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.     

The uniqueness of the element of best mean approximation to a continuous function using splines with fixed nodes

Suppose that on the Interval [a, b] the nodes $$a = x_0< x_1< \ldots< x_m< x_{m + 1} = b$$ are given and the functions u₀(t)=ω₀(t) $$u_i (t) = \omega _0 (t)\smallint _0^t \omega _1 (\varepsilon _1 )d\varepsilon _1 \ldots \smallint _a^{\varepsilon _{\iota - 1} } \omega _1 (\varepsilon _1 )d\varepsilon _\iota ,\varepsilon _0 = t(i = 1,2, \ldots ,n)$$ where the functions ω_i(t)> 0 have continuous (n?i)-th derivatives (i=0, 1, ..., n). S_n,m will designate the subspace of functions that have continuous (n?1)-st derivatives on [a, b] and coincide on each of the intervals [x_j, x_j+1] (j=0, 1, ..., m) with some polynomial from the system {u_i(t)} _i=0 ⁿ .THEOREM. For every continuous function on [a, b] there exists in S_n,m a unique element of best mean approximation.     

Subalgebras of free products of algebras of the variety\mathfrak{u}_{m,n}

The variety \(\mathfrak{u}_{m,n} \) is defined by the system of n-ary operations ω_i,..., ω_m, the system of m-ary operations ?_i,..., ?_n, 1≤ m ≤ n, and the system of identities $$\begin{gathered} x_1 ...x_n \omega _1 ...x_1 ...x_n \omega _m \varphi _i = x_i (i = 1,...,n), \hfill \\ x_1 ...x_m \varphi _1 ...x_1 ...x_m \varphi _n \omega _j = x_j (i = 1,...,m), \hfill \\ \end{gathered} $$ It is proved in this paper that the subalgebra U of the free product \(\Pi _{i \in I}^* A_i \) of the algebras A_i (i ε I) can be expanded as the free product of nonempty intersections U ∩ A_i (i ε I) and a free algebra.     

Singular series in the problem of the representation of a system of numbers by a system of forms

A system of Diophantine equations is considered for integers n₁,...,₂, $$\phi ^{\left( k \right)} \left( {x_1 , \ldots ,x_s } \right) = n_k \left( {k = 1, \ldots ,2} \right)$$ , Ф^(k)(x₁,...,x_s)=n_k (k=1,...,ρ), where Ф^(k) are integral forms of degree d is s variables. The singular integral and singular series of this problem are investigated.     

Lower bounds of linear forms of values of G functions

Lower bounds are obtained for linear forms of values of Siegel's G functions. In particular, it is found that ifα ₁...,α _m are pairwise distinct nonzero rational numbers, then for any positive ? and a natural q>q₀(?,α ₁,...,α _m) we have for any nonzero set (x₀ x₁,..., x_m) of integers the inequality $$|x_0 + x_1 In(i + a_1 q^{ - 1} ) + ... + x_m In(i + a_m q^{ - 1} )|q^{ - \lambda } (h_1 ...h_m )^{ - 1 - \varepsilon } ,$$ where h_i=max(i, ¦x_i¦), andλ=λ (?,α ₁,...,α _m).     

On optimal extrapolation and interpolation of fuzzy analytic functions

Для класса ? аналитич еских в единичном кру ге функций, ограниченны х по модулю единицей, погрешност ью наилучшего прибли жения в точкеz ₀ по значениям в точкахz ₁,..., z_n, заданным с погрешнос тьюδ, называется вели чинаr(z ₀, z₁ z..., z_n, α)=inf sup sup ¦f(z₀)-S(f₁, ...f_n)¦, где нижняя грань бере тся по всевозможным ф ункциям S: Сⁿ→С. ДляE～((?1,1) иz ₀∈ ∈(-1,1)Е рассматривается задача о нахождении п орядка информативности мно жестваЕ, т.е. минимальногоп, на котором достигается нижняя грань в равенстве $$R(z_0 ,\delta ,E) = \mathop {\inf }\limits_n {\text{ }}\mathop {\inf }\limits_{z_1 , \ldots ,z_n \in E} {\text{ }}r(z_0 ,z_1 , \ldots ,z_n ,\delta ).$$ Кроме того, приδ, близ ких к 1, решена задача о нахождении величины $$r_n (\delta ,E) = \mathop {\inf }\limits_{z_1 , \ldots ,z_n \in Ez_0 \in E} \sup r(z_0 ,z_1 , \ldots ,z_n ,\delta )$$ и найдены узлы, на кото рых достигается нижн яя грань.     

An estimate of an incomplete linear form in several algebraic numbers

Letμ>m?1, letν be a rational number, and letω _k=b _k ^v , where b_k ≠ 0 are distinct numbers of an imaginary quadratic field K, which satisfy some additional conditions. Then $$\begin{gathered} |{}_1x_1 \omega _1 + ... + x_m \omega _m | > X^{ - \mu } , \hfill \\ X = \max |x_k | \geqslant X, > 0, \hfill \\ 1 \leqslant k \leqslant m \hfill \\ \end{gathered}$$ where x₁, ..., x_m are integers of the field K, and X₀ is an effective constant.     

Estimate of a complete rational trigonometric sum

Supposef is a polynomial of degree n≥3 with integral coefficientsa ₀,a ₁,...,a _n; q is a natural number; (a ₁,...,a _n, q)=1,f(0) = 0. It is proved that $$\left| {\sum\nolimits_{x = 1}^q {e^{2\pi if(x)/q} } } \right|< e^{5n^2 /\ln n} q^{1 - 1/n} $$ .     

Automorphisms of the tensor product of Abelian and Grassmannian algebras

We consider an algebraB _n,m , over the field R with n+m generators x_i,..., x_n, ξ₁,..., η_m, satisfying the following relations: (1') $$\left[ {x_k ,x_l } \right] \equiv x_k x_l - x_l x_k = 0,[x_k ,\xi _i ] = 0,$$ , (2') $$\left\{ {\xi _i ,\xi _j } \right\} \equiv \xi _i \xi _j + \xi _j \xi _i = 0$$ , where k,l =1, ..., n and i, j=1,..., m. In this algebra we define differentiation, integration, and also a group of automorphisms. We obtain an integration equation invariant with respect to this group, which coincides in the case m=0 with the equation for the change of variables in an integral, an equation whichis well known in ordinary analysis; in the case n=0 our equation coincides with F. A. Berezin's result [1, 3] for integration over a Grassman algebra.     

Weighted restriction theorems for Bessel potentials

Пустьk-мерное евклид ово пространствоR ^k рассматривается как подмножествоR ⁿ . Зафиксируемр, 1<р<∞ иα >(n?k)/p, α≠п. Как обычно, бесселев потенциалJαf обобщенной функции Шварцаf наR ⁿ определяется с помощ ью ее преобразования Фурь е \((\widehat{G_\alpha f})(\xi ) = (2\pi )^{ - n/2} [1 + |\xi |^2 ]^{\alpha /2} f(\xi ), \xi \in R^n .B\) , ξ∈R ⁿ . В работе характ еризуются положител ьные весовые функцииw(x ₁,...,x _k ), которые при продолжении наR ⁿ с помощью равенстваw(x ₁,...,x _k ,...,x _n )=w(x ₁, ...,x _k ) обладают с ледующим свойством: существует числос>0, не зависящее отf, такое, что $$\begin{gathered} \int\limits_{R^k } {|(G_\alpha f)(x_1 ,...,x_k ,0,...,0)w(x_1 ,...,x_k )|^p dx_1 ...dx_k \leqq } \hfill \\ \leqq C\int\limits_{R^n } {|f(x_1 ,...,x_n )w(x_1 ,...,x_n )|^p dx_1 ...dx_n } \hfill \\ \end{gathered} $$      

On the positive nonoscillatory solutions of the difference equation Xn＋1=α＋（xn-k/xn-m）^p

The aim of this paper is to show that the following difference equation：Xn＋1=α＋（xn-k/xn-m）^p, n=0,1,2,…,
where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k＋1,…,x-m,…,X-1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α＋1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.     

The average quality of greedy-algorithms for the Subset-Sum-Maximization Problem

This paper deals with the quality of approximative solutions for the Subset-Sum-Maximization-Problem maximize $$\sum\limits_{i = l}^n {a_i x_i } $$ subject to $$\sum\limits_{i = l}^n {a_i x_i } \leqslant b$$ wherea _l,...,a_n,bεR⁺ andx _l,...x_nε{0,1}. produced by certain heuristics of a Greedy-type. Every heuristic under consideration realizes a feasible solution (x ₁, ..., x_n) whose objective value is less or equal the optimal value, which is of course not greater thanb. We use the gap between capacityb and realized value as an upper bound for the error made by the heuristic and as a criterion for quality. Under the stochastic model:a ₁, ..., a_n, b independent,a ₁...,a_n uniformly distributed on [0, 1], b uniformly distributed on [0,n] we derive the gap-distributions and the expected size of the gaps. The analyzed algorithms include four algorithms which can be done in linear time and four heuristics which require sorting, which means that they are done inO(nlnn) time.     

Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain

We consider the solutions of the inequalityLu≤?(¦gradu¦), whereL is a uniformly elliptic homogeneous operator and ? is a function increasing faster than any linear function but not faster thanξ lnξ, in the unbounded domain $$\left\{ {x \in \mathbb{R}^n |\sum\limits_{i = 2}^n {x_i^2< (\psi (x_1 ))^2 ,} {\text{ }} - \infty< x_1< \infty } \right\},$$ , whereψ is a bounded function with bounded derivative. We estimate the growth of the solutions in terms of $\int_0^{x_1 } {(1/\psi (r))dr}$ . For the special case in which?(ξ)=aξ lnξ+C, the solutionsu(x ₁,x ₂,...,x _n ) grow as $\left( {\int_0^{x_1 } {(1/\psi (r))dr} } \right)^N$ , whereN is any given number anda=a(N).     

Permutation functions and (n,m)-commutativity of semigroups

By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ?⁺, the set of all positive integers) if $$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$ holds for allx ₁,...,x _n ,y ₁,...,y _m ∈S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′≥n andm′≥m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ?⁺ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Rational approximations of convex functions

The following inequalities are shown to hold for the least uniform rational deviations R_n(f) of a function f(x), continuous and convex in the interval [a, b]: $$R_n (f) \leqslant C(v)\Omega (f)n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n$$ (ν is an integer, C(ν) depends only on ν, and Ω(f) is the total oscillation of f); $$R_n (f) \leqslant C_1 n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n\mathop {\inf }\limits_{(b - a)\chi _n \leqslant \lambda< b - a} \left\{ {\omega (\lambda ,f) + M(f)n^{ - 1} \ln \frac{{b - a}}{\lambda }} \right\}$$ (ν is an integer, C₁(ν) depends only on ν, x_n = exp (-n/(500 In²n)), ω (δ,f) is the modulus of continuity of f, and M(f) = max¦f(x) ¦.     

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.     

Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

On global relations for hypergeometric series

Let [K:Q]=k, f_i εK[[z]], ξ εK, Q εK[y₁,...,y_m]. A relation Q(f₁(ξ),..., f_m(ξ))=0 is called global if it holds in any local field where all f_i(ξ) exist. The paper establishes that for series of the form $$\sum\limits_{n = 0}^\infty {\frac{{(\mu _1 )_n \ldots (\mu _p )_n }}{{(\lambda _1 )_n \ldots (\lambda _{q - 1} )_n n!}}\left( {\frac{{z^{p - q} }}{{q - p}}} \right)^n , p > q,} $$ with some natural hypotheses on parameters global relations do not exist. Bibliography: 9 titles.     

Dimension and superposition of continuous functions

We give a relatively short proof of the following theorem of Sternfeld: LetX be a compact metric space with dimX ≧ 2, and letX ?R ^m be an embedding such that everyf ∈C(X) can be represented as $$f(x_1 ,x_2 ,...,x_m ) = \sum\limits_{i = 1}^m {g_i (x_i ),} (x_1 ,x_2 ,...,x_m ) \in X,g_i \in $$ Thenm ≧ 2 dimX + 1.