a nonlinear optimum detection problem i theory an approximate log likelihood ratio for detecting a deterministic signal in linear and nonlinear gaussian noise is derived the new aspect of this detection problem is the inclusion of a nonlinear form of gaussian noise and a certain interaction between the signal and the linear noise such as modulation of the signal by the noise the derived log likelihood ratio consists of a modified version of the classical correlation detector and a new processor where the latter is an order of magnitude smaller than the former in the absence of the signal noise interaction the modification consists of subtracting the estimate of the nonlinear noise from the input to the correlation detector the nonlinear processor is a quadratic correlation processor where the correlation is not against the transformed signal but against the difference of two estimates of the nonlinear noise under the two hypotheses signal presence and signal absence presence of the signal noise interaction introduces further modification and complication on both processors

signal detection in nearly gaussian skewed noise the problem of detecting a constant signal in nearly gaussian skewed noise is studied in this paper sensitivity of both the linear detector and the sign detector are examined it is shown that the performance of the linear detector deteriorates when the noise distribution becomes skewed on the other hand the sign detector appears to be less sensitive to the noise skewness furthermore a modification of the linear detector is proposed this modified scheme when the noise skewness measure is small will behave almost like the linear detector in gaussian noise

semismoothness of solutions to generalized equations and the moreau yosida regularization we show that a locally lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point we present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible nonpolyhedral convex sets we prove that the semismoothness of solutions to the moreau yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function

directional derivative of a minimax function etant donne une fonction a valeur reelle l definie sur x u v ou x est un espace localement convexe et ou u et v sont deux espaces topologiques de hausdorff on etudie l existence et la caracterisation de la derivee directionnelle dh x 0 d a x 0 x dans la direction d x de la fonction minimax h x sup v v o inf u u o l x u v ou u o et v o sont deux ensembles non vides dans u et dans v respectivement

existence of weakly efficient solutions in nonsmooth vector optimization in this paper we study the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems we consider problems whose objective functions are defined between infinite and finite dimensional banach spaces our results are stated under hypotheses of generalized convexity and make use of variational like inequalities

stability and bifurcation for a delayed predator prey model and the effect of diffusion abstract we consider a predator prey system with one or two delays and a unique positive equilibrium e its dynamics are studied in terms of the local stability of e and of the description of the hopf bifurcation that is proven to exist as one of the delays taken as a parameter crosses some critical values we also consider a reaction diffusion system with neumann conditions resulting from adding one spatial variable and diffusion terms in the previous model the spectral and bifurcation analysis in the neighborhood of e now as a stationary point of this latter system is addressed and the results obtained for the case without diffusion are applied

robust matched filters the problem of designing robust matched filters for situations in which there is uncertainty in the signal structure or noise statistics is considered two general aspects of this problem are treated first maximin robust designs are characterized for a general hilbert space formulation of the matched filtering problem and explicit solutions are given for two intuitively appealing models for uncertainty these results are seen to generalize earlier results for specific matched filtering problems the second general aspect treated is the application of the theoretical maximin results to the particular problem of designing filters to combat uncertain nonlinear channel distortion channel distortion is modeled by considering an unknown received signal which may differ in l 2 norm from the transmitted or nominal signal by no more than a fixed amount the effect of the channel distortion is seen to be equivalent to that of adding a white noise to the channel whose spectral height depends on the degree of distortion general expressions are developed for the determination of the robust filter and its performance and numerical results are presented for the case of baseband detection in gauss markov noise

prevalence an addendum to prevalence a translation invariant almost every on infinite dimensional spaces bull amer math soc n s 27 1992 no 2 217 238 mr1161274 93k 28018

hopf bifurcation of a predator prey system with stage structure and harvesting abstract in this paper a two species predator prey system with stage structure and harvesting is investigated the existence of hopf bifurcations of the system is given and the stability and directions of hopf bifurcations are determined by applying the normal form theory and the center manifold theorem

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

subdifferential representation formula and subdifferential criteria for the behavior of nonsmooth functions abstract several kinds of behaviors of extended real valued lower semicontinuous functions are known to be equivalent to certain appropriate conditions in terms of the clarke subdifferential the paper provides a systematic study showing that any such condition with the clarke subdifferential is valid if and only if it holds with any operator representing the clarke subdifferential like in the subdifferential proximal formula

various lipschitz like properties for functions and sets i directional derivative and tangential characterizations in this work we introduce for extended real valued functions defined on a banach space x the concept of k directionally lipschitzian behavior where k is a bounded subset of x for different types of sets k e g zero singleton or compact the k directionally lipschitzian behavior recovers well known concepts in variational analysis locally lipschitzian directionally lipschitzian or compactly epi lipschitzian properties respectively characterizations of this notion are provided in terms of the lower dini subderivatives we also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the bouligand tangent cones the special case of convex functions and sets is also studied

detection in multivariate non gaussian noise the applications of multivariate edgeworth series and higher order statistics to the discrete time detection of a known constant signal in multivariate non gaussian noise are considered a technique to derive suboptimum detectors from the neyman pearson optimum and locally optimum detectors is described a numerical algorithm based on knowledge of the noise cumulants is presented in order to analyze the finite sample size performance of the suboptimum detectors as an example the performance of the detectors as compared with the linear detector in multivariate gaussian gaussian mixture noise is presented via receiver operating characteristic curves numerical results indicate that the suboptimum detectors when exploiting knowledge of the dependence structure of the noise can have very good performance with respect to the linear detector

local hopf bifurcation and global periodic solutions in a delayed predator prey system abstract we consider a delayed predator prey system we first consider the existence of local hopf bifurcations and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from hopf bifurcations using the normal form theory and center manifold argument special attention is paid to the global existence of periodic solutions bifurcating from hopf bifurcations by using a global hopf bifurcation result due to wu trans amer math soc 350 1998 4799 we show that the local hopf bifurcation implies the global hopf bifurcation after the second critical value of delay finally several numerical simulations supporting the theoretical analysis are also given

stability and bifurcation in a neural network model with two delays abstract a simple neural network model with two delays is considered linear stability of the model is investigated by analyzing the associated characteristic transcendental equation for the case without self connection it is found that the hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem an example is given and numerical simulations are performed to illustrate the obtained results

expansions of the likelihood ratio and applications the likelihood ratio receiver although optimal is oftentimes difficult to realize to overcome this difficulty suboptimal approximations have been studied in the past one of the simplest approaches is to expand the likelihood ratio and truncate this expansion under the weak signal assumption in this paper we first study the relations between the taylor expansion and the volterra expansion under a weak signal assumption we emphasize the links between the taylor expansion with respect to the snr and the detection criteria on one hand and with robustness on the other in particular using two criteria derived from deflection we show that the minimax robust filter is the matched filter finally an application to underwater signal detection is studied showing that the degradation resulting from the use of the locally optimal receiver is small

effect of predator cannibalism and prey growth on the dynamic behavior for a predator stage structured population model with diffusion abstract in this paper a predator prey model with predator stage structured and diffusion is concerned we deal with the system by endowing it with the homogeneous neumann boundary conditions we first give the a priori estimates of positive solutions for the reduced reaction diffusion system secondly we discuss the effects of predator cannibalism and prey growth on the stability of nonnegative constant steady states of the model in detail thirdly we investigate the nonexistence and existence of nonconstant positive solutions finally we discuss the hopf bifurcation created by diffusion

conic version of loewner john ellipsoid theorem we extend john s inscribed ellipsoid theorem as well as loewner s circumscribed ellipsoid theorem from convex bodies to proper cones to be more precise we prove that a proper cone k k in mathbb r n rn contains a unique ellipsoidal cone q mathrm in k qin k of maximal canonical volume and on the other hand it is enclosed by a unique ellipsoidal cone q mathrm circ k qcirc k of minimal canonical volume in addition we explain how to construct the inscribed ellipsoidal cone q mathrm in k qin k the circumscribed ellipsoidal cone q mathrm circ k qcirc k is then obtained by duality arguments the canonical volume of an ellipsoidal cone is defined as the usual n n dimensional volume of a certain truncation of the cone

integration of multivalued operators and cyclic submonotonicity we introduce a notion of cyclic submonotonicity for multivalued operators from a banach space x to its dual we show that if the clarke subdifferential of a locally lipschitz function is strictly submonotone on an open subset u of x then it is also maximal cyclically submonotone on u and conversely that every maximal cyclically submonotone operator on u is the clarke subdifferential of a locally lipschitz function which is unique up to a constant if u is connected in finite dimensions these functions are exactly the lower c 1 functions considered by spingarn and rockafellar

on the local representation of piecewise smooth equations as a lipschitz manifold we study systems of equations f x 0 given by piecewise differentiable functions f from r n to r k k n the focus is on the representability of the solution set locally as an n k dimensional lipschitz manifold for that nonsmooth versions of inverse function theorems are applied it turns out that their applicability depends on the choice of a particular basis to overcome this obstacle we introduce a strong full rank assumption sfra in terms of clarke s generalized jacobians the sfra claims the existence of a basis in which clarke s inverse function theorem can be applied aiming at a characterization of sfra we consider also a full rank assumption fra the fra insures the full rank of all matrices from the clarke s generalized jacobian the article is devoted to the conjectured equivalence of sfra and fra for min type functions we give reformulations of sfra and fra using orthogonal projections basis enlargements cross products dual variables as well as via exponentially many convex cones the equivalence of sfra and fra is shown to be true for min type functions in the new case k 3

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

frechet differentiability of regular locally lipschitzian functions abstract this paper considers frechet differentiability almost everywhere in the sense of category of regular locally lipschitzian real valued functions defined on open subsets of a banach space it is first shown that for separable banach spaces clarke s generalized gradient of such a function is a minimal convex and compact valued upper semicontinuous multifunction using a theorem of christensen and kenderov it is then shown that for separable asplund spaces such a function is frechet differentiable on a dense g subset of its domain

second order directional derviatives in parametric optimization problems we characterize the second order directional derivative in the sense of ben tal and zowe of a marginal function of the following general form x inf f x y y x this requires us to introduce some geometrical notions as for example the first and second order tangent directions to a multifunction the main results are then applied to some particular types of marginal functions

infinite dimensional generalized jacobian properties and calculus rules the extension to infinite dimensional domains of clarke s generalized jacobian is the focus of this paper first a generalization of a fabian preiss theorem to the infinite dimensional setting is obtained as a consequence a new formula relating the clarke s generalized jacobians corresponding to finite dimensional spaces k l with k l is established furthermore in the infinite dimensional case basic properties pertaining the generalized jacobian are developed and then an identification of this set valued map is produced applications of these results in the form of chain rules including sum and product rules and a computational formula for continuous selections are derived

shape sensitivity analysis for a robin problem via minimax differentiability this paper deals with shape sensitivity analysis for a robin problem the structure of eulerian derivative with respect to the shape of the variable domain for a cost functional is established by using differentiability of a minimax combing with function space parametrization and function space embedding finally we apply a gradient type algorithm to our problem numerical examples show that our theory is very useful for practical purpose and the algorithm is feasible

lipschitz functions with prescribed derivatives and subderivatives in general it is difficult to construct lipschitz functions which are not directly built up from either convex or distance functions one impediment to such constructions is that outside of the real line it is difficult to find anti derivatives the main result of this paper provides under suitable circumstances a technique for constructing such anti derivatives more precisely we show that if f 1 f 2 dots f n are continuously differentiable real valued locally lipschitz functions defined on a non empty open subset a of a separable banach space x then there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co nabla f 1 x nabla f 2 x dots nabla f n x this same construction also shows that for any finite family t 1 t 2 dots t n of maximal cyclically monotone mappings from a into non empty subsets of x there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co t 1 x t 2 x dots t n x moreover we show that g is convex if and only if t 1 t 2 cdots t n

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

parametrizations of kojima s system and relations to penalty and barrier functions we investigate two homotopies that perturb kojima s system for describing critical points of a nonlinear optimization problem in finite dimension each of them characterizes stationary points of a usual penalty and a new barrier function the latter is a continuous deformation of the objective symmetric to the penalty from a formal point of view stationary points of these functions appear as perturbed critical points and vice versa this permits new interpretations of the related solution methods and allows estimates of the solutions by using implicit function theorems for lipschitzian equations

threshold detection in narrow band non gaussian noise the middleton class a narrow band non gaussian noise model 9 12 is examined it is shown that this noise model which is known to fit closely a variety of non gaussian noises can itself be closely approximated by a computationally much simpler noise model it is then shown by numerical examples that for the problem of locally optimum detection the simplest form of this approximation yields nearly optimal asymptotic performance the performance of other simple suboptimal threshold detectors in class a noise is also examined finally a useful relationship between the class a model and the e mixture model is developed

integrability of subdifferentials of certain bivariate functions abstract we study integrability properties of bivariate functions defined on the product of banach spaces

subdifferentials of nonconvex vector valued functions abstract the formula of clarke s subdifferential for the sum of two real valued locally lipschitz functions has been extended by rockafellar to the case where one of the two functions is directionally lipschitzian we introduce in this paper a notion of cone related to the epigraph of a function and with the aid of this cone we extend the directionally lipschitzian behaviour to vector valued functions and we study the sum of two vector valued functions

semiregularity and generalized subdifferentials with applications to optimization the michel penot subdifferential of a locally lipschitzian function is the principal part of the clarke subdifferential it coincides with the g derivative at differentiable points a locally lipschitzian function can be determined by its michel penot subdifferential uniquely up to an additive constant though this cannot be done by its clarke subdifferential if the set of abnormal points is not negligible a set valued operator is the michel penot subdifferential of a locally lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition various calculus rules hold for the michel penot subdifferential equalities hold for these rules at a point under semiregularity which is weaker than regularity for a locally lipschitzian function in a separable banach space semiregularity holds everywhere except for a haar zero set applications in optimization are discussed

application of the volterra functional expansion in the detection of nonlinear functions of gaussian processes detection of a memoryless nonlinear functional of a gaussian process in additive gaussian white noise is considered the volterra functional expansion for the likelihood ratio and two examples of calculating the kernels are presented it is shown that kernels up to third order can be obtained for a hard limited gaussian process and for the absolute value of a gaussian process for the case of hard limiting the kernels are nonlinear functions of the autocorrelation of the gaussian process for the absolute value case the kernels are nonlinear functions of the kernel derived for the linear problem a monte carlo simulation of receiver performance is presented for the case of detection of the absolute value of a first order butterworth process in additive gaussian white noise the suboptimal detector is obtained by truncating the log likelihood ratio to second order

implicit multifunction theorems with positively homogeneous maps abstract we extend robinson s and ledyaev and zhu s implicit multifunction theorems using the language of generalized derivatives with positively homogeneous maps allowing us to obtain results that more closely resemble the classical single valued implicit function theorem we highlight that using linear openness instead of metric regularity gives simpler proofs and stronger results as part of our analysis we study perturbations of generalized linear openness and metric regularity finally we discuss how our methods may also be adapted to study generalized calmness

approximating a second order directional derivative for nonsmooth convex functions for a lower semicontinuous convex function f the approximate second order directional derivative d delta mapsto f varepsilon x 0 d delta is defined through the varepsilon directional derivative f varepsilon x d the function v d x mapsto v d x f varepsilon x d is for all varepsilon 0 locally lipschitz on inf domf and at those points where it is not differentiable v d admits a directional derivative v d x 0 delta for all delta which we precisely denote by f varepsilon x 0 d delta the objective of the present work is two fold to classify all the possible differentiability properties of v d according to the behavior of the function lambda mapsto f x 0 lambda d on r and to study the existence or nonexistence of the limit of f varepsilon x 0 d delta when varepsilon to 0

shape sensitivity analysis for an interface problem via minimax differentiability this paper deals with the shape sensitivity analysis for the conductivity problem with discontinuous coefficients we consider the geometric inverse problem of locating the interfaces where the jump of the coefficients occurs we propose a reconstruction method based on the minimization of the kohn vogelius functional the reformulation of the functional in a suitable saddle point problem form allows us to obtain the optimality conditions by using the differentiability of the minimax combined with the function space parametrization technique an important advantage of this approach is that it avoids the differentiability of the state variable with respect to the domain the reconstruction is then performed with an iterative algorithm using an appropriate quasi newton method we give some numerical examples showing the efficiency of the method

directional derivative estimates for the optimal value function of a quasidifferentiable programming problem this paper is concerned with the optimal value function arising in the primal decomposition of a quasidifferentiable programming problem in particular estimates for the upper dini directional derivative of this function are derived they involve certain lagrange multipliers occurring in the necessary minimum conditions to the lower level problems this study generalizes some previously published results on this subject

radial epiderivatives and asymptotic functions in nonconvex vector optimization the notions of lower and upper radial epiderivatives a radial notion different from that generally discussed for nonconvex vector valued functions are introduced many of their properties are established and some of the optimality conditions for a point to be an ideal pareto or weak pareto minimum involving these epiderivatives are also presented in particular a characterization for the ideal minima in terms of the lower radial epiderivative is proved such a result appears to be new in the literature under some mild assumptions on the given function it is proved that the asymptotic cone of its epigraph is the epigraph of its upper radial epiderivative moreover given a vector minimization problem we describe the asymptotic behavior of its solution set by introducing some cones of asymptotic directions of the function involved finally we define the lower and upper radial subdifferentials and express the optimality conditions by means of these subdifferentials certainly these optimality conditions subsume various necessary or sufficient conditions found in the literature for convex or nonconvex functions

partially ball weakly inf compact saddle functions we study on a product banach space the properties of a class of saddle functions called partially ball weakly inf compact for such a function we prove that the domain of the subdifferential is nonempty that the operator naturally associated with the subdifferential is maximal monotone and that the subdifferential of the function is integrable for a function in a large subclass of that class we prove the density of the domain of the subdifferential in the domain of the function

optimal linear detectors for additive noise channels in an attempt to gain insight into the design of linear detectors for additive white noise channels discrete time case we describe several procedures both optimal and suboptimal using the wiener representation for nonlinear systems we derive an ad hoc suboptimal design procedure exact designs are found when the noise amplitude s probability distribution is stable and when the noise is laplacian considering all the linear detectors thus derived no general form for the optimal linear detector s unit sample response becomes apparent performance analyses and simulations indicate substantial performance losses occur when linear detectors are used instead of optimal likelihood ratio ones