Soft set theory and uni–int decision making

We firstly redefine the operations of Molodtsov’s soft sets to make them more functional for improving several new results. We also define products of soft sets and uni–int decision function. By using these new definitions we then construct an uni–int decision making method which selects a set of optimum elements from the alternatives. We finally present an example which shows that the method can be successfully applied to many problems that contain uncertainties.     

Methods for solving the matrix equation TB + CT = −A and its generalizations

A variety of physical problems give rise to special cases of the matrix equation

$\text{A + FRB}{}^2\text{+ FERB + CFRB + CFER = 0, (}{}^1\text{)}$

where it is required to determine the solution matrix R. One important special case, obtained by setting T = FRB and E=0 in (¹), is the matrix equation TB + CT =?A. In this article we suggest an algorithm for solving the latter matrix equation, and we extend the ideas behind this algorithm to solve (¹) for R. Some suggestions for the efficient implementation of the algorithms are also given.     

The hyperelliptic integrals and π

Text

A class of hyperelliptic integrals are expressed through hypergeometric functions, like those of Gauss, Lauricella and Appell, namely multiple power series. Whenever they can on their own be reduced to elliptic integrals through an algebraic transformation, we obtain a two-fold representation of the same mathematical object, and then several completely new π determinations through the above special functions and/or Euler integrals. All our π formulae have been successfully tested by means of convenient Mathematica_®'s packages and enter in a wide historical/sound context of π-formulae quite far from being exhausted. Due to their structure, the formulae's practical value does not lie in computing π, but in allowing, through π, a benchmark for computing the involved special functions, particularly those less elementary.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=UHZIHgAeCUc.     

Involutions with fixed point set RP(2m) ⊔ P(2m, 2n + 1)

Let (M, T) be a smooth closed manifold with a smooth involution T whose fixed point set is a disjoint union of an even-dimensional real projective space and a Dold manifold. In some cases, the equivariant bordism classes of (M, T) are determined.     

Domb's numbers and Ramanujan-Sato type series for 1/π

In this article, we construct a general series for . We indicate that Ramanujan's -series are all special cases of this general series and we end the paper with a new class of -series. Our work is motivated by series recently discovered by Takeshi Sato.     

π and the hypergeometric functions of complex argument

Text

In this article we derive some new identities concerning π, algebraic radicals and some special occurrences of the Gauss hypergeometric function ₂F₁ in the analytic continuation. All of them have been derived by tackling some elliptic or hyperelliptic known integral, and looking for another representation of it by means of hypergeometric functions like those of Gauss, Appell or Lauricella. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots. Founding upon a special hyperelliptic reduction formula due to Hermite (1876) [6], π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. Furthermore, starting with a certain binomial integral, we succeed in providing as a ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hypergeometric function of two complex-conjugate arguments. Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica^®.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=rQqtVtAf-RQ.     

On the nonlinear wave equation Utt−B(t,‖Ux‖)Uxx=f(x,t,U,Ux,Ut) associated with the mixed homogeneous conditions

In this paper we consider the following nonlinear wave equation

(1)     

Slowly oscillating functions and closed left ideals of βS

Let S be an infinite discrete semigroup with a point s such that st≠t for any t∈S. We prove that every closed left ideal of βS, where φ is a filter with a countable basis, is determined by a function on S which is slowly oscillating in the direction φ.     

On Tauberian theorems for (A)(C, α) summability method

In this paper we introduce some Tauberian conditions for the (A)(C, α) summability method. These results extend and generalize some of the classical Tauberian theorems for the Abel summability method.     

Pruning method for optimal solutions of int–int decision making scheme

This paper aims to find a faster method for optimal solutions of Feng et al.’s int^m–intⁿ decision making scheme. We first give theoretical characterizations of optimal decision sets. Then we develop a pruning method which filters out those objects that cannot be elements of any optimal decision sets in the beginning. Experimental results have shown that our method has higher efficiency in computing the optimal solutions of this scheme, particularly when we are processing soft sets with a great quantity of data.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

A 3D non-linear k–ε turbulent model for prediction of flow and mass transport in channel with vegetation

The results from a 3D non-linear k–ε turbulence model with vegetation are presented to investigate the flow structure, the velocity distribution and mass transport process in a straight compound open channel and a curved open channel. The 3D numerical model for calculating flow is set up in non-orthogonal curvilinear coordinates in order to calculate the complex boundary channel. The finite volume method is used to disperse the governing equations and the SIMPLEC algorithm is applied to acquire the coupling of velocity and pressure. The non-linear k–ε turbulent model has good useful value because of taking into account the anisotropy and not increasing the computational time. The water level of this model is determined from 2D Poisson equation derived from 2D depth-averaged momentum equations. For concentration simulation, an expression for dispersion through vegetation is derived in the present work for the mixing due to flow over vegetation. The simulated results are in good agreement with available experimental data, which indicates that the developed 3D model can predict the flow structure and mass transport in the open channel with vegetation.     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

Sur l'équation divu=fOn the equation divu=f

The main result is the following. Let $\text{Ω}$ be a bounded Lipschitz domain in $\text{R}{}^{\mathrm{d}}$, d?2. Then for every $\text{f∈}\text{L}{}^{\mathrm{d}}\text{(Ω)}$ with ∫f=0, there exists a solution $\text{u∈}\text{C}{}^0\text{(}\text{Ω}\text{)∩}\text{W}{}^{\mathrm{1,d}}\text{(Ω)}$ of the equation divu=f in $\text{Ω}$, satisfying in addition u=0 on $\text{?Ω}$ and the estimate

where C depends only on $\text{Ω}$. However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976.     

On simultaneous approximation to (α,α) with α+kα−1=0

We show that the modified Jacobi-Perron algorithm gives the best simultaneous approximation to (α,α²) with α³+kα−1=0. We claim the following facts:

(1)

the limit set of become an ellipse, where (p_n,q_n,r_n) is the nth convergent (p_n/q_n,r_n/q_n) of (α,α²) by the modified Jacobi-Perron algorithm,

(2)

the limit set of belongs to outside of the ellipse in (1).

     

Heat kernel bounds, ancient κ solutions and the Poincaré conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3-dimensional ancient κ solutions to the Ricci flow. As an application, using the W entropy associated with the heat kernel, we give a different and much shorter proof of Perelman's classification of backward limits of these ancient solutions. The method is partly motivated by Cao (2007) [1] and Sesum (2006) [27]. The current paper or Chow and Lu (2004) [6] combined with Chen and Zhu (2006) [4] and Zhang (2009) [31] lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.     

The Fibonacci sequence modulo π, chaos and some rational recursive equations

In this paper, we provide the closed form solution to the inter-related equations