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

     

Embedding height balanced trees and Fibonacci trees in hypercubes

A height balanced tree is a rooted binary tree T in which for every vertex v∈V(T), the difference b _T (v) between the heights of the subtrees, rooted at the left and right child of v is at most one. We show that a height-balanced tree T _h of height h is a subtree of the hypercube Q _h+1 of dimension h+1, if T _h contains a path P from its root to a leaf such that $\mathbf{b}_{T_{h}}(v)=1$ , for every non-leaf vertex v in P. A Fibonacci tree $\mathbb{F}_{h}$ is a height balanced tree T _h of height h in which $\mathbf{b}_{\mathbb{F}_{h}}(v)=1$ , for every non-leaf vertex. $\mathbb{F}_{h}$ has f(h+2)?1 vertices where f(h+2) denotes the (h+2)th Fibonacci number. Since f(h)～2^0.694h , it follows that if $\mathbb{F}_{h}$ is a subtree of Q _n , then n is at least 0.694(h+2). We prove that $\mathbb{F}_{h}$ is a subtree of the hypercube of dimension approximately 0.75h.     

Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and meaningful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbers and Lucas numbers are obtained.     

Preferences,agreement, consensus — measuring,aggregation and control

This paper presents a model for analysis and a practical method of management of a session meant at attainment of consensus regarding preferences over a set of multi-aspect options, these preferences being expressed through pairwise comparisons or orderings. The model and the method accept and process valued or fuzzy preferences, while preserving an entirely crisp procedure throughout the approach. The aggregation method presented previously in Owsiski and Zadrony [17, 18] is referred to and expanded for the case of clusterwise preference aggregation and consensus measurement. The framework of the clustering approach of Owsiski [19] is applied, allowing for a natural definition of consensus.     

Euclid空间中2—调和等距浸入的一些不存在性定理

Riemann 流形间的2-调和映照 f:M→N 是其张力场满足方程(f) R~N(df(e_i),τ(f))df(e_i)=0的映照.本文证明了 Euclid 空间中2-调和等距浸入的一些不存在性定理.特别是证明了~3中不存在非极小的2-调和等距浸入的曲面.     

A very simple proof of Pascal’s hexagon theorem and some applications

In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic.     

Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities

Our main purpose is to describe the recurrence relation associated to the sum of diagonal elements laying along a finite ray crossing Pascal’s triangle. We precise the generating function of the sequence of described sums. We also answer a question of Horadam posed in his paper [Chebyshev and Pell connections, Fibonacci Quart. 43 (2005), 108–121]. Further, using Morgan-Voyce sequence, we establish the nice identity $F_{n + 1} - iF_n = i^n \sum\limits_{k = 0}^n {(_{2k}^{n + k} )( - 2 - i)^k } $ of Fibonacci numbers, where i is the imaginary unit. Finally, connections to continued fractions, bivariate polynomials and finite differences are given.     

探索以(Fn-k,Fn,Fn)为边长的Fibonacci三角形

Fibonacci三角形是边长为Fibonacci数、面积为整数的三角形.存在以(F<,n-k>,F<,n>.F<,n>)为边长的Fibonacci三角形的情形可以被划分为三类(k时,不存在边长为(F<,n-k>,F<,n>.F<,n>)的Fibonacci三角形.     

Deterministic and Stochastic Control of Navier—Stokes Equation with Linear, Monotone, and Hyperviscosities

This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models. Accepted 7 June 1999     

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Meta—projective Modules,Tensor Products and Limits

Inthispaper,allringsconsideredareassociativewithl#o,allmodulesareunital.Meta-projectivemodulesandmeta-injectivemeduleswerefirstintroductedbyChen-Tong[l]andFeng[2j.LetRbearing,andMaleftR-module.M,iscalledamaximalquotientprojectivesubmeduleofMifM,isamaximalsubmoduleandM/M'isprojective.Accordingto[2j,M,iscalledamaximalquotientinjectivesubmoduleofMifMiisamaximalsubmoduleandM/M,isinjective.Missaidtobemeta-projectiveiftheintersectionofMandallmaximal...quotientproJectivemodulesiso.Similarly,M…     

Bernstein functions, complete hyperexpansivity and subnormality—I

Special classes of functions on the classical semigroupN of non-negative integers, as defined using the classical backward and forward difference operators, get associated in a natural way with special classes of bounded linear operators on Hilbert spaces. In particular, the class of completely monotone functions, which is a subclass of the class of positive definite functions ofN, gets associated with subnormal operators, and the class of completely alternating functions, which is a subclass of the class of negative definite functions onN, with completely hyper-expansive operators. The interplay between the theories of completely monotone and completely alternating functions has previously been exploited to unravel some interesting connections between subnormals and completely hyperexpansive operators. For example, it is known that a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {1/_n}(n0). The present paper discovers some new connections between the two classes of operators by building upon some well-known results in the literature that relate positive and negative definite functions on cartesian products of arbitrary sets using Bernstein functions. In particular, it is observed that the weight sequence of a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {_n+1/_n}(n0). It is also established that the weight sequence of any completely hyperexpansive weighted shift is a Hausdorff moment sequence. Further, the connection of Bernstein functions with Stieltjes functions and generalizations thereof is exploited to link certain classes of subnormal weighted shifts to completely hyperexpansive ones.