localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

bound states of a one band model for 3d periodic medium we consider the bound states generated by impurities in a periodic three dimensional medium the computations of eigenstates and eigenfunctions for a one band model of the medium are carried out using numerical evaluation of the green s function for bound eigenmode representations we achieve high accuracy in the computation of the eigenmodes and eigenstates for a variety of defects for appropriately adjusted models these computations can be useful for numerical estimates of the minimal configuration of defects capable of introducing a bound state in a gap as well as estimating the rate of the exponential decay of the bound state

pendulum elliptic functions and relative cohomology classes revisiting canonical integration of the classical pendulum around its unstable equilibrium normal hyperbolic canonical coordinates are constructed and an identity between elliptic functions is found whose proof can be based on symplectic geometry and global relative cohomology alternatively it can be reduced to a well known identity between elliptic functions normal canonical action angle variables are also constructed around the stable equilibrium and a corresponding identity is exhibited

the best test of exponentiality against singly truncated normal alternatives abstract we show that the likelihood ratio test of exponentiality against singly truncated normal alternatives is the uniformly most powerful unbiased test and can be expressed in terms of the sampling coefficient of variation this test is closely related to greenwood s statistic for testing departures from the uniform distribution we provide a way to approximate the critical points of the test using saddlepoint methods that gives a high degree of accuracy

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

testing hypotheses about the shape parameter of a gamma distribution the purpose of this expository article is to provide greater access to critical values for testing hypotheses about the shape parameter of a gamma distribution among the possible tests we consider tests of exponentiality against gamma increasing failure rate and or decreasing failure rate alternatives extensive tables of critical values are given for individual sample sizes from 2 to 30 and values of the shape parameter in half integer increments from i to 10 median unbiased estimators are also given for the shape parameter to avoid small sample complications with maximum likelihood methods power curves are given for testing exponentiality against gamma alternatives these curves illustrate certain advantages of this parametric test over nonparametric alternatives

a note on overdispersed exponential families abstract the issue of creating overdispersion in a given one parameter one dimensional exponential family by extending it to a two parameter exponential family with the same support is considered an easily verifiable sufficient condition for this is derived it is shown that a large class of families satisfy this condition and that this class includes efron s 1986 and lindsay s 1986 family as special cases this class is also closely related to jorgensen s 1988 exponential dispersion models ump unbiased tests for testing overdispersion are exhibited and it is shown that in this context cox s overdispersion test is a special case a graphical display is developed to select a family of the general class which should be used with a given set of data to overdisperse a given target one parameter family two real data illustrations are given kr

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

an efficient chebyshev lanczos method for obtaining eigensolutions of the schrodinger equation on a grid a grid method for obtaining eigensolutions of bound systems is presented in this the block lanczos method is applied to a cheby shev approximation of exp h where is the range of eigenvalues we are interested in with this choice a preferential convergence of the eigenvectors corresponding to low lying eigenvalues ofhis achieved the method is used to solve a variety of one two and three dimensional problems to apply the kinetic energy operator we use the fast sine transform instead of the fast fourier transform thus fullfilling a priori the box boundary conditions we further extend the chebyshev approximation to treat general functions of matrices thus allowing its application to cases for which no analytical expressions of the expansion coefficients are available

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

rigid motions action angles relative cohomology and polynomials with roots on the unit circle revisiting canonical integration of the classical solid near a hyperbolic or elliptic uniform rotation normal canonical coordinates p q are constructed so that the hamiltonian becomes a function normal form of x pq or of x p2 q2 the two cases are treated simultaneously distinguishing them respectively by a label a in terms of various power series with coefficients which are shown to be polynomials in a variable ra2 depending on the inertia moments the normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without reference to the construction of the normal coordinates via elliptic integrals unlike the derivation of the normal coordinates p q results and conjectures also emerge about the properties of the above polynomials and the location of their roots in particular a class of polynomials with all roots on the unit circle arises

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

a remarkable property of definite integrals we show that the integral of f u is the same as the integral of f x both with respect to x for a variety of functions u in this note f will denote any integrable function and all integral signs denote the cauchy principal value the formula oo 00 f u dx f x dx 00 oo where 2 u x 1 x goes back at least as far as cauchy and can be proved as follows from a graph of the relation 2 one sees that the solution for x has two branches 1 x y u iju2 4 oo t oo thus fo or00 r f u dx f u dx f u xi x du oo j0 oo but x x i 1 1 1 ju2 4 iu2 4 j thus for example exp x2 1 x2 dx exp x l x dx o 2e oo 1 r00 r2 w e x dx 2e2 oo 2e2 equations 1 2 clearly represent a useful property of the definite integral that deserves to be widely known what is more remarkable is that 1 remains true when 2 is replaced by v aj 3 u x2 y r j x s received june 21 1982 1980 mathematics subject classification primary 26a06 26a42 1983 american mathematical society 0025 5718 82 0000 0898 01 50 561 license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

canonical partition functions of hamiltonian systems and the stationary phase formula we show that there is a symplectic action of a torus associated to harmonic flows on the cotangent bundle of a semi simple lie algebra this allows to obtain a completely classical proof of the gallavotti marchioro formula by the method of the stationary phase

algebraic linearization of dynamics of calogero type for any coxeter group calogero moser systems can be generalized for any root system including the noncrystallographic cases the algebraic linearization of the generalized calogero moser systems and of their quadratic respectively quartic perturbations are discussed

invariant exponential models applied to reliability theory and survival analysis abstract for a two parameter exponential model with increasing failure rate ifr or decreasing failure rate dfr distributions necessary and sufficient conditions of the existence of a solution of the likelihood equations are given also all of the scale invariant two parameter statistical models closed by raising to a power and by exponential tilting are introduced the conditions of existence of a solution of the likelihood equations are studied for these invariant models and the models are applied to obtain some uniformly most powerful unbiased tests of exponentially against alternatives in these models