stochastic equations for fields in complex manifolds the construction of the euclidean quantum mechanics on a manifold are generalised to two dimensional models of quantum field theory partial differential equations for euclidean fields are obtained which are random perturbations of instanton equations

common randomness in information theory and cryptography i secret sharing as the first part of a study of problems involving common randomness at distance locations information theoretic models of secret sharing generating a common random key at two terminals without letting an eavesdropper obtain information about this key are considered the concept of key capacity is defined single letter formulas of key capacity are obtained for several models and bounds to key capacity are derived for other models

covariant spdes and quantum field structures covariant stochastic partial differential equations spdes are studied in any dimension a special class of such equations is selected and it is proved that the solutions can be analytically continued to minkowski spacetime yielding tempered wightman distributions which are covariant obey the locality axiom and a weak form of the spectral axiom

feynman path integral and poisson processes with piecewise classical paths we prove the existence of a feynman integral formula for gentle perturbations of the harmonic oscillator this result is extended to bose relativistic theory

euclidean random fields obtained by convolution from generalized white noise we study euclidean random fields x over rd of the form x g f where f is a generalized white noise over rd and g is an integral kernel we give conditions for the existence of the characteristic functional and moment functions and we construct a convergent lattice approximation of x finally we perform the analytic continuation of the moment functions and the characteristic functional of x obtaining the corresponding relativistic functions on minkowski space

classical dirichlet forms on topological vector spaces closability and a cameron martin formula abstract we prove a sufficient condition for the closability of classical dirichlet forms on l 2 e which is also necessary if all components of the dirichlet form are closable here e is a locally convex topological vector space and a not necessarily quasi invariant probability measure on e the same condition is shown to imply the existence of a closed extension whose domain can be described explicitly in the special case where is quasi invariant with respect to certain vectors in e our result generalises previous theorems on closability in addition we prove a cameron martin type formula for a large class of measures if e is finite dimensional our characterisation of closability is the analogue of the corresponding one dimensional result applications to quantum fields and the connection with the well studied case of abstract wiener spaces are discussed

construction of convergent simplicial approximations of quantum fields on riemannian manifolds we construct simplicial approximations of random fields on riemannian manifolds of dimensiond we prove convergence of the fields to the continuum limit for arbitraryd in the gaussian case and ford 2 in the non gaussian case in particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on riemannian manifolds with exponential interaction

characterizing the statistical properties of mutual information in mimo channels we consider gaussian multiple input multiple output mimo frequency selective spatially correlated fading channels assuming that the channel is unknown at the transmitter and perfectly known at the receiver for gaussian codebooks using results from multivariate statistics we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal to noise ratio snr we show that our bound is tighter than previously reported analytical lower bounds and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity based on a closed form approximation of the variance of mutual information in correlated flat fading mimo channels we provide insights into the multiplexing diversity tradeoff for gaussian code books furthermore for a given total number of antennas we consider the problem of finding the optimal ergodic capacity maximizing number of transmit and receive antennas and we reveal the snr dependent nature of the maximization strategy finally we present numerical results and comparisons between our capacity bounds and previously reported bounds

uniqueness and global markov property for euclidean fields the case of general polynomial interactions we give a general method for proving uniqueness and global markov property for euclidean quantum fields the method is based on uniform continuity of local specifications proved by using potential theoretical tools and exploitation of a suitable fkg order structure we apply this method to give a proof of uniqueness and global markov property for the gibbs states and to study extremality of gibbs states also in the case of non uniqueness in particular we prove extremality for 2 4 also in the case of non uniqueness and global markov property for weak coupling 2 4 which solves a long standing problem uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

ym2 continuum expectations lattice convergence and lassos the two dimensional yang mills theory ym2 is analyzed in both the continuum and the lattice in the complete axial gauge the continuum theory may be defined in terms of a lie algebra valued white noise and parallel translation may be defined by stochastic differential equations this machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesc the expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group the time parameters of the heat kernels are determined by the areas enclosed by the collectionc and the arguments are determined by the crossing topologies of the curves inc the expectations for the wilson lattice models have a similar structure and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations it is also shown that the lasso variables advocated by l gross 36 exist and are sufficient to generate all the measurable functions on the ym2 measure space

verification of the global markov property in some class of strongly coupled exponential interactions we verify the global markov property in some class of strongly coupled exponential interactions in two dimensional space time to obtain this result we apply the albeverio and ho egh krohn strategy the basic ingredients we use in order to employ this strategy are the fortuin kastelyn ginibre correlation inequalities

on the ergodic secret key agreement over spatially correlated multiple antenna channels with public discussion we consider secret key agreement with public discussion over multiple input multiple output mimo rayleigh fast fading channels under correlated environment we assume that transmit legitimate receiver and eavesdropper antennas are correlated the legitimate receiver and the eavesdropper are assumed to have perfect channel knowledge while the transmitter has only knowledge of the correlation matrices first we derive the expression of the secret key capacity under the considered setup we prove that the optimal transmit strategy achieving the secret key capacity consists in transmitting independent gaussian signals along the eingenvectors of the transmit correlation matrix the powers allocated to each channel mode are determined as the solution to a numerical optimization problem a necessary and sufficient condition for beamforming i e transmitting along the strongest channel mode to be capacity achieving is derived moreover we analyze the impact of correlation matrices on the system performance finally we study the system s performance in the two extreme power regimes in the high power regime we provide closed form expressions of the gain loss due to correlation in the low signal to noise ratio snr regime we investigate the energy efficiency of the system by determining the minimum energy required for sharing a secret key bit and the wideband slope while highlighting the impact of correlation matrices

large deviations from classical paths hamiltonian flows as classical limits of quantum flows we prove that in the limit 0 the probability for the paths of the stochastic jump process associated to the quantum time evolution to be in a tublet around the classical trajectory is of order 1 exp a we give some applications of this result to the study of the classical limit of wigner functions

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

a cell decomposition of the space of real hankel matrices of rank n and some applications abstract we construct a cellular decomposition of the space of all infinite real hankel matrices of rank n it induces cell decompositions on each manifold h m of hankel matrices of fixed rank m n the relationship between the cellular decomposition of h n and the decomposition of rat n into continued fraction cells is determined finally the cell decomposition is applied to derive some topological properties of the manifolds h n and rat n

capacity scaling in mimo wireless systems under correlated fading previous studies have shown that single user systems employing n element antenna arrays at both the transmitter and the receiver can achieve a capacity proportional to n assuming independent rayleigh fading between antenna pairs we explore the capacity of dual antenna array systems under correlated fading via theoretical analysis and ray tracing simulations we derive and compare expressions for the asymptotic growth rate of capacity with n antennas for both independent and correlated fading cases the latter is derived under some assumptions about the scaling of the fading correlation structure in both cases the theoretic capacity growth is linear in n but the growth rate is 10 20 smaller in the presence of correlated fading we analyze our assumption of separable transmit receive correlations via simulations based on a ray tracing propagation model results show that empirical capacities converge to the limit capacity predicted from our asymptotic theory even at moderate n 16 we present results for both the cases when the transmitter does and does not know the channel realization

optimal transmission strategies and impact of correlation in multiantenna systems with different types of channel state information we study the optimal transmission strategy of a multiple input single output miso wireless communication link the receiver has perfect channel state information csi while the transmitter has different types of csi i e either perfect csi or no csi or long term knowledge of the channel covariance matrix for the case in which the transmitter knows the channel covariance matrix it was recently shown that the optimal eigenvectors of the transmit covariance matrix correspond with the eigenvectors of the channel covariance matrix however the optimal eigenvalues are difficult to compute we derive a characterization of the optimum power allocation furthermore we apply this result to provide an efficient algorithm which computes the optimum power allocation in addition to this we analyze the impact of correlation on the ergodic capacity of the miso system with different csi schemes at first we justify the belief that equal power allocation is optimal if the transmitter is uninformed and the transmit antennas are correlated next we show that the ergodic capacity with perfect csi and without csi at the transmitter is schur concave i e the more correlated the transmit antennas are the less capacity is achievable in addition we show that the ergodic capacity with covariance knowledge at the transmitter is schur convex with respect to the correlation properties these results completely characterize the impact of correlation on the ergodic capacity in miso systems furthermore the capacity loss or gain due to correlation is quantified for no csi and perfect csi at the transmitter the capacity loss due to correlation is bounded by some small constant whereas the capacity gain due to correlation grows unbounded with the number of transmit antennas in the case in which transmitter knows the channel covariance matrix finally we illustrate all theoretical results by numerical simulations

multiparameter martingale differential forms on utilise une approche similaire au depart a celle de withney 1057 pour construire des formes de martingales a parametres multiples

representation and construction of multiplicative noise 251 the basic assumption in this paper in which we continue expand and in certain respects bring to conclusion the investigations from 2 is as follows we consider exclusively noise with values in some topological monoid semigroup which are multiplicative in distribution at dif ferent points since multiplication of independent random variables with values in a monoid corresponds to convolution it is clear that this is an invariance assumption we will now give a brief discussion of our results generally speaking one should look upon them as generalizations to random fields and therefore to a more general time parameter of the classical levy khinchin formulae and their consequences section 3 corresponds to processes with stationary and independent increments otherwise stationarity is not assumed in section 1 we prove a kind of daniell kolmogorov theorem to settle the relevant questions on consistency and existence furthermore support properties are discussed we introduce an additional basic assumption our noise is continuous on its support in the sense that it is non atomic in section 2 we restrict ourselves to noise with values in an abelian group a complete characterisation via the fourier transform is afforded by theorem 2 5 in section 3 we assume a general kind of invariance not necessarily given by translations also in the domain of definition then at least formally our random field is of the form x p where x t

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

some examples concerning the global markov property we present examples of interactions of classical lattice systems whose extremal gibbs states fail to have the global markov property one of the examples is translation invariant

euclidean formulation of quantum field theory without positivity general properties of local quantum field theories qft without positivity are discussed in connection with their euclidean formulation modified euclidean axioms for local qft s without positivity are presented which allow us to recover by analytic continuation wightman functions satisfying the modified wightman axioms for indefinite metric qft s

connectivity properties of spaces of partial realizations abstract this paper addresses the problem of determining the number of connected components of the space of real scalar m n hankel matrices of rank n for n

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

stochastic multiplicative measures generalized markov semigroups and group valued stochastic processes and fields abstract we extend the well known relation between markov semigroups and processes with independent increments to the case of generalized group valued semigroups indexed by some measurable space and stochastic multiplicative measures we give a kolmogorov type theorem and a classification theorem connecting the stochastic multiplicative measures with vector valued levy khinchine fields the results have applications to the study of stochastic connections and differential equations for forms on lie groups

phase diagrams and cluster expansions for low temperature models i the phase diagram low temperature phase diagrams of two dimensional open image in new window quantum field models are constructed let open image in new window lie in an r 1 dimensional space of perturbations of a polynomial withr degenerate minima perform a scaling open image in new window and assume 1 we constructk distinct states on left begin array 20 c r k end array right hypersurfaces of codimensionk 1 in the space of perturbations an expansion is used to exhibit exponential clustering of the schwinger functions of each of these states at the core of the construction is a general technique for finding the thermodynamically stable phases from a collection of competing minima we draw on ideas of pirogov and sinai 24 for this problem

the stability and instability of partial realizations a finite or infinite sequence of real numbers is said to be stable if it admits minimal realization by a stable linear system it is shown that the preservation of stability as such a sequence is truncated or extended is not a generic property even among stable sequences this is of interest for identification of linear systems from partial data and for partial realization of random systems in which cases it constitutes a negative result certain finite sequences have infinitely many minimal realizations each having different stability properties in this case a graphical criterion in the spirit of the nyquist criterion is derived to exploit this lack of uniqueness in order to determine whether one can achieve a stable pole placement by a judicious choice of partial realization

local relativistic invariant flows for quantum fields for quantum fields with trigonometric interaction in arbitrary space dimension we construct a representation of the lorentz group by automorphisms on a banach space generated by the weyl algebra

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

a global parametrization of symptotically stable linear systems abstract in this paper we construct a homeomorphism from the set of p m transfer functions of mcmillan degree n onto the open subspace of asymptotically stable linear systems this homeomorphism yields a one to one correspondence between 1 i canonical forms for state space equivalence of minimal systems and 2 ii canonical forms for state space equivalence of asymptotically stable minimal systems implications for the topology of various sets of asymptotically stable systems are given

topology of the set of asymptotically stable minimal systems it is shown that the set of asymptotically stable and minimal single input single output systems of order n has n 1 connected components in the case of asymptotically stable and minimal multiple input multiple output systems however there is only one connected component

sewing yang mills measures and moduli spaces over compact surfaces consider a closed surface obtained by pasting two surfaces 1and 2along some of their boundary components a formula is derived expressing quantum yang mills functional integrals for in terms of the corresponding integrals for the surfaces i using this a formula is derived which expresses the symplectic volume measure on the moduli space of flatsu 2 connections over a closed riemann surface in terms of the corresponding volume measures for the surfaces 1and 2 when the latter have one boundary component each

the bruhat decomposition of finite hankel matrices abstract the bruhat decomposition of the space of n n hankel matrices is derived from the bruhat decomposition of the general linear group some new results concerning the hankel minors map h det h1 det hn are obtained det hi denotes the i th principal minor of h the adherence order of the bruhat decomposition is determined in the complex case

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

stochastic path ordered exponentials we prove convergence of an approximation of the stochastic product integral for conditional wiener paths to the solution of a certain stochastic integral equation this is used to establish the wiener ito representation for the kernel of the semigroup exp t a where a 1 a 2 for functions a with values in the space of anti hermitian matrices

physical properties of quantum field theory measures well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory the difference of typical configurations of free massive scalar field theories with different masses is studied we apply the same methods to study the ashtekar lewandowski al measure on spaces of connections in particular we prove that the diffeomorphism group acts ergodically with respect to the al measure on the ashtekar isham space of quantum connections modulo gauge transformations we also prove that a typical with respect to the al measure quantum connection restricted to a piecewise analytic curve leads to a parallel transport discontinuous at every point of the curve

a convergent lattice approximation for nonlinear electromagnetic fields in four dimensions a discrete version of the quaternionic cauchy riemann equation a f where f is gauss poisson white noise is discussed on the lattice z4 random variables f n and a n are constructed which approximate the corresponding random fields f and a respectively in the limit 0 in the gaussian case the random variables a n can be interpreted as the lattice approximation of the free electromagnetic euclidean potential field whereas in the non gaussian case one obtains an approximation of nonlinear interacting electromagnetic quantum fields convergence to the continuum limit is proven

ferromagneticity of simplicial fields on two dimensional compact manifolds smooth triangulations of a compact smooth connected two dimensional riemannian manifold m are considered the q simplicial fields are defined with values in the space of q cochains and a natural gaussian measure is defined giving their distribution with covariance defined essentially in terms of the combinatorial laplacian c in the continuum limit this measure for q 0 is the free quantum field measure over m in this case it is shown that for a certain collection of triangulations there exists a sequence of subdivisions of each triangulation such that the corresponding measure is ferromagnetic it is also shown that for sufficiently fine subdivisions m2i m 0 has nonpositive off diagonal elements the proofs are obtained by a result on triangulations by simplexes with acute angles it is also proven that the probability measures describing quantum fields on m with polynomial trigonometric or exponential interactions satisfy fkg inequalities

a cell structure for the set of autoregressive systems the set of autoregressive systems generalizes the set of transfer functionsin a natural way we describe a topology for the set of all autoregressive systems of fixed size and bounded mcmillan degree we show that this topological space has the structure of a finite cw complex

bruhat canonical form for linear systems based on a generalization of the classical bruhat factorization of nonsingular matrices to arbitrary rectangular matrices a new canonical form for state space equivalence of controllable and observable linear systems is introduced the proposed canonical form is shown to be closely related to a canonical form due to bosgra and van der weiden moreover in the single input single output case and up to minor details the proposed normal form is equivalent to the continued fraction canonical form introduced by kalman connections to a cell decomposition by fuhrmann and krishnaprasad are discussed as well discrete invariants appearing in the bruhat canonical form are shown to be invariants for restricted output feedback equivalence

approximating ito integrals of differential forms and geodesic deviation suppose x is a semimartingale on a differential manifold m with a linear connection the main purpose of this paper is to show that the ito integral with respect to of a differential form along the path of x is the limit in probability of certain riemann sums constructed in a natural way using the exponential map in differential geometry for this we study the deviation between the stochastic development of x in the tangent space at some point and the image of x under the inverse of the exponential map at the point

random fields as solutions of the inhomogeneous quaternionic cauchy riemann equation i invariance and analytic continuation we consider random fieldsa satisfying the quaternionic cauchy riemann equation a f wheref is white noise under appropriate conditions onf a is invariant under the proper euclidean group in four dimensions but in general not under time reflection the schwinger functions can be analytically continued to wightman functions satisfying the relativistic postulates on invariance specrral property and locality

phase diagrams and cluster expansions for low temperaturep φ 2 models we give a cluster expansion for the schwinger functions of the stable phases found in part i the wightman axioms the mass gap and asymptoticity of perturbation theory follow

uniqueness and the global markov property for euclidean fields the case of general exponential interaction the uniqueness and the global markov property for the regular gibbs measure corresponding to the interaction u lambda varphi lambda int limits lambda d 2 x int d varrho alpha e alpha varphi 0 x for 0 d a probability measure with support in 2 sqrt pi 2 sqrt pi is proved

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

the gibbs measures and partial differential equations i ideas and local aspects we investigate the connections of the gibbs measures which appear in euclidean field theory and the corresponding partial differential equations of classical euclidean field theory

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

some covariant representations of massless boson fields considers the c algebra of the canonical commutation relations ccr acted on by a group g of one particle symmetry transformations v a symplectic operator t defines a representation pi t pi f omicron t where pi f is the fock representation the automorphisms of the ccr algebra that are induced by g are shown to be continuously implemented in pi t if and only if a v g av g is a continuous hilbert schmidt 1 cocycle of g here a is related to t by t exp a a being a suitable bounded antilinear self adjoint operator some new examples of fully poincare covariant representations of massless fields in 1 1 dimensions are constructed

uniqueness of gibbs states for general p ϕ 2 weak coupling models by cluster expansion we consider quantum fields with weak coupling in two space time dimensions we prove that the set of their ultraregular gibbs states consists of only one point and this point is an extremal gibbs state

on the poisson integrals representation in the classical statistical mechanics of continuous systems description of the grand canonical gibbs ensemble for classical continuous systems in terms of the nonlocally perturbed infinite divisible generalized random fields is presented the equivalence of the traditional description with the ones presented here on the level of dlr equations is established the antiferromagnetism for the purely repulsive interactions has been observed finally the usefulness of our description for an analysis of the high temperature cluster expansion has been demonstrated

euclidean quantum vector fields we study quantum vector fields in euclidean space time these fields can be identified with generalized random vector fields which we study in terms of their covariance we prove that the conditions of translational invariance covariance with respect to some representation j of the orthogonal group o n where none of the irreducible components j is trivial and the condition of reflection positivity cannot be fulfilled at the same time unless the test function space is restricted by some gauge condition however if the representation is trivial i e if every matrix in o n is mapped to the identity we can explicitly write down covariance matrices which lead to gaussian fields which fulfill all conditions in the axiomatic framework

a feynman kac type formula for the deterministic and stochastic wave equations and other p d e s we establish a probabilistic representation for a wide class of linear deterministic p d e s with potential term including the wave equation in spatial dimensions 1 to 3 our representation applies to the heat equation where it is related to the classical feynman kac formula as well as to the telegraph and beam equations if the potential is a random spatially homogeneous gaussian noise then this formula leads to an expression for the moments of the solution

secure communication in the low snr regime secrecy capacity of a multiple antenna wiretap channel is studied in the low signal to noise ratio snr regime expressions for the first and second derivatives of the secrecy capacity with respect to snr at snr 0 are derived transmission strategies required to achieve these derivatives are identified in particular it is shown that it is optimal in the low snr regime to transmit in the maximal eigenvalue eigenspace of hm hm nm ne he he where hm and he denote the channel matrices associated with the legitimate receiver and eavesdropper respectively and nm and ne are the noise variances at the receiver and eavesdropper respectively energy efficiency is analyzed by finding the minimum bit energy required for secure and reliable communications and the wideband slope increased bit energy requirements under secrecy constraints are quantified finally the impact of fading is investigated and the benefits of fading in terms of energy efficiency are shown

toda flows inverse spectral transform and realization theory abstract in this paper we establish some new relationships between the solution techniques for completely integrable nonlinear differential equations of the toda type and topics in realization theory

the yang mills measure for s2 abstract the yang mills measure for gauge fields over the two sphere is constructed using a conditioned white noise process stochastic differential equations are used to study parallel translation under this measure expectation values of a broad class of wilson loop configurations are computed explicitly and it is shown that these values are invariant under area preserving diffeomorphisms of the sphere