quasi monte carlo rules for the approximation of integrals over the unit cube in r s and are generalizations of number theoretic rules introduced by korobov 1959 and hlawka 1962 themselves generalizations in a sense of rectangle rules for approximating one dimensional integrals and trapezoidal rules for periodic integrands error bounds for rank 1 rules are known for a variety of classes of integrands for periodic integrands with unit period in each variable these bounds are conveniently characterized by the figure of merit p which was originally introduced in the context of number theoretic rules the problem of finding good rules of order n that is having n nodes then becomes that of finding rules with large values of p this paper presents a new approach based on the theory of simultaneous diophantine approximation which uses a generalized continued fraction algorithm to construct rank 1 rules of high order

superrigidity theorems in positive characteristic abstract we generalize the superrigidity theorem of g a margulis 1991 discrete subgroups of semisimple lie groups vii 5 19 springer verlag new york berlin to the case of the more general non reductive target group f we also provide a proof of the cocycle superrigidity theorem in the case of algebraic groups defined over a local field of positive characteristic and the generalization of this proof to the case of the more general target group analogous to our generalization of the superrigidity theorem