first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

fuzzy differential equations and the extension principle we study the cauchy problem for differential equations considering its parameters and or initial conditions given by fuzzy sets these fuzzy differential equations are approached in two different ways a by using a family of differential inclusions and b the zadeh extension principle for the solution of the model we conclude that the solutions of the cauchy problem obtained by both are the same we also provide some illustrative examples

the reliability functions of the general source with fixed length coding the reliability function problems with fixed length source coding for the general source are studied for all rates r our fundamental philosophy in doing so is to convert all of the reliability function problems to the pertinent computation problems in the large derivation probability theory it turns out that this kind of new methodology which was previously developed by han see ibid vol 43 p 1145 64 1997 enables us to establish quite compact general formulas of the reliability function for general sources including all nonstationary and or nonergodic sources with countably infinite alphabet such general formulas are presented from the information spectrum point of view

on completeness of quadratic systems abstract a dynamical system is called complete if every solution of it exists for all t r let k be the dimension of the vector space of quadratic systems the set of complete quadratic systems is shown to contain a vector subspace of dimension 2 k 3 we provide two proofs one by the gronwall lemma and the second by compactification that is capable of showing incompleteness as well characterization of a vector subspace of complete quadratic systems is provided the celebrated lorenz system for all real ranges of its parameters is shown to belong to this subspace we also provide a sufficient condition for a system to be incomplete

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

general formulas for capacity of classical quantum channels the capacity of a classical quantum channel or in other words the classical capacity of a quantum channel is considered in the most general setting where no structural assumptions such as the stationary memoryless property are made on a channel a capacity formula as well as a characterization of the strong converse property is given just in parallel with the corresponding classical results of verdu han 1994 which are based on the so called information spectrum method the general results are applied to the stationary memoryless case with or without cost constraint on inputs whereby a deep relation between the channel coding theory and the hypothesis testing for two quantum states is elucidated

existence theorem to the cauchy problem of fuzzy differential equations under compactness type conditions kaleva fuzzy sets and systems 35 1990 389 396 has concluded that the peano theorem does not hold for fuzzy differential equations on en d this paper deals with the cauchy problem of fuzzy differential equations the existence theorems under compactness type conditions are obtained

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

error estimation in the approximation of the solution of nonlinear fuzzy fredholm integral equations using a fixed point technique the sequence of successive approximations and a recent quadrature formula for fuzzy number valued functions it is constructed a numerical method for the solution of nonlinear fuzzy fredholm integral equations moreover the error estimate of the method and a criterion to stop the corresponding algorithm are given

improved predictor corrector method for solving fuzzy initial value problems in this paper an improved predictor corrector ipc method is proposed to solve the fuzzy initial value problem the ipc method is generated by combining an explicit three step method and an implicit two step method the methods are compared with the methods discussed in t allahviranloo n ahmady e ahmady numerical solution of fuzzy differential equations by predictor corrector method information sciences 177 7 2007 1633 1647 and are shown to be more accurate the convergence and stability of the proposed methods are also presented in detail these methods are illustrated by solving some examples

on the triangle inequalities in fuzzy metric spaces this paper presents level forms of the triangle inequalities in fuzzy metric spaces x d l r to aid discussion a fuzzy pre metric condition is introduced it is first pointed out that under the fuzzy pre metric condition the first triangle inequality is always equivalent to its level form the second triangle inequality is equivalent to one level form when r is right continuous and to another level form also when further conditions are imposed on r in a fuzzy metric space the level form of the first triangle inequality and one of the level forms of the second triangle inequality are always valid the other level form of the second triangle inequality holds for all but at most countable a 0 1 finally a fixed point theorem for fuzzy metric spaces is derived as an application of the preceding results

large deviation analysis for quantum security via smoothing of renyi entropy of order 2 it is known that the security evaluation can be done by smoothing of renyi entropy of order 2 in the classical and quantum settings when we apply universal 2 hash functions using the smoothing of renyi entropy of order 2 we derive security bounds for l 1 distinguishability and modified mutual information criterion under the classical and quantum setting and have derived these exponential decreasing rates these results are extended to the case when we apply e almost dual universal 2 hash functions furthermore we apply this analysis to the secret key generation with error correction

numerical solution of fuzzy differential equations by predictor corrector method in this paper three numerical methods to solve the fuzzy ordinary differential equations are discussed these methods are adams bashforth adams moulton and predictor corrector predictor corrector is obtained by combining adams bashforth and adams moulton methods convergence and stability of the proposed methods are also proved in detail in addition these methods are illustrated by solving two fuzzy cauchy problems

monotone method for fuzzy differential equations we develop the monotone iterative technique to approximate the extremal solutions for the initial value problem relative to a fuzzy differential equation in a fuzzy functional interval to our procedure it is crucial to have at our disposal fuzzy comparison results and several properties of sets of fuzzy functions and fuzzy numbers including a compactness result in spaces of fuzzy functions which are useful from the point of view of applications

fuzzy quasilinear spaces and applications we introduce the concept of fuzzy quasilinear space and fuzzy quasilinear operator moreover we state some properties and give results which extend to the fuzzy context some results of linear functional analysis

toward the existence and uniqueness of solutions of second order fuzzy differential equations in this paper the existence and uniqueness of solutions for second order fuzzy differential equations with initial conditions under generalized h differentiability is proved to this end the concept of second order generalized differential equation is defined which is based on an enlargement of the class of differentiable fuzzy mappings

critical points at infinity and blow up of solutions of autonomous polynomial differential systems via compactification abstract critical points at infinity for autonomous differential systems are defined and used as an essential tool r n is mapped onto the unit ball by various mappings and the boundary points of the ball are used to distinguish between different directions at infinity these mappings are special cases of compactifications it is proved that the definition of the critical points at infinity is independent of the choice of the mapping to the unit ball we study the rate of blow up of solutions in autonomous polynomial differential systems of equations via compactification methods to this end we represent each solution as a quotient of a vector valued function which is a solution of an associated autonomous system by a scalar function which is a solution of a related scalar equation

existence and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

global existence of solutions to fuzzy differential equations abstract the local existence and uniqueness theorems are investigated by wu et al j math anal appl 202 1996 629 644 for the cauchy problem of the fuzzy valued mappings of a real variable whose values are in the fuzzy number space e n d in this paper we point out a variety of results which assure global existence of solutions to fuzzy differential equations

a class of linear differential dynamical systems with fuzzy initial condition this paper investigates linear first order fuzzy differential dynamical systems where the initial condition is described by a fuzzy number we use a complex number representation of the a level sets of the fuzzy system and prove theorems that provide the solutions under such representation which is applicable to practical computations and also has some implications for theory then the paper shows some properties of the two dimensional dynamical systems and their phase portraits are described by means of examples there may be a significant difference between the solutions according to whether the matrix is nonnegative or not finally the paper points out future research on the fuzzy dynamical systems

optimal sequence of quantum measurements in the sense of stein s lemma in quantum hypothesis testing we derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing we discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0 as an asymptotically optimal measurement we propose a projection measurement characterized by the irreducible representation theory of the special linear group sl especially in the spin 1 2 system it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction as a by product we obtain another proof of quantum stein s lemma in addition an asymptotically optimal measurement is constructed in the quantum gaussian case and it is physically meaningful

a quantum version of sanov s theorem we present a quantum version of sanov s theorem focussing on a hypothesis testing aspect of the theorem there exists a sequence of typical subspaces for a given set of stationary quantum product states asymptotically separating them from another fixed stationary product state analogously to the classical case the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set while in the classical case the separating subsets can be chosen universally in the sense that they depend only on the chosen set of i i d processes in the quantum case the choice of the separating subspaces depends additionally on the reference state

approximate solutions existence and uniqueness of the cauchy problem of fuzzy differential equations abstract the cauchy problem of fuzzy differential equations is investigated by the use of the concept of h differentiability due to m l puri and d a ralescu j math anal appl 91 1983 552 558 the existence and uniqueness theorem is obtained for the solution of the cauchy problem x t f t x t x t 0 x 0 for the fuzzy valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in r n where the function f satisfies the generalized lipschitz condition

artificial neural network approach for solving fuzzy differential equations the current research attempts to offer a novel method for solving fuzzy differential equations with initial conditions based on the use of feed forward neural networks first the fuzzy differential equation is replaced by a system of ordinary differential equations a trial solution of this system is written as a sum of two parts the first part satisfies the initial condition and contains no adjustable parameters the second part involves a feed forward neural network containing adjustable parameters the weights hence by construction the initial condition is satisfied and the network is trained to satisfy the differential equations this method in comparison with existing numerical methods shows that the use of neural networks provides solutions with good generalization and high accuracy the proposed method is illustrated by several examples

strong converse and stein s lemma in quantum hypothesis testing the hypothesis testing problem for two quantum states is treated we show a new inequality between the errors of the first kind and the second kind which complements the result of hiai and petz 1991 to establish the quantum version of stein s lemma the inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy which yields the strong converse in quantum hypothesis testing finally we discuss the relation between the bound and the power exponent derived by han and kobayashi 1989 in classical hypothesis testing

numerical validation of blow up solutions of ordinary differential equations abstract this paper focuses on blow up solutions of ordinary differential equations odes we present a method for validating blow up solutions and their blow up times which is based on compactifications and the lyapunov function validation method the necessary criteria for this construction can be verified using interval arithmetic techniques some numerical examples are presented to demonstrate the applicability of our method

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

on the converse theorem in statistical hypothesis testing simple statistical hypothesis testing is investigated by making use of the divergence geometric method the asymptotic behavior of the minimum value of the error probability of the second kind under the constraint that the error probability of the first kind is bounded above by exp rn is looked for where r is a given positive number if r is greater than the divergence of the two probability measures the so called converse theorem holds it is shown that the condition under which the converse theorem holds can be divided into two separate cases by analyzing the geodesic connecting the two probability measures and as a result an explanation is given for the han kobayashi linear function f sub t x

on peano theorem for fuzzy differential equations nieto s theorem states that every fuzzy differential equation with bounded and continuous right hand side admits a solution in this paper we show that the provided proof of this result is not correct

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

the cauchy problem for complex fuzzy differential equations abstract we discuss the existence of a solution to the cauchy problem for fuzzy differential equations that accommodates the notion of fuzzy sets defined by complex valued membership functions we first propose definitions of complex fuzzy sets and discuss entailed results which parallel those of regular fuzzy numbers we then give two existence results relevant to the cauchy problem for fuzzy differential equations in the case of bounded integral operators these results require either holder continuous or lipschitz continuous response functions

moderate deviation analysis for classical quantum channels and quantum hypothesis testing in this paper we study the tradeoffs between the error probabilities of classical quantum channels and the blocklength n when the transmission rates approach the channel capacity at a rate lower than 1 sqrt n a research topic known as moderate deviation analysis we show that the optimal error probability vanishes under this rate convergence our main technical contributions are a tight quantum sphere packing bound obtained via chaganty and sethuraman s concentration inequality in strong large deviation theory and asymptotic expansions of error exponent functions moderate deviation analysis for quantum hypothesis testing is also established the converse directly follows from our channel coding result while the achievability relies on a martingale inequality

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

global solutions of fuzzy integro differential equations under generalized differentiability by the method of upper and lower solutions this paper is devoted to introduce a new concept of upper and lower solutions for fuzzy integro differential equations that is defined on integral equations associated with the main problem this plays an essential role in proof of existence and uniqueness results under weak conditions by the method of upper and lower solutions existence and uniqueness of global solutions of an initial value problem for first order nonlinear fuzzy integro differential equation under generalized differentiability are given our technique is illustrated by some examples

asymptotics of quantum relative entropy from a representation theoretical viewpoint in this paper it is proved that the quantum relative entropy d can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive operator valued measures povms the sequence of povms depends on but is independent of the choice of

asymptotic equilibrium and stability of fuzzy differential equations the local existence and uniqueness theorems and the global existence of solutions were investigated in 1 3 respectively for the cauchy problem of fuzzy valued functions of a real variable whose values are in the fuzzy number space e n d in this paper we first study the asymptotic equilibrium for fuzzy evolution equations then the stability properties of the trivial fuzzy solution of the perturbed semilinear fuzzy evolution equations are investigated by extending the lyapunov s direct method

general formulas for fixed length quantum entanglement concentration in this paper we derive general formulas for the amount of entanglement that can be concentrated from general sequences of partially entangled bipartite pure states the formulas are obtained by using an information spectrum approach the formulas obtained express the optimal rate under constant error constraints or exponential error constraints in the asymptotic framework since the formulas treat general sequences they can be applied to i i d sequences as well as sequences with correlations

simulation of random processes and rate distortion theory we study the randomness necessary for the simulation of a random process with given distributions on terms of the finite precision resolvability of the process finite precision resolvability is defined as the minimal random bit rate required by the simulator as a function of the accuracy with which the distributions are replicated the accuracy is quantified by means of various measures variational distance divergence orstein 1973 prohorov 1956 and related measures of distance between the distributions of random process in the case of ornstein prohorov and other distances of the kantorovich vasershtein type we show that the finite precision resolvability is equal to the rate distortion function with a fidelity criterion derived from the accuracy measure this connection leads to new results on nonstationary rate distortion theory in the case of variational distance the resolvability of stationary ergodic processes is shown to equal entropy rate regardless of the allowed accuracy in the case of normalized divergence explicit expressions for finite precision resolvability are obtained in many cases of interest and connections with data compression with minimum probability of block error are shown

solving fuzzy differential equations by differential transformation method an extension of the differential transformation method dtm which is an analytical numerical method for solving the fuzzy differential equation fde is given the concept of generalised h differentiability is used this concept is based on an enlargement of the class of differentiable fuzzy mappings to define this the lateral hukuhara derivatives are considered the proposed algorithm is illustrated by numerical examples and some error comparisons are made with other methods for solving a fde

geometric treatments and a common mechanism in finite time singularities for autonomous odes abstract geometric treatments of blow up solutions for autonomous ordinary differential equations and their blow up rates are concerned our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces and dynamics on their center stable manifolds in particular we show that dynamics on center stable manifolds of invariant sets at infinity with appropriate time scale desingularizations as well as blowing up of singularities characterize dynamics of blow up solutions as well as their rigorous blow up rates similarities for characterizing finite time extinction and asymptotic behavior of compacton traveling waves to blow up solutions are also shown

comparison results for fuzzy differential equations in this paper we present several comparison results for the solutions of fuzzy differential equations by using the hukuhara derivative these results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations besides some side results about the hukuhara difference and the partial orderings are given we illustrate the applicability of the new results by showing several examples

euler polygonal method for metric dynamical systems following some ideas developed for equations with a nonlinear differential we consider the existence and uniqueness of solution for the metric dynamical system f x 0 and we study the approximation under appropriate hypotheses of the unique solution using the euler polygonal approximation method in relation with function f

general formulas for the neyman pearson type ii error exponent subject to fixed and exponential type i error bounds the general formulas for the neyman pearson type ii error exponent subject to two different type i error constraints as indicated in the title of the correspondence are established as revealed in the formulas the type ii error exponents are fully determined by the ultimate statistical characteristic of the normalized log likelihood ratio evaluated under the null hypothesis distribution applications of the general formulas to distributed neyman pearson detection and the channel reliability function are also demonstrated

opportunistic detection rules finite and asymptotic analysis opportunistic detection rules odrs are variants of fixed sample size detection rules in which the statistician is allowed to make an early decision on the alternative hypothesis opportunistically based on the sequentially observed samples from a sequential decision perspective odrs are also mixtures of one sided and truncated sequential detection rules several results regarding odrs are established in this paper in the finite regime the maximum sample size is modeled either as a fixed finite number or a geometric random variable with a fixed finite mean for both cases the corresponding bayesian formulations are investigated the former case is a slight variation of the well known finite length sequential hypothesis testing procedure in the literature whereas the latter case is new for which the bayesian optimal odr is shown to be a sequence of likelihood ratio threshold tests with two different thresholds a running threshold which is determined by solving a stationary state equation is used when future samples are still available and a terminal threshold simply the ratio between the priors scaled by costs is used when the statistician reaches the final sample and thus has to make a decision immediately in the asymptotic regime the tradeoff among the exponents of the false alarm and miss error probabilities and the normalized expected stopping time under the alternative hypothesis is completely characterized and proved to be tight via an information theoretic argument within the tradeoff region one noteworthy fact is that the performance of the stein chernoff lemma is attainable by odrs

numerical validation of blow up solutions with quasi homogeneous compactifications we provide a numerical validation method of blow up solutions for finite dimensional vector fields admitting asymptotic quasi homogeneity at infinity our methodology is based on quasi homogeneous compactifications containing quasi parabolic type and directional type compactifications divergent solutions including blow up solutions then correspond to global trajectories of associated vector fields with appropriate time variable transformation tending to equilibria on invariant manifolds representing infinity we combine standard methodology of rigorous numerical integration of differential equations with lyapunov function validations around equilibria corresponding to divergent directions which yields rigorous upper and lower bounds of blow up time as well as rigorous profile enclosures of blow up solutions

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

global existence and uniqueness of solutions for fuzzy differential equations under dissipative type conditions in this paper using the properties of a differential and integral calculus for fuzzy set valued mappings and completeness of metric space of fuzzy numbers the global existence uniqueness and the continuous dependence of a solution on a fuzzy differential equation are derived under the dissipative type conditions we also present the global existence and uniqueness of solutions for a fuzzy differential equation on a closed convex subset of fuzzy number space

existence and uniqueness of cauchy problem for fuzzy differential equations under dissipative conditions the existence theorem of peano for the fuzzy differential equation x t f t x t x t 0 x 0 does not hold in general except in the special case where the fuzzy number space e n d is finite dimensional 1 or f is assumed to be continuous and bounded 2 in this paper the dissipative type conditions which guarantee the existence theorem for peano are specified based on the existence theorem of approximate solutions to above cauchy problem

an information spectrum approach to classical and quantum hypothesis testing for simple hypotheses the information spectrum analysis made by han for classical hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and quantum hypothesis testing the results are also applied to fixed length source coding when loosening the normalizing condition for probability distributions and for quantum states we establish general formulas for several quantities relating to the asymptotic optimality of tests codes in terms of classical and quantum information spectra

blow up solutions of cubic differential systems abstract we investigate the existence problem for blow up solutions of cubic differential systems we find sets of initial values of the blow up solutions we also discuss a method of finding upper estimates for the blow up time of these solutions our approach can be applied to systems of partial differential equations we apply this approach to the cauchy dirichlet problem for systems of semilinear heat equations with cubic nonlinearities

analytical and numerical solutions of fuzzy differential equations in this paper we study analytical and numerical solutions of fuzzy differential equations based on the extension principle for linear fuzzy differential equations we state some results on the behaviour of the solutions and study their relationship with the generalised hukuhara derivative in order to approximate the solutions of linear and non linear fuzzy differential equations we propose a new fuzzification of the classical euler method and then incorporate an unconstrained optimisation technique this combination offers a powerful tool to tackle uncertainty in any numerical method an efficient computational algorithm is also provided to guarantee the convexity of fuzzy solutions on the time domain several illustrative examples are given

the strong converse theorem for hypothesis testing the authors present the answer to a so called converse problem by giving the explicit form of the power exponent this strong converse should be regarded as completing a weak converse previously obtained by r e blahut 1974

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

initial value problem for fuzzy differential equations under dissipative conditions a new concept of inner product on the fuzzy space e n d is introduced studied and used to prove several theorems stating the existence uniqueness and boundedness of solutions of fuzzy differential equations a stability result is also proved in the same context

quantum sphere packing bounds with polynomial prefactors we study lower bounds on the optimal error probability in classical coding over classical quantum channels at rates below the capacity commonly termed quantum sphere packing bounds winter and dalai have derived such bounds for classical quantum channels however the exponents in their bounds only coincide when the channel is classical in this paper we show that these two exponents admit a variational representation and are related by the golden thompson inequality reaffirming that dalai s expression is stronger in general classical quantum channels second we establish a finite blocklength sphere packing bound for classical quantum channels which significantly improves dalai s prefactor from the order of subexponential to polynomial furthermore the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o log n n indicating our sphere packing bound is almost exact in the high rate regime finally for a special class of symmetric classical quantum channels we can completely characterize its optimal error probability without the constant composition code assumption the main technical contributions are two converse hoeffding bounds for quantum hypothesis testing and the saddle point properties of error exponent functions

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

hypothesis testing with the general source the asymptotically optimal hypothesis testing problem with general sources as the null and alternative hypotheses is studied under exponential type error constraints on the first kind of error probability our fundamental philosophy is to convert all of the hypothesis testing problems to the pertinent computation problems in the large deviation probability theory this methodologically new approach enables us to establish compact general formulas of the optimal exponents of the second kind of error and correct testing probabilities for the general sources including all nonstationary and or nonergodic sources with arbitrary abstract alphabet countable or uncountable these general formulas are presented from the information spectrum point of view

nonlinear problems with blow up solutions numerical integration based on differential and nonlocal transformations and differential constraints abstract several new methods of numerical integration of cauchy problems with blow up solutions for nonlinear ordinary differential equations of the first and second order are described solutions of such problems have singularities whose positions are unknown a priori for this reason the standard numerical methods for solving problems with blow up solutions can lead to significant errors the first proposed method is based on the transition to an equivalent system of equations by introducing a new independent variable chosen as the first derivative t y x where x and y are independent and dependent variables in the original equation the second method is based on introducing a new auxiliary nonlocal variable of the form x 0 x g x y y x d x with the subsequent transformation to the cauchy problem for the corresponding system of odes the third method is based on adding to the original equation of a differential constraint which is an auxiliary ode connecting the given variables and a new variable the proposed methods lead to problems whose solutions are represented in parametric form and do not have blowing up singular points therefore the transformed problems admit the application of standard fixed step numerical methods the efficiency of these methods is illustrated by solving a number of test problems that admit an exact analytical solution it is shown that i the methods based on nonlocal transformations of a special kind are more efficient than several other methods namely the method based on the hodograph transformation the method of the arc length transformation and the method based on the differential transformation and ii among the proposed methods the most general method is the method based on the differential constraints some examples of nonclassical blow up problems are considered in which the right hand side of equations has fixed singular points or zeros simple theoretical estimates are derived for the critical value of an independent variable bounding the domain of existence of the solution it is shown by numerical integration that the first and the second painleve equations with suitable initial conditions have non monotonic blow up solutions it is demonstrated that the method based on a nonlocal transformation of the general form as well as the method based on the differential constraints admit generalizations to the nth order odes and systems of coupled odes

fuzzy laplace transform based on the henstock integral and its applications in discontinuous fuzzy systems abstract the existing results on the fuzzy laplace transform and their applications were based on zaheh s decomposition theorem and were formally characterized by the integrals of real valued functions directly that is the existence of the fuzzy laplace transform in essence has not been solved in this article the fuzzy laplace transform is incorporated into the framework of the henstock integral and proposed by use of fuzzy henstock integrals on infinite intervals in addition as a theoretical basis the existence and the basic properties of the fuzzy laplace transform are investigated the convolution of fuzzy valued functions and real valued functions is defined and the convolution theorem of the fuzzy laplace transform is given finally discontinuous fuzzy initial value problems and two kinds of fuzzy volterra integral equations are discussed with use of the fuzzy laplace transform presented in this article

second order asymptotics in fixed length source coding and intrinsic randomness there is a difference between the optimal rates of fixed length source coding and intrinsic randomness when we care about the second order asymptotics we prove this difference for general information sources and then investigate independent and identically distributed i i d random variables and markovian variables as examples the difference is demonstrated through an investigation of the second order asymptotic behavior of the rates a universal fixed length source code attaining the second order optimal rate is also proposed the difference between the rates of fixed length source coding and intrinsic randomness proves that the outputs of fixed length source codes are not uniformly distributed

nth order fuzzy linear differential equations in this paper a numerical method for solving nth order linear differential equations with fuzzy initial conditions is considered the idea is based on the collocation method the existence theorem of the fuzzy solution is considered this method is illustrated by solving several examples

a peano theorem for fuzzy differential equations with evolving membership grade abstract the peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations the difficulty appears to be due to the standard use of the supremum metric defined by the supremum over the hausdorff metric between the level sets of the fuzzy sets another may have been the classical formulation of fuzzy differential equations in terms of the hukuhara derivative of the level sets here a peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions with aubin s morphological equations which allow non local set evolution a major difference from previous publications is the use of the endograph metric end essentially the hausdorff metric between the endographs in r n 0 1 of fuzzy sets instead of the supremum metric another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations the result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity compact supports or even normality

comparation between some approaches to solve fuzzy differential equations in this paper we study the class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via zadeh s extension principle we get a fuzzy solution for this class of fuzzy differential equations and several illustrative examples are presented we also give some properties and we show the relationships with others interpretation finally we propose a numerical procedure for obtaining the fuzzy solution

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

numerical solutions for linear system of first order fuzzy differential equations with fuzzy constant coefficients in this paper a novel iterative method is proposed to obtain approximate analytical solutions for the linear systems of first order fuzzy differential equations fdes with fuzzy constant coefficients fccs while avoiding the complexities of eigen value computations a theorem for the convergence and the validity of the approach is also presented in detail numerical experiments and comparisons with exact solutions reveal that the proposed method is capable of generating accurate results

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

global existence of solutions for fuzzy second order differential equations under generalized h differentiability in this paper we study the global existence of solutions for second order fuzzy differential equations with initial conditions under generalized h differentiability second derivative of the h difference of two functions under generalized h differentiability is obtained two theorems which assure global existence of solutions for second order fuzzy differential equations are given and proved some examples are given to illustrate these results

one shot rates for entanglement manipulation under non entangling maps we obtain expressions for the optimal rates of one shot entanglement manipulation under operations which generate a negligible amount of entanglement as the optimal rates for entanglement distillation and dilution in this paradigm we obtain the max and min relative entropies of entanglement the two logarithmic robustnesses of entanglement and smoothed versions thereof this gives a new operational meaning to these entanglement measures moreover by considering the limit of many identical copies of the shared entangled state we partially recover the recently found reversibility of entanglement manipulation under the class of operations which asymptotically do not generate entanglement

on fuzzy differential equations in the quotient space of fuzzy numbers since addition possesses an inverse in the quotient space of fuzzy numbers based on mares equivalence relation in this paper we introduce a metric on the quotient space of fuzzy numbers and then deal with fuzzy mappings of a real variable whose values are equivalence classes of fuzzy numbers we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation finally some examples are given to illustrate the main theorems