两类广泛递归序列的动力特性

We investigate the dynamics of two extensive classes of recursive sequences:xn+1=c∑ k ∑xn-ioxn-i1…xn-i2j+f(xn-io,xn-i1,…,xn-i2k)j=0(i0,i1,…,i2j)∈A2j/c∑ k ∑xn-ioxn-i1…xn-i2j-1+c+f(xn-io,xn-i1,…,xn-i2k)j=1(i0,i1,…,i2j)∈A2j-1 and xn+1=c∑ k ∑xn-ioxn-i1…xn-i2j-1+c+f(xn-io,xn-i1,…,xn-i2k)j=1(i0,i1,…,i2j)∈A2j-1/c∑ k ∑xn-ioxn-i1…xn-i2j+f(xn-io,xn-i1,…,xn-i2k)j=0(i0,i1,…,i2j)∈A2j We prove that their unique positive equilibrium x = 1 is globally asymptotically stable.And a new access is presented to study the theory of recursive sequences.     

A Partitioned Version of the Erdös–Szekeres Theorem for Quadrilaterals

We prove a partitioned version of the Erdös–Szekeres theorem for the case $k = 4$: any finite set $X \subset \bbbr^2$ of points in general position can be partitioned into sets $X_0, X_{ij}$ where $i=1,2,3,4$ and $j=1,\ldots,26$, so that $|X_{1j}|=|X_{2j}|=|X_{3j}|=|X_{4j}|$, $|X_0|\leq 4$ and for all $j$ every transversal $\{x_1,x_2,x_3,x_4\}$, $x_1 \in X_{1j}, x_2 \in X_{2j},x_3 \in X_{3j}, x_4 \in X_{4j}$, is in convex position. In order to prove this, we show another theorem, the partitioned version of the same type lemma, which was proved by Bárány and Valtr.     

Some boundedness results for systems of two rational difference equations

We study k ^th order systems of two rational difference equations

On the average sum of the kth divisor function over values of quadratic polynomials

The Ramanujan Journal - Let $$F(x) \in \mathbb {Z}[x_1 , x_2 ,\ldots , x_n ]$$ , $$n\ge 3$$ , be an n-variable quadratic polynomial with a nonsingular quadratic part. Using the circle method we...     

On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let be the Euclidean Dirac operator in the n-dimensional flat space the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials. Received: 27 October 2005     

Polar Decompositions of C 0(N) Contractions

Let A be a bounded linear operator on a complex separable Hilbert space H. We show that A is a C₀(N) contraction if and only if , where U is a singular unitary operator with multiplicity and x₁, . . . , x_d are orthonormal vectors satisfying . For a C₀(N) contraction, this gives a complete characterization of its polar decompositions with unitary factors.     

Hölder estimates of solutions to a degenerate diffusion equation

This paper is concerned with the Hölder estimates of weak solutions of the Cauchy problem for the general degenerate parabolic equations

with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.

     

TIME-VARYING VECTOR FIELDS ON A COMPACT RIEMANNIAN MANIFOLD (I)

This paper studies critical points of a time-varying vector field $\[f:M \times R \to TM\]$ on a compact Riemannian manifold M. It is shown that if a critical point $\[{x_0}\]$ admits an exponential dichotomy, then there are two families of manifolds,stable manifold family and unstable manifold family of $f$ through $\[{x_0}\]$ in some open neighborhood $V$ of $\[{x_0}\]$, moreover , the critical point $\[{x_0}\]$ is isolated. Also it is shown that the solution curve family of the perturbed time-varying vector field yielded by a small change of $f$ is qualitatively the same as that of $f$.     

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms and . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .     

Stability of functional equations of <Emphasis Type="Italic">n</Emphasis>-Apollonius type in fuzzy ternary Banach algebras

Using the fixed point method, we investigate the generalized Hyers–Ulam stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras for the additive functional equation of n-Apollonius type, namely

$${\sum_{i=1}^{n} f(z-x_{i}) = -\frac{1}{n} \sum_{1 \leq i < j \leq n} f(x_{i}+x_{j}) + n f (z-\frac{1}{n^{2}} \sum_{i=1}^{n}x_{i}),}$$

where \({n \geq 2}\) is a fixed positive integer.

     

On Singular Phenomena in Certain Time-Optimal Problem

The purpose of this paper is to present the solution of time-optimal problem ofthe controlled object the dynamics of which is given by: , , where and motion resistance function if ,f(x)=-A if x > 0 where . That model describes dynamicsof a very important class of industrial installations. As the time-optimalproblem will be understood a transfer of the initial state to the target state in a minimumtime . There has been shown that in the formula defining resistancefunction f(x)there exists a value that plays an essentialrole in time-optimal structure formation. Namely, if then thetime-optimal control process is typical, analogous as in classical case , i.e. there exists a switching curve formed by thetrajectories of time-optimal solutions reaching the target state and thetime-optimal process is formed by at most one switching operation. For the caseA>A_bwe will examine two following singular phenomena.(a) If the target state z₁=(0, 0) then there exists theswitching curve, dividing the state plane into two sets, however only one itsbranch is formed by the time-optimal solution reaching the target z₁=(0, 0) and generated by the control u=-1. None of solution formsthe second branch of switching curve. It is formed by a state-locus dependingon the value of Aonly. In dependency of the starting state z₀ thetime-optimal control process is generated by bang-bang control with none,one or two switching operations. This is the first singular phenomenon,because any small decrease of the value Aover A _b requires to change thestructure which would be able to generate the time-optimal process.(b) The paper shows, that if the target state z ₁(x_1, 0), x₁>0then there exists a set of the starting states from which there start twotrajectories reaching the target in the same minimum time. This is thesecond phenomenon.Finally, some suggestions as to practical applications have been given too.     

Nash equilibria on topological semilattices

A function ${u : X \to \mathbb{R}}$ defined on a partially ordered set is quasi-Leontief if, for all ${x \in X}$ , the upper level set ${\{x\prime \in X : u(x\prime) \geq u(x)\}}$ has a smallest element; such an element is an efficient point of u. An abstract game ${u_{i} : \prod^{n}_{j=1} X_j \to \mathbb{R}, i \in \{1, \ldots , n\}}$ , is a quasi-Leontief game if, for all i and all ${(x_{j})_{j \neq i} \in \prod_{j \neq i} X_{j}, u_{i}((x_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ is quasi-Leontief; a Nash equilibrium x* of an abstract game ${u_{i} :\prod^{n}_{j=1} X_{j} \to \mathbb{R}}$ is efficient if, for all ${i, x^{*}_{i}}$ is an efficient point of the partial function ${u_{i}((x^{*}_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ . We establish the existence of efficient Nash equilibria when the strategy spaces X _i are topological semilattices which are Peano continua and Lawson semilattices.     

Some boundary-value problems for linear multidimensional second-order hyperbolic equations

For the linear hyperbolic equations $$\sum\limits_{i,j = 1}^{m + 1} {a_{ij} \left( {x,x_{m + 1} } \right)u_{x_i x_j } + \sum\limits_{i = 1}^{m + 1} {a_i \left( {x,x_{m + 1} } \right)u_{x_i } + c\left( {x,x_{m + 1} } \right)u = 0,x = \left( {x_1 ,...,x_m } \right)} ,} m \geqslant 2,$$ the correctness of multidimensional analogues of the problems of Darboux and Goursat is established and a theorem on the uniqueness of a solution of the Cauchy characteristic problem is proved.     

A construction of multiresolution analysis by integral equations

In this paper we present a versatile construction of multiresolution analysis of two variables by means of eigenvalue problems of the integral equation, for . As a consequence we show that if is the solution of the equation with , then constructs a two-variable multiresolution analysis.

     

Hyperbolic reflection groups associated to the quadratic forms {-3x_0^2 + x_1^2 + \cdots + x_n^2}

We determine the maximal hyperbolic reflection groups associated to the quadratic forms ${-3x_0^2 + x_1^2 + \cdots + x_n^2, n \ge 2}$ , and present the Coxeter schemes of their fundamental polyhedra. These groups exist in dimensions up to 13, and a proof is given that in higher dimensions these quadratic forms are not reflective.     

NA样本部分线性模型估计的强相合性

考虑回归模型:Y~((j))(x_(in),t_(in))=t_(in)β+g(x_(in))+σ_(in)e~((j))(x_(in)),1≤j≤m,1≤i≤n,其中σ_(in)~2=f(u_(in)),(x_(in),t_(in),u_(in))为固定非随机设计点列,β是未知待估参数,g(·)和f(·)是未知函数,误差{e~((j))(x_(in))}是均值为零的NA变量.给出基于g(·)和f(·)一类非参数估计的β的最小二乘估计和加权最小二乘估计,并在适当条件下得到了它们的强相合性.     

On the optimal interpoint distance sum inequality

Let \({{\left\{x_{1}, \dots, x_{n}\right\}\subset \mathbb{R}^2}}\) be a set of points in the unit circle. It is shown that

$\sum\limits^{n}_{i=1}{\min_{j \neq i}{\left\|x_{i} - x_{j}\right\|^2}}\leq9,$

which is best possible and improves earlier results by Arpacioglu and Haas and Xia and Liu.

     

Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $f\left( {s,w_{00} \upsilon _0 ,...,\sum\limits_{j = 0}^n {w_{nj\upsilon _j } } } \right) = \sum\limits_{j = 0}^n {w_{n + 1j\upsilon j} + w_{n + 1n + 1} f\left( {x,\upsilon ,\upsilon _1 ,...,\upsilon _n } \right),}$ where $w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)$ for the given functions a _ij is solved on $\mathbb{R},u \ne {\text{0}}$ .     

CONVERGENCE RATE OF MULTIVARIATE K-NEAREST NEIGHBOR DENSITY ESTIMATES

Let be the collection of m-times continuously differentiable probability densities fon R~d such that 丨D~af(x_1)-D~af(x_2)丨≤M‖x_1-x_2‖~β for x_1,x_2∈R~d,[a]=m,where D~adenotes the differential operator defined by D~a=([a])/(x_1~a…x_d~a_d).Under rather weak conditionson K(x),the necessary and sufficient conditions for sup丨_n(x)-f(x)丨=0(((logn/n)~λ/(d+3λ),λ=m+β,f∈ are that ∫x~aK(xi)dx=0 for 0<[a]≤m.Finally the convergenco rate at apoint is given.     

On the \varpi_n-related elements in the stable homotopy groups of spheres

Let X = {x ₁, …, x _n } be a set of n points in the d-dimensional Euclidean space , with unit diameter. In this work we give the complete proof of a conjecture by Witsenhausen, stating that the maximum M(d, n) of in dimension d is attained if and only if the points are distributed as evenly as possible among the vertices of a regular d-dimensional simplex of edge-length 1. Received: 5 July 2007, Revised: 22 October 2007