共查询到20条相似文献,搜索用时 15 毫秒
1.
A surface Γ=(f 1(X1,..., xm),...,f n(x1,..., xm)) is said to be extremal if for almost all points of Γ the inequality $$\parallel a_1 f_1 (x_1 , \ldots ,x_m ) + \ldots + a_n f_n (x_1 , \ldots ,x_m )\parallel< H^{ - n - \varepsilon } ,$$ , where H=max(¦a i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa 1, ...,a n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf 1, ...,f n, the extremality of a class of surfaces in n-dimensional Euclidean space. 相似文献
2.
Получена оценка (в опр еделенном смысле неу лучшаемая) наилучшего приближе ния в метрикеL 1=L1(0,2π) 2π-перио дических функций кла сса WrHω = {f:f∈Cr,ω(f (r),δ) ≦ ω(δ)}, r = 0, 1, ..., (ω(δ) — выпуклый вверх мо дуль непрерывности) ф ункциями класса W 1 r+v N = {?: ?r+v?1)(t) —локально абсолютн о непрерывна, ∥?(r+v∥L1≦N}, v≧2. Доказано, что каждое п одпространство нече тной размерности, реализу ющее поперечник (по Колмогорову) класс а W 1 r+v в L1, обладает аналог ичным свойством относител ьно класса WrHω при любом выпуклом вверх ω(δ). 相似文献
3.
А. Хатамов 《Analysis Mathematica》1984,10(1):15-21
Пустьf - действительн означная конечная фу нкция на конечном отрезке Δ=[а, b] вещественной оси, |Δ|=b?a, M(f) = sup {|f(x)|: x∈Δ}, Rn(f,p Δ) = inf∥f?r∥Lp(Δ) (0 < p < ∞), где нижняя грань бере тся по всем рациональ ным функциямr порядка не вышеп, K(М, Δ) класс всех выпуклых на отре зке Δ функцийf, для кот орыхM(f)≦M. Теорема.При любом вещ ественном р, 0<р<∞ и вс ехп=1, 2, ... sup {Rn(f, p, Δ):f∈K(M, Δ)} ≦ C(p)M|Δ|1/pn?2,где С(р) - величина, зави сящая лишь от р. 相似文献
4.
Paul-Georg Becker 《Monatshefte für Mathematik》1992,114(3-4):183-198
Suppose thatf 1(z), ...f m(z) are algebraically independent functions of a complex variable satisfying $$f_i (z) = a_i (z)f_i (Tz) + b_i (z),$$ wherea i (z),b i (z) are rational functions andTz=p(z ?1)?1 for a polynomialp(z) of degree larger than 1. We show thatf 1(a), ...,f m (a) are algebraically independent under suitable conditions onf anda. As an application of our main result, we deduce three corollaries, which generalize earlier work by Davison and Shallit and by Tamura. 相似文献
5.
F. Móricz 《Analysis Mathematica》1980,6(4):327-341
Пустьd-натуральное ч исло,Z d — множество на боров k=(k 1, ...,k d ), состоящих из неотрицательных цел ыхk j ,Z + d =k∈Z d :k≧1. Предположи м, что системаf k (x):k∈Z + d ? ?L2(X,A, μ) и последовател ьностьa k :k∈Z + d . таковы, чт о для всех b∈Zd и m∈Z + d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z + d положительн а и не убывает. Например, есл иf k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {ak}. В работе получены оце нки порядка роста пря моугольных частных суммS m (x)= =∑ akfk(x) при maxmj→∞ как в случ ае {ak}∈l2, таки для {ak}l2. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными. 相似文献
6.
We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions sr(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–sr(f; x) ¦ and f(x)–sr(f; x)|c on the class WmH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971. 相似文献
7.
J. O. C. Ezeilo 《Annali di Matematica Pura ed Applicata》1971,88(1):135-142
Summary Consider the equation(1.2) in which a1, a2, ..., an−1 are constants and the functions fn(x) and p(t) (both continuous) together with
are all bodnded. A recent investigation by Reissig[1] shows that if fn(x)sgn x>0 (|x|≥h>0) then subject to certain conditions, which are stated explicitly in[1], the solutions of such an equation(1.2) are all ultimately bounded The object of the psesent paper is to generalize that result to the equation(1.1) in which φn is a bounded function depending on all the variables shown, and each coefficient ϕi (i=2,3,..., n−1) satisfies
as |ζ|→∞ for some constant ai.
Entrata in Redazione il 20 agosto 1970. 相似文献
8.
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,...,an,bεR+ andx l,...xnε{0,1}. produced by certain heuristics of a Greedy-type. Every heuristic under consideration realizes a feasible solution (x 1, ..., xn) 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, ..., an, b independent,a 1...,an 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. 相似文献
9.
N. A. Sapogov 《Journal of Mathematical Sciences》1981,16(2):925-933
The work contains some results pertaining to the solution ψj(x) of the functional equation $$\left| {\Sigma \Psi _j \left( {a_j^T t} \right)} \right| \leqslant \varepsilon ,$$ where a j T =(a1j, a2j, ..., apj)∈ ?p, all the coefficients aij are constant, t=(t1, t2, ..., tp) ∈ ?p, \(a_j^T t = \sum\limits_{i - 1}^p {a_{ij} t_i } ,p \geqslant 2\) and the relation (*) is satisfied for all Inequality (*) is connected with certain characterization theorems of probability theory and statistics. For simplicity, it is assumed that the ψj(x) are continuous functions, x∈?1. The following basic ressult is obtained. 相似文献
10.
Kh. A. Khachatryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2008,43(5):305-316
The paper deals with vector integro-differential equation of convolution type that have the form where \(\vec f\) = (f 1, f 2, ..., f N ) T is the unknown vector-function, a i are nonnegative numbers, \(\vec g\) = (g 1, g 2, ..., g N ) T ? L 1 ×N (0,+∞) ≡ L 1 (0,+∞) × ... × L (0,+∞) is the independent term of the equation with nonnegative components and 0 ≤ K ij ? L 1 (?∞,+∞), i, j = 1, 2, …,N are the kernel-functions. These equations have significant applications in the wave non-local interaction theory. Using some special factorization methods, solvability of the system is proved in different functional spaces.
相似文献
$ - \frac{{d^2 f_i }}{{dx^2 }} + a_i f_i (x) = g_i (x) + \sum\limits_{j = 1}^N {\int\limits_0^\infty {K_{ij} (x - t)f_j (t)dt, } i = 1,2, \ldots ,N,} $
11.
David M. Bressoud 《Proceedings Mathematical Sciences》1987,97(1-3):61-66
Given a basic hypergeometric series with numerator parametersa
1,a
2, ...,a
r and denominator parametersb
2, ...,b
r, we say it isalmost poised ifb
i, =a
1
q
δ,i
a
i,δi = 0, 1 or 2, for 2 ≤i ≤r. Identities are given for almost poised series withr = 3 andr = 5 when a1, =q
−2n.
Partially supported by N.S.F. Grant No. DMS-8521580. 相似文献
12.
P. V. Galkin 《Mathematical Notes》1974,15(1):3-8
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 ωi(t)> 0 have continuous (n?i)-th derivatives (i=0, 1, ..., n). Sn,m will designate the subspace of functions that have continuous (n?1)-st derivatives on [a, b] and coincide on each of the intervals [xj, xj+1] (j=0, 1, ..., m) with some polynomial from the system {ui(t)} i=0 n .THEOREM. For every continuous function on [a, b] there exists in Sn,m a unique element of best mean approximation. 相似文献
13.
V. I. Nechaev 《Mathematical Notes》1975,17(6):504-511
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 $$\left| {\sum\nolimits_{x = 1}^q {e^{2\pi if(x)/q} } } \right|< e^{5n^2 /\ln n} q^{1 - 1/n} $$ . 相似文献
14.
《Applied Mathematics Letters》2004,17(10):1147-1152
The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝd ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝd and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L2-norm by elements of the set Gn = {Σi=0staggeredn cig(〈ai, x〉 + bi) : aiℝd, bi, ciℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝd6t6ed|u(t)| > ∞ 相似文献
15.
For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log+ x = max{log x, 0}, log + 1 x = log + x, log + n+1 x = log + log + n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ M,n+1(f) ≤ α if and only if σ M,n (a j ) ≤ α for all j = 0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ M,n+1(f) = σ M,n (a q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given. 相似文献
16.
D. A. Mit'kin 《Mathematical Notes》1973,14(1):597-602
Let n≥4 be even, p > (n2?2n)/2 be simple odd, andf(x)=a 0+a 1+...+a nxn be a polynomial with integral coefficients that are not quadratic over the residue field modulo p, (a n, p)=1. The following inequality is proved: $$\left| {\sum\nolimits_{x = 1}^p {\left( {\frac{{f(x)}}{p}} \right)} } \right| \leqslant (n - 2)\sqrt {p + 1 - \frac{{n(n - 4)}}{4}} + 1.$$ 相似文献
17.
The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let xi, for i=1, 2, ..., n+2, be defined recursively by xi+1=f(xi), where x1 is an arbitrary real number and f is a polynomial of degree n. Let xi+1?xi≧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[xi, ..., xi+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 {xi}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation xi+1=f(xi) 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 δn?1=|2n?1/n!an|. Then, if n is even and an<0, there do not exist n + 1 consecutive increments Δxi=xi+1?xi in the solution {xi} with Δxi≧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. pn?2(n)≦(n!)(n?2)!/2. 相似文献
18.
If a 1,a 2,a 3,… are nonnegative real numbers and $f_{j}(x) = \sqrt{a_{j}+x}$ , then lim n→∞ f 1°f 2°?°f n (0) is a continued radical with terms a 1,a 2,a 3,…. The set of real numbers representable as a continued radical whose terms a i are all from a set S={a,b} of two natural numbers is a Cantor set. We investigate the thickness, measure, and sums of such Cantor sets. 相似文献
19.
Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d W (·, ·) associated to the generating set {a, b, a ?1, b ?1}. Letting B n = {x ∈ ?: d W (x, e ?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ?1 b ?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B n → X is at least a constant multiple of (log n)1/q , an asymptotically optimal estimate as n → ∞. 相似文献
20.
A. B. Voronetskii 《Journal of Mathematical Sciences》1988,43(5):2635-2645
Let f1, ..., fh be representatives of classes in the genus of a positive quadratic form f, ri(k) the number of representations of k by the form fi, (k) the value of the singular Hardy-Littlewood series, Ei the number of integer automorphisms of fi. We derive an expression for in terms of the values of singular series of genus 1 and 2. This expression is evaluated for the simplest model f(x)=x
1
2
+...+x
m
2
, m1 (mod 8), k is a prime in-integer, k=1(mod 8).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 26–39, 1986. 相似文献