On the possibilistic mean value and variance of multiplication of fuzzy numbers

In this paper, we introduce the definitions of the possibilistic mean, variance and covariance of multiplication of fuzzy numbers, and show some properties of these definitions. Then, we apply these definitions to build the possibilistic models of portfolio selection under the situations involving uncertainty over the time horizon, by considering the portfolio selection problem from the point of view of possibilistic analysis. Moreover, numerical experiments with real market data indicate that our approach results in better portfolio performance.     

On the ordering of graphs with respect to their matching numbers

The number of k-matchings in a graph G is denoted by m(G,k). If for two graphs G₁ and G₂, m(G₁, k) ≥m(G₂,k) for all k, then we write G₁ > G₂. Six classes of graphs are ordered with respect to the relation >.     

Elliptic complex numbers with dual multiplication

Investigated is a number system in which the square of a basis number: (w)², and the square of its additive inverse: (−w)², are not equal. Termed W space, a vector space over the reals, this number system will be introduced by restating defining relations for complex space C, then changing a defining conjugacy relation from conj(z) + z = 0 in the complexes to conj(z) + z = 1 for W space. This change produces a dual-represented vector space consisting of two dual, isomorphic fields, which are unified under one “context-sensitive” multiplication. Fundamental algebraic and geometric properties will be investigated. W space can be interpreted as a generalization of the complexes but is characterized by an interacting duality which seems to produce two of everything: two representations, two multiplications, two norm values, and two solutions to a linear equation. W space will be compared to a previous suggestion of a similar algebra, and then possible applications will be offered, including a W space fractal.     

Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.     

Total ordering for intuitionistic fuzzy numbers

Intuitionistic fuzzy set plays a vital role in decision making, data analysis, and artificial intelligence. Many decision‐making problems consist of different types of datum, where fuzzy set theoretical approaches may fail to obtain the optimal decision. Numerous approaches for intuitionistic fuzzy decision‐making problem have been introduced in the literature to overcome these short comings. But there is no single approach that can be used to solve all kinds of problems because of the partial ordering defined on the collection of intuitionistic fuzzy numbers (IFNs). Even though ranking of fuzzy numbers have been studied from early seventies in the last century, a total order on the entire class of fuzzy numbers has been introduced by Wang and Wang (Fuzzy Sets Syst 2014, 243, 131–141) only on 2014. A total order on the collection of all IFN is an open problem till today. In this article, a total order on the entire class of IFN using upper lower dense sequence in the interval [0, 1] is proposed and compared with existing techniques using illustrative examples, further an algorithm (which is problem independent) for solving any intuitionistic fuzzy multicriteria decision‐making problem (Intuitionistic fuzzy MCDM) is introduced. This new total ordering on IFNs generalizes the total ordering defined in Wang and Wang ( 22 ) for fuzzy numbers. © 2016 Wiley Periodicals, Inc. Complexity 21: 54–66, 2016     

Generic sequences,transducers and multiplication of normal numbers

Existence of a topological joining between two subshiftsX andX′ defines a relation between points of the two. Supposex ∈X is generic for an invariant measureμ onX; when is a relatedx′ ∈X′ also generic for some corresponding measureμ′ onX′? We prove this property holds in several situations for bounded-to-one joinings: whenμ andμ′ are the measures with maximal entropy on intrinsically ergodicX andX′, and also whenμ has a unique preimage on the joining, a property for which several sufficient conditions are given. In the latter case it is also possible to prove that the nearer a point is to genericity with respect toμ, the nearer to genericity with respect toμ′ a related point is. Bounded-to-one joinings may be defined by nonambiguous rational transductions. This provides several applications, most of them in Number Theory. It is proven that transducers performing multiplication by integers have the suitable properties: this implies multiplication by a rational preserves near normality; so does addition of a rational. An application to Markov measures, and sufficient conditions for a transducer to map normality to the basep to normality to the basep′, p′≠p, are given.     

Intrinsic ordering,combinatorial numbers and reliability engineering

A new algorithm for evaluating the top event probability of large fault trees (FTs) is presented. This algorithm does not require any previous qualitative analysis of the FT. Indeed, its efficiency is independent of the FT logic, and it only depends on the number n of basic system components and on their failure probabilities. Our method provides exact lower and upper bounds on the top event probability by using new properties of the intrinsic order relation between binary strings. The intrinsic order enables one to select binary n  -tuples with large occurrence probabilities without necessity to evaluate them. This drastically reduces the complexity of the problem from exponential (2ⁿ$2^n$ binary n-tuples) to linear (n Boolean variables). Our algorithm is mainly based on a recursive formula for rapidly computing the sum of the occurrence probabilities of all binary n-tuples with weight m whose 1s are placed among the k right-most positions. This formula, as well as the balance between accuracy and computational cost, is closely related to the famous Pascal’s triangle.     

On the mantissa distribution of powers of natural and prime numbers

Given a fixed integer exponent r≧1, the mantissa sequences of (n ^r ) _n and of ${(p_{n}^{r})}_{n}$ , where p _n denotes the nth prime number, are known not to admit any distribution with respect to the natural density. In this paper however, we show that, when r goes to infinity, these mantissa sequences tend to be distributed following Benford’s law in an appropriate sense, and we provide convergence speed estimates. In contrast, with respect to the log-density and the loglog-density, it is known that the mantissa sequences of (n ^r ) _n and of ${(p_{n}^{r})}_{n}$ are distributed following Benford’s law. Here again, we provide previously unavailable convergence speed estimates for these phenomena. Our main tool is the Erd?s–Turán inequality.     

On perfect hashing of numbers with sparse digit representation via multiplication by a constant

Consider the set of vectors over a field having non-zero coefficients only in a fixed sparse set and multiplication defined by convolution, or the set of integers having non-zero digits (in some base b) in a fixed sparse set. We show the existence of an optimal (or almost-optimal, in the latter case) ‘magic’ multiplier constant that provides a perfect hash function which transfers the information from the given sparse coefficients into consecutive digits. Studying the convolution case we also obtain a result of non-degeneracy for Schur functions as polynomials in the elementary symmetric functions in positive characteristic.     

Rapid multiplication modulo the sum and difference of highly composite numbers

We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form where is small. In particular this allows rapid computation modulo numbers of the form .

In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages.

     

On a 1-1-correspondence between rooted trees and natural numbers

To each rooted tree, a natural number is assigned. It is shown that the correspondence is 1-to-1. An appendix gives the rooted trees corresponding to the numbers 1, 2,…, 45.     

On multiplication and comultiplication modules

This article deals with some results concerning multiplication and comultiplication modules over a commutative ring.     

On the set of natural numbers which only yield orders of Abelian groups

Let A denote the set of all natural numbers n such that every group of order n is Abelian. Let C denote the set of all natural numbers n such that every group of order n is cyclic. We prove that Σ_{n ≤ x,n?A?C}1 has roughly the order of magnitude x(log log x)^?1.     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

On unknotting numbers and four-dimensional clasp numbers of links

In this paper, we estimate the unknotting number and the four-dimensional clasp number of a link, considering the greatest euler characteristic for an oriented two-manifold in the four-ball bounded by the link. Combining with a result due to Rudolph, we prove that an inequality stronger than the Bennequin unknotting inequality actually holds for any link diagram. As an application we show the equality conjectured by Boileau and Weber for a closed positive braid diagram.