On the enumeration of distinct fuzzy switching functions

This paper entails the development of bound equations on the number of minimized fuzzy switching functions. A triangular matrix representing the transitivity between fundamental fuzzy phrases is discussed and applied in the enumeration process. Improved upper and lower bounds have been developed by use of the transitivity matrix. In addition, an algorithm is described to calculate the number of distinct fuzzy switching functions for any number of variables.     

On lower and upper bounds of matrices

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in lp when p?1 and its dual version, the upper bounds when 0<p?1. We also determine such bounds explicitly for some families of matrices.     

On the near optimality of the stochastic approximation of smooth functions by neural networks

We consider the problem of approximating the Sobolev class of functions by neural networks with a single hidden layer, establishing both upper and lower bounds. The upper bound uses a probabilistic approach, based on the Radon and wavelet transforms, and yields similar rates to those derived recently under more restrictive conditions on the activation function. Moreover, the construction using the Radon and wavelet transforms seems very natural to the problem. Additionally, geometrical arguments are used to establish lower bounds for two types of commonly used activation functions. The results demonstrate the tightness of the bounds, up to a factor logarithmic in the number of nodes of the neural network. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the nearest parametric approximation of a fuzzy number

Many nearest parametric approximation methods of fuzzy sets are proposed in the literature. It is clear that the specific approximations may lead to the loss of information about fuzziness. To overcome this problem, most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation. But these approximations mostly are not flexible to the decision maker's choice. Hence, in this paper, we offer a parametric fuzzy approximation method based on the decision maker's strategy as an extension of trapezoidal approximation of a fuzzy number. This method comprises the selection of the form of the parametric membership function and its evaluation.     

Parameterized approximation of fuzzy number with minimum variance weighting functions

In the parameterized fuzzy number expectation and fuzzy number interval approximation with the weighting function method, we once considered the weighting function form with maximum entropy, which has some interesting properties. In the present paper, we will propose another kind of parameterized weighting function with minimum variance, and apply it to the fuzzy number expectation and fuzzy number interval approximation problems. It shows that the minimum variance weighting function also has some similar interesting properties, and can be used to express the decision maker’s preference information in the fuzzy number defuzzification process.     

On the properties of a certain subclass of close-to-convex functions

In the present paper, we introduce an interesting subclass Kps (h)of analytic functions in the open unit disk U. For functions belonging to theclass Kps (h), basic properties such as the coefficient bounds, the distortionand growth theorems are derived. The results presented here would provideextensions of those given by Q.-H. Xu et al. [2].     

Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

In this paper, the stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in the literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones. Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinement of the polynomial Lyapunov function so as to make it share the fuzzy structure of the model proves advantageous. Conditions thus obtained are tested via available SOS software.     

On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety

We extend the lower bounds on the complexity of computing Betti numbers proved in [P. Bürgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic and semialgebraic sets, J. Complexity 22 (2006) 147–191] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACE-hard. Then we prove PSPACE-hardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer.     

On the worst case complexity of potential reduction algorithms for linear programming

There are several classes of interior point algorithms that solve linear programming problems in O(√nL) iterations. Among them, several potential reduction algorithms combine both theoretical (O(√nL) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(√nL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least Θ(√nL) iterations to find an optimal solution. The research of the author was partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory.     

On fuzzy multi-valued functions, part 4: Fuzzy economic models

An application to fuzzy economic systems of the fuzzy multi-valued functions theory elaborated by both of us in our previous papers is presented. First, stationary and dynamic fuzzy economic models are defined. Next the optimality of fuzzy trajectories of these models is introduced and investigated.     

Subclasses of bi-univalent functions defined by convolution

In this paper, we introduced two new subclasses of the function class Σ of bi-univalent functions analytic in the open unit disc defined by convolution. Furthermore, we find estimates on the coefficients ∣a₂∣ and ∣a₃∣ for functions in these new subclasses.     

On Some Special Classes of Sequential Dynamical Systems

Sequential Dynamical Systems (SDSs) are mathematical models for analyzing simulation systems. We investigate phase space properties of some special classes of SDSs obtained by restricting the local transition functions used at the nodes. We show that any SDS over the Boolean domain with symmetric Boolean local transition functions can be efficiently simulated by another SDS which uses only simple threshold and simple inverted threshold functions, where the same threshold value is used at each node and the underlying graph is d-regular for some integer d. We establish tight or nearly tight upper and lower bounds on the number of steps needed for SDSs over the Boolean domain with 1-, 2- or 3-threshold functions at each of the nodes to reach a fixed point. When the domain is a unitary semiring and each node computes a linear combination of its inputs, we present a polynomial time algorithm to determine whether such an SDS reaches a fixed point. We also show (through an explicit construction) that there are Boolean SDSs with the NOR function at each node such that their phase spaces contain directed cycles whose length is exponential in the number of nodes of the underlying graph of the SDS.AMS Subject Classification: 68Q10, 68Q17, 68Q80.     

On finding compromise solutions in multicriteria problems using the fuzzy min-operator

This paper presents an application of fuzzy approaches to the linear vectormaximum problem. It shows that using the fuzzy min-operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem.     

On the number of distinct multinomial coefficients

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial coefficients as monomials.     

On lower bounds for integration of multivariate permutation-invariant functions

In this note we study multivariate integration for permutation-invariant functions from a certain Banach space _Ed,α$E_{d,\alpha }$ of Korobov type in the worst case setting. We present a lower error bound which particularly implies that in dimension d$d$ every cubature rule which reduces the initial error necessarily uses at least d+1$d+1$ function values. Since this holds independently of the number of permutation-invariant coordinates, this shows that the integration problem can never be strongly polynomially tractable in this setting. Our assertions generalize results due to Sloan and Wo?niakowski (1997) [3]. Moreover, for large smoothness parameters α$\alpha$ our bound cannot be improved. Finally, we extend our results to the case of permutation-invariant functions from Korobov-type spaces equipped with product weights.     

一类具有上下界的均衡问题

本文获得一类具有上、下界的抽象均衡问题解的存在性条件, 回答了Isac, Sehgal和Singh提出的一个公开问题,进一步研究了具上、下界及约束条件的抽象均 衡问题,获得了更为深刻的结果. 最后,把上述公开问题推广到两个空间的情形.     

Two-sided bounds uniform in the real argument and the index for modified Bessel functions

Bounds uniform in the real argument and the index for the functionsa _ν (x)=xI′ _ν (x)/I′ _ν (x) andb _ν (x)=xK′ _ν (x)/K _ν (x), as well as for the modified Bessel functionsI _ν(x) andK _ν(x), are established in the quadrantx>0, ν≥0, except for some neighborhoods of the pointx=0, ν=0. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 681–692, May, 1999.