existence of variationally defined curves with higher order elliptic lagrangians abstract we present a method for proving the existence of solutions to a class of one dimensional variational problems namely those where the lagrangian is strongly elliptic the method is demonstrated by some examples of optimal interpolation problems which are motivated by applications to the mechanics and control of rigid bodies in each case the key step is to show that the variational problem satisfies the palais smale condition we do so in a general setting showing that the number of initial conditions required depends on the higher order energy bounding properties of the lagrangian

central limit theorems for bounded random variables under belief measures abstract recently a new type of central limit theorem for belief functions was given in epstein et al 9 in this paper we generalize the central limit theorem in epstein et al 9 to accommodate general bounded random variables these results are a natural extension of the classical central limit theory for additive probability measures

on the existence of geodesics on stationary lorentz manifolds with convex boundary abstract in this paper we consider the problem of the existence and multiplicity for geodesics not touching the boundary of a stationary lorentz manifold having convex boundary a physical example of a stationary and nonstatic lorentz manifold having convex boundary is the stationary axisymmetric asymptotically flat gravitational field outside a rotating massive object whenever its angular speed is small and its mean radius is close to the schwarzschild radius

orthogonal trajectories on riemannian manifolds and applications to plane wave type spacetimes abstract we present a result on trajectories of a lagrangian system joining two given submanifolds of a riemannian manifold under the action of an unbounded potential as an application we consider geodesics in a class of semi riemannian manifolds the plane wave type spacetimes

an existence theory for relativistic brachistochrones in stationary space times we use a variational principle proven by giannoni et al j math phys 38 6367 6381 1997 for relativistic brachistochrones and techniques of global analysis to develop an existence theory for brachistochrones in stationary space times we also develop a ljusternik schnirelman theory for brachistochrones obtaining multiplicity results depending on the topology of the space time

nonlocal transform domain filter for volumetric data denoising and reconstruction we present an extension of the bm3d filter to volumetric data the proposed algorithm bm4d implements the grouping and collaborative filtering paradigm where mutually similar d dimensional patches are stacked together in a d 1 dimensional array and jointly filtered in transform domain while in bm3d the basic data patches are blocks of pixels in bm4d we utilize cubes of voxels which are stacked into a 4 d group the 4 d transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes thus the spectrum of the group is highly sparse leading to very effective separation of signal and noise through coefficient shrinkage after inverse transformation we obtain estimates of each grouped cube which are then adaptively aggregated at their original locations we evaluate the algorithm on denoising of volumetric data corrupted by gaussian and rician noise as well as on reconstruction of volumetric phantom data with non zero phase from noisy and incomplete fourier domain k space measurements experimental results demonstrate the state of the art denoising performance of bm4d and its effectiveness when exploited as a regularizer in volumetric data reconstruction

stationary lower probabilities and unstable averages this paper explores the possibilities for probability like models of stationary nondeterministic phenomena that possess divergent but bounded time averages a random sequence described by a stationary probability measure must have almost surely convergent time averages whenever it has almost surely bounded time averages hence no measure can provide the mathematical model we desire in turning to lower probability based models we first explore the relationships between divergence stationarity and monotone continuity and those between monotone continuity and unicity of extensions we then construct several examples of stationary lower probabilities for sequences of uniformly bounded random variables such that divergence of time averages occurs with lower probability one we conclude with some remarks on the problem of estimating lower probability models on the basis of cylinder set observations

trajectories in godel type space times under the action of a vector field abstract in this paper we consider a godel type lorentz manifold m and look for curves 0 1 m joining two fixed points in m or having prescribed period y t satisfying the system d s s 1 2 curl a s s we find timelike trajectories of these problems

some quasilinear parabolic equations this paper is a continuation of earlier works l 5 concerning quasilinear problems we are interested in the existence of solutions for parabolic equations find u e lq o t w q sz satisfying a t u f u du s u 0 is given a t is a parabolic operator of a type u div i t x u du firstly when s is in lp o t w sp s2 but the growth of f is unlimited with respect to u and the solution u is unbounded this kind of problem was treated in the elliptic case for instance in 5 the method of relative rearrangement was used to control g secondly when s u a bounded radon measure on q 0 t x cl and u 0 is also a bounded radon measure on 0 recent results in this direction have been obtained in 6 see also references therein the main differences with our results are as follows i the method that we use is new and different especially to get a priori estimates and compactness results see lemmas 2 2 and 2 3 ii the operator is just strictly monotonic some dependence with respect to the solution u is allowed

two views of belief belief as generalized probability and belief as evidence abstract belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the kolmogorov axioms defining probability functions we argue that there are at least two useful and quite different ways of understanding belief functions the first is as a generalized probability function which technically corresponds to the inner measure induced by a probability function the second is as a way of representing evidence evidence in turn can be understood as a mapping from probability functions to probability functions it makes sense to think of updating a belief if we think of it as a generalized probability on the other hand it makes sense to combine two beliefs using say dempster s rule of combination only if we think of the belief functions as representing evidence many previous papers have pointed out problems with the belief function approach the claim of this paper is that these problems can be explained as a consequence of confounding these two views of belief functions

on the energy functional on finsler manifolds and applications to stationary spacetimes in this paper we first study some global properties of the energy functional on a non reversible finsler manifold in particular we present a fully detailed proof of the palais smale condition under the completeness of the finsler metric moreover we define a finsler metric of randers type which we call fermat metric associated to a conformally standard stationary spacetime we shall study the influence of the fermat metric on the causal properties of the spacetime mainly the global hyperbolicity moreover we study the relations between the energy functional of the fermat metric and the fermat principle for the light rays in the spacetime this allows one to obtain existence and multiplicity results for light rays using the finsler theory finally the case of timelike geodesics with fixed energy is considered

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

lightlike rays in stationary spacetimes with critical growth abstract the aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary lorentzian manifolds whose metric coefficients have an optimal growth

on p convex sets and geodesics on etudie les geodesiques sur une variete contrainte m a bord irregulier on considere une classe d ensembles irreguliers m qui permet d obtenir des resultats d existence et de multiplicite pour les geodesiques

on a class of singular second order hamiltonian systems with infinitely many homoclinic solutions abstract we show existence of infinitely many homoclinic orbits at the origin for a class of singular second order hamiltonian systems u v u t u 0 t we use variational methods under the assumption that v t u satisfies the so called strong force condition

classification of fuzzy measures the aim of this paper is to study several classes of fuzzy measures and the relationships among them this classification is based on the one due to banon but it has some particularities first new kinds of fuzzy measures are considered but the most important feature is that the classification is not directly applied to fuzzy measures but to pairs of dual fuzzy measures

on a multiplication and a theory of integration for belief and plausibility functions abstract belief and plausibility functions have been introduced as generalizations of probability measures which abandon the axiom of additivity it turns out that elementwise multiplication is a binary operation on the set of belief functions if the set functions of the type considered here are defined on a locally compact and separable space x a theorem by choquet ensures that they can be represented by a probability measure on the space containing the closed subsets of x the so called basic probability assignment this is basic for defining two new types of integrals one of them may be used to measure the degree of non additivity of the belief or plausibility function the other one is a generalization of the lebesgue integral the latter is compared with choquet s and sugeno s integrals for non additive set functions

bayes like decision making with upper and lower probabilities abstract we consider the use of interval valued probabilities to represent the support lent to the hypothesis that the parameter value lies in a subset a of the parameter set when we observe x know the likelihoods f e and have some prior information concerning the parameter our model for prior information is that of a salient prior distribution in which we have little confidence although we have much less confidence in any alternative prior we consider notions of acceptable and coherent decision making as well as notions of being able to achieve a bayes rule and least commitment throughout we are motivated to preserve some of the elements of bayesian decision making without thereby committing ourselves to unwarranted claims of knowledge

periodic solutions for dynamical systems on non complete riemannian manifolds abstract we look for periodic solutions in a dynamical system on a non complete riemannian manifold if the potential is bounded and suitable convexity assumptions hold the existence of infinitely many solutions can be proved by means of variational methods penalization arguments and classical ljusternik schnirelman theory

finite dimensional distributions and tail behavior in stationary interval valued probability models we consider the relationship between the finite dimensional distributions of a stationary time series model and its asymptotic behavior in the framework of interval valued probability ivp a simple generalization of additive probability measures by caratheodory s theorem the specification of a countably additive probability measure on the algebra of cylinders c uniquely defines its behavior on c containing the tail events if the measure is stationary then the ergodic theorem indicates that its extension assigns zero probability to the tail event consisting of all sequences for which the time averages diverge the divergence event this link between the marginals and the tail behavior is no longer valid in ivp and we can reconcile arbitrary finite dimensional distributions and tail behavior for stationary ivp based models the lining mechanism between the marginals of a time series model and its asymptotic behavior turns out to be continuity not stationarity or even additivity we prove that any stationary finitely additive probability charge defined on cylinders has a stationary charge extension that can assign the divergence event any prescribed probability moreover on the space of binary sequences we consider ivp models that incorporate i stationarity ii continuity along c iii almost sure support for divergence iv estimability of the divergence event from cylinders v nearly additive finite dimensional distributions we enhance the previous constructions of ivp s satisfying i iv so that they satisfy v by agreeing with a stationary measure either exactly on one dimensional cylinders or arbitrarily closely on a given clans of bounded dimensional cylinders our time series constructions follow from the observation that the algebra of cylinders and the tail algebra are mutually nonsingular we use the same idea to prove the existence of joints for general marginal ivp s these constructions have implications for frequentist interpretations of probability

multiple critical points for indefinite functionals and applications abstract in this paper we prove a multiplicity result for critical points of an indefinite functional on a manifold by using a generalization of the ljusternik schnirelman category the abstract result is applied to prove the existence of multiple timelike trajectories for lorentzian manifolds of splitting type

products of capacities on separate spaces through additive measures an approach to product of capacities is proposed it extends our previous approach which works only for a special class of capacities the main advantage of our approach as opposed to most approaches given so far is that it is designed for general i e not necessarily discrete monotone capacities the product is not defined uniquely but rather a lower and upper bound are given which in case where both capacities are additive measures both coincide with the usual additive product

on the geodesical connectedness for a class of semi riemannian manifolds we prove a variational principle for geodesics on a semi riemannian manifold m of arbitrary index k and possessing k linearly independent killing vector fields that generate a timelike distribution on m using such a principle and a suitable completeness condition for m we prove some existence and multiplicity results for geodesics joining two fixed points of m

category of loop spaces of open subsets in euclidean space has been studied in several recent papers e g l 2 5 6 91 in situations where the potential function v is singular in the sense that it is defined and c only on an open subset u of euclidean k space rk and not on the whole of rk if e wgt2 r rk is the sobolev space of t periodic functions from r to rk and e u is the subset of e of functions which map into u then the topology of e u plays a crucial role in these studies in particular the knowledge that the lusternik schnirelmann category of e u is infinite when used in conjunction with min max methods implies the existence of infinitely many critical points another subset e u which consists of those functions in e u which take the origin to a base point in u plays a dual role first of all in problems concerning the existence of geodesics in u from one subset a of u to another b of u see e g 3 the category of e u plays a role secondly the general inequality cat e u 2 cat e u see 6 when u is connected and simply connected can be applied to show that cat e u cat e u thus providing a tool for computing cat e u the topological analogue of e u is the space of free loops au c s u of continuous maps from the circle s to u and the analogue of e u is the space of based loops ru which consists of maps in au which take a base point in s to a base point in u au has the same homotopy type as e u and s2u has the same homotopy type as e u thus homotopy type invariants e g category of e u and e u may be studied through au and szu respectively the following classical theorem of serre l l is useful in this regard and is stated in terms of a general space x

symmetric upper probabilities as a first step toward developing statistical models based on upper and lower probabilities we study upper probabilities and upper expectations on the unit interval that are symmetric by which we mean invariant with respect to equimeasurability these upper probabilities are generalizations of uniform probability measures we give some characterizations of these upper probabilities specifically we show that symmetry of the upper expectation functional is equivalent to the underlying set of densities being closed under majorization we also show that a function is the upper distribution for a symmetric upper probability if and only if its lower graph is star shaped with respect to the origin and to the point 1 1 we derive inner and outer approximations to symmetric classes of probabilities based on the upper probability the class of symmetric upper expectations that are completely determined by their values on the indicator functions is characterized we provide a geometric characterization of a hierarchy of upper probabilities including fine s generalized upper probabilities and 2 alternating choquet capacities in particular we establish a 1 1 correspondence between symmetric 2 alternating capacities and nonincreasing density functions we prove that undominated generalized upper probabilities do not exist in the symmetric case examples from robust statistics are considered an example is given that shows that symmetry of upper probabilities does not imply symmetry of upper expectations a corollary is that symmetry of the choquet integral does not imply symmetry of the upper expectation functional

decision making with imprecise probabilities dempster shafer theory and application the information and data used to support decision making under uncertainty in water resources situations can often be characterized as being very limited or weak the representation of knowledge in conventional bayesian decision analysis is in the form of precisely specified distributions and is the same no matter how weak the information source for this knowledge a bayesian analysis therefore may inadvertently impart too much precision to the input information and to the results the concept of imprecise probability addresses this problem of excessive precision and a number of methods incorporating this concept have emerged one such method developed by dempster and shafer accommodates greater imprecision by allowing the specification of probabilities on intervals theoretical aspects of the dempster shafer methodology and its application to inference and decision analysis are described a water resources example of an application of the dempster shafer approach is presented and the results contrasted with those obtained from the closest equivalent bayesian scheme

image denoising by sparse 3 d transform domain collaborative filtering we propose a novel image denoising strategy based on an enhanced sparse representation in transform domain the enhancement of the sparsity is achieved by grouping similar 2d image fragments e g blocks into 3d data arrays which we call groups collaborative altering is a special procedure developed to deal with these 3d groups we realize it using the three successive steps 3d transformation of a group shrinkage of the transform spectrum and inverse 3d transformation the result is a 3d estimate that consists of the jointly filtered grouped image blocks by attenuating the noise the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block the filtered blocks are then returned to their original positions because these blocks are overlapping for each pixel we obtain many different estimates which need to be combined aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy a significant improvement is obtained by a specially developed collaborative wiener filtering an algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail an extension to color image denoising is also developed the experimental results demonstrate that this computationally scalable algorithm achieves state of the art denoising performance in terms of both peak signal to noise ratio and subjective visual quality

geodesics in stationary spacetimes and classical lagrangian systems abstract we state a fundamental correspondence between geodesics on stationary spacetimes and the equations of classical particles on riemannian manifolds accelerated by a potential and a magnetic field by variational methods we prove some existence and multiplicity theorems for fixed energy solutions joining two points or periodic of the above described riemannian equation as a consequence we obtain existence and multiplicity results for geodesics with fixed energy connecting a point to a line or periodic trajectories in standard stationary spacetimes

conditional extremals in complete riemannian manifolds abstract conditional extremal curves in a complete riemannian manifold m are defined as the critical points of the squared l 2 distance between the tangent vector field of a curve and a so called prior vector field we prove that this l 2 distance satisfies the palais smale condition on the space of absolutely continuous curves joining two submanifolds of m and thus establish the existence of critical points we also prove a morse index theorem in the case where the two submanifolds are single points and use the morse inequalities to place lower bounds on the number of critical points of each index

on modeling of linguistic information using random sets this paper discusses the formal connection between possibility distributions zadeh 21 and the theory of random sets via choquet s theorem based upon these relationships it is suggested that plausible inferences and modeling of common sense can be derived from the statistics of random sets the analysis of subjectivity in meaning representation of natural languages can be carried out by taking account of covariates of individuals as in the statistical analysis of survival data

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measures of uncertainty in expert systems abstract this paper compares four measures that have been advocated as models for uncertainty in expert systems the measures are additive probabilities used in the bayesian theory coherent lower or upper previsions belief functions used in the dempster shafer theory and possibility measures fuzzy logic special emphasis is given to the theory of coherent lower previsions in which upper and lower probabilities expectations and conditional probabilities are constructed from initial assessments through a technique of natural extension mathematically all the measures can be regarded as types of coherent lower or upper previsions and this perspective gives some insight into the properties of belief functions and possibility measures the measures are evaluated according to six criteria clarity of interpretation ability to model partial information and imprecise assessments especially judgements expressed in natural language rules for combining and updating uncertainty and their justification consistency of models and inferences feasibility of assessment and feasibility of computations each of the four measures seems to be useful in special kinds of problems but only lower and upper previsions appear to be sufficiently general to model the most common types of uncertainty

comments on image denoising by sparse 3 d transform domain collaborative filtering in order to resolve the problem that the denoising performance has a sharp drop when noise standard deviation reaches 40 proposed to replace the wavelet transform by the dct in this comment we argue that this replacement is unnecessary and that the problem can be solved by adjusting some numerical parameters we also present this parameter modification approach here experimental results demonstrate that the proposed modification achieves better results in terms of both peak signal to noise ratio and subjective visual quality than the original method for strong noise

dominance of capacities by k additive belief functions in this paper we deal with the set of k additive belief functions dominating a given capacity we follow the line introduced by chateauneuf and jaffray for dominating probabilities and continued by grabisch for general k additive measures first we show that the conditions for the general k additive case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid on the other hand we show that these conditions cannot be improved we solve this situation by imposing additional constraints on the dominating functions then we consider the more restrictive case of k additive belief functions in this case a similar result with stronger conditions is proved although better this result is not completely satisfactory and as before the conditions cannot be strengthened however when the initial capacity is a belief function we find a subfamily of the set of dominating k additive belief functions from which it is possible to derive any other dominant k additive belief function and such that the conditions are even more restrictive obtaining the natural extension of the result for probabilities finally we apply these results in the fields of social welfare theory and decision under risk