universal randomized switching in this paper we consider a competitive approach to sequential decision problems suitable for a variety of signal processing applications where at each of a succession of times a selection must be made from among a fixed set of strategies or outcomes for each such decision and outcome pair loss is incurred and it is the time accumulation of these losses that is sought to be minimized rather than using a statistical performance measure our goal in this pursuit is to sequentially accumulate loss that is no larger than that of the best loss that could be obtained through a partitioning of the sequence of observations into an arbitrary fixed number of segments and independently selecting a different strategy for each segment for this purpose we introduce a randomized sequential algorithm built upon that of kozat and singer that asymptotically achieves the performance of a noncausal algorithm that would be able to choose the number of segments and the best algorithm for each segment based on observing the whole observation process a priori in addition to improving upon the bounds of kozat and singer as well as gyorgy the results we provide hold for more general loss functions than the square error loss studied therein

a zero delay sequential scheme for lossy coding of individual sequences we consider adaptive sequential lossy coding of bounded individual sequences when the performance is measured by the sequentially accumulated mean squared distortion the encoder and the decoder are connected via a noiseless channel of capacity r and both are assumed to have zero delay no probabilistic assumptions are made on how the sequence to be encoded is generated for any bounded sequence of length n the distortion redundancy is defined as the normalized cumulative distortion of the sequential scheme minus the normalized cumulative distortion of the best scalar quantizer of rate r which is matched to this particular sequence we demonstrate the existence of a zero delay sequential scheme which uses common randomization in the encoder and the decoder such that the normalized maximum distortion redundancy converges to zero at a rate n sup 1 5 log n as the length of the encoded sequence n increases without bound

eigenvalue inequalities for the buckling problem of the drifting laplacian on ricci solitons abstract in this paper we investigate the buckling problem of the drifting laplacian and get a general inequality for its eigenvalues on a bounded connected domain in complete ricci solitons supporting a special function by applying this general inequality we obtain some universal inequalities for eigenvalues of the same problem on bounded connected domains in the gaussian shrinking solitons and some general product solitons

gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces abstract let m n g e d v be a smooth metric measure space in this paper we derive a series of gradient estimates and a harnack inequality for positive solutions of a nonlinear parabolic partial differential equation t u q u a u ln u in m n where q c 2 1 m n and a r

remarks on non compact gradient ricci solitons in this paper we show how techniques coming from stochastic analysis such as stochastic completeness in the form of the weak maximum principle at infinity parabolicity and lp liouville type results for the weighted laplacian associated to the potential may be used to obtain triviality rigidity results and scalar curvature estimates for gradient ricci solitons under lp conditions on the relevant quantities

on prediction of individual sequences sequential randomized prediction of an arbitrary binary sequence isinvestigated no assumption is made on the mechanism of generating the bit sequence the goal of the predictor is to minimize its relative loss i e to make almost as few mistakes as the best expert in a fixed possibly infinite set of experts we point out a surprising connection between this prediction problem and empirical process theory first in the special case of static memoryless experts we completely characterize the minimax relative loss in terms of the maximum of an associated rademacher process then we show general upper and lower bounds on the minimaxrelative loss in terms of the geometry of the class of experts as main examples we determine the exact order of magnitude of the minimax relative loss for the class of autoregressive linear predictors and for the class of markov experts

potential function estimates for quasi einstein metrics abstract in this paper we derive the growth pinching estimates for potential functions of quasi einstein metrics on complete noncompact connected manifolds based on the estimates for the scalar curvature and the using of the weighted measure comparison theorem our results show that the estimates for potential functions rely on the sign of constants and

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

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

local parabolic and elliptic gradient estimates for a generalized heat type equation under the yamabe flow abstract in this paper we investigate local parabolic li yau s type and j li s type and local elliptic hamilton s type shouplet zhang s type gradient estimates for positive solutions to a generalized heat type equation of t u h u q under the yamabe flow as applications of local parabolic gradient estimates we establish associated harnack inequalities

universal multiple description scalar quantization analysis and design this paper introduces a new high rate analysis of the multiple description scalar quantizer mdsq with balanced descriptions the analysis provides insight into the structure of the mdsq suggesting the nonoptimality of uniform central quantizer cell lengths as well as a method to approximate optimal cell lengths for both level constrained and entropy constrained mdsq new upper bounds on the granular distortion for sources with smooth probability density functions pdfs are derived under the mean squared error measure which are 0 4 db lower than previous results based on the insights a universal multiple description scalar quantizer umdsq is proposed which at high rate can achieve nearly the same performance as the fully optimized entropy constrained mdsq ecmdsq without requiring extensive training the proposed umdsq has only two control parameters and a continuum of tradeoff points between the central and side distortions can be achieved as the two parameters are varied

riesz transforms for bounded laplacians on graphs we study several problems related to the ell p boundedness of riesz transforms for graphs endowed with so called bounded laplacians introducing a proper notion of the gradient of a function we prove for p in 1 2 an ell p estimate for the gradient of the continuous time heat semigroup an ell p interpolation inequality as well as the ell p boundedness of the modified littlewood paley stein function for a graph with bounded laplacian this yields an analogue to dungey s results in 21 while removing some additional assumptions coming back to the classical notion of the gradient we give a counterexample to the interpolation inequality and hence to the boundedness of riesz transforms for bounded laplacians for 1 p 2 finally we prove the boundedness of the riesz transform for 1 p infty under the assumption of positive spectral gap

operator content of n state quantum chains in the c 1 region the authors conjecture the operator content for n state quantum chains n or 5 in the domains of the coupling constants where the central charge of the virasoro algebra is equal to one free boundary conditions as well as boundary conditions compatible with the torus are considered the conjectured operator content is compared with finite size scaling estimates

liouville theorems and gradient estimates for a nonlinear elliptic equation in this paper we establish gradient estimates for positive solutions to the equation 0 1 fup u on any smooth metric measure space whose m bakry emery curvature is bounded from below by m 1 k with k 0 these estimates imply liouville theorems for 0 1 when p 1 our main theorem reduces to the gradient estimate of wang 2010 9 as applications several harnack inequalities are obtained

the c 2 3 minimal n 1 superconformal system and its realisation in the critical o 2 gaussian model the structure of the 2 3 minimal n 1 superconformal system is analysed in detail the primary operators are constructed as operators in the critical o 2 gaussian model at some specific fixed radius the operator algebra is verified explicitly operator product coefficients and some superspace correlation functions are calculated exactly

vanishing theorems for f harmonic forms on smooth metric measure spaces abstract in this paper we first establish a monotonicity formula for vector bundle valued f harmonic p forms on a smooth metric measure space provided f r is less that an explicit constant as applications we get some vanishing theorems for l 2 f harmonic forms on concrete geometric models in the second part for a metric measure space with nonnegative bakry emery ricci curvature and with moderate volume growth we prove that any bounded f harmonic 1 form must be parallel moreover some vanishing theorems under nonnegative m bakry emery ricci curvature assumption are also proved finally we consider smooth metric measure spaces with weighted poincare inequality and show some vanishing theorems for l q f harmonic 1 forms

bit error resilient index assignment for multiple description scalar quantizers this paper addresses the problem of increasing the robustness to bit errors for two description scalar quantizers our approach is to start with an m diagonal index assignment and further apply a permutation to the indexes of each description to increase the minimum hamming distance d rm min of the set of valid index pairs in particular we show how to construct linear permutation pairs achieving d rm min geq 3 and establish a lower bound in terms of the description rate r for the highest value of m for which such permutations exist

new volume comparison results and applications to degeneration of riemannian metrics abstract we consider a condition on the ricci curvature involving vector fields which is broader than the bakry emery ricci condition under this condition volume comparison laplacian comparison isoperimetric inequality and gradient bounds are proven on the manifold specializing to the bakry emery ricci curvature condition we initiate a short cut approach to work on the original manifold which yields under a weaker than usual assumption the results mentioned above for the original manifold these results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead applications on convergence and degeneration of riemannian metrics under this curvature condition are given to this effect in particular for the bakry emery ricci curvature condition the gradient of the potential function is allowed to have singularity of order close to 1 which is broader than those allowed by the traditional weighted method in general coupling with some new techniques in constructing model functions this approach enables us to extend some of the results in the papers 14 11 36 29 and 31 the condition also covers general ricci solitons instead of just gradient ricci solitons

discrete denoising with shifts we introduce s dude a new algorithm for denoising discrete memoryless channel dmc corrupted data the algorithm which generalizes the recently introduced dude discrete universal denoiser aims to compete with a genie that has access in addition to the noisy data also to the underlying clean data and that can choose to switch up to m times between sliding window denoisers in a way that minimizes the overall loss when the underlying data form an individual sequence we show that the s dude performs essentially as well as this genie provided that m is sublinear in the size of the data when the clean data are emitted by a piecewise stationary process we show that the s dude achieves the optimum distribution dependent performance provided that the same sublinearity condition is imposed on the number of switches to further substantiate the universal optimality of the s dude we show that when the number of switches is allowed to grow linearly with the size of the data any sequence of scheme s fails to compete in the above sense using dynamic programming we derive an efficient implementation of the s dude which has complexity time and memory growing linearly with the data size and the number of switches m preliminary experimental results are presented suggesting that s dude has the capacity to improve on the performance attained by the original dude in applications where the nature of the data abruptly changes in time or space as is often the case in practice

partition functions of the two dimensional ashkin teller model on the critical line by mapping the ashkin teller model onto a scalar free field theory the author derives the continuum limit of the partition functions in various finite geometries he justifies expressions which have been conjectured recently explaining in particular why the free field lives on an orbifold and why the spin should be described by a twist operator

operator content of the multicritical magnetic hard square model the transfer matrix spectra of the magnetic hard squares along the multicritical line are analysed using the data for the strip width n up to 12 the operator content of the model is exactly the same as that of the ashkin teller model when n is odd the spectrum is accounted for by the continuum theory in which the antiperiodic boundary condition is imposed to one of the ising variables of the ashkin teller model

universal zero delay joint source channel coding we consider zero delay joint source channel coding of individual source sequences for a general known channel given an arbitrary finite set of schemes with finite memory not necessarily time invariant decoders a scheme is devised that does essentially as well as the best in the set on all individual source sequences using this scheme we construct a universal zero delay joint source channel coding scheme that is guaranteed to achieve asymptotically the performance of the best zero delay encoding decoding scheme with a finite state encoder and a markov decoder on all individual sequences for the case where the channel is a discrete memoryless channel dmc we construct an implementable zero delay joint source channel coding scheme that is based on the follow the perturbed leader scheme of gyoumlrgy for lossy source coding of individual sequences our scheme is guaranteed to attain asymptotically the performance of the best in the set of all encoding decoding schemes with a symbol by symbol decoder and arbitrary encoder on all individual sequences

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

operator content of the ashkin teller quantum chain superconformal and zamolodchikov fateev invariance ii boundary conditions compatible with the torus for pt i see ibid vol 20 no 8 p l479 85 1987 the authors present the operator content of the ashkin teller quantum chain with eight boundary conditions corresponding to the eight elements of the group d4 which gives the global symmetry of the system this operator content is a conjecture based on an extensive numerical study and was proven to be correct at the ising decoupling point for the values of the coupling constant where n 2 superconformal invariance was seen for free boundary conditions the situation is now more complex one finds sectors which have n 2 and n 1 superconformal invariance and sectors which have no superconformal invariance at all if however they combine the wrong sectors of the spectra which correspond to two different values of the coupling constant a new symmetry shows up they also analyse the operator content at the fateev zamolodchikov points here the wrong sectors can be described by a few new primary fields

efficient adaptive algorithms and minimax bounds for zero delay lossy source coding zero delay lossy source coding schemes are considered for both individual sequences and random sources performance is measured by the distortion redundancy which is defined as the difference between the normalized cumulative mean squared distortion of the scheme and the normalized cumulative distortion of the best scalar quantizer of the same rate that is matched to the entire sequence to be encoded by improving and generalizing a scheme of linder and lugosi weissman and merhav showed the existence of a randomized scheme that for any bounded individual sequence of length n achieves a distortion redundancy o n sup 1 3 logn however both schemes have prohibitive complexity both space and time which makes practical implementation infeasible in this paper we present an algorithm that computes weissman and merhav s scheme efficiently in particular we introduce an algorithm with encoding complexity o n sup 4 3 and distortion redundancy o n sup 1 3 logn the complexity can be made linear in the sequence length n at the price of increasing the distortion redundancy to o n sup 1 4 spl radic logn we also consider the problem of minimax distortion redundancy in zero delay lossy coding of random sources by introducing a simplistic scheme and proving a lower bound we show that for the class of bounded memoryless sources the minimax expected distortion redundancy is upper and lower bounded by constant multiples of n sup 1 2

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

on the generalization of natural type selection to multiple description coding natural type selection was originally proposed by zamir and rose for universal single description coding in this paper we generalize this principle to universal multiple description coding mdc two schemes based on random codebooks are proposed one is of fixed distortion and the other is of fixed weight their operational sum rate distortion functions are derived which coincide with the egc el gamal cover sum rate bound if the parameters of the schemes are optimized it is also shown that in both schemes the joint type of reconstruction codewords can be used to improve the rate distortion r d performance based on our theoretical results a practical universal scheme is proposed by leveraging the mdc methods based on low density generator matrix ldgm codes the performance of this scheme is compared experimentally with the egc bound which shows its effectiveness

diameter estimate for compact quasi einstein metrics in this paper we prove a gradient estimate for the potential function of a nontrivial compact quasi einstein metric with constant 0 based on this gradient estimate we derive the lower bound estimate for the diameter

operator content of the ising model with three spin coupling the ising quantum chain with staggered three spin coupling is studied for free and toroidal boundary conditions the operator content of the finite size limit of the spectra is conjectured it is found to be related to the operator content of the ashkin teller model 1943

operator content of the four state potts quantum chain with z3 boundary conditions the four state potts quantum chain with a toroidal boundary condition leaving a global z3 symmetry of the hamiltonian is investigated at the critical point the infinite system shows su 2 kac moody symmetry the conjectured operator content is confirmed by numerical finite size calculations

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

on properties of locally optimal multiple description scalar quantizers with convex cells it is known that the generalized lloyd method is applicable to locally optimal multiple description scalar quantizer mdsq design however it remains unsettled when the resulting mdsq is also globally optimal we partially answer the above question by proving that for a fixed index assignment there is a unique locally optimal fixed rate mdsq of convex cells under trushkin s sufficient conditions for the uniqueness of locally optimal fixed rate single description scalar quantizer this result holds for fixed rate multiresolution scalar quantizer mrsq of convex cells as well thus the well known log concave probability density function pdf condition can be extended to the multiple description and multiresolution cases

geometry of manifolds with densities abstract we study the geometry of complete riemannian manifolds endowed with a weighted measure where the weight function is of quadratic growth assuming the associated bakry emery curvature is bounded from below we derive a new laplacian comparison theorem and establish various sharp volume upper and lower bounds we also obtain some splitting type results by analyzing the busemann functions in particular we show that a complete manifold with nonnegative bakry emery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions while some of our results are new even for the gradient ricci solitons the novelty here is that only a lower bound of the bakry emery curvature is involved in our analysis

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

boundary conditions and inversion indentities for solvable lattice models with a sublattice symmetry the transfer matrices of solvable lattice spin models are considered under general boundary conditions associated with the symmetry of the models it is shown how the full symmetry can be exploited for systems having a sublattice symmetry by considering an extended model where the states distinguish the sublattices in this extended scheme the parity of the number of sites is tied to the boundary conditions the inversion identities can be extended and be used to obtain the full operator content of the critical models as an application the complete spectra of the transfer matrices for the eight vertex model at the decoupling point are calculated

tracking the best linear predictor in most on line learning research the total on line loss of the algorithm is compared to the total loss of the best off line predictor u from a comparison class of predictors we call such bounds static bounds the interesting feature of these bounds is that they hold for an arbitrary sequence of examples recently some work has been done where the predictor ut at each trial t is allowed to change with time and the total on line loss of the algorithm is compared to the sum of the losses of ut at each trial plus the total cost for shifting to successive predictors this is to model situations in which the examples change over time and different predictors from the comparison class are best for different segments of the sequence of examples we call such bounds shifting bounds they hold for arbitrary sequences of examples and arbitrary sequences of predictors naturally shifting bounds are much harder to prove the only known bounds are for the case when the comparison class consists of a sequences of experts or boolean disjunctions in this paper we develop the methodology for lifting known static bounds to the shifting case in particular we obtain bounds when the comparison class consists of linear neurons linear combinations of experts our essential technique is to project the hypothesis of the static algorithm at the end of each trial into a suitably chosen convex region this keeps the hypothesis of the algorithm well behaved and the static bounds can be converted to shifting bounds

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

λ hypersurfaces on shrinking gradient ricci solitons abstract we consider an orientable smooth non totally geodesic hypersurface properly immersed on the cylinder shrinking soliton s 2 k 1 k r l g f where g is the product metric and the weight function f is defined by f p x x 2 4 if has weighted weak index less than or equal to l 1 we show that splits off a linear space also we classify the f stable orientable complete hypersurfaces properly immersed on s 2 k 1 k r g f with f p t t 2 4 obtaining as result s 2 k 1 k t or s 2 k 1 k 1 r

li yau type and souplet zhang type gradient estimates of a parabolic equation for the v laplacian abstract in this paper we study a parabolic equation concerning v laplacian on complete riemannian manifold m v q x t t u x t a u x t on m 0 t where v is a vector field on m and v v we give a gradient estimate of li yau type for positive solutions of this equation as corollary we obtain a new gradient estimate in the case that a u a u log u we also prove a souplet zhang type gradient estimate for positive bounded solutions of this equation as corollary we obtain a gradient estimate for positive bounded solution of v q x t t u x t a u x t 1

on optimal causal coding of partially observed markov sources in single and multiterminal settings the optimal causal zero delay coding of a partially observed markov process is studied where the cost to be minimized is a bounded nonnegative additive measurable single letter function of the source and the receiver output a structural result is obtained extending witsenhausen s and walrand varaiya s structural results on optimal causal coders to more general state spaces and to a partially observed setting the decentralized multiterminal setup is also considered for the case where the source is an i i d process it is shown that an optimal solution to the decentralized causal coding of correlated observations problem is memoryless for markov sources a counterexample to a natural separation conjecture is presented

near optimal rates for limited delay universal lossy source coding we consider the problem of limited delay lossy coding of individual sequences here the goal is to design fixed rate compression schemes to minimize the normalized expected distortion redundancy relative to a reference class of coding schemes measured as the difference between the average distortion of the algorithm and that of the best coding scheme in the reference class in compressing a sequence of length t the best schemes available in the literature achieve an o t 1 3 normalized distortion redundancy relative to finite reference classes of limited delay and limited memory and the same redundancy is achievable up to logarithmic factors when the reference class is the set of scalar quantizers it has also been shown that the distortion redundancy is at least of order 1 t in the latter case and the lower bound can easily be extended to sufficiently powerful possibly finite reference coding schemes in this paper we narrow the gap between the upper and lower bounds and give a compression scheme whose normalized distortion redundancy is o ln t t relative to any finite class of reference schemes only a logarithmic factor larger than the lower bound the method is based on the recently introduced shrinking dartboard prediction algorithm a variant of exponentially weighted average prediction the algorithm is also extended to the problem of joint source channel coding over a known stochastic noisy channel and to the case when side information is also available to the decoder the wyner ziv setting the same improvements are obtained for these settings as in the case of a noiseless channel our method is also applied to the problem of zero delay scalar quantization where o ln t 1 t normalized distortion redundancy is achieved relative to the infinite class of scalar quantizers of a given rate almost achieving the known lower bound of order 1 t the computationally efficient algorithms known for scalar quantization and the wyner ziv setting carry over to our improved coding schemes presented in this paper

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

symmetry respecting real space renormalization for the quantum ashkin teller model we use a simple real space renormalization group approach to investigate the critical behavior of the quantum ashkin teller model a one dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously this approach which is based on exploiting the on site symmetry of the model has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the quantum transverse field ising model our investigation explores this approach in more generality in a model in which the critical behavior has a richer structure but which reduces to the simpler ising case at a special point we demonstrate that the correlation length critical exponent as predicted from this real space renormalization group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality near the ising special point the error in the estimated critical exponent from this simple method is comparable to that of numerically intensive simulations based on much more sophisticated methods although the accuracy decreases away from the decoupled ising model point

asymptotically optimal low complexity sequential lossless coding for piecewise stationary memoryless sources i the regular case the lower bound on the redundancy for lossless universal coding of regular memoryless sources with a bounded number of abrupt changes in the statistics is shown to be asymptotically achievable using a fixed per letter computational complexity sequential compression scheme with fixed storage complexity the scheme which outperforms any other known fixed complexity scheme when regularity conditions hold is presented and its redundancy is upper bounded although the upper bounds are merely asymptotic simulation results show that even for relatively short sequences the redundancy obtained by asymptotically optimal schemes of higher complexity can still be achieved with fixed per letter complexity furthermore in practice a fixed complexity scheme based on the proposed scheme can in most cases achieve optimal redundancy even when the regularity conditions do not hold

monotonicity formulas via the bakry emery curvature abstract we prove several monotone formulas for smooth metric measure spaces with nonnegative m bakry emery curvature first we define a f r and v f r for the weighted green function and establish the monotone formulas these formulas are parallel to the weighted volume comparison theorem on smooth metric measure spaces with nonnegative m bakry emery curvature second we define the weighted nash entropy fisher information and perelman s entropy then we get the monotone formulas along the weighted heat equation via the m bakry emery curvature

the bakry emery ricci tensor and its applications to some compactness theorems let m g be a complete and connected riemannian manifold of dimension n by using the bakry emery ricci curvature tensor on m we prove two theorems which correspond to the myers compactness theorem

on the design of optimal noisy channel scalar quantizer with random index assignment the general approach in a noisy channel scalar quantizer design is an iterative descent algorithm which guarantees only a locally optimal solution while sufficient conditions under which the local optimum becomes a global optimum are known in the noiseless channel case such sufficient conditions were not derived for the noisy counterpart moreover efficient globally optimal design techniques for general discrete distributions in the noiseless case exist however they seem not to extend to the noisy scenario when a fixed index assignment is assumed recently the design of noisy channel scalar quantizer with random index assignment ria was proposed using a locally optimal iterative algorithm in this paper we derive sufficient conditions for the uniqueness of a local optimum which thus guarantee the global optimality of the solution these sufficient conditions are satisfied for a log concave probability density function which is additionally symmetric around its mean furthermore we show that assuming an ria the globally optimal design for general discrete sources can also be carried out efficiently

toroidal boundary conditions in n state quantum chains the author considers one dimensional n state quantum chains having as symmetry an abelian group a of order n generalised ising chain these hamiltonians depend on 2 n 1 coupling constants for special values of the coupling constants the symmetry of the quantum chains is larger than a corresponding to a in general non abelian group g with m elements m n for each element of g he finds explicitly a boundary condition which leaves the hamiltonian translationally invariant this problem is relevant in finite size scaling studies at the critical point of a second order phase transition

quantizer monotonicities and globally optimal scalar quantizer design new monotonicity properties of optimal scalar quantizers are discussed these monotonicities reveal a globally optimal scalar quantizer structure depending on the probability mass functions and on the number of quantizer levels by incorporating the monotone quantizer structure into a dynamic programming process the time complexities of previous algorithms for designing globally optimal scalar quantizers can be significantly reduced for very general classes of distortion measures

efficient on line schemes for encoding individual sequences with side information at the decoder we present adaptive on line schemes for lossy encoding of individual sequences under the conditions of the wyner ziv wz problem in the first part of this paper a set of fixed rate scalar source codes with zero delay is presented we propose a randomized on line coding scheme which achieves asymptotically and with high probability the performance of the best source code in the set uniformly over all source sequences efficient algorithms for implementing this scheme for small and large sets of encoders are presented in the second part of this work we generalize our results to the case of variable rate coding a set of variable rate scalar source codes is presented this time the performance is measured by the lagrangian cost lc which is defined as a weighted sum of the distortion and the length of the encoded sequence efficient algorithms for implementing the generalized on line coding scheme are presented we then consider the special case of lossless variable rate coding an on line scheme which uses huffman codes is presented we show that this scheme can be implemented efficiently using the same graphic methods from the first part finally combining the results from former sections we build a generalized efficient algorithm for a structured set of variable rate encoders

bakry emery curvature and model spaces in sub riemannian geometry we prove comparison theorems for the sub riemannian distortion coefficients appearing in interpolation inequalities these results which are equivalent to a sub laplacian comparison theorem for the sub riemannian distance are obtained by introducing a suitable notion of sub riemannian bakry emery curvature the model spaces for comparison are variational problems coming from optimal control theory as an application we establish the sharp measure contraction property for 3 sasakian manifolds satisfying a suitable curvature bound

conformal invariants associated to a measure in this note we study some conformal invariants of a riemannian manifold mn g equipped with a smooth measure m in particular we show that there is a natural definition of the ricci and scalar curvatures associated to such a space both of which are conformally invariant we also adapt the methods of fefferman and graham fefferman c graham c r 1985 asterisque numero hors serie 95 116 and graham jenne mason and sparling graham c r jenne r mason l j sparling g a j 1992 j london math soc 46 557 565 to construct families of conformally covariant operators defined on these spaces certain variational problems in this setting are considered including a generalization of the einstein hilbert action

twisted and conical kahler ricci solitons on fano manifolds in this paper we first study the relationship between the existence of twisted kahler ricci solitons and the properness of modified twisted k energy approximating by a sequence of smooth twisted kahler ricci solitons we obtain an existence result of conical kahler ricci solitons

the caffarelli kohn nirenberg inequalities and manifolds with nonnegative weighted ricci curvature we prove that n dimensional n 3 complete and non compact metric measure spaces with non negative weighted ricci curvature in which some caffarelli kohn nirenberg type inequality holds are close to the model metric measure n space i e the euclidean metric n space

on lfp spectrum and τ quasi einstein metric abstract in this paper we study l f p spectrum estimates for some smooth potential function f on complete noncompact riemannian manifolds via the bakry emery curvature as its applications we give the upper bound estimate of the l f 2 spectrum for the complete noncompact quasi einstein metric we give an example to show the sharpness of our estimate we also get a lower bound estimate when 0 1 and 0

the weighted isoperimetric type and sobolev type inequalities for hypersurfaces in carnot groups abstract this paper mainly studies hypersurfaces immersed in a sub riemannian step k carnot group g with a density e f where f is a smooth function on g we give the first variation formula about the weighted horizontal area of hypersurfaces furthermore on the base of some work we derive the weighted isoperimetric type and sobolev type inequalities for hypersurfaces

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

how to use expert advice we analyze algorithms that predict a binary value by combining the predictions of several prediction strategies called experts our analysis is for worst case situations i e we make no assumptions about the way the sequence of bits to be predicted is generated we measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence where the expectation is taken with respect to the randomization in the predictins we show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert and we give efficient algorithms that achieve this our upper and lower bounds have matching leading constants in most cases we then show how this leads to certain kinds of pattern recognition learning algorithms with performance bounds that improve on the best results currently know in this context we also compare our analysis to the case in which log loss is used instead of the expected number of mistakes

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

fast encoder optimization for multi resolution scalar quantizer design the design of optimal multi resolution scalar quantizers using the generalized lloyd method was proposed by brunk and farvardin for the case of squared error distortion since the algorithm details heavily rely on the quadratic expression of the error function its extension to general error functions faces some challenges especially at the encoder optimization step in this work we show how these challenges can be overcome for any convex difference distortion measure under the assumption that all quantizer cells are convex i e intervals and present an efficient algorithm for optimal encoder partition computation the proposed algorithm is faster than the algorithm used by brunk and farvardin moreover it can also be applied to channel optimized and to multiple description scalar quantizer design with squared error distortion and it outperforms in speed the previous encoder optimization algorithms proposed for these problems

coding for a binary independent piecewise identically distributed source two weighting procedures are presented for compaction of output sequences generated by binary independent sources whose unknown parameter may occasionally change the resulting codes need no knowledge of the sequence length t i e they are strongly sequential and also the number of parameter changes is unrestricted the additional transition redundancy of the first method was shown to achieve the merhav lower bound i e log t bits per transition for the second method we could prove that additional transition redundancy is not more than 3 2 log t bits per transition which is more than the merhav bound however the storage and computational complexity of this method are also more interesting than those of the first method simulations show that the difference in redundancy performance between the two methods is negligible

design of optimal scalar quantizer for sequential coding of correlated sources this paper addresses the design of a sequential scalar quantizer ssq for finite alphabet correlated sources in the fixed rate fr and entropy constrained ec cases the optimization problem is formulated as the minimization of a weighted sum of distortions and rates the proposed solution is globally optimal for the class of ssqs with convex cells and is based on solving the minimum weight path mwp problem in the ec case respectively a length constrained mwp problem in the fr case in a series of weighted directed acyclic graphs the asymptotic time complexity is o k 1 2 k 2 2 where k 1 and k 2 are the respective sizes of the alphabets of the two sources additionally it is proved that by applying the proposed algorithms to discretizations of correlated sources with continuous joint probability density function the performance approaches that of the optimal ec ssq respectively fr ssq with convex cells for the original sources as the accuracy of the discretization increases extensive experiments performed with correlated gaussian sources validate the effectiveness in practice of the proposed approach in approximating the optimal ssq for the case of continuous alphabet sources

isoperimetric rigidity and distributions of 1 lipschitz functions abstract we prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal then the space is foliated by minimal geodesics where the observable variance is defined to be the supremum of the variance of 1 lipschitz functions on the space our result can be considered as a variant of cheeger gromoll s splitting theorem and also of cheng s maximal diameter theorem as an application we obtain a new isometric splitting theorem for a complete weighted riemannian manifold with a positive bakry emery ricci curvature

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

perelman s entropy formula for the witten laplacian on riemannian manifolds via bakry emery ricci curvature in this paper we study perelman s mathcal w entropy formula for the heat equation associated with the witten laplacian on complete riemannian manifolds via the bakry emery ricci curvature under the assumption that the m dimensional bakry emery ricci curvature is bounded from below we prove an analogue of perelman s and ni s entropy formula for the mathcal w entropy of the heat kernel of the witten laplacian on complete riemannian manifolds with some natural geometric conditions in particular we prove a monotonicity theorem and a rigidity theorem for the mathcal w entropy on complete riemannian manifolds with non negative m dimensional bakry emery ricci curvature moreover we give a probabilistic interpretation of the mathcal w entropy for the heat equation of the witten laplacian on complete riemannian manifolds and for the ricci flow on compact riemannian manifolds

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

splitting theorems for finsler manifolds of nonnegative ricci curvature we investigate the structure of a finsler manifold of nonnegative weighted ricci curvature including a straight line and extend the classical cheeger gromolllichnerowicz splitting theorem such a space admits a diffeomorphic measurepreserving splitting in general as for a special class of berwald spaces we can perform the isometric splitting in the sense that there is a one parameter family of isometries generated from the gradient vector field of the busemann function a betti number estimate is also given for berwald spaces

conformal invariance studies of the baxter wu model and a related site colouring problem the partition function of the baxter wu model is exactly related to the generating function of a site colouring problem on a hexagonal lattice we extend the original bethe ansatz solution of these models in order to obtain the eigenspectra of their transfer matrices in finite geometries and general toroidal boundary conditions the operator content of these models is studied by solving numerically the bethe ansatz equations and by exploring conformal invariance since the eigenspectra are calculated for large lattices the corrections to finite size scaling are also calculated

integral pinched gradient shrinking ρ einstein solitons abstract the gradient shrinking einstein soliton is a triple m n g f such that r i j f i j r g i j where m n g is a riemannian manifold 0 r 0 and f is the potential function on m n in this paper using algebraic curvature estimates and the yamabe sobolev inequality we prove some integral pinching rigidity results for compact gradient shrinking einstein solitons

conformal invariance and line defects in the two dimensional ising model the quantum ising chain with p equidistant defects is studied the exact form of the hamiltonian is found for an infinite number of sites using conformal invariance generalised corner exponents are computed and found to depend continuously on the defect strengths realisations of extended conformal algebras in the hamiltonian spectrum are obtained