random walk approach to the analytic solution of random systems with multiplicative noise the anderson localization problem abstract we discuss here in detail a new analytical random walk approach to calculating the phase diagram for spatially extended systems with multiplicative noise we use the anderson localization problem as an example the transition from delocalized to localized states is treated as a generalized diffusion with a noise induced first order phase transition the generalized diffusion manifests itself in the divergence of averages of wavefunctions correlators this divergence is controlled by the lyapunov exponent which is the inverse of the localization length 1 the appearance of the generalized diffusion arises due to the instability of a fundamental mode corresponding to correlators the generalized diffusion can be described in terms of signal theory which operates with the concepts of input and output signals and the filter function delocalized states correspond to bounded output signals and localized states to unbounded output signals respectively the transition from bounded to unbounded signals is defined uniquely by the filter function h z

minimal representations geometric quantization and unitarity in the framework of geometric quantization we explicitly construct in a uniform fashion a unitary minimal representation pio of every simply connected real lie group go such that the maximal compact subgroup of go has finite center and go admits some minimal representation we obtain algebraic and analytic results about pio we give several results on the algebraic and symplectic geometry of the minimal nilpotent orbits and then quantize these results to obtain the corresponding representations we assume lie go c is simple

on twistor transformations and invariant differential operator of simple lie group g2 2 the twistor transformations associated to the simple lie group g2 are described explicitly we consider the double fibration g2 p2 g2 b g2 p1 where p1 and p2 are two parabolic subgroups of g2 and b is a borel subgroup and its local version h2 f h1 where h1 is the heisenberg group of dimension 5 embedded in the coset space g2 p1 f cp1 h1 and h2 contains the nilpotent lie group h2 of step three the baker campbell hausdorff formula is used to parametrize the coset spaces coordinates charts their transition functions and the fibers of the projection as complex curves we write down the relative de rham sequence on f along the fibers and push it down to h1 to get a family of matrix valued differential operators dk then we establish a kind of penrose correspondence for g2 the kernel of dk is isomorphic to the first cohomology of the sheaf o k over h2 we also give the penrose type integral transformation u pf for f o k which gives solutions to equations dku 0 when restricted to the re

exact generalized lyapounov exponents for one dimensional disordered tight binding models a conceptually simple procedure is given for computing the moments of the distribution of states of one dimensional disordered tight binding models from which generalized lyapounov exponents are defined

classification of minimal representations of real simple lie groups based on an idea by gan and savin represent theory 9 46 93 2005 we give a classification of minimal representations of connected simple real lie groups not of type a actually we prove that there exist no new minimal representations up to infinitesimal equivalence

improved confidence intervals for the difference between binomial proportions based on paired data existing methods for setting confidence intervals for the difference theta between binomial proportions based on paired data perform inadequately the asymptotic method can produce limits outside the range of validity the exact conditional method can yield an interval which is effectively only one sided both these methods also have poor coverage properties better methods are described based on the profile likelihood obtained by conditionally maximizing the proportion of discordant pairs a refinement methods 5 and 6 which aligns 1 alpha with an aggregate of tail areas produces appropriate coverage properties a computationally simpler method based on the score interval for the single proportion also performs well method 10

binomial confidence intervals abstract for x with binomial n p distribution section 1 gives a one page table of 95 and 99 confidence intervals for p for n 1 2 30 this interval is equivariant under x n x and p 1 p has approximately equal probability tails is approximately unbiased has crow s property of minimizing the sum of the n 1 possible lengths and each of its ends is increasing in x and decreasing in n with about as regular steps as possible sections 2 and 3 consider the usual approximate confidence intervals calculations and asymptotic results show the need for the continuity correction in these even when n is large because of the nonuniformity of the binomial normal convergence as p 0 these intervals fail to have their stated asymptotic confidence coefficients simple corrections are given for this in the approximate interval x n c n x n 1 x n 1 2n it is shown that the factor c n should be replaced by c n c 2 2c n 1 n

nilpotent orbits and some small unitary representations of indefinite orthogonal groups abstract for 2 m l 2 let g be a simply connected lie group with g 0 so 2m 2l 2m as lie algebra let g k p be the complexification of the usual cartan decomposition let k be the analytic subgroup with lie algebra k g 0 and let u g be the universal enveloping algebra of g this work examines the unitarity and k spectrum of representations in the analytic continuation of discrete series of g relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g the roots with respect to the usual compact cartan subalgebra are all ei ej with 1 i q l u be the parabolic subalgebra of g for which em em 1 contributes to u and the other simple roots contribute to l let l be the analytic subgroup of g with lie algebra l g 0 let l c int g l let 2 u be the sum of the roots contributing to u and let q l u be the parabolic subalgebra opposite to q the members of u p are nilpotent members of g the group l c acts on u p with finitely many orbits and the topological closure of each orbit is an irreducible algebraic variety if y is one of these varieties let r y be the dual coordinate ring of y this is a quotient of the algebra of symmetric tensors on u p that carries a fully reducible representation of l c for s z let s k 1 m l s 2 e k then s defines a one dimensional l l module c s extend this to a q l module by having u act by 0 and define n s 2 u u g q c s 2 u let n s 2 u be the unique irreducible quotient of n s 2 u the representations under study are s s n s 2 u and s s n s 2 u where s dim u k and s is the sth derived bernstein functor for s 2l 2 it is known that s s and that s is in the discrete series enright parthsarathy wallach and wolf showed for m s 2l 2 that s s and that s is still unitary the present paper shows that s is unitary for 0 s m 1 even though s s and it relates the k spectrum of the representations s to the representation of l c on a suitable r y with y depending on s use of a branching formula of d e littlewood allows one to obtain an explicit multiplicity formula for each k type in s the variety y is indispensable in the proof the chief tools involved are an idea of b gross and wallach a geometric interpretation of littlewood s theorem and some estimates of norms it is shown further that the natural invariant hermitian form on s does not make s unitary for s l c on any r y a final section of the paper treats in similar fashion the simply connected lie group with lie algebra g 0 so 2m 2l 2m 1 2 m l 2

approximate binomial confidence limits abstract this article examines the accuracy of normal approximations to confidence limits for the binomial n p parameter p the method of getting exact values from an f table is too slow for practical use and useless for large n the two usual approximations have unsatisfactory accuracy both for small n and asymptotically and the simpler one lacks an invariance property of the confidence limit hall s 1982 approximation also lacks invariance and even when symmetrized has unsatisfactory accuracy pratt s 1968 approximation has excellent accuracy for all n inversion of a molenaar 1973 normal approximation to the binomial gives better asymptotic accuracy than pratt s approximation but poor values for observations close to n

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

multiple scattering of waves in random media a transfer matrix approach transfer matrices address multiple scattering of waves in dense media by relating properties of a thick slice of medium to the corresponding properties of a slice thin enough for multiple scattering to be absent hence scattering properties can easily be calculated we present a new method for deriving a whole hierarchy of transfer matrices which can be applied to studying statistics of waves in random media under conditions of extreme multiple scattering the new method based on matrix power series expansions confirms early results and carries them forward into new areas allowing us to study density of states fluctuations and transport phenomena

multifractal wavefunctions at the mobility edge the author describes the critical behaviour of wavefunctions at the anderson metal insulator transition in terms of fractal measures at the mobility edge the wavefunctions have structure on all lengths and they scale with a continuous set of non related exponents dq the results are compared with field theoretic calculations by wegner 1980 in d 2 in

maximal fluctuations a new phenomenon in disordered systems abstract rigorous results for 1d systems combined with more intuitive results for higher dimensions indicate that there is a universal theorem on fluctuations in the transmission coefficient of disordered systems recognising that the transmission coefficient for waves traversing a disordered system can be no greater than unity and no less than zero the theorem states that in each channel either t 2 1 or t 2 0 with the probability set by the conductance this is the most extreme excursion of the distribution possible hence the name maximal fluctuations the effect is distinct from the universal fluctuations which refer to the sensitivity of t to changes in the medium

exact one sided confidence limits for the difference between two correlated proportions we construct exact and optimal one sided upper and lower confidence bounds for the difference between two probabilities based on matched binary pairs using well established optimality theory of buehler starting with five different approximate lower and upper limits we adjust them to have coverage probability exactly equal to the desired nominal level and then compare the resulting exact limits by their mean size exact limits based on the signed root likelihood ratio statistic are preferred and recommended for practical use

the smallest representations of nonlinear covers of odd orthogonal groups we construct the smallest genuine representations of a nonlinear cover of the group rm so 0 p q where p q is odd we determine correspondences of infinitesimal characters arising from restricting the smallest representations to dual pairs frak s frak o p a oplus frak s frak o b where a b q

bloch waves in periodic multi layered acoustic waveguides the band spectrum and associated floquet bloch eigensolutions arising in acoustic and electromagnetic waveguides which have periodic structure along the guide while remaining of finite width are found homogeneous dirichlet or neumann conditions along the guide walls or an alternation of them are taken importantly in some cases a total stop band at zero frequency is identified providing space for low frequency localized modes geometric defects in the structured waveguide also create these modes numerical and asymptotic techniques identify dispersion curves and trapped modes some cases demonstrate maxima and minima of the spectral edges within the brillouin zone and also allow for ultraslow light or sound imaging applications using anomalous dispersion to generate subwavelength resolution are possible and are demonstrated

unipotent representations attached to spherical nilpotent orbits a coadjoint nilpotent orbit inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of a complex semisimple lie group g is said to be spherical if it contains an open orbit of a borel subgroup we determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups our results actually extend to a larger class of nilpotent coadjoint orbits

generalized lyapunov exponents in high dimensional chaotic dynamics and products of large random matrices we study the behavior of the generalized lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsd for products of random matrices without any particular structure the generalized lyapunov exponents become equal in this limit and the value of one of the generalized lyapunov exponents is obtained by simple arguments on the contrary for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized lyapunov exponents remains different ford indicating that high dimensionality cannot always destroy intermittency

construction process for simple lie algebras abstract we give here a construction process for the complex simple lie algebras and the non hermitian type real forms which intersect the minimal nilpotent complex adjoint orbit using a finite dimensional irreducible representation of the conformal group or of some two fold covering of it with highest weight vector a semi invariant of degree four this process leads to a five graded simple complex lie algebra and the underlying semi invariant is intimately related to the structure of the minimal nilpotent orbit we also describe a similar construction process for the simple real lie algebras of hermitian type

universality classes and fluctuations in disordered systems the medium we apply this result to trace t tl m where tl is the amplitude transmission matrix the eigenfunctions of tl t define a set of channels through which the current flows and the eigenvalues are the corresponding transmission coefficients we prove that these coefficients are either o 0 or 1 as l increases more channels are shut down this is the maximal fluctuation theorem fluctuations cannot be greater than this we expect that our classification scheme will prove of further value in proving theorems about limiting distributions we show by numerical simulations that our theorem holds good for a wide variety of systems in one two and three dimensions disordered media scatter waves incident upon them and induce in the scattered wavefield a degree of disorder which is far more extreme than the physical disorder of the medium itself the fluctuations are brought about by multiple scattering of the wavefield and its ability to interfere constructively or destructively with itself the most extreme instances are found in the absence of absorption when multiple scattering can have free rein as a specific example we consider the system shown in figure 1 in which waves are incident on a slab of disordered material of finite thickness in one direction but effectively infinite in the other directions we shall assume that the slab is statistically homogeneous the waves can be either transmitted through or reflected from the slab statistics in the transmission coefficient pose particular challenges as the thickness l of the slab is increased fluctuations in the transmission coefficient far from settling down to some average value become more extreme the question is what can we say about the limiting scattering properties of a thick slab of material as far as we are aware this question has only been addressed for special cases the results we derive here apply in very general circumstances

score and profile likelihood confidence intervals for contingency table parameters a straightforward approach to computing score and profile likelihood confidence intervals for contingency table parameters is described the computational approach herein avoids two main limitations of existing methods 1 compared with existing methods this paper s approach is applicable to a much broader class of parameters 2 unlike existing methods this paper s approach is not case specific and hence lends itself to a general yet very simple computational algorithm this paper describes the sliding quadratic computational algorithm and illustrates its use on examples that have not been previously considered in the literature a small scale simulation study highlights the advantage of using score and profile likelihood intervals rather than wald intervals

confidence intervals from the difference between two correlated proportions abstract when data arise as matched binary pairs it is often of interest to test whether or not the probabilities of an odd pair of each type are equal construction of a small sample confidence interval for the difference in these probabilities is more difficult and either approximate or conservative methods need be resorted to two methods are suggested in this article one a natural asymptotic method and the other a more generally applicable conservative method the coverage probability and average width are studied for small and moderate sample sizes

simple improved confidence intervals for comparing matched proportions for binary matched pairs data this article discusses interval estimation of the difference of probabilities and an odds ratio for comparing success probabilities we present simple improvements of the commonly used wald confidence intervals for these parameters the improvement of the interval for the difference of probabilities is to add two observations to each sample before applying it the improvement for estimating an odds ratio transforms a confidence interval for a single proportion

fock model and segal bargmann transform for minimal representations of hermitian lie groups abstract for any hermitian lie group g of tube type we construct a fock model of its minimal representation the fock space is defined on the minimal nilpotent k c orbit x in p c and the l 2 inner product involves a k bessel function as density here k g is a maximal compact subgroup and g c k c p c is a complexified cartan decomposition in this realization the space of k finite vectors consists of holomorphic polynomials on x the reproducing kernel of the fock space is calculated explicitly in terms of an i bessel function we further find an explicit formula of a generalized segal bargmann transform which intertwines the schrodinger and fock model its kernel involves the same i bessel function using the segal bargmann transform we also determine the integral kernel of the unitary inversion operator in the schrodinger model which is given by a j bessel function

anomalies in the one dimensional anderson model at weak disorder we show that at the special energiese 2cos p q the invariant measure the lyapunov exponent and the density of states can be extended to zero disorder as c functions in the disorder parameter in particular we obtain asymptotic series in the disorder parameter this gives a rigorous proof of the existence of the anomalies originally discovered by kappus and wegner and studied by derrida and gardner and by bovier and klein