degenerate principal series and local theta correspondence iii the case of complex groups abstract let g g be one of the following complex dual pairs o m c sp 2 n c gl m c gl 2 n c sp 2 m c o 4 n c we determine the relationship between the space of g coinvariants and certain degenerate principal series representations of g we also show that the space of g invariant distributions is generated by the dirac distribution as a g representation

on certain distinguished unitary representations supported on null cones let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i or inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i and let g u p q inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i be the isometry group of a inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i hermitian form of signature p q for 2 n min p q we consider the action of g on v n the direct sum of n copies of the standard module v inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i and the associated action of g on the regular part of the null cone denoted by x 00 we show that there is a commuting set of g invariant differential operators acting on the space of c functions on x 00 which transform according to a distinguished gl n inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i character and the resulting kernel is an irreducible unitary representation of g our result can be interpreted as providing a geometric construction of the theta lift of the characters from the group g u n n or o 4 n the construction and approach here follow a previous work of zhu and huang representation theory 1 1997 where the group concerned is g o p q with p q even

boolean functions projection operators and quantum error correcting codes this paper describes a fundamental correspondence between boolean functions and projection operators in hilbert space the correspondence is widely applicable and it is used in this paper to provide a common mathematical framework for the design of both additive and nonadditive quantum error correcting codes the new framework leads to the construction of a variety of codes including an infinite class of codes that extend the original 5 6 2 code found by rains it also extends to operator quantum error correcting codes

on some degenerate principal series representations of u n n we study the infinitesimal action of u n n on the degenerate principal series representations of u n n associated with a parabolic subgroup which stabilizes a maximal isotropic subspace of c2n relative to the indefinite hermitian form with signature n n this allows us in particular to determine the reducibility socle series and unitarity of these representations

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

high performance single error correcting quantum codes for amplitude damping we construct families of high performance quantum amplitude damping codes all of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension one family is built from nonlinear error correcting codes for classical asymmetric channels with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block length n 8 except n 2r 1 we generalize this construction to employ classical codes over gf 3 with which we numerically obtain better performing codes up to length 14 because the resulting codes are of the codeword stabilized cws type conceptually simple though potentially computationally expensive encoding and decoding circuits are available

one dimensional representations of u p q and the howe correspondence abstract we explicitly determine the theta lifts of all one dimensional representations of u p q in terms of langlands parameters and determine exactly which lifts are unitary moreover we show that such a lift is unitary if and only if it is a weakly fair derived functor module of the form a q finally we show that the correspondence of unitary representations behaves well with respect to associated cycles

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

exploring spiral defect chaos in generalized swift hohenberg models with mean flow we explore the phenomenon of spiral defect chaos in two types of generalized swift hohenberg model equations that include the effects of long range drift velocity or mean flow we use spatially extended domains and integrate the equations for very long times to study the pattern dynamics as the magnitude of the mean flow is varied the magnitude of the mean flow is adjusted via a real and continuous parameter that accounts for the fluid boundary conditions on the horizontal surfaces in a convecting layer for weak values of the mean flow we find that the patterns exhibit a slow coarsening to a state dominated by large and very slowly moving target defects for strong enough mean flow we identify the existence of spatiotemporal chaos which is indicated by a positive leading order lyapunov exponent we compare the spatial features of the mean flow field with that of rayleigh b enard convection and quantify their differences in the neighborhood of spiral defects

spiral defect chaos swift hohenberg model versus boussinesq equations spiral defect chaos sdc in rayleigh benard convection is a well established spatio temporal complex pattern which competes with stationary rolls near the onset of convection the characteristic properties of sdc are accurately described on the basis of the standard three dimensional boussinesq equations as a much simpler and attractive two dimensional model for sdc generalized swift hohenberg sh equations have been extensively used in the literature from the early beginning here we show that the description of sdc by sh models has to be considered with care especially regarding its long time dynamics for parameters used in previous sh simulations sdc occurs only as a transient in contrast to the experiments and the rigorous solutions of the boussinesq equations the small scale structure of the vorticity field at the spiral cores which might be crucial for persistent sdc is presumably not perfectly captured in the sh model

theta lifting of unitary lowest weight modules and their associated cycles we consider a reductive dual pair g g in the stable range with g the smaller member and of hermitian symmetric type we study theta lifting of holomorphic nilpotent k c orbits in relation to theta lifting of unitary lowest weight modules of g we determine the associated cycles of all such representations in particular we prove that the multiplicity in the associated cycle is preserved under theta lifting

on howe s duality theorem abstract we adopt the langlands classification to the context of real reductive dual pairs and prove that howe s duality correspondence maps hermitian representations to hermitian representations

the regularized siegel weil formula the second term identity and the rallis inner product formula in this paper we establish the second term identity of the siegel weil formula in full generality and derive the rallis inner product formula for global theta lifts for any dual pair as a corollary we resolve the non vanishing problem of global theta lifts initiated by steve rallis

singular unitary representations of classical groups on presente une description de la partie relativement singuliere du dual unitaire d un groupe classique de type i sur un corps local f de caracteristique autre que 2

simple family of nonadditive quantum codes most known quantum codes are additive meaning the codespace can be described as the simultaneous eigenspace of an abelian subgroup of the pauli group while in some scenarios such codes are strictly suboptimal very little is understood about how to construct nonadditive codes with good performance here we present a family of nonadditive quantum codes for all odd blocklengths n that has a particularly simple form our codes correct single qubit erasures while encoding a higher dimensional space than is possible with an additive code or for n of 11 or greater any previous codes

a new construction of quantum error correcting codes in this paper we present a characterization of binary and non binary quantum error correcting codes based on this characterization we introduce a method to construct p ary quantum codes using boolean functions satisfying a system of certain quadratic relations as a consequence of the construction we are able to construct quantum codes of minimum distance 2 in particular we produce a class of binary quantum n 2 n 2 1 2 n 1 n 1 2 2 codes for odd length 5 for n 11 this improves the result by rains in quantum codes of minimal distance two 1999 showing the existence of binary quantum n 3 2 n 4 2 codes for odd n 5 moreover our binary quantum n 2 n 2 1 2 n 1 n 1 2 2 codes of odd length achieve the singleton bound asymptotically finally based on our characterization some propagation rules of quantum codes are proposed and the rules are similar to those in classical coding theory it turns out that some new quantum codes are found through these propagation rules

codeword stabilized quantum codes we present a unifying approach to quantum error correcting code design that encompasses additive stabilizer codes as well as all known examples of nonadditive codes with good parameters we use this framework to generate new codes with superior parameters to any previously known in particular we find 10 18 3 and 10 20 3 codes we also show how to construct encoding circuits for all codes within our framework

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

on the siegel weil formula for unitary groups following s s kudla and s rallis we extend the siegel weil formula for unitary groups which relates a value of a siegel eisenstein series to the convergent integral of a theta function

characterizing dynamics with covariant lyapunov vectors a general method to determine covariant lyapunov vectors in both discrete and continuous time dynamical systems is introduced this allows us to address fundamental questions such as the degree of hyperbolicity which can be quantified in terms of the transversality of these intrinsic vectors for spatially extended systems the covariant lyapunov vectors have localization properties and spatial fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the lyapunov exponents

polynomial invariants of quantum codes the weight enumerators shor and laflamme 1997 of a quantum code are quite powerful tools for exploring its structure as the weight enumerators are quadratic invariants of the code this suggests the consideration of higher degree polynomial invariants we show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one to one correspondence with s sub k sup n we then present a number of equations and inequalities in these invariants in particular we give a higher order generalization of the shadow enumerator of a code and prove that its coefficients are nonnegative we also prove that the quartic invariants of a 4 4 2 sub 2 code are uniquely determined an important step in a proof that any 4 4 2 sub 2 code is additive rains 1999

codeword stabilized quantum codes algorithm and structure the codeword stabilized cws quantum code formalism presents a unifying approach to both additive and nonadditive quantum error correcting codes ieee trans inf theory 55 433 2009 this formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state finding such a classical code can be very difficult here we consider an algorithm which maps the search for cws codes to a problem of identifying maximum cliques in a graph while solving this problem is in general very hard we provide three structure theorems which reduce the search space specifying certain admissible and optimal n k d additive codes in particular we find that the re does not exist any 7 3 3 cws code though the linear programming bound does not rule it out the complexity of the cws search algorithm is compared with the contrasting method introduced by aggarwal and calderbank ieee trans inf theory 54 1700 2008

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

graph concatenation for quantum codes graphs are closely related to quantum error correcting codes every stabilizer code is locally equivalent to a graph code and every codeword stabilized code can be described by a graph and a classical code for the construction of good quantum codes of relatively large block length concatenated quantum codes and their generalizations play an important role we develop a systematic method for constructing concatenated quantum codes based on graph concatenation where graphs representing the inner and outer codes are concatenated via a simple graph operation called generalized local complementation our method applies to both binary and nonbinary concatenated quantum codes as well as their generalizations

optimal binary one error correcting codes of length 10 have 72 codewords the maximum number of codewords in a binary code with length n and minimum distance d is denoted by a n d by construction it is known that a 10 3 spl ges 72 and a 11 3 spl ges 144 these bounds have long been conjectured to be the exact values this is here proved by classifying various codes of smaller length and lengthening these using backtracking and isomorphism rejection there are 562 inequivalent codes attaining a 10 3 72 and 7398 inequivalent codes attaining a 11 3 144

on the siegel weil formula for quaternionic unitary groups we extend the siegel weil formula to all quaternion dual pairs applications include the classification problem of skew hermitian forms over a quaternion algebra over a number field and a product formula for the weighted average of the representation numbers of a skew hermitian form by another skew hermitian form

a nonadditive quantum code every good quantum error correcting code discovered thus far such as those known as stabilizer or additive codes has had the structure of an eigenspace of an abelian group generated by tensor products of pauli matrices in this letter we present the first example of a code that is better than any code of this type it encodes six states in five qubits and can correct the erasure of any single qubit

microextensive chaos of a spatially extended system by analyzing chaotic states of the one dimensional kuramoto sivashinsky equation for system sizes l in the range 79l ll 93 we show that the lyapunov fractal dimension d scales microextensively increasing linearly with l even for increments ensuremath delta l that are small compared to the average cell size of 9 and to various correlation lengths this suggests that a spatially homogeneous chaotic system does not have to increase its size by some characteristic amount to increase its dynamical complexity

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

green forms and the arithmetic siegel weil formula we construct natural green forms for special cycles in orthogonal and unitary shimura varieties in all codimensions and for compact shimura varieties of type mathrm o p 2 and mathrm u p 1 we show that the resulting local archimedean height pairings are related to special values of derivatives of siegel eisentein series a conjecture put forward by kudla relates these derivatives to arithmetic intersections of special cycles and our results settle the part of his conjecture involving local archimedean heights

a regularized siegel weil formula for unitary groups for the quasi split unitary group u n n we prove a refined analogue of the result of s s kudla and s rallis on an identity a regularized siegel weil formula between a residue of an eisenstein series and the integral of a certain theta function

on shalika periods and a theorem of jacquet martin let pi be a cuspidal automorphic representation of gl 4 bbb a with central character mu 2 it is known that pi has shalika period with respect to mu if and only if the l function l s s pi bigwedge 2 otimes mu 1 has a pole at s 1 jacquet and martin considered the analogous question for cuspidal representations pi d of the inner form gl 2 d bbb a and obtained a partial result via the relative trace formula in this paper we provide a complete solution to this problem via the method of theta correspondence and give necessary and sufficient conditions for the existence of shalika period for pi d we also resolve the analogous question in the local setting

permutationally invariant codes for quantum error correction abstract a permutationally invariant n bit code for quantum error correction can be realized as a subspace stabilized by the non abelian group sn the code is spanned by bases for the trivial representation and all other irreducible representations both those of higher dimension and orthogonal bases for the trivial representation are available for error correction a number of new non additive binary codes are obtained including two new 7 bit codes and a large family of new 9 bit codes it is shown that the degeneracy arising from permutational symmetry facilitates the correction of certain types of two bit errors the correction of two bit errors of the same type is considered in detail but is shown not to be compatible with single bit error correction using 9 bit codes

nonadditive quantum error correcting code we report the first nonadditive quantum error correcting code namely a 9 12 3 code which is a 12 dimensional subspace within a 9 qubit hilbert space that outperforms the optimal stabilizer code of the same length by encoding more levels while correcting arbitrary single qubit errors taking advantage of the graph states we construct explicitly a complete encoding decoding circuit for the proposed nonadditive error correcting code

a notion of rank for unitary representations of general linear groups a notion of rank for unitary representations of general linear groups over a locally compact nondiscrete field is defined rank measures how singular a representation is when restricted to the unipotent radical of a maximal parabolic subgroup irreducible representations of small rank are classified it is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations i definition and basic properties of rank 0 introduction the unitary dual of gln f f a locally compact nondiscrete field has been classified by d vogan v in the archimedean case and up to the construction of all cuspidal representations by m tadic in the non archimedean case much remains to be done however in terms of the study of specific properties of the unitary representations involved the inspiration for this paper comes form the following well known theorem of howe and moore theorem 0 1 hm let g be a zariski connected reductive algebraic group over a nondiscrete locally compact field f and let p be a unitary representation of dimension greater than 1 such that the center z of g acts by a character then the absolute values of the matrix coefficients of p approach zero at infinity on g z an easy consequence of this is corollary 0 2 in the hypotheses for theorem 0 1 if n is a subgroup of g which is not compact modulo z any vector that is invariant under n transforms according to a character of g o in particular if p is irreducible and has a nontrivial n invariant vector p must be a character received by the editors august 15 1988 1980 mathematics subject classification 1985 revisiont primary 22e46 22e50 this paper was submitted in a different form in partial fulfillment of the requirements for doctor of philosophy in the department of mathematics of yale university the author acknowledges support from the consiglio nazionale delle ricerche italy 1990 american mathematical society 0002 9947 90 1 00 25 per page

eigendistributions for orthogonal groups and representations of symplectic groups we consider the action of h o p q on the matrix space mp q n r we study a certain orbit o of h in the null cone n mp q n r which supports an eigendistribution d o for h using some identities of capelli type developed in the appendix we determine the structure of g sp 2n r cyclic module generated by d o under the oscillator representation of g the metaplectic cover of g sp 2n p q r applications to local theta correspondence and a generalized huygens principle are given

generalized graph states based on hadamard matrices graph states are widely used in quantum information theory including entanglement theory quantum error correction and one way quantum computing graph states have a nice structure related to a certain graph which is given by either a stabilizer group or an encoding circuit both can be directly given by the graph to generalize graph states whose stabilizer groups are abelian subgroups of the pauli group one approach taken is to study non abelian stabilizers in this work we propose to generalize graph states based on the encoding circuit which is completely determined by the graph and a hadamard matrix we study the entanglement structures of these generalized graph states and show that they are all maximally mixed locally we also explore the relationship between the equivalence of hadamard matrices and local equivalence of the corresponding generalized graph states this leads to a natural generalization of the pauli x z pairs which characterizes the local symmetries of these generalized graph states our approach is also naturally generalized to construct graph quantum codes which are beyond stabilizer codes

degenerate principal series and local theta correspondence following our previous paper lz which deals with the groupu n n we study the structure of certain howe quotients p q and p q 1 which are natural sp 2n r modules arising from the oscillator representation associated with the dual pair o p q sp 2n r by embedding them into the degenerate principal series representations of sp 2n r studied in l2

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

structure of characteristic lyapunov vectors in spatiotemporal chaos we study lyapunov vectors lvs corresponding to the largest lyapunov exponents in systems with spatiotemporal chaos we focus on characteristic lvs and compare the results with backward lvs obtained via successive gram schmidt orthonormalizations systems of a very different nature such as coupled map lattices and the continuous time lorenz 96 model exhibit the same features in quantitative and qualitative terms additionally we propose a minimal stochastic model that reproduces the results for chaotic systems our work supports the claims about universality of our earlier results i g szendro phys rev e 76 025202 r 2007 for a specific coupled map lattice

degenerate principal series representations of gln c and gln r let f be the field c or r for 1 k n 2 let pk be the maximal parabolic subgroup of gln f with its levi factor isomorphic to gln k f glk f in this paper we explicitly calculate the action of the enveloping algebra of gln f on the degenerate principal series representations induced from characters of pk we address composition series k types and unitarity