second order and moderate deviations asymptotics for successive refinement we derive the optimal second order coding region and moderate deviations constant for successive refinement source coding with a joint excess distortion probability constraint we consider two scenarios 1 a discrete memoryless source dms and arbitrary distortion measures at the decoders and 2 a gaussian memoryless source gms and quadratic distortion measures at the decoders for a dms with arbitrary distortion measures we prove an achievable second order coding region using type covering lemmas by kanlis and narayan and by no ingber and weissman we prove the converse using the perturbation approach by gu and effros when the dms is successively refinable the expressions for the second order coding region and the moderate deviations constant are simplified and easily computable for this case we also obtain new insights on the second order behavior compared with the scenario where separate excess distortion proabilities are considered for example we describe a dms for which the optimal second order region transitions from being characterizable by a bivariate gaussian to a univariate gaussian as the distortion levels are varied we then consider a gms with quadratic distortion measures to prove the direct part we make use of the sphere covering theorem by verger gaugry together with appropriately defined gaussian type classes to prove the converse we generalize kostina and verdu s one shot converse bound for point to point lossy source coding we remark that this proof is applicable to general successively refinable sources in the proofs of the moderate deviations results for both scenarios we follow a strategy similar to that for the second order asymptotics and use the moderate deviations principle

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

a modular functor which is universal for quantum computation we show that the topological modular functor from witten chern simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor s state space a computational model based on chern simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model the chief technical advance the density of the irreducible sectors of the jones representation has topological implications which will be considered elsewhere

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

failure of successive refinement for symmetric gaussian mixtures we show that information refinement fails for a continuous alphabet source whose input distribution is the symmetric mixture of normal densities with means spl mu and spl mu and common variance spl sigma sup 2

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

combinatorial quantization of the hamiltonian chern simons theory ii motivated by a recent paper of fock and rosly 6 we describe a mathematically precise quantization of the hamiltonian chern simons theory we introduce the chern simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly the lattice model enjoys the symmetry with respect to a quantum gauge group using this fact we construct the algebra of observables of the hamiltonian chern simons theory equipped with a operation and a positive inner product

on the curvature of piecewise flat spaces we consider analogs of the lipschitz killing curvatures of smooth riemannian manifolds for piecewise flat spaces in the special case of scalar curvature the definition is due to t regge considerations in this spirit date back to j steiner we show that if a piecewise flat space approximates a smooth space in a suitable sense then the corresponding curvatures are close in the sense of measures

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

nonasymptotic and second order achievability bounds for coding with side information we present a novel nonasymptotic or finite blocklength achievability bounds for three side information problems in network information theory these include 1 the wyner ahlswede korner wak problem of almost lossless source coding with rate limited side information 2 the wyner ziv wz problem of lossy source coding with side information at the decoder and 3 the gel fand pinsker gp problem of channel coding with noncausal state information available at the encoder the bounds are proved using ideas from channel simulation and channel resolvability our bounds for all three problems improve on all previous nonasymptotic bounds on the error probability of the wak wz and gp problems in particular those derived by verdu using our novel nonasymptotic bounds we recover the general formulas for the optimal rates of these side information problems finally we also present achievable second order coding rates by applying the multidimensional berry esseen theorem to our new nonasymptotic bounds numerical results show that the second order coding rates obtained using our nonasymptotic achievability bounds are superior to those obtained using existing finite blocklength bounds

successive refinement of information characterization of the achievable rates let r spl middot be the rate distortion function assume that we want to describe a source with distortion no larger than spl delta sub 1 from the rate distortion theory we know that we need to do so at a rate r sub 1 no smaller than r spl delta sub 1 spl lsqb bits symbol spl rsqb if it turns out that a more accurate description at distortion spl delta sub 2 spl delta sub 2

the absolute continuity of convolutions of orbital measures in symmetric spaces abstract we characterize the absolute continuity of convolution products of orbital measures on the classical irreducible riemannian symmetric spaces g k of cartan type iii where g is a non compact connected lie group and k is a compact connected subgroup by the orbital measures we mean the uniform measures supported on the double cosets kzk in g the characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z a consequence of our work is to show that the convolution product of any rank g k continuous k bi invariant measures is absolutely continuous in any of these symmetric spaces other than those whose restricted root system is type a n or d 3 when rank g k 1 is needed

the smoothness of convolutions of zonal measures on compact symmetric spaces abstract we prove that for every irreducible compact symmetric space g c k of rank r the convolution of any 2 r 1 continuous k bi invariant measures is absolutely continuous with respect to the haar measure on g c we also prove that the convolution of r 1 continuous k invariant measures on the 1 eigenspace in the cartan decomposition of the lie algebra of g c is absolutely continuous with respect to lebesgue measure these results are nearly sharp

a quantum group version of quantum gauge theories in two dimensions for the special case of the quantum group slq 2 c q exp pi i r r or 3 the authors present an alternative approach to quantum gauge theories in two dimensions they exhibit the similarities to witten s combinatorial approach which is based on the ideas of migdal 1970 the main ingredient is the turaev viro combinatorial construction of topological invariants of closed compact 3 manifolds and its extension to arbitrary compact 3 manifolds as given by the authors in collaboration with w muller 1992

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

state sum invariants of compact 3 manifolds with boundary and 6j symbols the authors extend the combinatorial construction of invariants of smooth compact closed 3 manifolds as given by turaev and viro 1991 to obtain invariants of 3 manifolds and with a boundary the techniques uses quantum 6j symbols associated to the quantized universal enveloping algebra uq sl 2 c q a root of unity to give a construction of a state sum for given triangulation this state sum is then invariant under subdivisions and isotopes of the boundary their methods also lead to a simplified proof of the main result of turaev and viro that the state sum is independent of the triangulation and hence gives rise to an invariant of the manifold they use surgery to calculate the state sum for some closed manifolds their results also confirm a finding of turaev 1990 which relates this theory to the topological quantum field theory with a chern simons action in the sense of witten 1989

multiple description coding with many channels an achievable region for the l channel multiple description coding problem is presented this region generalizes two channel results of el gamal and cover 1982 and of zhang and berger 1987 it further generalizes three channel results of gray and wyner 1974 and of zhang and berger a source that is successively refinable on chains is shown to be successively refinable on trees a new outer bound on the rate distortion rd region for memoryless gaussian sources with mean squared error distortion is also derived the achievable region meets this outer bound for certain symmetric cases

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

l2 singular dichotomy for orbital measures of classical compact lie groups abstract we prove that for any classical compact simple connected lie group g the g invariant orbital measures supported on non trivial conjugacy classes satisfy a surprising l 2 singular dichotomy either h k l 2 g or h k is singular to the haar measure on g the minimum exponent k for which h k l 2 is specified it depends on lie properties of the element h g as a corollary we complete the solution to a classical problem to determine the minimum exponent k such that k l 1 g for all central continuous measures on g our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes

information spectrum approach to second order coding rate in channel coding in this paper second order coding rate of channel coding is discussed for general sequence of channels the optimum second order transmission rate with a constant error constraint epsiv is obtained by using the information spectrum method we apply this result to the discrete memoryless case the discrete memoryless case with a cost constraint the additive markovian case and the gaussian channel case with an energy constraint we also clarify that the gallager bound does not give the optimum evaluation in the second order coding rate

moderate deviations in channel coding we consider block codes whose rate converges to the channel capacity with increasing blocklength at a certain speed and examine the best possible decay of the probability of error for discrete memoryless channels we prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes

observables in the turaev viro and crane yetter models we define an invariant of graphs embedded in a 3 manifold and a partition function for 2 complexes embedded in a triangulated 4 manifold by specifying the values of variables in the turaev viro and crane yetter state sum models in the case of the three dimensional invariant we prove a duality formula relating its fourier transform to another invariant defined via the colored jones polynomial in the case of the four dimensional partition function we give a formula for it in terms of a regular neighborhood of the 2 complex and the signature of its complement some examples are computed which show that the partition function determines an invariant which can detect non locally flat surfaces in a 4 manifold

new results in binary multiple descriptions an encoder whose input is a binary equiprobable memoryless source produces one output of rate r 1 and another of rate r 2 let d 1 d 2 and d 0 respectively denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate r l only the encoder output of rate r 2 only and both encoder outputs the two descriptions problem is to determine the region r of all quintuples r 1 r 2 d 1 d 2 d 0 that are achievable in thc usual shannon sense let r d 1 d log 2 d 1 d log 2 1 d denote the error frequency rate distortion function of the source the no excess rate case prevails when r 1 r 2 r d 0 and the excess rate case when r 1 r 2 r d 0 denote the section of r at r 1 r 2 d 0 by d r 1 r 2 d 0 d 1 d 2 r 1 r 2 d 1 d 2 d 0 in r in the no excess rate case we show that a portion of the boundary of d r 1 r 2 d 0 coincides with the curve frac 1 2 d 1 2d 0 frac 1 2 d 2 2d 0 frac 1 2 1 2d 0 2 this curve is an extension of witsenhausen s hyperbola bound to the case d 0 0 it follows that the projection of r onto the d 1 d 2 plane at fixed d 0 consists of all d 1 geq d 0 and d 2 geq d 0 that lie on or above this hyperbola in the excess rate case we show by counterexample that the achievable region of el gamal and cover is not tight

discrete lossy gray wyner revisited second order asymptotics large and moderate deviations in this paper we revisit the discrete lossy gray wyner problem in particular we derive its optimal second order coding rate region its error exponent reliability function and its moderate deviations constant under mild conditions on the source to obtain the second order asymptotics we extend some ideas from watanabe s work in particular we leverage the properties of an appropriate generalization of the conditional distortion tilted information density which was first introduced by kostina and verdu the converse part uses a perturbation argument by gu and effros in their strong converse proof of the discrete gray wyner problem the achievability part uses two novel elements 1 a generalization of various type covering lemmas and 2 the uniform continuity of the conditional rate distortion function in both the source joint distribution and the distortion level to obtain the error exponent for the achievability part we use the same generalized type covering lemma and for the converse we use the strong converse together with a change of measure technique finally to obtain the moderate deviations constant we apply the moderate deviations theorem to probabilities defined in terms of information spectrum quantities

lattice topological field theory on nonorientable surfaces the lattice definition of the two dimensional topological quantum field theory fukuma et al commun math phys 161 157 1994 is generalized to arbitrary not necessarily orientable compact surfaces it is shown that there is a one to one correspondence between real associative algebras and the topological state sum invariants defined on such surfaces the partition and n point functions on all two dimensional surfaces connected sums of the klein bottle or projective plane and g tori are defined and computed for arbitrary algebras in general and for the group ring a r g of discrete groups g in particular

on multiple descriptions and team guessing witsenhausen s hyperbola bound for the multiple description problem without excess rate in case of a binary source is not tight for exact joint reproductions however this bound is tight for almost exact joint reproductions theorem 1 conjectured by witsenhausen the proof is based on an em approximative form of the team guessing lemma for em sequences of random variables this result may be of interest also for team guessing the hyperbola bound is also tight for exact joint reproductions and arbitrarily small but positive excess rate theorem 2 the proof of this result uses our covering lemma

strong successive refinability and rate distortion complexity tradeoff we investigate the second order asymptotics source dispersion of the successive refinement problem similar to the classical definition of a successively refinable source we say that a source is strongly successively refinable if successive refinement coding can achieve the second order optimum rate including the dispersion terms at both decoders we establish a sufficient condition for strong successive refinability we show that any discrete source under hamming distortion and the gaussian source under quadratic distortion are strongly successively refinable we also demonstrate how successive refinement ideas can be used in point to point lossy compression problems in order to reduce complexity we give two examples the binary hamming and gaussian quadratic cases in which a layered code construction results in a low complexity scheme that attains optimal performance for example when the number of layers grows with the block length n we show how to design an o n log n algorithm that asymptotically achieves the rate distortion bound

computation and analysis of the n layer scalable rate distortion function methods for determining and computing the rate distortion rd bound for n layer scalable source coding of a finite memoryless source are considered optimality conditions were previously derived for two layers in terms of the reproduction distributions q sub y1 and q sub y2 y sub 1 however the ignored and seemingly insignificant boundary cases where q sub y1 0 and q sub y2 y sub 1 is undefined have major implications on the solution and its practical application we demonstrate that once the gap is filled and the result is extended to n layers it is in general impractical to validate a tentative solution as one has to verify the conditions for all conceivable q sub yi 1 y sub n y sub 1 y sub i at each y sub 1 y sub i such that q sub y1 y sub i 0 as an alternative computational approach we propose an iterative algorithm that converges to the optimal joint reproduction distribution q sub y1 y sub n if initialized with q sub y1 y sub n 0 everywhere for nonscalable coding n 1 the algorithm specializes to the blahut arimoto 1972 algorithm the algorithm may be used to directly compute the rd bound or as an optimality testing procedure by applying it to a perturbed tentative solution q we address two additional difficulties due to the higher dimensionality of the rd surface in the scalable n 1 case namely identifying the sufficient set of lagrangian parameters to span the entire rd bound and the problem of efficient navigation on the rd surface to compute a particular rd point

rate distortion function when side information may be present at the decoder a discrete memoryless source x sub k is to be coded into a binary stream of rate r bits symbol such that x sub k can be recovered with minimum possible distortion the system is to be optimized for best performance with two decoders one of which has access to side information about the source for given levels of average distortion for these two decoders the minimum achievable rate r in the usual shannon theory sense is given as a per letter minimization of information theoretic quantities the problem is solved for two cases where the encoder is and is not informed of the side information

l2 singular dichotomy for orbital measures of classical simple lie algebras we prove that the g invariant orbital measures supported on adjoint orbits in the lie algebra of a classical compact connected simple lie group satisfy a smoothness dichotomy either k is singular to lebesgue measure or k l 2 the minimum k for which k l 2 is specified and is also the minimum k such that the k fold sum of the orbit has positive measure

abelian chern simons theory and linking numbers via oscillatory integrals a rigorous mathematical model of abelian chern simons theory based on the theory of infinite dimensional oscillatory integrals developed by albeverio and ho egh krohn is introduced a gauge fixed chern simons path integral is constructed as a fresnel integral in a certain hilbert space wilson loop variables are defined as fresnel integrable functions and it is shown in this context that the expectation value of products of wilson loops with respect to the chern simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops as conjectured by witten furthermore a lattice chern simons action is proposed which converges to the continuum limit

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

second order region for gray wyner network the coding problem over the gray wyner network is studied from the second order coding rates perspective a tilted information density for this network is introduced in the spirit of kostina verdu and under a certain regularity condition the second order region is characterized in terms of the variance of this tilted information density and the tangent vector of the first order region the second order region is proved by the type method the achievability part is proved by the type covering argument and the converse part is proved by a refinement of the perturbation approach that was used by gu effros to show the strong converse of the gray wyner network this is the first instance that the second order region is characterized for a multi terminal problem where the characterization of the first order region involves an auxiliary random variable

fixed length lossy compression in the finite blocklength regime this paper studies the minimum achievable source coding rate as a function of blocklength n and probability that the distortion exceeds a given level d tight general achievability and converse bounds are derived that hold at arbitrary fixed blocklength for stationary memoryless sources with separable distortion the minimum rate achievable is shown to be closely approximated by r d v d n q 1 where r d is the rate distortion function v d is the rate dispersion a characteristic of the source which measures its stochastic variability and q 1 is the inverse of the standard gaussian complementary cumulative distribution function

lie group valued moment maps we develop a theory of quasi hamiltonian g spaces for which the moment map takes values in the group g itself rather than in the dual of the lie algebra the theory includes counterparts of hamiltonian reductions the guillemin sternberg symplectic cro

two dimensional lattice gauge theory based on a quantum group in this article we analyse a two dimensional lattice gauge theory based on a quantum group the algebra generated by gauge fields is the lattice algebra introduced recently by a yu alekseev h grosse and v schomerus in 1 we define and study wilson loops this theory is quasi topological as in the classical case which allows us to compute the correlation functions of this theory on an arbitrary surface

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

topological quantum field theory and invariants of graphs for quantum groups on the basis of generalized 6j symbols we give a formulation of topological quantum field theories for 3 manifolds including observables in the form of coloured graphs it is shown that the 6j symbols associated with deformations of the classical groups at primitive even roots of unity provide examples of this construction calculational methods are developed which in particular yield the dimensions of the state spaces as well as a rather simple proof of the relation previously found by turaev and walker for the case ofu q sl 2 c between these models and corresponding ones based on the ribbon graph construction of reshetikhin and turaev

non asymptotic converse bounds and refined asymptotics for two source coding problems in this paper we revisit two multi terminal lossy source coding problems the lossy source coding problem with side information available at the encoder and one of the two decoders which we term as the kaspi problem kaspi 1994 and the multiple description coding problem with one semi deterministic distortion measure which we refer to as the fu yeung problem fu and yeung 2002 for the kaspi problem we first present the properties of optimal test channels subsequently we generalize the notion of the distortion tilted information density for the lossy source coding problem to the kaspi problem and prove a non asymptotic converse bound using the properties of optimal test channels and the well defined distortion tilted information density finally for discrete memoryless sources we derive refined asymptotics which includes the second order large and moderate deviations asymptotics in the converse proof of second order asymptotics we apply the berry esseen theorem to the derived non asymptotic converse bound the achievability proof follows by first proving a type covering lemma tailored to the kaspi problem then properly taylor expanding the well defined distortion tilted information densities and finally applying the berry esseen theorem we then generalize the methods used in the kaspi problem to the fu yeung problem as a result we obtain the properties of optimal test channels for the minimum sum rate function a non asymptotic converse bound and refined asymptotics for discrete memoryless sources since the successive refinement problem is a special case of the fu yeung problem as a by product we obtain a non asymptotic converse bound for the successive refinement problem which is a strict generalization of the non asymptotic converse bound for successively refinable sources zhou tan and motani 2017

sums of adjoint orbits and l2 singular dichotomy for su m abstract let g be a real compact connected simple lie group and g its lie algebra we study the problem of determining from root data when a sum of adjoint orbits in g or a product of conjugacy classes in g contains an open set our general methods allow us to determine exactly which sums of adjoint orbits in su m and products of conjugacy classes in su m contain an open set in terms of the highest multiplicities of eigenvalues for su m and su m we show l 2 singular dichotomy the convolution of invariant measures on adjoint orbits or conjugacy classes is either singular to haar measure or in l 2

wigner symbols and combinatorial invariants of three manifolds with boundary in this paper we generalize the partition function proposed by ponzano and regge in 1968 to the case of a compact 3 dimensional simplicial pair m m the resulting state sum z m m contains both wigner 6j symbols associated with tetrahedra and wigner 3jm symbols associated with triangular faces lying in m in order to show the invariance of z m m under pl homeomorphisms we exploit some results due to pachner on the equivalence of n dimensional pl pairs both under bistellar moves on n simplices in the interior of m and under elementary boundary operations shellings and inverse shellings acting on n simplices which have some component in m we find in particular the algebraic identities involving a suitable number of wigner symbols which realize the complete set of pachner s boundary operations in n 3

rate distortion when side information may be absent the problem is considered of encoding a discrete memoryless source when correlated side information may or may not be available to the decoder it is assumed that the side information is not available to the encoder the rate distortion function r d l d 2 is determined where d 1 is the distortion achieved with side information and d 2 is the distortion achieved without it a generalization is made to the case of m decoders each of which is privy to its own side information an appropriately defined d admissible rate for this general case is shown to equal r d when the side information sources satisfy a specified degradedness condition explicit results are obtained in the quadratic gaussian case and in the binary hamming case

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

second order asymptotics in fixed length source coding and intrinsic randomness there is a difference between the optimal rates of fixed length source coding and intrinsic randomness when we care about the second order asymptotics we prove this difference for general information sources and then investigate independent and identically distributed i i d random variables and markovian variables as examples the difference is demonstrated through an investigation of the second order asymptotic behavior of the rates a universal fixed length source code attaining the second order optimal rate is also proposed the difference between the rates of fixed length source coding and intrinsic randomness proves that the outputs of fixed length source codes are not uniformly distributed

tensor products of quantized tilting modules letuk denote the quantized enveloping algebra corresponding to a finite dimensional simple complex lie algebral assume that the quantum parameter is a root of unity ink of order at least the coxeter number forl also assume that this order is odd and not divisible by 3 if typeg2 occurs we demonstrate how one can define a reduced tensor product on the familyf consisting of those finite dimensional simpleuk modules which are deformations of simplel and which have non zero quantum dimension this together with the work of reshetikhin turaev and turaev wenzl prove that uk f is a modular hopf algebra and hence produces invariants of 3 manifolds also by recent work of duurhus jakobsen and nest it leads to a general topological quantum field theory the method of proof explores quantized analogues of tilting modules for algebraic groups

invariants of spin networks embedded in three manifolds we study the invariants of spin networks embedded in a three dimensional manifold which are based on the path integral for su 2 bf theory these invariants appear naturally in loop quantum gravity and have been defined as spin foam state sums by using the chain mail technique we give a more general definition of these invariants and show that the state sum definition is a special case this provides a rigorous proof that the state sum invariants of spin networks are topological invariants we derive various results about the bf theory spin network invariants and we find a relation with the corresponding invariants defined from chern simons theory i e the witten reshetikhin turaev invariants we also prove that the bf theory spin network invariants coincide with v turaev s definition of invariants of coloured graphs embedded in 3 manifolds and thick surfaces constructed by using shadow world evaluations our framework therefore provides a unified view of these invariants

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

dichromatic state sum models for four manifolds from pivotal functors a family of invariants of smooth oriented four dimensional manifolds is defined via handle decompositions and the kirby calculus of framed link diagrams the invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category a state sum formula for the invariant is constructed via the chain mail procedure so a large class of topological state sum models can be expressed as link invariants most prominently the crane yetter state sum over an arbitrary ribbon fusion category is recovered including the nonmodular case it is shown that the crane yetter invariant for nonmodular categories is stronger than signature and euler invariant a special case is the four dimensional untwisted dijkgraaf witten model derivations of state space dimensions of tqfts arising from the state sum model agree with recent calculations of ground state degeneracies in walker wang models relations to different approaches to quantum gravity such as cartan geometry and teleparallel gravity are also discussed

kitaev lattice models as a hopf algebra gauge theory we prove that kitaev s lattice model for a finite dimensional semisimple hopf algebra h is equivalent to the combinatorial quantisation of chern simons theory for the drinfeld double d h this shows that kitaev models are a special case of the older and more general combinatorial models this equivalence is an analogue of the relation between turaev viro and reshetikhin turaev tqfts and relates them to the quantisation of moduli spaces of flat connections we show that the topological invariants of the two models the algebra of operators acting on the protected space of the kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism are isomorphic this is established in a gauge theoretical picture in which both models appear as hopf algebra valued lattice gauge theories we first prove that the triangle operators of a kitaev model form a module algebra over a hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism both algebras can be viewed as the algebra of functions on gauge fields in a hopf algebra gauge theory the isomorphism between them induces an algebra isomorphism between their subalgebras of invariants which are interpreted as gauge invariant functions or observables it also relates the curvatures in the two models which are given as holonomies around the faces of the lattice this yields an isomorphism between the subalgebras obtained by projecting out curvatures which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models

s4 symmetry of 6j symbols and frobenius schur indicators in rigid monoidal c categories we show that a left rigid monoidal c category with irreducible monoidal unit is also a sovereign and spherical category defining a frobenius schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects as a main result we prove s4 invariance of 6j symbols in such a category