the fourier transform of semi simple coadjoint orbits letmbe a closed coadjoint orbit of a real connected semi simple lie groupg and letfm c g gbe its fourier transform in this paper we compute the restriction offmto the lie algebra k of a maximal compact subgroupkofg using our technique of localization in equivariant cohomology developed recently we extend previous results by m duflo g heckman m vergne and i sengupta

the propagator for quantum mechanics on a group manifold from an infinite dimensional analogue of the duistermaat heckman integration formula an exact expression for the quantum mechanical propagator for a particle moving on a group manifold is shown to arise from the application of an infinite dimensional version of the duistermaat heckman integration formula to a suitable path integral over the based loop group of the group in an appendix the equivalence of this expression to a spectral representation of the propagator is demonstrated by means of a poisson resummation

tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

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

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

aperiodic propagation criteria for boolean functions we characterise the aperiodic autocorrelation for a boolean function f and define the aperiodic propagation criteria apc of degree l and order q we establish the strong similarity between apc and the extended propagation criteria as defined by preneel et al in 1991 although the criteria are not identical we also show how aperiodic autocorrelation can be related to the first derivative of f we further propose the metric apc distance and show that quantum error correcting codes are natural candidates for boolean functions with favourable apc distance

the duistermaat heckman integration formula on flag manifolds an exposition is given of various geometrical properties of flag manifolds and of the duistermaat heckman integration formula as applied to flag manifolds

quantifying entanglement resources we present an overview of the quantitative theory of single copy entanglement in finite dimensional quantum systems in particular we emphasize the point of view that different entanglement measures quantify different types of resources which leads to a natural interdependence of entanglement classification and quantification apart from the theoretical basis we outline various methods for obtaining quantitative results on arbitrary mixed states this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

towards human consistent multistage decision making and control models using fuzzy sets and fuzzy logic abstract fuzzy multistage decision making control models are presented two generations of them are distinguished the first consists of conventional models derived from bellman and zadeh s general framework fuzzy constraints goals decision etc the second consists of newer approaches which additionally allow for a more general aggregation of control stage scores outcomes e g at most of the control stages instead of at all conventionally used both generations have roots in papers co authored by bellman they are further steps towards human consistent soft models whose implementation should be much easier

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

quantifiers in the formulation of multiple objective decision functions abstract we discuss the interpretation of linguistically quantified statements using fuzzy sets we indicate that there are two methods of interpreting these statements the algebraic approach and the substitution approach we then use quantified statements to express overall decision functions involving multiple objectives these statements are of the type a solution should satisfy many of the objectives using the algebraic and substitution approach we develop a whole family of forms for combining multiple objectives we show that algebraic approach corresponds to situations in which we are unconcerned about the allocation of satisfaction amongst the objectives but just interested in getting much satisfaction the substitution approach on the other hand leads to forms which are dependent upon the allocation of satisfaction between the objectives

a soft measure of consensus in the setting of partial fuzzy preferences abstract consensus as traditionally meant to be a full and unanimous agreement is often not reachable in practice a degree of consensus for indicating how far a particular group of individuals is from consensus may be therefore very useful we propose a new measure degree of consensus which is more human consistent in the sense that it better reflects a real human perception of the essence of consensus in practice basically our consensus measure expresses the degree to which say most of the important individuals agree as to their testimonies concerning almost all of the relevant options the point of departure is the set of individual testimonies which are here the individual fuzzy preference relations as a formal tool we use a fuzzy logic based calculus of linguistically quantified propositions

the hypoelliptic laplacian on a compact lie group abstract let g be a compact lie group and let g be its lie algebra in this paper we produce a hypoelliptic laplacian on g g which interpolates between the classical laplacian of g and the geodesic flow this deformation is obtained by producing a suitable deformation of the dirac operator of kostant we show that various poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory the paper was motivated by papers by atiyah and frenkel in connection with localization formulas in equivariant cohomology and with kac s character formulas for affine lie algebras in a companion paper we will use similar methods in the context of selberg s trace formula

wall crossing for symplectic vortices and quantum cohomology we derive a wall crossing formula for the symplectic vortex invariants of toric manifolds as an application we give a proof of batyrev s formula for the quantum cohomology of a monotone toric manifold with minimal chern number at least two

quantify entanglement by concurrence hierarchy we define the concurrence hierarchy as d 1i ndependent invariants under local unitary transformations in d level quantum system the first one is the original concurrence defined by wootters 1998 phys rev lett 80 2245 and hill and wootters 1997 phys rev l ett 78 5022 in a two level quantum system and generalized to the d level pure quantum state case we propose to use this concurrence hierarchy as a measurement of entanglement this measurement does not increase under local quantum operations and classical communication

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

large deviation bounds for k designs we present a technique for de randomizing large deviation bounds of functions on the unitary group we replace the haar measure with a pseudo random distribution a k design k designs have the first k moments equal to those of the haar measure the advantage of this is that approximate k designs can be implemented efficiently whereas haar random unitaries cannot we find large deviation bounds for unitaries chosen from a k design and then illustrate this general technique with three applications we first show that the von neumann entropy of a pseudo random state is almost maximal then we show that if the dynamics of the universe produces a k design then suitably sized subsystems will be in the canonical state as predicted by statistical mechanics finally we show that pseudo random states are useless for measurement based quantum computation

the kirwan map equivariant kirwan maps and their kernels consider a hamiltonian action of a compact lie group k on a compact symplectic manifold we find descriptions of the kernel of the kirwan map corresponding to a regular value of the moment map k we start with the case when k is a torus t we determine the kernel of the equivariant kirwan map defined by goldin in go corresponding to a generic circle s t and show how to recover from this the kernel of t as described by tolman and weitsman in to we in the situation when the fixed point set of the torus action is finite similar results have been obtained in our previous papers je je ma for a compact nonabelian lie group k we will use the non abelian localization formula of je ki1 and je ki2 to establish relationships some of them obtained by tolman and weitsman in to we between ker k and ker t where t k is a maximal torus in the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of t

geometrical entanglement of highly symmetric multipartite states and the schmidt decomposition in a previous paper we examined a geometric measure of entanglement based on the minimum distance between the entangled target state of interest and the space of unnormalized product states here we present a detailed study of this entanglement measure for n qubit target states that are invariant under the permutation of any two qubits we analytically obtain the permutation invariant unnormalized direct product states that extremize the distance function we then solve for the hessian to show that up to the action of trivial symmetries the solutions correspond to local minima of the distance function in addition we show that the conditions that determine the extremal solutions for general target states can be obtained directly by parametrizing the product states via their schmidt decomposition

complex equivariant intersection excess normal bundles and bott chern currents the purpose of this paper is to establish an intersection formula in equivariant complex geometry in the presence of an excess normal bundle the contribution of the excess normal bundle to the formula appears through an additive genus k r in a forthcoming paper an infinite dimensional analogue of this formula will be shown to be the result of bismut lebeau on the behaviour of quillen metrics under complex immersions

index theorem and equivariant cohomology on the loop space in this paper extending ideas of witten and atiyah we describe some relations of equivariant cohomology on the loop space of a manifold to the path integral representation of the index of the dirac operator on a twisted spin complex in particular the natural extension of the chern character to the whole loop space is described also it is shown that the non zero free homotopy classes of the loop space have 0 measure for the index measures associated to the index problem

coherent states over grassmann manifolds and the wkb exactness in path integral u n coherent states over grassmann manifold gn n u n u n u n n are formulated to be able to argue the wkb exactness in the path integral representation of a character formula the phenomena is the so called localization of duistermaat heckman the exponent in the path integral formula is proportional to an integer k labeling the u n representation thus when k a usual semiclassical approximation by regarding k 1 can be performed to yield a desired conclusion the mechanism of the localization is uncovered by the help of the generalized schwinger boson technique the discussion on the feynman kernel is also presented

equivariant de rham torsions the purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de rham theory this difference is the sum of the integral of a chern simons current and of a new invariant the v invariant of an odd dimensional manifold equipped with an action of a compact lie group the v invariant localizes on the critical manifolds of invariant morse bott functions the results in this paper are shown to be compatible with results of bunke and also our with previous results on analytic torsion forms

bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication we show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite states locally the number of pure states that can be discriminated by local operations and classical communication is bounded by the total dimension over the average entanglement a similar general condition is also shown for pure and mixed states these results offer a rare operational interpretation for three abstractly defined distancelike measures of multipartite entanglement

a complete set of local invariants for a family of multipartite mixed states we study the equivalence of quantum states under local unitary transformations by using the singular value decomposition a complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in composite systems two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices

symplectic manifolds coherent states and semiclassical approximation the symplectic structure and hamiltonian dynamics for a class of grassmannian manifolds are described here using the two dimensional sphere s2 and disc d2 as illustrative cases we write their path integral representations using coherent state techniques these path integrals can be evaluated exactly by semiclassical methods thus providing examples of the localization formula along the way we also give a local coordinate description for a class of grassmannians

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

canonical decompositions of n qubit quantum computations and concurrence the two qubit canonical decomposition su 4 su 2 su 2 su 2 su 2 writes any two qubit unitary operator as a composition of a local unitary a relative phasing of bell states and a second local unitary using lie theory we generalize this to an n qubit decomposition the concurrence canonical decomposition ccd su 2n kak the group k fixes a bilinear form related to the concurrence and in particular any unitary in k preserves the tangle i 1y i ny 2 for n even thus the ccd shows that any n qubit unitary is a composition of a unitary operator preserving this n tangle a unitary operator in a which applies relative phases to a set of ghz states and a second unitary operator which preserves the tangle as an application we study the extent to which a large random unitary may change concurrence the result states that for a randomly chosen a a su 22p the probability that a carries a state of tangle 0 to a state of maximum tangle approaches 1 as the even number of qubits approach

toward an understanding of entanglement for generalized n qubit w states we solve stationarity equations of the geometric measure of entanglement for multi qubit w type states in this way we compute analytically the maximal overlap of one parameter n qubit and two parameter four qubit w type states and their nearest product states possible extensions to arbitrary w type states and geometrical interpretations of these results are discussed in detail

superconnection currents and complex immersions let i m m be an immersion of complex manifolds and let be a complex of holomorphic hermitian vector bundles on m which provides a projective resolution of the sheaf of sections of a holomorphic vector bundle on m we study the convergence as u of a class of superconnection currents u introduced by quillen and we calculate the limit microlocal estimates of the speed of convergence are also given

thom polynomials of morin singularities we prove a formula for thom polynomials of a d singularities in any codimension we use a combination of the test curve model of porteous and the localization methods in equivariant cohomology our formulas are independent of the codimension and are computationally effective up to d 6

softer optimization and control models via fuzzy linguistic quantifiers a softening of a basic formulation of multicriterion optimization and control multistage decision making is presented for optimization instead of seeking an optimal solution that best satisfies all the fuzzy objectives as has been done so far we seek an optimal solution that best satisfies most much more than 50 etc a linguistic quantifier in general of the fuzzy objectives for control we seek in turn an optimal sequence of controls that best satisfies the fuzzy constraints and fuzzy goals at most much more than 50 etc of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used some applications to softer evidence aggregation in expert systems are also indicated

a wall crossing formula for the signature of symplectic quotients we use symplectic cobordisrn and the localization result of ginzburg guillemin and karshon to find a wall crossing formula for the signature of regular symplectic quotients of hamiltonian torus actions the formula is recursive depending ultimately on fixed point data in the case of a circle action we obtain a formula for the signature of singular quotients as well we also show how formulas for the poincar6 polynomial and the euler characteristic equivalernt to those of kirwan can be expressed in the same recursive manner

nekrasov functions from exact bohr sommerfeld periods the case of su n we suggested in 2009 that the nekrasov function with one non vanishing deformation parameter is obtained by the standard seiberg witten sw contour integral construction the only difference is that the sw differential pdx is substituted by its quantized version for the corresponding integrable system and contour integrals become exact monodromies of the wavefunction this provides an explicit formulation of the earlier guess by nekrasov and shatashvili in 2009 in this paper we successfully check this suggestion in the first order in 2 and the first order in instanton expansion for the su n model where the consistency of the so deformed sw equations is already non trivial

the genus and a regularization of an s1 equivariant euler class we show that a new multiplicative genus in the sense of hirzebruch can be obtained by generalizing a calculation due to atiyah and witten we introduce this as the genus compute its value for some examples and highlight some of its interesting properties we also indicate a connection with the study of multiple zeta values which gives an algebraic interpretation for our proposed regularization procedure

geometric measures of entanglement and the schmidt decomposition in the standard geometric approach the entanglement of a pure state is sin 2 where is the angle between the entangled state and the closest separable state of products of normalized qubit states we consider here a generalization of this notion by considering separable states that consist of products of unnormalized states of different dimensions the distance between the target entangled state and the closest unnormalized product state can be interpreted as a measure of the entanglement of the target state the components of the closest product state and its norm have an interpretation in terms of respectively the eigenvectors and eigenvalues of the reduced density matrices arising in the schmidt decomposition of the state vector for several cases where the target state has a large degree of symmetry we solve the system of equations analytically and look specifically at the limit where the number of qubits is large

monotones and invariants for multi particle quantum states we introduce new entanglement monotones which generalize to the case of many parties those which give rise to the majorization based partial ordering of bipartite states entanglement we give some examples of restrictions they impose on deterministic and probabilistic conversion between multipartite states via local actions and classical communication these include restrictions which do not follow from any bipartite considerations we derive supermultiplicativity relations between each state s monotones and the monotones for collective processing when the parties share several states we also investigate polynomial invariants under local unitary transformations and show that a large class of these are invariant under collective unitary processing and also multiplicative putting restrictions for example on the exact conversion of multiple copies of one state to multiple copies of another

centered complexity one hamiltonian torus actions we consider symplectic manifolds with hamiltonian torus actions which are almost but not quite completely integrable the dimension of the torus is one less than half the dimension of the manifold we provide a complete set of invariants for such spaces when they are centered and the moment map is proper in particular this classifies the preimages under the moment map of all sufficiently small open sets which is an important step towards global classification as an application we construct a full packing of each of the grassmannians gr 2 r5 and gr 2 r6 by two equal symplectic balls

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

the bando calabi futaki character as an obstruction to semistability the purpose of this note is to correct some errors in b and c of main theorem in mn and to make an additional observation in particular we shall give a first nonvanishing example of the obstruction obstrasymp m l introduced in m2

reduction in stages and complete quantization of the mic kepler problem the one parameter deformation family of the standard kepler problem known as the mic kepler problem is completely quantized using the explicit momentum mapping of the torus actions on some toric manifolds and some equivariant cohomology theory these manifolds appear as symplectic faces of the system during reduction process

revised geometric measure of entanglement we present a revised geometric measure of entanglement rgme which is just certain generalization of geometric measure of entanglement gme the revised version is an entanglement monotone some useful inequalities about rgme are deduced for example we give the formulae of rgme for the two parameter class of states in a 2 n quantum system the two particles high dimensional maximally entangled mixed state the isotropic state including the n particle d level case and two multipartite bound entangled states the results show there is a relation and then rgme is an appropriate measure of entanglement

bott chern currents and complex immersions in the whole paper we will say that i m m is an immersion of smooth manifolds if m is a submanifold of m and if i is the corresponding injection map in particular the topology of m is the topology induced by the topology of m in differential geometry such maps i m m are also called embeddings

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

matrix permanent and quantum entanglement of permutation invariant states we point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states this connection allows us to interpret the permanent as an angle between vectors by employing a recently introduced permanent inequality by carlen et al methods appl anal 13 1 2006 we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way

an entanglement measure based on two order minors in this paper a new entanglement measure called emm the entanglement measure based on minors has been constructed by the convex roof method and proved to be a good entanglement measure according to the axiomatic point of view computation of emm can be finished directly by the two order minors of a coefficient matrix instead of the eigenvalues of the density operator required by most of the other entanglement measures so it is very fast and very easy on the other hand emm is related to the modified bures distance between and the closest separable state so it has geometry meaning we also investigate the relations between emm and the entanglement of formation negativity and logarithmic negativity and discover that emm is always smaller than or equal to them emm is equivalent to concurrence however their definitions and methods of proof are completely different emm will improve the efficiency of searching for maximally entangled multipartite states

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

time reversal and n qubit canonical decompositions on pure states of n quantum bits the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spin flip the monotone vanishes for n odd but for n even there is an explicit formula for its value on mixed states i e a closed form expression computes the minimum over all ensemble decompositions of a given density for n even a matrix decomposition k1ak2 of the unitary group is explicitly computable and allows for study of the monotone s dynamics the side factors k1 and k2 of this concurrence canonical decomposition ccd are concurrence symmetries so the dynamics reduce to consideration of the a factor this unitary a phases a basis of entangled states and the concurrence dynamics of u are determined by these relative phases in this work we provide an explicit numerical algorithm computing k1ak2 for n odd further in the odd case we lift the monotone to a two argument function the concurrence capacity of according to the double argument lift may be n

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

quotient of a loop group and witten genus we define a dirac ramond operator over the quotient lg h of a loop group by a subgroup h of the compact lie group g we state the conjecture that its equivariant index with respect of the natural circle action over the quotient of the loop group is equal to the witten genus of the homogeneous manifold g h we motivate the conjecture by a short time argument which allows to come back to a dirac ramond operator at the manner of taubes on a limit model where all the computations can be done by hand

entanglement of pure states for a single copy an optimal local conversion strategy between any two pure states of a bipartite system is presented it is optimal in that the probability of success is the largest achievable if the parties which share the system and which can communicate classically are only allowed to act locally on it the study of optimal local conversions sheds some light on the entanglement of a single copy of a pure state we propose a quantification of such an entanglement by means of a finite minimal set of new measures from which the optimal probability of conversion follows

on aggregation functions based on linguistically quantified propositions and finitely additive set functions we study aggregation functions defined as convex combinations of the linguistically quantified propositions at least k criteria are satisfied our approach is similar to the towa function in spirit but instead of using a maxitive measure we propose to rely on a real valued finitely additive set function this assumption leads to a new framework however it is combinatorial by nature and in general it produces functions with high computational costs therefore we analyze some particular settings and introduce new aggregation functions which can alleviate the combinatorial burden thanks to several combinatorial identities these methods have interesting features and in particular some of them make it possible to set different types of relationships between criteria by allowing the use of different t norms the interest of our proposals is illustrated with a famous example which cannot be modeled by classical aggregation functions such as the choquet integral

integrability of zero dimensional replica field theories at β 1 building on insights from the theory of integrable lattices the integrability is claimed for nonlinear replica sigma models derived in the context of real symmetric random matrices specifically the fermionic and the bosonic replica partition functions are proven to form a single supersymmetric pfaff kp hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density density eigenvalue correlation function in the infinite dimensional gaussian orthogonal ensemble implications of the formalism outlined are briefly discussed

the geometry of entanglement and grover s algorithm a measure of entanglement with respect to a bipartite partition of n qubit has been defined and studied from the viewpoint of riemannian geometry iwai 2007 j phys a math theor 40 12161 this paper has two aims one is to study further the geometry of entanglement and the other is to investigate grover s search algorithms both the original and the fixed point ones in reference with entanglement as the distance between the maximally entangled states and the separable states is known already in the previous paper this paper determines the set of maximally entangled states nearest to a typical separable state which is used as an initial state in grover s search algorithms and to find geodesic segments which realize the above mentioned distance as for grover s algorithms it is already known that while the initial and the target states are separable the algorithms generate sequences of entangled states this fact is confirmed also in the entanglement measure proposed in the previous paper and then a split grover algorithm is proposed which generates sequences of separable states only with respect to the bipartite partition

aggregating evidence using quantified statements abstract starting with a collection of experts opinions as to the value of some unknown variable in the form of fuzzy subsets we are concerned with the problem of aggregating the opinions with regard to what values for the unknown variable are possible certain or probable we first show that linguistically quantified propositions provide a framework in which to aggregate the opinions we provide two methodologies for implementing the aggregation in terms of quantified statements the first method involves a consensory aggregation and is based upon an algebraic interpretation of a quantified statement the second involves a competitive type aggregation and is based upon substitution interpretation of a quantified statement we also provide for the ability to include differing importances to the experts as well as differing degrees of confidence in an opinion

some induced ordered weighted averaging operators and their use for solving group decision making problems based on fuzzy preference relations abstract in r r yager d p filev operations for granular computing mixing words and numbers in proceedings of the fuzz ieee world congress on computational intelligence anchorage 1998 pp 123 128 yager and filev introduced the induced ordered weighted averaging iowa operator in this paper we provide some iowa operators to aggregate fuzzy preference relations in group decision making gdm problems these iowa operators when guided by fuzzy linguistic quantifiers allow the introduction of some semantics or meaning in the aggregation and therefore allow for a better control over the aggregation stage developed in the resolution process of the gdm problems in particular we present the importance iowa i iowa operator which applies the ordering of the argument values based upon the importance of the information sources the consistency iowa c iowa operator which applies the ordering of the argument values based upon the consistency of the information sources and the preference iowa p iowa operator which applies the ordering of the argument values based upon the relative preference values associated to each one of them we provide a procedure to deal with ties in respect to the ordering induced by the application of one of these iowa operators it consists of a sequential application of the above iowa operators we also present a selection process for gdm problems based on the concept of fuzzy majority and the above three iowa operators finally we analyse the reciprocity and consistency properties of the collective fuzzy preference relations obtained using iowa operators

twisted n 2 supersymmetry with central charge and equivariant cohomology we present an equivariant extension of the thom form with respect to a vector field action in the framework of the mathai quillen formalism the associated topological quantum field theories correspond to twisted n 2 supersymmetric theories with a central charge we analyze in detail two different cases topological sigma models and non abelian monopoles on four manifolds

path integrals for spinning particles stationary phase and the duistermaat heckmann theorem we examine the problem of the evaluation of both the propagator and of the partition function of a spinning particle in an external field at the classical as well as the quantum level in connection with the asserted exactness of the stationary phase approximation at the classical level we argue that exactness of this approximation stems from the fact that the dynamics on the two sphere s2 of a spinning particle in a magnetic field is the reduction from r4 to s2 of a linear dynamical system on r4 at the quantum level however and within the path integral approach the restriction inherent to the use of the stationary phase approximation to regular paths clashes with the fact that no regulators are present in the action that enters the path integral this is shown to lead to a prefactor for the path integral that is strictly divergent except in the classical limit a critical comparison is made with the various approaches that have been presented in the literature the validity of a formula given i

intersection pairings in the n fold reduced product of adjoint orbits in previous work we computed the symplectic volume of the symplectic reduced space of the product of n adjoint orbits of a compact lie group in this paper we compute the intersection pairings in the same object in previous work we computed the symplectic volume of the symplectic reduced space of the product of n adjoint orbits of a compact lie group in this paper we compute the intersection pairings in the same object

on the construction of certain 6 dimensional symplectic manifolds with hamiltonian circle actions let m w be a connected compact 6 dimensional symplectic manifold equipped with a semi free hamiltonian s 1 action such that the fixed point set consists of isolated points or surfaces assume dim h 2 m 3 in an earlier paper we defined a certain invariant of such spaces which consists of fixed point data and twist type and we divided the possible values of these invariants into six types in this paper we construct such manifolds with these types as a consequence we have a precise list of the values of these invariants

logic based neurons extensions uncertainty representation and development of fuzzy controllers abstract the paper concentrates on further generalizations of logic oriented neurons as introduced in pedrycz 1993 by furnishing them with additional numerical facilities some relationships between these new constructs and the concepts linguistic modifiers are studied furthermore we point at some essential links between fuzzy neurons and a well known owa aggregation operator in sequel we bring up new architectural aspects of fuzzy neural networks including those aimed at representing and processing uncertainty associated with the input data a fuzzy controller is studied in this context with a main emphasis focused on its performance achieved in presence of a genuine linguistic information coming from the environment under control

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

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

correlation functions of harish chandra integrals over the orthogonal and the symplectic groups the harish chandra correlation functions i e integrals over compact groups of invariant monomials prod operatorname tr x p 1 varomega y q 1 varomega dagger x p 2 cdots with the weight exp tr x y are computed for the orthogonal and symplectic groups we proceed in two steps first the integral over the compact group is recast into a gaussian integral over strictly upper triangular complex matrices with some additional symmetries supplemented by a summation over the weyl group this result follows from the study of loop equations in an associated two matrix integral and may be viewed as the adequate version of duistermaat heckman s theorem for our correlation function integrals secondly the gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions

multi periodic coherent states and the wkb exactness we construct the path integral formula in terms of a multi periodic coherent state as an extension of the nielsen rohrlich formula for spin we make an exact calculation of the formula and show that when a parameter corresponding to the magnitude of spin becomes large the leading order term of the expansion coincides with the exact result we also give an explicit correspondence between the trace formula in the multi periodic coherent state and the one in the generalized coherent state

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

replica field theories painleve transcendents and exact correlation functions exact solvability is claimed for nonlinear replica ensuremath sigma models derived in the context of random matrix theories contrary to other approaches reported in the literature the framework outlined does not rely on traditional replica symmetry breaking but rests on a previously unnoticed exact relation between replica partition functions and painlev e transcendents while expected to be applicable to matrix models of arbitrary symmetries the method is used to treat fermionic replicas for the gaussian unitary ensemble gue chiral gue symmetry classes a and aiii in cartan classification and ginibre s ensemble of complex non hermitian random matrices further applications are briefly discussed

on certain infinite dimensional aspects of arakelov intersection theory letk y x be an embedding of compact complex manifolds bismut and lebeau have calculated the quillen norm of the canonical isomorphism identifying the determinant of the cohomology of a holomorphic vector bundle overy and the determinant of the cohomology of a resolution by a complex of holomorhic vector bundles overx the purpose of this paper is to show that the formula of bismut lebeau can be viewed as an equivariant intersection formula over the loop space of the considered manifolds in the presence of an infinite dimensional excess normal bundle this excess normal bundle is responsible for the appearance of the additive genusr of gillet and soule in the formula of bismut and lebeau