a satellite of the grand furuta inequality and its application abstract the grand furuta inequality has the following satellite sgf t 0 1 given as a mean theoretic expression a b 0 t 0 1 a r t 1 t r p t s r a t s b p b for r t p s 1 where is the geometric mean and s s 0 1 is a formal extension of it is shown that sgf t 0 1 has the lowner heinz property i e sgf t 1 implies sgf t for every t 0 1 furthermore we show that a recent further extension of gfi by furuta himself has also the lowner heinz property

on operator order and chaotic operator order for two operators abstract in this paper we characterize operator order a b o and chaotic operator order log a log b for positive and invertible operators a and b in terms of operator inequalities via the furuta inequality and operator equalities due to the douglas s majorization and factorization related results are obtained which include generalizations and characterizations of some well known results

nonlinear decision weights in choice under uncertainty in most real world decisions consequences are tied explicitly to the outcome of events previous studies of decision making under uncertainty have indicated that the psychological weight attached to an event called a decision weight usually differs from the probability of that event we investigate two sources of nonlinearity of decision weights subadditivity of probability judgments and the overweighting of small probabilities and underweighting of medium and large probabilities these two sources of nonli nearity are combined into a two stage model of choice under uncertainty in the first stage events are taken into subjective probability judgments and the second stage takes probability judgments into decision weights we then characterize the curvature of the decision weights by extending a condition employed by wu and gonzalez 1996 in the domain of risk to the domain of uncertainty and show that the nonlinearity of decision weights can be decomposed into subadditivity of probability judgments and the curvature of the probability weighting function empirical tests support the proposed two stage model and indicate that decision weights are concave then convex more specifically our results lend support for a new property of subjective probability ju dgments interior additivity subadditive at the boundaries but additive away from the boundaries and show that the probability weighting function is inverse s shaped as in wu and gonzalez 1996

indirect and direct gaussian distributed source coding problems we consider the distributed source coding system of l correlated gaussian sources y l l 1 2 ldots l which are noisy observations of correlated gaussian remote sources x k k 1 2 ldots k we assume that y l t y 1 y 2 ldots y l is an observation of the source vector x k t x 1 x 2 ldots x k having the form y l ax k n l where a is a l times k matrix and n l t n 1 n 2 ldots n l is a vector of l independent gaussian random variables also independent of x k in this system l correlated gaussian observations are separately compressed by l encoders and sent to the information processing center we study the remote source coding problem where the decoder at the center attempts to reconstruct the remote source x k we consider three distortion criteria based on the covariance matrix of the estimation error on x k for each of those three criteria we derive explicit inner and outer bounds of the rate distortion region next in the case of k l and a i l we study the multiterminal source coding problem where the decoder wishes to reconstruct the observation y l x l n l to investigate this problem we shall establish a result that provides a strong connection between the remote source coding problem and multiterminal source coding problem using this result we derive several new partial solutions to the multiterminal source coding problem

the gaussian ceo problem for scalar sources with arbitrary memory in this paper we consider the achievable sum rate distortion tradeoff for the gaussian central estimation officer ceo problem with a scalar source having arbitrary memory we describe how the arbitrary memory problem can be fully characterized by using known results for the vector ceo problem and then we formulate the variational problem of minimizing the sum rate subject to a distortion constraint to solve the problem we extend the conventional lagrange method and show that if the solution exists it should consist of a zero part and a non zero part where the non zero part is determined by solving a set of euler equations by calculating the second variation of the min sum rate problem a sufficient condition is also found that can be used to determine if the necessary solution results in the minimal sum rate the special case of two terminals is examined in detail and it is shown that an analytical solution is possible in this case analysis and discussion with examples are provided to illustrate the theoretical results the general solution obtained in this paper is shown to be compatible with the previous results for cases such as the problem of rate evaluation for sources without memory

on a trace inequality abstract let a and b be positive operators and p s 0 assume either 1 a b and max p 2 1 2 or 2 a and b are invertible with log a log b and then for any continuous increasing function on with 0 0 the trace inequality tr a a bpa sa tr a p 2 s 2 holds this generalizes both a trace inequality due to kosaki and one due to furuta

spectral inequalities for matrix exponentials abstract this note generalizes an inequality of bernstein as follows if c is an n n complex matrix and c k is the k th compound of c 1 k n n n k and if the eigenvalues of c k are labeled in order of decreasing magnitude 1 c k 2 c k n c k define the partial trace tr k i c by tr k i c h 1 i h c k i 1 n then for any complex n n matrix a tr k i e a e a tr k i e a a i 1 n with equality if a is normal or k n a spectral inequality of k fan is also generalized through the use of compound matrices

the ceo problem multiterminal source coding we consider a new problem in multiterminal source coding motivated by the following decentralized communication estimation task a firm s chief executive officer ceo is interested in the data sequence x t sub t 1 sup spl infin which cannot be observed directly perhaps because it represents tactical decisions by a competing firm the ceo deploys a team of l agents who observe independently corrupted versions of x t sub t 1 sup spl infin because x t is only one among many pressing matters to which the ceo must attend the combined data rate at which the agents may communicate information about their observations to the ceo is limited to say r bits per second if the agents were permitted to confer and pool their data then in the limit as l spl rarr spl infin they usually would be able to smooth out their independent observation noises entirely then they could use their r bits per second to provide the ceo with a representation of x t with fidelity d r where d spl middot is the distortion rate function of x t in particular with such data pooling d can be made arbitrarily small if r exceeds the entropy rate h of x t suppose however that the agents are not permitted to convene agent i having to send data based solely on his own noisy observations y sub i t we show that then there does not exist a finite value of r for which even infinitely many agents can make d arbitrarily small furthermore in this isolated agents case we determine the asymptotic behavior of the minimal error frequency in the limit as l and then r tend to infinity

an araki lieb thirring inequality for geometrically concave and geometrically convex functions for positive definite matrices a and b the araki lieb thirring inequality amounts to an eigenvalue log submajorisation relation for fractional powers atbt w log t ab 0 t 1 while for t 1 the reversed inequality holds in this paper i generalise this inequality replacing the fractional powers xt by a larger class of functions i prove that a continuous non negative function f with domain dom f 0 x0 for some positive x0 possibly infinity satisfies f a f b w log f2 1 2 ab for all positive semidefinite a and b with spectrum in dom f if and only if f is geometrically concave and 0 xf x f x for all x dom f the reversed inequality holds for a continuous non negative function f if and only if f is geometrically convex and xf x f x for all x dom f as an application i obtain a complementary inequality to the golden thompson inequality

interior additivity and subjective probability assessment of continuous variables one of the goals of psychological research on subjective probability judgment is to develop prescriptive procedures that can improve such judgments in this paper our aim is to reduce partition dependence a judgmental bias that arises from the particular way in which a state space is partitioned for the purposes of making probability judgments we explore a property of subjective probabilities called interior additivity ia our story begins with a psychological model of subjective probability judgment that exhibits ia the model is a linear combination of underlying support for the event in question and a term that reflects a prior belief that all elements in the state space partition are equally likely the model is consistent with known properties of subjective probabilities such as binary complementarity subadditivity and partition dependence and has several additional properties related to ia we present experimental evidence to support our model the model further suggests a simple prescriptive method based on ia that decision and risk analysts can use to reduce partition dependence and we present preliminary empirical evidence demonstrating the effectiveness of the method

hermite hadamard s type inequalities for operator convex functions some hermite hadamard s type inequalities for operator convex functions of selfadjoint operators in hilbert spaces are given applications for particular cases of interest are also provided

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

on the sum rate of gaussian multiterminal source coding new proofs and results we show that the lower bound on the sum rate of the direct and indirect gaussian multiterminal source coding problems can be derived in a unified manner by exploiting the semidefinite partial order of the distortion covariance matrices associated with the minimum mean squared error mmse estimation and the so called reduced optimal linear estimation through which an intimate connection between the lower bound and the berger tung upper bound is revealed we give a new proof of the minimum sum rate of the indirect gaussian multiterminal source coding problem i e the gaussian ceo problem for the direct gaussian multiterminal source coding problem we derive a general lower bound on the sum rate and establish a set of sufficient conditions under which the lower bound coincides with the berger tung upper bound we show that the sufficient conditions are satisfied for a class of sources and distortion constraints in particular they hold for arbitrary positive definite source covariance matrices in the high resolution regime in contrast with the existing proofs the new method does not rely on shannon s entropy power inequality

function order of positive operators based on the mond pecaric method we shall show function order preserving operator inequalities under general setting based on kantorovich type inequalities for convex functions due to mond pecaric let a and b be positive operators on a hilbert space h satisfying mi b mi 0 let f t be a continuous convex function on m m if g t is a continuous increasing convex function on m m sp a then for a given 0 a b 0implies g a i f b where maxm t m f m f m f m t m m m g t as applications we shall extend kantorovich type operator inequalities by furuta yamazaki and yanagida and present operator inequalities on the usual order and the chaotic order via ky fan furuta constant among others we show the following inequality if a b 0 and mi b mi 0 then mp 1mq 1aq q 1 q 1qq mp mp q m m mmp mmp q 1aq bp holds for all p 1 and q 1 such that qmp 1 mp mpm m qmp 1

the non regular ceo problem we consider the ceo problem for non regular source distributions such as uniform or truncated gaussian a group of agents observe independently corrupted versions of data and transmit coded versions over rate limited links to a ceo the ceo then estimates the underlying data based on the received coded observations agents are not allowed to convene before transmitting their observations this formulation is motivated by the practical problem of a firm s ceo estimating non regular beliefs about a sequence of events before acting on them agents observations are modeled as jointly distributed with the underlying data through a given conditional probability density function we study the asymptotic behavior of the minimum achievable mean squared error distortion at the ceo in the limit when the number of agents l and the sum rate r tend to infinity we establish a 1 r 2 convergence of the distortion an intermediate regime of performance between the exponential behavior in discrete ceo problems berger zhang and viswanathan 1996 and the 1 r behavior in gaussian ceo problems viswanathan and berger 1997 achievability is proved by a layered architecture with scalar quantization distributed entropy coding and midrange estimation the converse is proved using the bayesian chazan zakai ziv bound

the off diagonal block of a ppt matrix abstract we give an operator inequality involving the off diagonal block of a positive partial transpose matrix and the geometric mean of its diagonal blocks some new eigenvalue inequalities are derived nicely completing the existing ones

extension of the furuta inequality and ando hiai log majorization abstract ando and hiai have obtained many excellent log majorization results for power means of positive semidefinite matrices one of these results ensures matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson inequality firstly we show an extension of the furuta inequality by using itself secondary by using techniques due to ando and hiai and also using this extension of the furuta inequality we extend some results on log majorization

a general method for comparing probability assessors a probability assessor or forecaster is a person who assigns subjective probabilities to events which will eventually occur or not occur in this paper a general method for comparing forecasters after a finite number of trials is introduced the general method is proven to include calculating all proper scoring rules as special cases it also includes comparison of forecasters in all simple two decision problems as special cases the relationship between the general method and calibration is also explored the general method is also translated into a method for deciding who will give better forecasts in the future

an order inequality characterizing invariant barycenters on symmetric cones abstract this paper is concerned with invariant and contractive barycenters on the wasserstein space of probability measures on metric spaces of non positive curvature where the center of gravity also called the cartan barycenter is the canonical barycenter on hadamard spaces we establish an order inequality of probability measures on partially ordered symmetric spaces of non compact type namely symmetric cones self dual homogeneous cones characterizing the cartan barycenter among other invariant and contractive barycenters the derived inequality and partially ordered structures on the probability measure space lead also to significant results on norm inequalities including the ando hiai inequality for probability measures on symmetric cones

furuta s inequality and its application to ando s theorem abstract as a continuation of preceding notes we discuss furuta s inequality under the chaotic order defined by log a log b for positive invertible operators a and b which is applied to a generalization of ando s theorem consequently we obtain furuta s inequality under the chaotic order

rate region of the quadratic gaussian two encoder source coding problem we determine the rate region of the quadratic gaussian two encoder source coding problem this rate region is achieved by a simple architecture that separates the analog and digital aspects of the compression furthermore this architecture requires higher rates to send a gaussian source than it does to send any other source with the same covariance our techniques can also be used to determine the sum rate of some generalizations of this classical problem our approach involves coupling the problem to a quadratic gaussian ldquoceo problem rdquo

hermite hadamard s type inequalities for convex functions of selfadjoint operators in hilbert spaces abstract some hermite hadamard s type inequalities for convex functions of selfadjoint operators in hilbert spaces under suitable assumptions for the involved operators are given applications in relation with the celebrated holder mccarthy s inequality for positive operators and ky fan s inequality for real numbers are given as well

extensions of inequalities for unitarily invariant norms via log majorization abstract a capital letter means n n matrix t is said to be positive definite denoted by t 0 if t is positive semidefinite and invertible we shall show the following central results via log majorization obtained by an order preserving operator inequality theorem if a 0 and b 0 then for 0 1 t 0 1 and r t a 1 t 2 a t b a 1 t 2 s log w 2 a r b s a w 2 holds for 1 r t 1 t 1 s 1 where w 1 s r 1 t s our result extends the following recent elegant inequality by matharu and aujlia let a b be positive definite and 0 1 then i 1 k j a 1 b i 1 k j a b 1 k n also some results associated with log majorization are shown

ando hiai inequality and furuta inequality abstract let a b 0 the positive operators on a hilbert space ando hiai inequality ah and furuta inequality f are given as follows ah a b 0 a b i a r b r i for r 1 and f a b 0 a r 1 r p r b p b a we show these inequalities imply each other

operator inequality implying generalized bebiano lemos providencia one a capital letter means n n matrix very recently fujii et al show for every a b 0 and p 1 a12as2bp sas21pa12p 1 s p s log a1 s2b1 sa1 s2 holds for any s 0 in fact yields bebiano lemos providencia inequality a12 as2bsas2 tsa12 log a1 t2bta1 t2 for s t 0 as an extension of we show the following result the following i and ii hold and they are equivalent i for every a 0 b 0 0 1 and each t 0 1 and any real number q 0 aq2a12ba12 aq2h log aq 1 t r 2a qr2a1 qt2ba1 qt2sa qr2 aq 1 t r 2 holds for s 1 and r t where 1 t r 1 t s r and h 1 t r s 1 t s r ii if a b 0 with a 0 then for t 0 1 and p 1 a1 t r ar2a t2bpa t2sar21 t r p t s r for s 1 and r t i implies in fact put t 0 s 1 qr 1 in i then aq2 a12ba12 aq2 1 q q log a1 q2b 1 q qa1 q2 holds and finally replace b by bp s a by as by 1p and also replace q by 1s then we have the desired

trace inequalities for logarithms and powers of j hermitian matrices abstract spectral inequalities are stated for the trace of the exponential or the logarithmic of certain j hermitian matrices j i r i n r 0 r n the obtained inequalities are established in the context of indefinite inner product spaces and they are known to be valid for hilbert space operators or operator algebras

on an operator kantorovich inequality for positive linear maps abstract we improve the operator kantorovich inequality as follows let a be a positive operator on a hilbert space with 0 m a m then for every unital positive linear map a 1 2 m m 2 4 m m 2 a 2 as a consequence a 1 a a a 1 m m 2 2 m m

distributed source coding of correlated gaussian remote sources we consider the distributed source coding system for l correlated memoryless gaussian remote sources specified with l correlated gaussian random variables xi i 1 2 l we deal with the case where each of l distributed encoders cannot directly observe the source xi but its noisy version yi xi ni here ni i 1 2 l are independent additive l gaussian noises also independent of xi i 1 2 l on this coding system the determination problem of the rate distortion region remains open in general in this paper we derive explicit outer and inner bounds of the rate distortion region we further find an explicit sufficient condition for those two bounds to match we also study the sum rate part of the rate distortion region when the correlation has some symmetrical property we derive an explicit upper bound of the sum rate part from the inner bounds of the rate distortion region on a lower bound we derive a new explicit bound by making full use of the symmetrical property of the correlation we further derive an explicit necessary and sufficient condition for the lower bound to coincide with the upper bound

operator inequalities from a general theorem to concrete inequalities abstract the aim of this paper is to give a method to extract concrete inequalities from a general theorem which is established by making use of majorization relation between functions by this method we can get a lot of inequalities among others we extend furuta inequality as follows let f i g j be positive operator monotone functions on 0 and put k t t r 0 f 1 t r 1 f m t r m h t t p 0 g 1 t p 1 g n t p n where p 0 1 and r i 0 p j 0 then 0 a c b implies for 0 1 r 0 p r 0 with p p 0 p n k c 1 2 h a k c 1 2 k c 1 2 h c k c 1 2 k c 1 2 h b k c 1 2 moreover we show log c 1 2 e a c 1 2 log c 1 2 e c c 1 2 log c 1 2 e b c 1 2 provided c is invertible we also refer to operator geometric mean

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

some ways of constructing furuta type inequalities abstract let h be an operator mean associated with an operator monotone function h and let a b be positive operators we investigate the following furuta type inequality for some fixed continuous function h a b 0 a r r x h r b r 1 where x h is the positive solution of h x b 1 the map h r r plays a central role in constructing a furuta type inequality similar to the role of the map p r r 1 p r in the following part of the furuta inequality a b 0 p r 1 b r 2 a p b r 2 1 r p r b p r 1 r p r we obtain this map h r r from the following ando hiai type inequality a b 0 a h b 1 a r h b r 1 r 1

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

a complement of the ando hiai inequality abstract in this paper we present a complement of a generalized ando hiai inequality due to fujii and kamei m fujii e kamei ando hiai inequality and furuta inequality linear algebra appl 416 2006 541 545 let a and b be positive operators on a hilbert space h such that 0 m 1 a m 1 and 0 m 2 b m 2 for some scalars m i m i i 1 2 and let 0 1 put h i m i m i for i 1 2 then for each 0 r 1 and s 1 a r r 1 s r b s k h 1 s h 2 s r s 1 s 1 s r h 2 1 s s r 1 s r a b rs 1 s r where a b a 1 2 a 1 2 ba 1 2 a 1 2 is the geometric mean and a generalized kantorovich constant k h p is defined for h 0 as k h p h p h p 1 h 1 p 1 p h p 1 h p h p for all real numbers p r

generalizations of kosaki trace inequalities and related trace inequalities on chaotic order abstract by using an extension of the furuta inequality and following kosaki s nice technique we show generalizations of trace inequalities by kosaki and related trace inequalities on chaotic order i e log a log b

on some operator inequalities a real valued continuous function f 2 defined on a finite or infinite interval of the real line is said to be operator monotone if for any pair of bounded selfadjoint operators a b on an infinite dimensional hilbert space a b implies f a f b here f a and f b are defined according to the usual functional calculus and refers to the order relation induced by the cone of positive semi definite operators operator monotony is quite a restrictive condition in fact for p 0 the function f 2 2 p is operator monotone on the half line 0 c if and only if 0 1 the function f 2 2 p possesses some property akin to operator monotony in this direction furuta i 3 recently observed that if r 0 p 0 q 1 satisfies 1 2r q p 2r then a b 0 implies a tp 2 q arbpa r l if p 2r and q 2 the requirement is automatically satisfied hence a p ap 2bpap 2 i 2 whenever a b 0 when a is invertible this means that

operator functions associated with furuta s inequality abstract we strengthen a recent monotonicity theorem on an operator function due to furuta which is an extension of an operator inequality known as furuta s inequality this is done by using means of operators it leads to an alternative proof of ando s result on the geometric mean

a survey of jensen type inequalities for functions of selfadjoint operators in hilbert spaces the main aim of the present paper is to survey different inequalities for convex functions of selfadjoint operators in hilbert spaces which have appeared recently in the literature applications for some particular convex functions of interest are also given

subjective probability assessment in decision analysis partition dependence and bias toward the ignorance prior decision and risk analysts have considerable discretion in designing procedures for eliciting subjective probabilities one of the most popular approaches is to specify a particular set of exclusive and exhaustive events for which the assessor provides such judgments we show that assessed probabilities are systematically biased toward a uniform distribution over all events into which the relevant state space happens to be partitioned so that probabilities are partition dependent we surmise that a typical assessor begins with an ignorance prior distribution that assigns equal probabilities to all specified events then adjusts those probabilities insufficiently to reflect his or her beliefs concerning how the likelihoods of the events differ in five studies we demonstrate partition dependence for both discrete events and continuous variables studies 1 and 2 show that the bias decreases with increased domain knowledge studies 3 and 4 and that top experts in decision analysis are susceptible to this bias study 5 we relate our work to previous research on the pruning bias in fault tree assessment e g fischhoff et al 1978 and show that previous explanations of pruning bias enhanced availability of events that are explicitly specified ambiguity in interpreting event categories and demand effects cannot fully account for partition dependence we conclude by discussing implications for decision analysis practice

distributed compression of linear functions partial sum rate tightness and gap to optimal sum rate we consider the problem of distributed compression of the difference z y1 cy2 of two jointly gaussian sources y1 and y2 under an mse distortion constraint d on z the rate region for this problem is unknown if the correlation coefficient and the weighting factor c satisfy c 0 inspired by ahlswede and han s scheme for the problem of distributed compression of the modulo 2 sum of two binary sources we first propose a hybrid random structured coding scheme that is capable of saving the sum rate over both the random quantize and bin qb coding scheme and krithivasan and pradhan s structured lattice coding scheme the main idea is to use a random coding component in the first layer to adjust the source correlation so that the structured coding component in the second layer can be more efficient with the outputs from the first layer as decoder side information we then provide a new sum rate lower bound for the problem in hand by connecting it to the gaussian two terminal source coding problem with covariance matrix distortion constraint our lower bound not only improves existing bounds in many cases but also allows us to prove sum rate tightness of the qb scheme when c is either relatively small or large and d is larger than some threshold furthermore our lower bound enables us to show that our new hybrid scheme performs within two b s from the optimal sum rate for all values of c and d

operator monotone functions which are defined implicitly and operator inequalities the function t 0 1 is operator monotone on 0 t this is known as the lowner heinz inequality however not too many examples of concrete operator monotone functions are known so far we will systematically seek operator monotone functions which are defined implicitly this investigation is new and our method seems to be powerful we will actually find a family of operator monotone functions which includes t 0 1 moreover by constructing one parameter families of operator monotone functions we will get many operator inequalities especially we will extend the furuta inequality and the exponential inequality of ando

on the generalized gaussian ceo problem this paper considers a distributed source coding dsc problem where l encoders observe noisy linear combinations of k correlated remote gaussian sources and separately transmit the compressed observations to the decoder to reconstruct the remote sources subject to a sum distortion constraint this dsc problem is referred to as the generalized gaussian ceo problem since it can be viewed as a generalization of the quadratic gaussian ceo problem where the number of remote source k 1 first we provide a new outer region obtained using the entropy power inequality and an equivalent argument in the sense of having the same rate distortion region and berger tung inner region among a certain class of generalized gaussian ceo problems we then give two sufficient conditions for our new outer region to match the inner region achieved by berger tung schemes where the second matching condition implies that in the low distortion regime the berger tung inner rate region is always tight while in the high distortion regime the same region is tight if a certain condition holds the sum rate part of the outer region is also studied and shown to meet the berger tung sum rate upper bound under a certain condition which is obtained using the karush kuhn tucker conditions of the underlying convex semidefinite optimization problem and is in general weaker than the aforesaid two for rate region tightness

the rate distortion function for the quadratic gaussian ceo problem a new multiterminal source coding problem called the ceo problem was presented and investigated by berger zhang and viswanathan recently viswanathan and berger have investigated an extension of the ceo problem to gaussian sources and call it the quadratic gaussian ceo problem they considered this problem from a statistical viewpoint deriving some interesting results in this paper we consider the quadratic gaussian ceo problem from a standpoint of multiterminal rate distortion theory we regard the ceo problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given this viewpoint leads us to a complete solution to the problem we determine the tradeoff between the total amount of rate and squared distortion deriving an explicit formula of the rate distortion function the derived function has the form of a sum of two nonnegative functions one is a classical rate distortion function for single gaussian source and the other is a new rate distortion function which dominates the performance of the system for a relatively small distortion it follows immediately from our result that the conjecture of viswanathan and berger on the asymptotic behavior of minimum squared distortion for large rates is true

rate distortion theory for gaussian multiterminal source coding systems with several side informations at the decoder in this paper we consider the separate coding problem for l 1 correlated gaussian memoryless sources we deal with the case where l sources work as partial side information at the decoder for the reconstruction of the remaining source the determination problem of the rate distortion region for this system is the so called many help one problem and it has been known as a highly challenging problem for almost 20 years in this paper we give a partial solution to this problem we determine the rate distortion region in the case where the l sources working as partial side information are conditionally independent if the remaining source we wish to reconstruct is given the additive white gaussian noise ceo problem is a special case of this we also discuss the relation of the result previous results of ours

tsallis entropies and matrix trace inequalities in quantum statistical mechanics extensions of trace inequalities arising in statistical mechanics are derived in a straightforward and unified way from a variational characterization of the tsallis entropy namely one parameter extension of the thermodynamic inequality is presented and its equivalence to a generalized peierls bogoliubov inequality is stated in the sense that one may be obtained from the other and vice versa

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

inequalities for the wasserstein mean of positive definite matrices abstract we prove majorization inequalities for different means of positive definite matrices these include the cartan mean the karcher mean the log euclidean mean the wasserstein mean and the power mean

a proof via operator means of an order preserving inequality abstract we give an alternative and simplified proof via operator means of an order preserving inequality on a and b in the case a b 0 where an operator is a bounded linear operator on a hilbert space

a new majorization between functions polynomials and operator inequalities let p be the set of all non negative operator monotone functions defined on 0 and put p 1 h h 1 p then p p 1 p 1 and p 1 p 1 p 1 for a function h t and a strictly increasing function h we write

strong majorization for hermitian matrices we introduce and study an additive relation and a multiplicative relation for hermitian matrices the relation g a f means that f is the real part of some matrix similar to g the relation g m f expresses the property that the positive definite matrix f is the modulus of some matrix similar to the positive definite matrix g there is a close connection with the additive and multiplicative majorization relations for the corresponding spectra

a belief based account of decision under uncertainty we develop a belief based account of decision under uncertainty this model predicts decisions under uncertainty from i judgments of probability which are assumed to satisfy support theory and ii decisions under risk which are assumed to satisfy prospect theory in two experiments subjects evaluated uncertain prospects and assessed the probability of the respective events study 1 involved the 1995 professional basketball playoffs study 2 involved the movement of economic indicators in a simulated economy the results of both studies are consistent with the belief based account but violate the partition inequality implied by the classical theory of decision under uncertainty

mixed matrix operator inequalities we will show that if 0 a b c and s s t t 1 then a s bs t ct implies that ar 2 s bss t cst ar 2 ar 2 s bts t ctt ar 2 s r t r fort 1 t s 0 r 0 and a1 r ar 2 s bts t ctt ar 2 1 r t r fort 1 r 0 these are the extensions of inequalities shown by t furuta proc amer math soc 101 1987 85 111 1991 511 moreover we will give an inequality say mixed inequality which includes the above inequalities and the inequality due to hansen pedersen math ann 258 1982 229

matrix valued jensen s inequality and its application to eigenvalue analysis abstract we first establish jensen s inequality for lower semicontinuous convex functions from a hadamard space into an ordered hadamard space we apply jensen s inequality to matrix valued integrable functions on probability measure spaces and provide new results on matrix analysis and eigenvalue analysis by discovering a variety of lipschitz convex functions related to eigenvalue maps and lowner ordering of positive definite matrices

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

on a question of furuta on chaotic order the chaotic order a b among positive invertible operators a b 0 on a hilbert space is introduced by loga logb uchiyama s method brings us the furuta inequality for the chaotic order from the furuta inequality related to this furuta posed the following question for a b 0 a b if and only if q ar t ar 2a t 2bpa t 2sar 2 r t p t s r holds for all p 1 r t s 1 and t 0 1 recently he gave a counterexample to the only if part in this note we point out that condition q characterizes the operator order a b moreover q characterizes the spectral order by extending the bounds of t

monotonicity of order preserving operator functions abstract we discuss monotonicity of order preserving operator functions and related order preserving operator inequalities let a b 0 with a 0 t 0 1 and p 1 let f a 2 a 2 a t 2 b p a t 2 a 2 1 t p t a 2 we show that i f r w f r 1 f r s f r s for any s s 1 r t and 1 t p t w 1 ii f q s f t s f r s f r s for any r r t s 1 and t 1 q t these imply the following recent inequality due to kamei a t 1 t p t b p a t 2 f r s a t 2 for r t and s 1

the sum rate bound for a new class of quadratic gaussian multiterminal source coding problems in this paper we show tightness of the berger tung bt sum rate bound for a new class of quadratic gaussian multiterminal mt source coding problems dubbed bi eigen equal variance with equal distortion beev ed where the l l source covariance matrix has equal diagonal elements with two distinct eigenvalues and the l target distortions are equal let k k l be the number of repetitions of the larger eigenvalue the beev covariance structure allows us to connect k i i d virtual gaussian sources with the l given mt sources via an l k semiorthogonal transform whose rows have equal euclidean norm plus additive i i d gaussian noises resulting in the two sets of sources being mutually conditional i i d by relating the given mt source coding problem to a generalized gaussian ceo problem with the k virtual sources as remote sources and the l mt sources as observations we obtain a lower bound on the mt sum rate and show its achievability by bt schemes under the equal distortion constraints our beev ed class of quadratic gaussian mt source coding problems subsumes both the positive symmetric case considered by wagner et al and the negative symmetric case other examples including a subclass of sources with be circulant symmetric covariance matrices and equal distortion constraints are also provided to highlight tightness of the sum rate bound