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

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

numerically solution of fuzzy differential equations by adomian method in this paper a method for numerical approximation of fuzzy first order initial value problem is presented buckly and feruing fuzzy sets syst 110 2000 43 have solved fuzzy first order initial value problem they solved problem in crisp case and then extend the solution to fuzzy case in this paper we use adomian method to find the approximate solution in crisp case and extend it to fuzzy case using this method we solve problems that the classical methods could not be applied for them in crisp case

embedding problem of noncompact fuzzy number space e abstract in this paper we use the embedding j wu and zhang from noncompact fuzzy number space e into the concrete frechet space c 0 1 c 0 1 to discuss some properties of the e valued functions such as measurability integrability etc

on bounded queries and approximation this paper investigates the computational complexity of approximating several np optimization problems using the number of queries to an np oracle as a complexity measure the results show a tradeoff between the closeness of the approximation and the number of queries required for an approximation factor k n log log k n n queries to an np oracle can be used to approximate the maximum clique size of a graph within a factor of k n however this approximation cannot be achieved using fewer than log log k n n c queries to any oracle unless pe np where c is a constant that does not depend on k these results hold for approximation factors k n geq 2 that belong to a class of functions which includes any integer constant function log n log a n and n 1 a similar results are obtained for graph coloring set cover and other np optimization problems

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

bounded queries approximations and the boolean hierarchy this paper investigates nondeterministic bounded query classes in relation to the complexity of np hard approximation problems and the boolean hierarchy nondeterministic bounded query classes turn out be rather suitable for describing the complexity of np hard approximation problems the results in this paper take advantage of this machine based model to prove that in many cases np approximation problems have the upward collapse property that is a reduction between np approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level for example if maxclique reduces to log n approximating maxclique using many one reductions then the traveling salesman problem tsp is equivalent to maxclique under many one reductions several upward collapse theorems are presented in this paper the proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes in fact these results depend on a surprising connection between the boolean hierarchy and nondeterministic bounded query classes

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

parametric representation of fuzzy numbers and application to fuzzy calculus we present several models to obtain simple parametric representations of the fuzzy numbers or intervals based on the use of piecewise monotonic functions of different forms the representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers we use the standard cuts setting but also the membership functions are obtained immediately and can be used directly to obtain error controlled approximations of the fuzzy calculus in terms of a finite set of parameters the general setting is the hermite type interpolation where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces and of the parameters by which the results are obtained the representations are designed to model the lower and the upper extremal values of each cut compact intervals of the fuzzy numbers and are able to produce almost any possible configuration differentiable continuous or with a finite number of discontinuity points by using parametric monotonic functions of different types we show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems including the numerical solution of fuzzy differential equations the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics the proposed model is called the lower upper representation and we denote the associated fuzzy numbers or intervals as lu fuzzy

the multivariate skew normal distribution summary the paper extends earlier work on the so called skew normal distribution a family of distributions including the normal but with an extra parameter to regulate skewness the present work introduces a multivariate parametric family such that the marginal densities are scalar skew normal and studies its properties with special emphasis on the bivariate case

asymptotic permutation tests in general factorial designs summary in general factorial designs where no homoscedasticity or a particular error distribution is assumed the well known wald type statistic is a simple asymptotically valid procedure however it is well known that it suffers from a poor finite sample approximation since the convergence to its 2 limit distribution is quite slow this becomes even worse with an increasing number of factor levels the aim of the paper is to improve the small sample behaviour of the wald type statistic maintaining its applicability to general settings as crossed or hierarchically nested designs by applying a modified permutation approach in particular it is shown that this approach approximates the null distribution of the wald type statistic not only under the null hypothesis but also under the alternative yielding an asymptotically valid permutation test which is even finitely exact under exchangeability finally its small sample behaviour is compared with competing procedures in an extensive simulation study

the cauchy problem for continuous fuzzy differential equations abstract we prove a version of the classical peano s theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the hausdorff distances between level sets

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

the boolean hierarchy and the polynomial hierarchy a closer connection we show that if the boolean hierarchy collapses to level k then the polynomial hierarchy collapses to bh 3 k where bh 3 k is the k th level of the boolean hierarchy over sigma p 2 this is an improvement over the known results which show that the polynomial hierarchy would collapse to p np np o log n this result is significant in two ways first the theorem says that a deeper collapse of the boolean hierarchy implies a deeper collapse of the polynomial hierarchy also this result points to some previously unexplored connections between the boolean and query hierarchies of delta p 2 and delta p 3 namely begin eqnarray bh k rm co bh k implies bh 3 k rm co bh 3 k jot p rm np k p rm np k 1 implies p rm np rm np k 1 p rm np rm np k 2 end eqnarray

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

ip pspace in this paper it is proven that when both randomization and interaction are allowed the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space

initial value problems for higher order fuzzy differential equations we consider nth order fuzzy differential equations with initial value conditions we prove the existence and uniqueness of solution for nonlinearities satisfying a lipschitz condition we apply the obtained results to the particular case of linear fuzzy problems

numerical methods forfuzzy differential inclusions in this paper a numerical method for solving fuzzy differential inclusions is considered fuzzy reachable set can be approximated by proposed method with complete error analysis which is discussed in detail the method is illustrated by solving some linear and nonlinear fuzzy initial value problems

pros and cons of permutation tests in clinical trials hypothesis testing in which the null hypothesis specifies no difference between treatment groups is an important tool in the assessment of new medical interventions for randomized clinical trials permutation tests that reflect the actual randomization are design based analyses for such hypotheses this means that only such design based permutation tests can ensure internal validity without which external validity is irrelevant however because of the conservatism of permutation tests the virtues of permutation tests continue to be debated in the literature and conclusions are generally of the type that permutation tests should always be used or permutation tests should never be used a better conclusion might be that there are situations in which permutation tests should be used and other situations in which permutation tests should not be used this approach opens the door to broader agreement but begs the obvious question of when to use permutation tests we consider this issue from a variety of perspectives and conclude that permutation tests are ideal to study efficacy in a randomized clinical trial which compares in a heterogeneous patient population two or more treatments each of which may be most effective in some patients when the primary analysis does not adjust for covariates we propose the p value interval as a novel measure of the conservatism of a permutation test that can be defined independently of the significance level this p value interval can be used to ensure that the permutation test have both good global power and an acceptable degree of conservatism copyright 2000 john wiley sons ltd

structure in approximation classes the study of the approximability properties of np hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking the last important breakthrough proves the apx completeness of several important optimization problems and thus reconciles two distinct views of approximation classes syntactic and computational s khanna et al in proc 35th ieee symp on foundations of computer science ieee computer society press los alamitos ca 1994 pp 819 830 in this paper we obtain new results on the structure of several computationally defined approximation classes in particular after defining a new approximation preserving reducibility to be used for as many approximation classes as possible we give the first examples of natural npo complete problems and the first examples of natural apx intermediate problems moreover we state new connections between the approximability properties and the query complexity of npo problems

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

fuzzy valued mappings with finite variation fuzzy valued measures and fuzzy valued lebesgue stieltjes integrals the fuzzy valued mapping with finite variation the distribution function of a fuzzy valued measure and the fuzzy valued lebesgue stieltjes integral are defined the correlative properties with them are established finally the fuzzy valued stochastic lebesgue stieltjes integral is introduced

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

fuzzy integral equations abstract we introduce a definition of the integral of a fuzzy valued function that is only slightly different from the usual one yet that is more intuitive and that can be applied to a larger class of functions we show that our definition is equivalent to the extension principle for functions of this class we then apply integration theory of fuzzy valued functions to integral equations we define a measure of the fuzziness of a fuzzy valued function and show that using this measure we can view the integration of fuzzy functions as a fuzzy linear transformation from a certain subspace of fuzzy valued functions to the space of fuzzy numbers we then show how this can be used to study fredholm integral equations of the second kind

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

revisiting fuzzy differential equations abstract we revisit the original formulation of fuzzy differential equations fdes and provide examples to show that the theory of fdes is richer than the corresponding theory of ordinary differential equations odes and therefore need not be rejected the alternative formulation in terms of differential inclusions does not always have advantage over odes as claimed we also discuss a stability result via lyapunov functions in this setup

polynomial terse sets abstract let a be a set and k n be such that we wish to know the answers to x1 a x2 a xk a for various k tuples x1 x2 xk if this problem requires k queries to a in order to be solved in polynomial time then a is called polynomial terse or pterse we show the existence of both arbitrarily complex pterse and non pterse sets and that p np iff every np complete set is pterse we also show connections with p immunity p selective p generic sets and the boolean hierarchy in our framework unique satisfiability and a variation of it called ksat is in some sense close to satisfiability

numerical solutions of fuzzy differential equations abstract numerical algorithms for solving fuzzy ordinary differential equations fode are considered a scheme based on the classical euler method is discussed in detail and this is followed by a complete error analysis the algorithm is illustrated by solving several linear and nonlinear fuzzy cauchy problems

a note on two point boundary value problems associated with non linear fuzzy differential equations in recent papers lakshmikantham et al and o regan et al assert that two point boundary value problem for a fuzzy differential equation is equivalent to an integral equation and the proofs of the main results in the above cited papers use this equivalence we prove by a counterexample that by using h differentiability or hukuhara differentiability this assertion does not hold moreover it is proved that a large class of two point boundary value problems have no solutions at all under h differentiability or hukuhara differentiability

erratum the polynomial time hierarchy collapses if the boolean hierarchy collapses it is shown that if the boolean hierarchy bh collapses then there exists a sparse set s such that text co np subseteq text np s and therefore the polynomial time hierarchy ph col

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

analysis of small sample size studies using nonparametric bootstrap test with pooled resampling method experimental studies in biomedical research frequently pose analytical problems related to small sample size in such studies there are conflicting findings regarding the choice of parametric and nonparametric analysis especially with non normal data in such instances some methodologists questioned the validity of parametric tests and suggested nonparametric tests in contrast other methodologists found nonparametric tests to be too conservative and less powerful and thus preferred using parametric tests some researchers have recommended using a bootstrap test however this method also has small sample size limitation we used a pooled method in nonparametric bootstrap test that may overcome the problem related with small samples in hypothesis testing the present study compared nonparametric bootstrap test with pooled resampling method corresponding to parametric nonparametric and permutation tests through extensive simulations under various conditions and using real data examples the nonparametric pooled bootstrap t test provided equal or greater power for comparing two means as compared with unpaired t test welch t test wilcoxon rank sum test and permutation test while maintaining type i error probability for any conditions except for cauchy and extreme variable lognormal distributions in such cases we suggest using an exact wilcoxon rank sum test nonparametric bootstrap paired t test also provided better performance than other alternatives nonparametric bootstrap test provided benefit over exact kruskal wallis test we suggest using nonparametric bootstrap test with pooled resampling method for comparing paired or unpaired means and for validating the one way analysis of variance test results for non normal data in small sample size studies copyright 2017 john wiley sons ltd

a note on fuzzy integral equations in this paper existence theorems for certain volterra integral equations and fredholm integral equation for the fuzzy set valued mappings are obtained

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

quadrature rules for integrals of fuzzy number valued functions in this paper we introduce some quadrature rules for the henstock integral of fuzzy number valued mappings by giving error bounds for mappings of bounded variation and of lipschitz type we also consider generalizations of classical quadrature rules such as midpoint type trapezoidal and three point type quadrature finally we study fine quadrature rules and we present some numerical applications

on the existence and uniqueness of solutions of fuzzy volterra fredholm integral equations abstract in this paper we study the problem of existence and uniqueness of solutions of volterra fredholm integral equation for the fuzzy set valued mappings

bounded queries to sat and the boolean hierarchy we study the complexity of decision problems that can be solved by a polynomial time turing machine that makes a bounded number of queries to an np oracle depending on whether we allow some queries to depend on the results of other queries we obtain two probably different hierarchies we present several results relating the bounded np query hierarchies to each other and to the boolean hierarchy we also consider the similarly defined hierarchies of functions that can be computed by a polynomial time turing machine that makes a bounded number of queries to an np oracle we present relations among these two hierarchies and the boolean hierarchy in particular we show for all k that there are functions computable with 2k parallel queries to an np set that are not computable in polynomial time with k serial queries to any oracle unless p np as a corollary k 1 parallel queries to an np set allow us to compute more functions than are computable with only k parallel queries to an np set unless p np the same is true of serial queries similar results hold for all tt self reducible sets using a mind change technique we show that 2k 1 parallel queries to an np set allow us to accept in polynomial time exactly the same sets as can be accepted in polynomial time with k serial queries to an np set in fact the same is true for any class in place of np that is closed under polynomial time positive bounded truth table reductions this contrasts with the expected result for function computations with an np oracle beigel 1988 in addition we show that the boolean hierarchy and the bounded query hierarchies of languages either stand or collapse together finally we show that if the boolean hierarchy collapses to any level but the zeroth deterministic polynomial time then for all k there are functions computable in polynomial time with k parallel queries to an np set that are not computable in polynomial time with k 1 serial queries to any set np complete sets are p superterse