Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

Approximate <Emphasis Type="Italic">C</Emphasis>*-ternary ring homomorphisms

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation

一类对称函数的Schur凸性及其应用

对x=(x_1,…,x_n)∈[0,1)~n∪(1,+∞o)~n,定义对称函数■其中r∈N,i_1,i_2,…,i_n为非负整数.研究了F_n(x,r)的Schur凸性、Schur乘性凸性和Schur调和凸性.作为应用,用控制理论建立了一些不等式,特别地,给出了高维空间的一些新的几何不等式.     

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.     

Complexity of self-correcting circuits for some sequence of Boolean functions

k-Self-correcting circuits of functional elements in the basis {x ₁&x ₂, $ \bar x $ \bar x } are considered. It is assumed that constant faults on outputs of functional elements are of the same type. Inverters are supposed to be reliable in service. The weight of each inverter is equal to 1. Conjunctors can be reliable in service, or not reliable. Each reliable conjunctor implements a conjunction of two variables and has a weight p > k + 2. Each unreliable conjunctor implements a conjunction in its correct state and implements a Boolean constant δ (δ ∈ {0, 1}) otherwise. Each unreliable conjunctor has the weight 1. It is stated that the complexity of realization of monotone threshold symmetric functions $ f_2^n \left( {x_1 ,...,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n} x_1 x_j ,n = 3,4 $ f_2^n \left( {x_1 ,...,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n} x_1 x_j ,n = 3,4 , ... by such circuits asymptotically equals (k + 3)n.     

Estimate of a multiple sum involving the legendre symbol

Supposef(x₁,..., x_n) is a polynomial of even degree d having coefficients in the finite field k=[q] and satisfying certain natural conditions, and let χ be the quadratic character of k. Then $$\left| {\sum {x_1 , \ldots ,} x_n \in k\chi (f(x_1 , \ldots ,x_n ))} \right| \leqslant Cq^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where the constant C depends only on d and n.     

Directional approximation of functions of two variables in the spaces ℒ p (R2) and ℒ p (R + 2 )

Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R ², R ₊ ² }, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ? E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊ ² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote In this paper, we investigate the relation between the behavior of the quantity as n → ∞ (here, E ⊂ R, G ∈ {R ², R ₊²}, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ⊂ E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and then the relations are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and Then the relations and are equivalent. Original Russian Text ? N.Yu. Dodonov, V.V. Zhuk, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 23–33.     

On analytic continuation of a hypergeometric series using the Euler-Knopp transformation

The Euler-Knopp transformation is considered in terms of the problems of regularity and acceleration of the rate of convergence. The object of study is the hypergeometric series

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model ,where yi's are responses, xi = (xi1, xi2,…,xip)' and ti ∈T are known and nonrandom design points, T is a compact set in the real line is an unknown parameter vector, g(·) is an unknown function and {Ei} isa linear process, i.e., random variables with zeromean and variance o2e. Drawing upon B-spline estimation of g(·) and least squares estimation of 0, we construct estimators of the autocovariances of {Ei}- The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {Ei} are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.     

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let {Xni} be an array of rowwise negatively associated random variables and Tnk=k∑i=1 i^a Xni for a ≥ -1, Snk =∑｜i｜≤k Ф（i/nη）1/nη Xni for η∈（0,1],where Ф is some function. The author studies necessary and sufficient conditions of ∞∑n=1 AnP（max 1≤k≤n｜Tnk｜〉εBn）〈∞ and ∞∑n=1 CnP（max 0≤k≤mn｜Snk｜〉εDn）〈∞ for all ε 〉 0, where An, Bn, Cn and Dn are some positive constants, mn ∈ N with mn /nη →∞. The results of Lanzinger and Stadtmfiller in 2003 are extended from the i.i.d, case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.     

Caratterizzazione delle tracce per spazi con peso

Sunto Si prova un teorema di tracce per spazi di funzioni, definite su R ₊ ⁿ , con norme del tipo . Entrata in Redazione il 23 luglio 1969. Lavoro eseguito nell'ambito dei gruppi di ricerca del Comitato Nazionale per la Matematica del C.N.R.     

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P _C is a metric projection. We consider the iteration process {x _n } of C defined by x ₁ = x ∈ C is arbitrary and

Small deviations of modified sums of independent random variables

Let S_n = X₁ + · · · + X _n , n ≥ 1, and S ₀ = 0, where X ₁, X ₂, . . . are independent, identically distributed random variables such that the distribution of S _n/B _n converges weakly to a nondeoenerate distribution F _α as n → ∞ for some positive B _n . We study asymptotic behavior of sums of the form

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

ON LINEAR AND NONLINEAR RIEMANN-HILBERT PROBLEMS FOR REGULAR FUNCTION WITH VALUES IN A CLIFFORD ALGEBRA

This paper deals with the boundary value problems for regular function with valuesin a Clifford algebra: ()W=O, x∈R~n\Г, w~+(x)=G(x)W~-(x)+λf(x, W~+(x), W~-(x)), x∈Г; W~-(∞)=0,where Г is a Liapunov surface in R~n the differential operator ()=()/()x_1+()/()x_2+…+()/()x_ne_n, W(x) =∑_A, ()_AW_A(x) are unknown functions with values in a Clifford algebra ()_n Undersome hypotheses, it is proved that the linear baundary value problem (where λf(x, W~+(x),W~-(x)) =g(x)) has a unique solution and the nonlinear boundary value problem has atleast one solution.     

Optimal methods for the approximate calculation of functional on classesW r L ∞

Пусть T_n(f)={L₁(f), ..., L_n(f)} — набор линейных функционал ов, заданных на простран стве \(C_{(r - 1)} (\parallel f\parallel _{C_{(r - 1)} } = \mathop {\max }\limits_{0 \leqq i \leqq r - 1} \parallel f^{(i)} \parallel _C );A_{n,r}\) — множество всех так их наборов функцио налов; С_{2n, 2} — множество всех н аборов из 2n функциона лов вида $$T_{2n} (f) = \{ f(x_1 ), \ldots ,f(x_n ),f'(x_1 ), \ldots ,f'(x_n )\}$$ и s: Е_n→Е₁. Доказано, что е слиW _∞ ^r множество всех 2π-периодических функ цийfεW∞0, 2π_r, то приr=1,2,3,... ирε(1, ∞) и $$\begin{gathered} \mathop {\inf }\limits_{T_{2n} \in A_{2n,r} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \varphi _{n,r} \parallel _p \hfill \\ \mathop {\inf }\limits_{T_{2n} \in C_{2n,2} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \parallel \varphi _{n,r} \parallel _\infty - \varphi _{n,r} \parallel _p , \hfill \\ \end{gathered}$$ где ?_n,r —r-й периодичес кий интеграл, в средне м равный нулю на периоде, от фун кции ?_n, 0t=sign sinnt. При этом указан ы оптимальные методы приближенного вычис ления.     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.