complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

probabilistic metrization of fuzzy uniformities abstract modifying the structure of fuzzy uniform spaces introduced by lowen 5 we propose a further concept of fuzzy uniformities so called fuzzy t

an outer bound for the vector gaussian ceo problem we study the vector gaussian ceo problem where there are arbitrary number of agents each having a noisy observation of a vector gaussian source the goal of the agents is to describe the source to a central unit which wants to reconstruct the source within a given distortion the rate distortion region of the vector gaussian ceo problem is unknown in general here we provide an outer bound for the rate distortion region of the vector gaussian ceo problem we obtain our outer bound by evaluating an outer bound for the multiterminal source coding problem by means of a technique relying on the de bruijn identity and properties of the fisher information next we investigate the tightness of our outer bound although our outer bound is tight for certain cases we show that our outer bound does not provide the exact rate distortion region in general to this end we provide an example and show that the rate distortion region is strictly contained in our outer bound for this example

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

capacity region of gaussian mimo broadcast channels with common and confidential messages we study the two user gaussian multiple input multiple output mimo broadcast channel with common and confidential messages in this channel the transmitter sends a common message to both users and a confidential message to each user which needs to be kept perfectly secret from the other user we obtain the entire capacity region of this channel we also explore the connections between the capacity region we obtain for the gaussian mimo broadcast channel with common and confidential messages and the capacity region of its nonconfidential counterpart i e the gaussian mimo broadcast channel with common and private messages which is not known completely

secret key generation using correlated sources and channels we study the secret key capacity in a joint source channel coding setup the terminals are connected over a discrete memoryless channel and have access to side information modelled as a pair of discrete memoryless source sequences as our main result we establish the upper and lower bounds on the secret key capacity in the lower bound expression the equivocation terms of the source and channel components are functionally additive even though the coding scheme generates a single secret key by jointly taking into account the source and channel equivocations our bounds coincide thus establishing the capacity when the underlying wiretap channel can be decomposed into a set of independent parallel and reversely degraded channels for the case of parallel gaussian channels and jointly gaussian sources we show that gaussian codebooks achieve the secret key capacity in addition when the eavesdropper also observes a correlated side information sequence we establish the secret key capacity when both the source and channel of the eavesdropper are a degraded version of the legitimate receiver we finally also treat the case when a public discussion channel is available propose a separation based coding scheme and establish its optimality when the channel output symbols of the legitimate receiver and eavesdropper are conditionally independent given the input

fuzzy smooth preproximity spaces abstract the concept of a fuzzy smooth preproximity space is introduced and studied also some particular constructions of fuzzy smooth preproximity are given a natural link is established between this concept and the concept of fuzzy pretopological spaces and fuzzy preuniform spaces

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

inflation algorithm for loop free non negative edge bipartite graphs of corank at least two we continue the study of finite connected loop free edge bipartite graphs with m 3m 3 vertices a class of signed graphs we started in simson 2013 48 and m g siorek et al 2016 19 by means of the non symmetric gram matrix g mm z of its symmetric gram matrix g 12 g g tr mm 12z and the gram quadratic form q zm zq zm z in particular we study connected non negative edge bipartite graphs with n r 3n r 3 vertices of corank r 2r 2 in the sense that the symmetric gram matrix g mn r z g mn r z of is positive semi definite of rank n 1n 1 the edge bipartite graphs of corank r 2r 2 are studied up to the weak gram zz congruence z z where z z means that g btr g bg btr g b for some b mn r z b mn r z such that det b 1det b 1 our main result of the paper asserts that given a connected edge bipartite graph with n r 3n r 3 vertices of corank r 2r 2 there exists a suitably chosen sequence t of the inflation operators of the form tab such that the composite operator t reduces to a connected bigraph d n r such that zd n r and d n r is one of the canonical r vertex extensions a n r n 1n 1 d n r n 4n 4 e 6 r e 7 r and e 8 r with n rn r vertices of the simply laced dynkin diagrams an dn e6 e7 e8an dn e6 e7 e8 with n 1n 1 vertices the algorithm constructs also a matrix b mn r z b mn r z with det b 1det b 1 defining the weak gram zz congruence zd n r that is satisfying the equation gd n r btr g b

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

a coxeter type classification of one peak principal posets abstract we present a complete coxeter spectral classification of finite one peak posets i t that are principal in the sense that their symmetric gram matrix g i 1 2 c i c i t r m i q is positive semi definite of rank i 1 where c i m i z is the incidence matrix of i equivalently posets t with the quadratic tits form q t z t 1 z in the sense of drozd non negative and the kernel ker q t v z t 1 q t v 0 z t 1 infinite cyclic we continue the study started in m g siorek d simson 2012 11 we show that up to the strong gram z congruence and the weak gram z congruence every one peak principal poset is uniquely determined by its complex coxeter spectrum specc i c moreover we calculate the exact number of all such posets of a given size up to isomorphism and present their hasse quivers we finish the paper with conjecture on the exact number of principal positive posets according to our knowledge no formula is known for the number of non isomorphic posets in a general case

the worst additive noise under a covariance constraint the maximum entropy noise under a lag p autocorrelation constraint is known by burg s theorem to be the pth order gauss markov process satisfying these constraints the question is what is the worst additive noise for a communication channel given these constraints is it the maximum entropy noise the problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints r sub 0 r sub p for high signal powers the worst additive noise is gauss markov of order p as expected but for low powers the worst additive noise is gaussian with a covariance matrix in a convex set which depends on the signal power

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

multi receiver wiretap channel with public and confidential messages we study the multi receiver wiretap channel mr wc with public and confidential messages in this channel there is a transmitter that wishes to communicate with two legitimate users in the presence of an external eavesdropper the transmitter sends a pair of public and confidential messages to each legitimate user while there are no secrecy constraints on the public messages confidential messages need to be transmitted in perfect secrecy we study the discrete memoryless mr wc as well as its gaussian multi input multi output mimo counterpart first we propose an inner bound for the general not necessarily degraded discrete memoryless mr wc by using marton s inner bound and rate splitting in conjunction with superposition coding and binning second we specialize this inner bound for the degraded discrete memoryless case this specialized form of the inner bound can be obtained by using superposition coding and binning only next we obtain an outer bound for the capacity region of the degraded channel which matches the inner bound for some special cases third we consider the degraded gaussian mimo channel and show that to evaluate both the inner and outer bounds considering only jointly gaussian auxiliary random variables and channel input is sufficient similar to the discrete memoryless case for the gaussian mimo case as well these bounds match for some special cases

a vector generalization of costa s entropy power inequality with applications this paper considers an entropy power inequality epi of costa and presents a natural vector generalization with a real positive semidefinite matrix parameter the new inequality is proved using a perturbation approach via a fundamental relationship between the derivative of mutual information and the minimum mean square error mmse estimate in linear vector gaussian channels as an application a new extremal entropy inequality is derived from the generalized costa epi and then used to establish the secrecy capacity regions of the degraded vector gaussian broadcast channel with layered confidential messages

on mmse crossing properties and implications in parallel vector gaussian channels the scalar additive gaussian noise channel has the single crossing point property between the minimum mean square error mmse in the estimation of the input given the channel output assuming a gaussian input to the channel and the mmse assuming an arbitrary input this paper extends the result to the parallel vector additive gaussian channel in three phases the channel matrix is the identity matrix and we limit the gaussian input to a vector of gaussian i i d elements the single crossing point property is with respect to the signal to noise ratio as in the scalar case the channel matrix is arbitrary and the gaussian input is limited to an independent gaussian input a single crossing point property is derived for each diagonal element of the mmse matrix the gaussian input is allowed to be an arbitrary gaussian random vector a single crossing point property is derived for each eigenvalue of the difference matrix between the two mmse matrices these three extensions are then translated to new information theoretic properties on the mutual information using the i mmse relationship a fundamental relationship between estimation theory and information theory revealed by guo and coworkers the results of the last phase are also translated to a new property of fisher information finally the applicability of all three extensions on information theoretic problems is demonstrated through a proof of a special case of shannon s vector entropy power inequality a converse proof of the capacity region of the parallel degraded broadcast channel bc under an input per antenna power constraint and under an input covariance constraint and a converse proof of the capacity region of the compound parallel degraded bc under an input covariance constraint

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

secure transmission with multiple antennas part ii the mimome wiretap channel the capacity of the gaussian wiretap channel model is analyzed when there are multiple antennas at the sender intended receiver and eavesdropper the associated channel matrices are fixed and known to all the terminals a computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem the converse is based on a sato type argument used in other broadcast settings and the coding theorem is based on gaussian wiretap codebooks at high signal to noise ratio snr the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition and independently coding across the resulting parallel channels the associated capacity is expressed in terms of the corresponding generalized singular values it is shown that a semi blind masked multi input multi output mimo transmission strategy that sends information along directions in which there is gain to the intended receiver and synthetic noise along directions in which there is not can be arbitrarily far from capacity in this regime necessary and sufficient conditions for the secrecy capacity to be zero are provided which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed the resulting scaling laws establish that to prevent secure communication the eavesdropper needs three times as many antennas as the sender and intended receiver have jointly and that the optimum division of antennas between sender and intended receiver is in the ratio of 2 1

secure diversity multiplexing tradeoff of zero forcing transmit scheme at finite snr in this paper we address the finite signal to noise ratio snr diversity multiplexing tradeoff dmt of the multiple input multiple output mimo wiretap channel where a zero forcing zf transmit scheme that intends to send the secret information in the orthogonal space of the eavesdropper channel is used first we introduce the secrecy multiplexing gain at finite snr that generalizes the definition at high snr then we provide upper and lower bounds on the outage probability under secrecy constraint from which secrecy diversity gain estimates of zf are derived through asymptotic analysis we show that the upper bound underestimates the secrecy diversity gain whereas the lower bound is tight at high snr and thus its related diversity gain estimate is equal to the actual asymptotic secrecy diversity gain of the mimo wiretap channel

smooth preuniform and preproximity spaces abstract fuzzy sets and fuzzy logic are very suited to interpret every day life sentences fuzzy interpretations are defined with the use of membership functions whose role is simply to give a kind of graphical visual support to help us to capture the meaning we think that generally membership values are not very important but only some characteristics of membership functions have to be kept we think that monotonic relations between degrees with which some properties are satisfied is an important characteristic we translate in this sense axioms corresponding to preuniform structures and preproximity structures contrary to fuzzy extensions like fuzzy topology for instance which generally keep the original axioms but adapt few of them to deal with fuzzy sets our approach induce much more transformations finally we study how classic constructions relating these concepts can be translated in this framework

comparison of topological and uniform structures for fuzzy numbers and the fixed point problem abstract the goal of this paper is to compare certain topological and uniform structures on sets of fuzzy numbers since fuzzy sets can be viewed as a kind of subsets they can be endowed with uniform structures in ways similar to those used for classic subsets the hausdorff construction is one such approach the structures we study are the myope structure the uniform convergence of the cuts in the sense of hausdorff and a structure induced by a metric defined by goetschel and voxman we show how these structures are related to fixed point problems for some classes of fuzzy numbers

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

counterexample to the vector generalization of costa s entropy power inequality and partial resolution we give a counterexample to the vector generalization of costa s entropy power inequality due to liu et al in particular the claimed inequality can fail if the matrix valued parameter in the convex combination does not commute with the covariance of the additive gaussian noise conversely the inequality holds if these two matrices commute

p critical integral quadratic forms and positive unit forms an algorithmic approach abstract we introduce the class of p critical integral unit forms q z n z containing the critical forms in the sense of ovsienko 13 several characterisations of p critical forms are given in particular it is proved that q is p critical if and only if there is a uniquely determined extended dynkin diagram a n d n e 6 e 7 e 8 and a special group isomorphism t z n z n such that q t is the quadratic form q z n z n 0 of the diagram a correspondence between positive forms p z n 1 z with a sincere root and p critical forms q z n z is described and efficient linear algebra algorithms for computing p critical unit forms and positive forms are constructed

fuzzy pretopological spaces and their representation today it is obvious that topology and uniformity and the many related theories have and will have a fundamental play in applied sciences every representation of real entities in a mathematical language necessarily implies a topological study of its goodness it is a problem of linguistic translation continuity as defined by hammer 71 this translation must preserved some important features of the original the power of actual digital computers enables us to use new ways in automatic pattern recognition but at the same time it shows us the many difficulties we encounter in searching a proper formulation of the topological concepts that everyone has in mind and uses in a naturally and more or less conscientious fashion one of the first axiomatisations of the concept of topological space was done by hausdorff and it is not so old it is only recently that some authors have thought if necessary to generalize these concepts and have obtained practical applications for instance appert 11 hammer 6 71 brissaud 3 mougeot auray duru 2 this has been especially motivated by the necessity of finding new concepts which well apply to finite sets and approx imations as hammer remarks topologists actually had one model in ind the real continuum and its generated euclidean spaces but not approximation and certainly not finite sets we think that the effort of interpretation and application to many fields such as linguistics system theory and education made by hammer is of great importance 7 and its references an important discovery in system theory which has been inforpally stated as a principle by zadeh 111 is as the complexity of a system increases our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics this principle is very close to limit problems and paradoxes encountered by physicians in particular the similitude with the heisenberg uncertainty prin ciple is clear it is certainly a motivation for the use of fuzzy set theory but these two aspects pretopology or extended topology as named by hammer

secure lossy transmission of vector gaussian sources we study the secure lossy transmission of a vector gaussian source to a legitimate user in the presence of an eavesdropper where both the legitimate user and the eavesdropper have vector gaussian side information the aim of the transmitter is to describe the source to the legitimate user in a way that the legitimate user can reconstruct the source within a certain distortion level while the eavesdropper is kept ignorant of the source as much as possible as measured by the equivocation we obtain an outer bound for the rate equivocation and distortion region of this secure lossy transmission problem this outer bound is tight when the transmission rate constraint is removed in other words we obtain the maximum equivocation at the eavesdropper when the legitimate user needs to reconstruct the source within a fixed distortion level while there is no constraint on the transmission rate this characterization of the maximum equivocation involves two auxiliary random variables we show that a nontrivial selection for both random variables may be necessary in general the necessity of two auxiliary random variables also implies that in general wyner ziv coding is suboptimal in the presence of an eavesdropper in addition we show that even when there is no rate constraint on the legitimate link uncoded transmission deterministic or stochastic is suboptimal the presence of an eavesdropper necessitates the use of a coded scheme to attain the maximum equivocation

gradient of mutual information in linear vector gaussian channels this paper considers a general linear vector gaussian channel with arbitrary signaling and pursues two closely related goals i closed form expressions for the gradient of the mutual information with respect to arbitrary parameters of the system and ii fundamental connections between information theory and estimation theory generalizing the fundamental relationship recently unveiled by guo shamai and verdu spl acute we show that the gradient of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the best estimate of the input given the output gradients and derivatives with respect to other parameters are then found via the differentiation chain rule

capacity equivocation region of the gaussian mimo wiretap channel we study the gaussian multiple input multiple output mimo wiretap channel which consists of a transmitter a legitimate user and an eavesdropper in this channel the transmitter sends a common message to both the legitimate user and the eavesdropper in addition to this common message the legitimate user receives a private message which is desired to be kept hidden as much as possible from the eavesdropper we obtain the entire capacity equivocation region of the gaussian mimo wiretap channel this region contains all achievable common message private message and private message s equivocation secrecy rates in particular we show the sufficiency of jointly gaussian auxiliary random variables and channel input to evaluate the existing single letter description of the capacity equivocation region due to csiszar korner

an alternative proof for the capacity region of the degraded gaussian mimo broadcast channel we provide an alternative proof for the capacity region of the degraded gaussian multiple input multiple output mimo broadcast channel our proof does not use the channel enhancement technique as opposed to the original proof of weingertan and the alternative proof of liu et al our proof starts with the single letter description of the capacity region of the degraded broadcast channel and by using it obtains a tight i e achievable outer bound for the capacity region of the degraded gaussian mimo broadcast channel by using two main technical tools the first one is the generalized de bruijn identity due to palomar which provides a connection between the differential entropy and the fisher information matrix the second tool we use is an inequality due to dembo which lower bounds the differential entropy in terms of the fisher information matrix

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

smooth pretopogenous structures abstract in this paper we present smooth extension of pretopogenous structure which is obtained by weakening the axioms of the corresponding classical structure this weakening consists in the embedding of the structure in a multiple valued logic context a natural link is established between these concepts and the concept of fuzzy pretopological spaces smooth preproximity spaces and fuzzy preuniform spaces the notation of product smooth pretopogenous structures is also introduced

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

sum capacity of gaussian vector broadcast channels this paper characterizes the sum capacity of a class of potentially nondegraded gaussian vector broadcast channels where a single transmitter with multiple transmit terminals sends independent information to multiple receivers coordination is allowed among the transmit terminals but not among the receive terminals the sum capacity is shown to be a saddle point of a gaussian mutual information game where a signal player chooses a transmit covariance matrix to maximize the mutual information and a fictitious noise player chooses a noise correlation to minimize the mutual information the sum capacity is achieved using a precoding strategy for gaussian channels with additive side information noncausally known at the transmitter the optimal precoding structure is shown to correspond to a decision feedback equalizer that decomposes the broadcast channel into a series of single user channels with interference pre subtracted at the transmitter

fuzzy preuniform structures and the structures they induce 1 main results with the increasing complexity of the systems that men have created or have tried to understand it has appeared that the classic quantitative tools are not well adapted we must take into account the imprecision inherent to every modelling of complex systems on this problem zadeh 131 stated the following principle as the complexity of a system increases our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics for this reason qualitative tools have been developed topological and connected theories qualitative dynamics etc and new theories have been built up fuzzy sets and possibility theories in fact the modelling problem is a continuity like one in the sense that it must carry and preserve certain important features of the original real world it is certainly this problem that thorn has in mind when he says that qualitative dynamics consiste a admettre a priori l existence d un modele differentiel sous jacent au processus etudie et faute de connaitre explicitement ce modele a deduire de la seule supposition de son existence des conclusions relatives a la nature des singularitis du processus 121 in this paper we generalize the concept of uniform space by defining preuniform spaces and a typology on them at the same time we bring this theory together with that of fuzzy sets in order to get a very powerful tool we have used the zadeh definition of fuzzy sets but the major part of our results can be translated into the case of l fuzzy sets when

vector gaussian multiterminal source coding we derive an outer bound of the rate region of the vector gaussian l terminal ceo problem by establishing a lower bound on each supporting hyperplane of the rate region to this end we prove a new extremal inequality by exploiting the connection between differential entropy and fisher information as well as some fundamental estimation theoretic inequalities it is shown that the outer bound matches the berger tung inner bound in the high resolution regime we then derive a lower bound on each supporting hyperplane of the rate region of the direct vector gaussian l terminal source coding problem by coupling it with the ceo problem through a limiting argument the tightness of this lower bound in the high resolution regime and the weak dependence regime is also proved

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

an extremal inequality motivated by multiterminal information theoretic problems we prove a new extremal inequality motivated by the vector gaussian broadcast channel and the distributed source coding with a single quadratic distortion constraint problems as a corollary this inequality yields a generalization of the classical entropy power inequality epi as another corollary this inequality sheds insight into maximizing the differential entropy of the sum of two dependent random variables

mimo multiple access channel with an arbitrarily varying eavesdropper secrecy degrees of freedom a two transmitter gaussian multiple access wiretap channel with multiple antennas at each of the nodes is investigated the channel matrices of the legitimate users are fixed and revealed to all the terminals whereas the channel matrices of the eavesdropper are arbitrarily varying and only known to the eavesdropper the secrecy degrees of freedom s d o f region under a strong secrecy constraint is characterized a transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver and uses a single user wiretap code for each user is shown to achieve the s d o f region the converse involves establishing an upper bound on a weighted sum rate expression this is accomplished by using induction where at each step one combines the secrecy and multiple access constraints associated with an adversary eavesdropping a carefully selected group of sub channels

the secrecy capacity of the mimo wiretap channel we consider the mimo wiretap channel that is a mimo broadcast channel where the transmitter sends some confidential information to one user which is a legitimate receiver while the other user is an eavesdropper perfect secrecy is achieved when the transmitter and the legitimate receiver can communicate at some positive rate while insuring that the eavesdropper gets zero bits of information in this paper we compute the perfect secrecy capacity of the multiple antenna mimo broadcast channel where the number of antennas is arbitrary for both the transmitter and the two receivers our technique involves a careful study of a sato like upper bound via the solution of a certain algebraic riccati equation

the capacity region of the degraded multiple input multiple output compound broadcast channel the capacity region of a compound multiple antenna broadcast channel is characterized when the users exhibit a certain degradedness order the channel under consideration has two users each user has a finite set of possible realizations the transmitter transmits two messages one for each user in such a manner that regardless of the actual realizations both users will be able to decode their messages correctly an alternative view of this channel is that of a broadcast channel with two common messages each common message is intended to a different set of users the degradedness order between the two sets of realizations users is defined through an additional fictitious user whose channel is degraded with respect to all realizations users from one set while all realizations users from the other set are degraded with respect to him

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

fuzzy preproximity spaces abstract the concept of proximity is generalized by defining fuzzy preproximity some particular constructions of fuzzy preproximity spaces are given a natural link is established between these concepts and the concept of fuzzy pretopological spaces and fuzzy preuniform spaces in the last section the notion of product fuzzy preproximity is introduced

the capacity region of the class of three receiver gaussian mimo multilevel broadcast channels with two degraded message sets nair and el gamal established the capacity region of the three receiver multilevel broadcast channel mbc with two degraded message sets for the three receiver mbc with two degraded message sets the output at receiver 2 is a degraded version of the output at receiver 1 however no order of degradedness is imposed on the output at receiver 3 the transmitter sends a common message to all three receivers and a private message to receiver 1 by considering a specific discrete memoryless example nair and el gamal showed that a direct extension of the korner marton region for the general two receiver broadcast channel with degraded message sets is strictly suboptimal they also considered a three receiver gaussian product mbc and showed that restricted to gaussian inputs the direct extension of the korner marton region is again strictly suboptimal however whether gaussian inputs are optimal remained unresolved in this paper we show that gaussian inputs along with time sharing between rate points obtained with gaussian inputs achieve the capacity region of the three receiver gaussian multiple input multiple output mbc this includes the three receiver gaussian product mbc considered by nair and el gamal with two degraded message sets our proof relies on the channel enhancement technique introduced by weingarten as well as the perturbation approach employed by liu and viswanath

mimo broadcast channel with an unknown eavesdropper secrecy degrees of freedom we study a multi antenna broadcast channel with two legitimate receivers and an external eavesdropper we assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint as our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages we show that a direct extension of the single user wiretap codebook does not achieve the secrecy degrees of freedom our proposed optimal scheme involves decomposing the signal space into a common subspace which can be observed by both receivers and private subspaces which can be observed by only one of the receivers and carefully transmitting a subset of messages in each subspace we also consider the case when each user s private message must additionally remain confidential from the other legitimate receiver and characterize the s d o f region in this case

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power

the secrecy capacity region of the gaussian mimo multi receiver wiretap channel in this paper we consider the gaussian multiple input multiple output mimo multi receiver wiretap channel in which a transmitter wants to have confidential communication with an arbitrary number of users in the presence of an external eavesdropper we derive the secrecy capacity region of this channel for the most general case we first show that even for the single input single output siso case existing converse techniques for the gaussian scalar broadcast channel cannot be extended to this secrecy context to emphasize the need for a new proof technique our new proof technique makes use of the relationships between the minimum mean square error and the mutual information and equivalently the relationships between the fisher information and the differential entropy using the intuition gained from the converse proof of the siso channel we first prove the secrecy capacity region of the degraded mimo channel in which all receivers have the same number of antennas and the noise covariance matrices can be arranged according to a positive semi definite order we then generalize this result to the aligned case in which all receivers have the same number of antennas however there is no order among the noise covariance matrices we accomplish this task by using the channel enhancement technique finally we find the secrecy capacity region of the general mimo channel by using some limiting arguments on the secrecy capacity region of the aligned mimo channel we show that the capacity achieving coding scheme is a variant of dirty paper coding with gaussian signals