extensions of the i mmse relationship to gaussian channels with feedback and memory unveiling a fundamental link between information theory and estimation theory the i mmse relationship by guo et al together with its numerous extensions has great theoretical significance and various practical applications on the other hand its influences to date have been restricted to channels without feedback or memory due to the absence of its extensions to such channels in this paper we propose the extensions of the i mmse relationship to discrete time and continuous time gaussian channels with feedback and or memory our approach is based on a very simple observation which can be applied to other scenarios such as a simple and direct proof of the classical de bruijn s identity

uniqueness of non linear ground states for fractional laplacians in mathbb r we prove uniqueness of ground state solutions q q x 0 of the non linear equation sq q q 1 0inr s q q q 1 0 i n r where 0 s 1 and 0 4s 1 2s for s 12 s 1 2 and 0 for s 12 s 1 2 here s denotes the fractional laplacian in one dimension in particular we answer affirmatively an open question recently raised by kenig martel robbiano and we generalize by completely different techniques the specific uniqueness result obtained by amick and toland for s 12 s 1 2 and 1 in 5 for the benjamin ono equation as a technical key result in this paper we show that the associated linearized operator l s 1 1 q is non degenerate i e its kernel satisfies ker l span q this result about l proves a spectral assumption which plays a central role for the stability of solitary waves and blowup analysis for non linear dispersive pdes with fractional laplacians such as the generalized benjamin ono bo and benjamin bona mahony bbm water wave equations

bimeasure algebras on locally compact groups for locally compact groups g and h let bm g h denote the banach space of bounded bilinear forms on c0 g c0 h using a consequence of the fundamental inequality of a grothendieck a multiplication and an adjoint operation are introduced on bm g h which generalize the convolution structure of m g h and which make bm g h into a kg2 banach algebra where kg is grothendieck s universal constant various topics relating to the ideal structure of bm g h and the lifting of unitary representations of g h to representations of bm g h are investigated

malliavin calculus in abstract wiener space using infinitesimals abstract using infinitesimals we develop malliavin calculus on spaces which result from the classical wiener space c 0 1 r by replacing r with any abstract wiener space h b we start from a brownian motion b on a loeb probability space with values in the banach space b b is the standard part of a finite dimensional brownian motion b then we define iterated ito integrals as standard parts of internal iterated ito integrals the integrator of the internal integrals is b and the values of the integrands are multilinear forms on f n where f is a finite dimensional linear space over r between the hilbert space h and its extension h in the first part we prove a chaos decomposition theorem for l2 functionals on that are measurable with respect to the algebra generated by b this result yields a chaos decomposition of l2 functionals with respect to the wiener measure on the standard space c b of b valued continuous functions on 0 1 in the second part we define the malliavin derivative and the skorohod integral as standard parts of internal operators defined on finite dimensional spaces in an application we use the transformation rule for finite dimensional euclidean spaces to study time anticipating and non anticipating shifts of brownian motion by bochner integrals girsanov transformations

embedding the abstract wiener space in a probability space abstract in the first part of this paper it is pointed out that for certain applications of the stochastic calculus of variations it is useful to replace the classical domain of definition the wiener space with a general probability space in which the wiener space is embedded this yields a certain conditional malliavin calculus and is applicable to signal and noise problems in a somewhat analogous way it is pointed out in the second part of the paper that formulating the ito calculus in a setup of an abstract wiener space embedded in a general probability space endowed with a filtration has certain useful applications in particular it enables the formulation and derivation of a dimension free form of the girsanov theorem as well as a dimension free form of the representation of l p wiener functionals as ito integrals

multiple wiener ito integrals possessing a continuous extension letf w be a wiener functional defined byf w in f wherein f denotes the multiple wiener ito integral of ordern of the symmetricl2 0 1 n kernelf we show that a necessary and sufficient condition for the existence of a continuous extension off i e the existence of a function o from the continuous functions on 0 1 which are zero at zero to which is continuous in the supremum norms and for which o w f w a s is that there exists a multimeasure dt1 dt n on 0 1 n such thatf t1 t n t1 1 t n 1 a e lebesgue on 0 1 n recall that a multimeasure a1 a n is for every fixedi and every fixedai ai 1 ai 1 an a signed measure ina i and there exists multimeasures which are not measures it is furthermore shown that iff t1 t2 t n t1 1 t n 1 then all the tracesf k k leqq left frac n 2 right off exist eachf k induces ann 2k multimeasure denoted by k the following relation holds i n f sum limits k 0 n 2 left frac 1 2 right k frac n k n 2k int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k cdot mu k dt 1 dt n 2k and each of the integrals in the above expression equals the multiple stratonovich or ogawa type integral of the tracef k namely int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k mu k dt 1 dt n 2k i n 2k circ f k

multiplicity results for some elliptic problems with concave nonlinearities implies that the variational functionalassociated with 1 is bounded below and coercive coercive ellipticproblems of this type with 0 have been studied by ambrosetti andmancini 3 struwe 9 ambrosetti and lupo 2 and other authors here we prove the existence of more solutions for 0 sufficiently small

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

bounds on the information rate of intertransition time restricted binary signaling over an awgn channel upper and lower bounds on the capacity of a continuous time additive white gaussian noise awgn channel with bilevel spl plusmn spl radic p input signals subjected to a minimum inter transition time t sub min constraint are derived the channel model and input constraints reflect basic features of certain magnetic recording systems the upper bounds are based on duncan s relation between the average mutual information in an awgn regime and the mean square error mse of an optimal causal estimator evaluation or upper bounding the mse of suboptimal causal estimators yields the desired upper bounds the lower bound is found by invoking the extended mrs gerber s lemma and asymptotic properties of the entropy of max entropic bipolar d k codes asymptotic results indicate that at low snr pt sub min n sub 0 with n sub 0 designating the noise one sided power spectral density the capacity tends to p n sub 0 nats per second nats s i e it equals the capacity in the simplest average power limited case at high snr the capacity in the simplest average power limited case at high snr the capacity behaves asymptotically as t sub min sup 1 ln snr ln snr nats s demonstrating the degradation relatively to t sub avg sup 1 lnsnr which is the asymptotic known behavior of the capacity with a bilevel average intertransition time t sub avg restricted channel input additional lower bounds are obtained by considering specific signaling formats such as pulsewidth modulation the effect of mild channel filtering on the lower bounds on capacity is also addressed and novel techniques to lower bound the capacity in this case are introduced

the construction of filtrations on abstract wiener space abstract the abstract wiener space unlike the classical wiener space does not possess a natural notion of time the introduction of such a notion through a resolution of the identity on the cameron martin space associated with the abstract wiener space is considered in this paper the notion of vector fields adapted to this filtration and causal linear operators are introduced and geometric characterizations of these notions are derived the notion of equivalence between two abstract wiener spaces endowed with filtrations is defined it is shown that under some general conditions every abstract wiener space endowed with a filtration is equivalent to a classical wiener space of a finite or countably infinite collection of independent one dimensional wiener processes let u be an adapted vector field and u the divergence of u some sufficient conditions for the exponential integrability of u are derived

mismatched estimation and relative entropy a random variable with distribution p is observed in gaussian noise and is estimated by a mismatched minimum mean square estimator that assumes that the distribution is q instead of p this paper shows that the integral over all signal to noise ratios snrs of the excess mean square estimation error incurred by the mismatched estimator is twice the relative entropy d p q in nats this representation of relative entropy can be generalized to nonreal valued random variables and can be particularized to give new general representations of mutual information in terms of conditional means inspired by the new representation we also propose a definition of free relative entropy which fills a gap in and is consistent with the literature on free probability

the sard inequality on wiener space lett w w u w be a cameron martin perturbation of the identity the formal infinite dimensional extension of the sard inequality ta a lambda d is shown to hold and applications to absolute continuity on wiener space are presented

on mutual information likelihood ratios and estimation error for the additive gaussian channel this paper considers the model of an arbitrarily distributed signal x observed through an added independent white gaussian noise w y x w new relations between the minimal mean square error of the noncausal estimator and the likelihood ratio between y and w are derived this is followed by an extended version of a recently derived relation between the mutual information i x y and the minimal mean square error these results are applied to derive infinite dimensional versions of the fisher information and the de bruijn identity a comparison between the causal and noncausal estimation errors yields a restricted form of the logarithmic sobolev inequality the derivation of the results is based on the malliavin calculus

multiple solutions for a neumann system involving subquadratic nonlinearities abstract in this paper we consider the model semilinear neumann system n u a x u c x f u u v in v b x v c x f v u v in u v 0 on where r n is a smooth open bounded domain denotes the outward unit normal to 0 is a parameter a b c l 0 and f c 1 r 2 r 0 is a nonnegative function which is subquadratic at infinity two nearby numbers are determined in explicit forms and with 0 such that for every 0 system n has only the trivial pair of solution while for every system n has at least two distinct nonzero pairs of solutions

infinitely many nonradial solutions of a euclidean scalar field equation abstract we study the equation u b x u x u x r n u h 1 r n the existence of a nonradial solution has been an open problem for some time even in the autonomous case under suitable hypotheses mainly on we find an unbounded sequence of nonradial solutions if n 4 or n 6 we also obtain infinitely many solutions if b and are not rotationally symmetric with respect to x but satisfy only a weaker symmetry

mutual information for stochastic signals and fractional brownian motion the mutual information between a stochastic signal and this signal plus a fractional brownian motion described as an additive fractional gaussian noise channel is expressed as the error of an estimation problem that can be naturally associated with this model if the stochastic signal with the additive fractional brownian motion occurs multiplied by a scalar parameter then the rate of change of the mutual information with respect to this parameter is described by the error of another related estimation problem these results generalize some results for a model where the fractional brownian motion is a brownian motion to a model with an arbitrary fractional brownian motion

second order asymptotics of mutual information a formula for the second order expansion of the input output mutual information of multidimensional channels as the signal to noise ratio snr goes to zero is obtained while the additive noise is assumed to be gaussian we deal with very general classes of input and channel distributions as special cases these channel models include fading channels channels with random parameters and channels with almost gaussian noise when the channel is unknown at the receiver the second term in the asymptotic expansion depends not only on the covariance matrix of the input signal but also on the fourth mixed moments of its components the study of the second order asymptotics of mutual information finds application in the analysis of the bandwidth power tradeoff achieved by various signaling strategies in the wideband regime

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

fading channels how perfect need perfect side information be the analysis of flat fading channels is often performed under the assumption that the additive noise is white and gaussian and that the receiver has precise knowledge of the realization of the fading process these assumptions imply the optimality of gaussian codebooks and of scaled nearest neighbor decoding here we study the robustness of this communication scheme with respect to errors in the estimation of the fading process we quantify the degradation in performance that results from such estimation errors and demonstrate the lack of robustness of this scheme for some situations we suggest the rule of thumb that in order to avoid degradation the estimation error should be negligible compared to the reciprocal of the signal to noise ratio snr

a further three critical points theorem abstract if x is a real banach space we denote by w x the class of all functionals x r possessing the following property if u n is a sequence in x converging weakly to u x and lim inf n u n u then u n has a subsequence converging strongly to u in this paper we prove the following result let x be a separable and reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional belonging to w x bounded on each bounded subset of x and whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that for each i the functional j is coercive and has a strict local not global minimum say x then for each compact interval a b i for which sup a b x j x there exists r 0 with the following property for every a b and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x has at least three solutions whose norms are less than r

a critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations abstract for an even functional on a banach space the symmetric mountain pass lemma gives a sequence of critical values which converges to zero under the same assumptions on the functional this paper establishes a new critical point theorem which provides a sequence of critical points converging to zero the theorem is applied to sublinear elliptic equations then a sequence of solutions converging to zero is obtained

mutual information relative entropy and estimation in the poisson channel let be a nonnegative random variable and let the conditional distribution of a random variable given be poisson for a parameter we identify a natural loss function such that 1 the derivative of the mutual information between and with respect to is equal to the minimum mean loss in estimating based on regardless of the distribution of 2 when is estimated based on by a mismatched estimator that would have minimized the expected loss had the integral over all values of of the excess mean loss is equal to the relative entropy between and for a continuous time setting where is a nonnegative stochastic process and the conditional law of given is that of a non homogeneous poisson process with intensity function under the same loss function 1 the minimum mean loss in causal filtering when is equal to the expected value of the minimum mean loss in noncausal filtering smoothing achieved with a channel whose parameter is uniformly distributed between 0 and bridging the two quantities is the mutual information between and 2 this relationship between the mean losses in causal and noncausal filtering holds also in the case where the filters employed are mismatched i e optimized assuming a law on which is not the true one bridging the two quantities in this case is the sum of the mutual information and the relative entropy between the true and the mismatched distribution of thus relative entropy quantifies the excess estimation loss due to mismatch in this setting these results are parallel to those recently found for the gaussian channel the i mmse relationship of guo the relative entropy and mismatched estimation relationship of verdu and the relationship between causal and noncasual mismatched estimation of weissman

malliavin derivatives and derivatives of functionals of the wiener process with respect to a scale parameter let f c 0w be a functional of the wiener process with variance parameter c 2 0 and let f cw be an extension of f c 0w to f cw c in 0 c 0 relations are derived between the malliavin derivatives between the derivatives with respect to the scale parameter partial f rho cw partial rho p 1 and noncoherent derivatives such as de f cw sqrt varepsilon c tilde w mid w d varepsilon varepsilon 0 where tilde w is another wiener process independent of w and between the generator of the nontime homogeneous ornstein uhlenbeck process

elliptic problems on complete non compact riemannian manifolds with asymptotically non negative ricci curvature abstract in this paper we discuss the existence and non existence of weak solutions to parametric equations involving the laplace beltrami operator g in a complete non compact d dimensional d 3 riemannian manifold m g with asymptotically non negative ricci curvature and intrinsic metric d g namely our simple model is the following problem g w v w f w in m w 0 in m where v is a positive coercive potential is a positive bounded function is a real parameter and f is a suitable continuous nonlinear term the existence of at least two non trivial bounded weak solutions is established for large value of the parameter requiring that the nonlinear term f is non trivial continuous superlinear at zero and sublinear at infinity our approach is based on variational methods no assumptions on the sectional curvature as well as symmetry theoretical arguments are requested in our approach

pointwise relations between information and estimation in gaussian noise many of the classical and recent relations between information and estimation in the presence of gaussian noise can be viewed as identities between expectations of random quantities these include the relationship between mutual information and minimum mean square error i mmse of guo the relative entropy and mismatched estimation relationship of verdu the relationship between causal estimation and mutual information of duncan and its extension to the presence of feedback by kadota the relationship between causal and non casual estimation of guo and its mismatched version of weissman we dispense with the expectations and explore the nature of the pointwise relations between the respective random quantities the pointwise relations that we find are as succinctly stated as and give considerable insight into the original expectation identities as an illustration of our results consider duncan s 1970 discovery that the mutual information is equal to the causal mmse in the additive white gaussian noise channel which can equivalently be expressed saying that the difference between the input output information density and half the causal estimation error is a zero mean random variable regardless of the distribution of the channel input we characterize this random variable explicitly rather than merely its expectation classical estimation and information theoretic quantities emerge with new and surprising roles for example the variance of this random variable turns out to be given by the causal mmse which in turn is equal to twice the mutual information by duncan s result

number fields with solvable galois groups and small galois root discriminants we apply class field theory to compute complete tables of number fields with galois root discriminant less than 8 e this includes all solvable galois groups which appear in degree less than 10 groups of order less than 24 and all dihedral groups dp where p is prime many people have studied questions of constructing complete lists of number fields subject to conditions on degree and possibly galois group with a goal of determining complete lists of such fields with discriminant less than a fixed bound this study can be phrased in those terms with the principle distinction being that our discriminant bounds apply to galois fields rather than to particular stem fields for a finite group g and bound b 0 let k g b be the set of number fields in c which are galois over q and which have root discriminant b it is a classical theorem that each set k g b is finite the first author and roberts jr07a considered these sets with b 8 e 44 7632 a constant introduced by serre ser86 they conjecture that there are only finitely many such fields in the union of the k g with g running through all finite groups they determine all sets k a where a is an abelian group and most sets k g for groups g which appear as galois groups of irreducible polynomials of degree at most 6 here we extend those results using class field theory we complete their study of galois groups which appear in degree 6 all solvable extensions in degrees 7 9 as well as for all groups g with g 23 finally we determine the sets k d where is an odd prime the primary theoretical difficulty in carrying out these computations is to deduce bounds which effectively screen out conductors for our extensions section 1 sets notation and describes some of the methods used before section 2 addresses this question and describes our overall process section 3 summarizes the results of our computations 1 notation and conventions 1 1 groups we let cn denote the cyclic group of order n and dn the dihedral group of order 2n direct products are indicated by juxtaposition so g1 g2 will be denoted by simply g1g2 and g1 g1 will be written g1 many groups will be semidirect products and n h will denote a semidirect product with normal subgroup n and complimentary subgroup h we will write g1 g2 g3 to mean g1 g2 g3 if h sn then the wreath product is denoted g oh g h an extension of g1 by g2 will be denoted by g1 g2 2010 mathematics subject classification primary 11r21 secondary 11r37

likelihood ratios and inference for poisson channels in recent years infinite dimensional methods have been introduced for gaussian channel estimation the aim of this paper is to study the application of similar methods to poisson channels in particular we compute the noncausal conditional mean estimator of a poisson channel using the likelihood ratio and the discrete malliavin gradient this algorithm is suitable for numerical implementation via the monte carlo scheme as an application we provide a new proof of a very deep and remarkable formula in information theory obtained recently in the literature and relating the derivatives of the input output mutual information of a general poisson channel and the conditional mean estimator of the input regardless the distribution of the latter the use of the aforementioned stochastic analysis techniques allows us to extend these results to more general channels such as mixed gaussian poisson channels

on nonlinear schrodinger equations on the hyperbolic space abstract we study existence of weak solutions for certain classes of nonlinear schrodinger equations on the poincare ball model b n n 3 by using the palais principle of symmetric criticality and suitable group theoretical arguments we establish the existence of a nontrivial weak solution

septic fields with discriminant 2 a 3 b we classify septic number fields which are unramified outside of 2 3 by a targeted hunter search there are exactly 10 such fields all with galois group s7 we also describe separate computations which strongly suggest that none of these fields come from specializing septic genus zero three point covers

hessian and concavity of mutual information differential entropy and entropy power in linear vector gaussian channels within the framework of linear vector gaussian channels with arbitrary signaling the jacobian of the minimum mean square error and fisher information matrices with respect to arbitrary parameters of the system are calculated in this paper capitalizing on prior research where the minimum mean square error and fisher information matrices were linked to information theoretic quantities through differentiation the hessian of the mutual information and the entropy are derived these expressions are then used to assess the concavity properties of mutual information and entropy under different channel conditions and also to derive a multivariate version of an entropy power inequality due to costa

a targeted martinet search constructing number fields with prescribed ramification is an important problem in computational number theory in this paper we consider the problem of computing all imprimitive number fields of a given degree which are unramified outside of a given finite set of primes s by combining the techniques of targeted hunter searches with martinet s relative version of hunter s theorem we then carry out this algorithm to generate complete tables of imprimitive number fields for degrees 4 through 10 and certain sets s of small primes

divergence and minimum mean square error in continuous time additive white gaussian noise channels new relations between the divergence of a process measure in the function space relative to the measure of the gaussian process with the same covariance and both the causal minimum mean square error cmmse and the noncausal minimum mean square error mmse are proved as an example a special type of signal which is composed of a sum of independent narrow band waves the divergence and its derivative relative to the signal energy to the white gaussian noise wgn energy ratio is calculated the results are used to evaluate the cmmse and the mmse of the signal optimum estimation since the divergence takes also a significant role in the derivation of the capacity of the additive non gaussian channel we use the calculation that was made for the divergence in the example to evaluate the capacity of an additive noise channel where the noise is composed of the sum of wgn and narrow band interferes

on the structure of independence on wiener space motives par des resultats de travaux anterieurs sur la caracterisation de l independance de variables aleatoires dans l espace de wiener les auteurs etudient ici la relation entre l independance de certaines classes de sous algebres et l orthogonalite des h derivees faibles des variables aleatoires qui engendrent ces sous algebres ou c h est l espace de wiener classique

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

turbo space time processing to improve wireless channel capacity by deriving a generalized shannon capacity formula for multiple input multiple output rayleigh fading channels and by suggesting a layered space time architecture concept that attains a tight lower bound on the capacity achievable foschini see wireless pers commun vol 6 no 3 p 311 35 1998 has shown a potential enormous increase in the information capacity of a wireless system employing multiple element antenna arrays at both the transmitter and receiver the layered space time architecture allows signal processing complexity to grow linearly rather than exponentially with the promised capacity increase this paper includes two important contributions first we show that foschini s lower bound is in fact the shannon bound when the output signal to noise ratio snr of the space time processing in each layer is represented by the corresponding matched filter bound this proves the optimality of the layered space time concept second we present an embodiment of this concept for a coded system operating at a low average snr and in the presence of possible intersymbol interference this embodiment utilizes the already advanced space time filtering coding and turbo processing techniques to provide yet a practical solution to the processing needed performance results are provided for quasi static rayleigh fading channels with no channel estimation errors we see for the first time that the shannon capacity for wireless communications can be both increased by n times where n is the number of the antenna elements at the transmitter and receiver and achieved within about 3 db in average snr about 2 db of which is a loss due to the practical coding scheme we assume the layered space time processing itself is nearly information lossless

on information transfer by envelope constrained signals over the awgn channel using t e duncan s theorem 1970 on the relation between mutual information and the mean square error of the optimum causal estimator of a random signal in additive white gaussian noise awgn the maximum achievable information transfer over the awgn channel is derived with the random telegraph wave input the information transfer is bounded and symptotically determined for the wiener phase modulated process input at large signal to noise ratio snr both results are compared to the information transfer for the capacity achieving gauss markov input process for both the wiener phase modulated and the gauss markov processes the information transfer increases asymptotically as the square root of snr but for the random telegraph wave it increases only as its logarithm

a general variational principle and some of its applications abstract in this paper given a reflexive real banach space x and two sequentially weakly lower semicontinuous functionals on x with strongly continuous and coercive we are mainly interested in the existence of infinitely many local minima of the functional for each sufficiently large r

a statistical mechanics approach to large system analysis of cdma multiuser detectors we present a theory based on statistical mechanics to evaluate analytically the performance of uncoded fully synchronous randomly spread code division multiple access cdma multiuser detectors with additive white gaussian noise awgn channel under perfect power control and in the large system limit application of the replica method a tool developed in the literature of statistical mechanics allows us to derive analytical expressions for the bit error rate as well as the multiuser efficiency of the individually optimum io and jointly optimum jo multiuser detectors over the whole range of noise levels the information theoretic capacity of the randomly spread cdma channel and the performance of decorrelating and linear minimum mean square error mmse detectors are also derived in the same replica formulation thereby demonstrating validity of the statistical mechanical approach

on the relation between the stratonovich and ogawa integrals l auteur montre que l existence de l integrale d ogawa implique l existence de l integrale de stratonovich et que les deux sont egales sous condition d une definition convenable de l integrale stochastique de stratonovich non adaptee

the clark ocone formula for vector valued wiener functionals abstract the classical representation of random variables as the ito integral of nonanticipative integrands is extended to include banach space valued random variables on an abstract wiener space equipped with a filtration induced by a resolution of the identity on the cameron martin space the ito integral is replaced in this case by an extension of the divergence to random operators and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense a complete characterization of measure preserving transformations in wiener space is presented as an application of this generalized clark ocone formula

invertibility of adapted perturbations of the identity on abstract wiener space abstract let w h be an abstract wiener space it is well known that a continuously increasing sequence of projections on h enables to define the notion of adapted shift under the assumption that such a sequence exists we study the invertibility of adapted shifts on abstract wiener space in particular we extend a recent result of ustunel which relates the invertibility of an adapted perturbation of the identity on the classical wiener space to the equality between the energy of the signal and the relative entropy of the measure it induces we also extend this result to a probability absolutely continuous but not necessarily equivalent to the wiener measure with finite entropy finally we relate this theorem both to the monge problem and to the innovation conjecture

a characterization of the kernels associated with the multiple integral representation of some functional of the wiener process in this paper we present a characterization of those wiener functionals that are the likelihood ratio for a signal plus independent noise model the characterization is expressed in terms of the representation of such functionals in a series of multiple wiener integrals

configuration spaces transfer and 2 nodal solutions of a semiclassical nonlinear schrodinger equation we establish new lower bounds for the number of nodal bound states for the semiclassical nonlinear schrodinger equation varepsilon 2 delta u a x u u p 2 u with bounded and uniformly continuous potential a the solutions we obtain have precisely two nodal domains and their positive and negative parts concentrate near the set of minimum points of a our approach is independent of penalization techniques and yields in some cases the existence of infinitely many nodal solutions for fixed varepsilon via a dynamical systems approach we exhibit positively invariant sets of sign changing functions for the negative gradient flow of the associated energy functional we analyze these sets on the cohomology level with the help of dold s fixed point transfer in particular we estimate their cuplength in terms of the cuplength of equivariant configuration spaces of subsets of mathbb r n we also provide new estimates of the cuplength of configuration spaces

the generalized area theorem and some of its consequences there is a fundamental relationship between belief propagation bp and maximum a posteriori decoding the case of transmission over the binary erasure channel was investigated in detail in a companion paper c meacuteasson a montanari and r urbanke maxwell s construction the hidden bridge between iterative and maximum a posteriori decoding ieee transactions on information theory submitted for publication this paper investigates the extension to general memoryless channels paying special attention to the binary case an area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed we show that this area theorem gives rise to an upper bound on the maximum a posteriori threshold for sparse graph codes in situations where this bound is tight the extrinsic soft bit estimates delivered by the bp decoder coincide with the correct a posteriori probabilities above the maximum a posteriori threshold more generally it is conjectured that the fundamental relationship between the maximum a posteriori probability map and the bp decoder which was observed for transmission over the binary erasure channel carries over to the general case we finally demonstrate that in order for the design rate of an ensemble to approach the capacity under bp decoding the component codes have to be perfectly matched a statement which is well known for the special case of transmission over the binary erasure channel

detection of stochastic processes this paper reviews two streams of development from the 1940 s to the present in signal detection theory the structure of the likelihood ratio for detecting signals in noise and the role of dynamic optimization in detection problems involving either very large signal sets or the joint optimization of observation time and performance this treatment deals exclusively with basic results developed for the situation in which the observations are modeled as continuous time stochastic processes the mathematics and intuition behind such developments as the matched filter the rake receiver the estimator correlator maximum likelihood sequence detectors multiuser detectors sequential probability ratio tests and cumulative sum quickest detectors are described

an mse based transfer chart for analyzing iterative decoding schemes using a gaussian approximation an alternative to extrinsic information transfer exit charts called mean square error mse charts that use a measure related to the mse instead of mutual information is proposed using the relationship between mutual information and minimum mean square error mmse for the additive white gaussian noise awgn channel a relationship between the rate of any code and the area under a plot of mmse versus signal to noise ratio snr is obtained when the a priori log likelihood ratio llr is from a binary input gaussian channel using this result a justification is provided for designing concatenated codes by matching the exit curves of the inner and outer decoder when the llrs are assumed to be gaussian which is also the typical assumption used for code design using exit charts even though the gaussian assumption is almost never true the results presented in this paper represent a step toward the analysis of iterative decoding schemes using a single parameter finally for the special case of awgn channel it is shown that any capacity achieving code has an exit curve that is a step function

on the clark ocone formula for the abstract wiener space abstract fix an abstract wiener space h b where h is a separable hilbert space densely embedded into a banach space b a pathwise construction of the ito integral as a continuous square integrable martingale is given where the integrands are h valued processes and the integrator is a b valued brownian motion we use this approach to the vector integral to prove that each malliavin differentiable functional defined on the space c b of continuous b valued functions on 0 1 endowed with the wiener measure can be decomposed into the sum of the expected value of and the ito integral of the conditional expectation of the malliavin derivative of with respect to the brownian filtration the malliavin derivative of is an h valued stochastic process in a second application it is shown that the iterated ito integral defined as a process on c b 0 1 is a continuous square integrable martingale

the relationship between causal and noncausal mismatched estimation in continuous time awgn channels a continuous time finite power process with distribution p is observed through an awgn channel at a given signal to noise ratio snr and is estimated by an estimator that would have minimized the mean square error if the process had distribution q we show that the causal filtering mean square error mse achieved at snr level snr is equal to the average value of the noncausal smoothing mse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr emerging as the bridge for equating these two quantities are mutual information and relative entropy our result generalizes that of guo shamai and verdu 2005 from the nonmismatched case where p q to general p and q among our intermediate results is an extension of duncan s theorem that relates mutual information and causal mmse to the case of mismatched estimation some further extensions and implications are discussed key to our findings is the recent result of verdu on mismatched estimation and relative entropy

a diffusion approximation result for two parameter processes we consider a one dimensional linear wave equation with a small mean zero dissipative field and with the boundary condition imposed by the so called goursat problem in order to observe the effect of the randomness on the solution we perform a space time rescaling and we rewrite the problem in a diffusion approximation form for two parameter processes we prove that the solution converges in distribution toward the solution of a two parameter stochastic differential equation which we identify the diffusion approximation results for oneparameter processes are well known and well understood in fact the solution of the one parameter analog of the problem we consider here is immediate unfortunately the situation is much more complicated for two parameter processes and we believe that our result is the first one of its kind

multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity some multiplicity results are presented for the eigenvalue problem div x 2a u x 2b f u x 2c g u in u 0o n

multiple existence of non radial solutions with group invariance for sublinear elliptic equations abstract we deal with sublinear elliptic equations in a ball and prove the existence of infinitely many solutions which are not radially symmetric but g invariant here g is any closed subgroup of the orthogonal group and is not transitive on the unit sphere

information estimation and lookahead in the gaussian channel we consider mean squared estimation with lookahead of a continuous time signal corrupted by additive white gaussian noise we show that the mutual information rate function i e the mutual information rate as function of the signal to noise ratio snr does not in general determine the minimum mean squared error mmse with fixed finite lookahead in contrast to the special cases with 0 and infinite lookahead filtering and smoothing errors respectively which were previously established in the literature further we investigate the simple class of continuous time stationary gauss markov processes ornstein uhlenbeck processes as channel inputs and explicitly characterize the behavior of the minimum mean squared error mmse with finite lookahead and signal to noise ratio snr we extend our results to mixtures of ornstein uhlenbeck processes and use the insight gained to present lower and upper bounds on the mmse with lookahead for a class of stationary gaussian input processes whose spectrum can be expressed as a mixture of ornstein uhlenbeck spectra

fixed point smoothing of scalar diffusions ii the error of the optimal smoother the problem of fixed point smoothing of a scalar diffusion process consists of estimating the initial value of the process given its noisy measurements as a function of time an asymptotic expansion of the joint filtering smoothing conditional density function is constructed in the limit of small measurements noise the approximate optimal nonlinear fixed point smoother of part i siam j appl math 54 1994 pp 833 853 is rederived from the expansion a detailed analysis of the conditional mean square estimation error cmsee of the optimal fixed point smoother and of its leading order approximation is presented it is shown that if the initial error is small e g if asymptotically optimal filtering is used first the leading order approximation to the optimal smoother is three dimensional and thus simpler than the four dimensional extended kalman smoother furthermore nonlinear fixed point smoothing can reduce the cmsee relative to that of filtering by a factor of frac 1 2 within smoothin

mutual information for stochastic signals and levy processes in this paper some relations between estimation and mutual information are given by expressing two mutual information calculations in terms of two distinct estimation errors specifically the mutual information between a stochastic signal and a pure jump levy process whose rate function depends on the signal is expressed in terms of a filtering error and the rate of change of this mutual information with respect to a parameter multiplying the rate function of the levy process is expressed in terms of a smoothing error these results generalize the analogous mutual information results for some gaussian noise processes with additive stochastic signals

elliptic problems on the ball endowed with funk type metrics abstract we study sobolev spaces on the n dimensional unit ball b n 1 endowed with a parameter depending finsler metric f a a 0 1 which interpolates between the klein metric a 0 and funk metric a 1 respectively we show that the standard sobolev space defined on the finsler manifold b n 1 f a is a vector space if and only if a 0 1 furthermore by exploiting variational arguments we provide non existence and existence results for sublinear elliptic problems on b n 1 f a involving the finsler laplace operator whenever a 0 1