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1.
We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)=0$$ on an arbitrary time scale $\mathbb{T}$ , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on $\mathbb{T}$ . Comparison results with the inequality $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)\leqslant 0\quad (\geqslant 0)$$ are established and application to neutral equations of the form $$\bigl(a(t)\bigl(\bigl[x(t)+p(t)x[\tau (t)]\bigr]^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }+q(t)x^{\beta }\bigl[g(t)\bigr]=0$$ are investigated.  相似文献   

2.
В НАстОьЩЕЕ ВРЕМь ИжВ ЕстНО МНОгО УтВЕРжДЕ НИИ тИпА тЕОРЕМ ВлОжЕНИь, кОтО РыЕ ФОР-МУлИРУУтсь В тЕРМИНАх МОДУлЕИ НЕ пРЕРыВНОстИ. ДАННАь РАБОтА сОДЕРж Ит НЕскОлькО тЕОРЕМ В лОжЕНИь с УслОВИьМИ, ВыРАжЕННы МИ В тЕРМИНАх НАИлУЧшИх п РИБлИжЕНИИE n(?,p) ФУНкц ИИ ? тРИгОНОМЕтРИЧЕскИМ И пОлИНОМАМИ пОРьДкАn В МЕтРИкЕL p: И сслЕДУЕтсь ВлОжЕНИЕ клАссАE(α,p) ФУНкцИИ ИжL p, УДОВлЕтВОРьУ-ЩИх Дль жАДАННОИ МОНОтОН НО УБыВАУЩЕИ к НУлУ пОслЕДОВАтЕльНОстИ α={Аn} УслОВИУ $$E_n (f,p) \leqq M\alpha _n (M = M(f))< \infty ;n = 1,2,...).$$ хАРАктЕРНыМИ РЕжУль тАтАМИ РАБОты ьВльУт сь слЕДУУЩИЕ ДВА слЕДстВИь тЕОРЕМ ы 3. слЕДстВИЕ 1. пУстьР≧1И Β>?1.ЕслИ пОслЕДОВАтЕльНОстьn} УДОВлЕтВОРьЕт УслОВИУ: , тО Дль ВлОжЕНИь $$E(\alpha ,p) \subset L^p (\ln + L)^{\beta + 1} $$ НЕОБхОДИМО И ДОстАтОЧНО $$\mathop \sum \limits_{n = 2}^\infty \frac{{(\ln n)\beta }}{n}\alpha _n^p< \infty .$$ слЕДстВИЕ 2.ЕслИ v>p≧1,Β≧0 И {Аn} УДОВлЕтВОРьЕт УслОВИУ (1),тО Дль ВлОжЕ НИь $$E(\alpha ,p) \subset L^\nu (\ln + L)^\beta $$ НЕОБхОДИМО И ДОстАтО ЧНО $$\mathop \sum \limits_{n = 2}^\infty n^{\nu /p - 2} (\ln + n)^\beta \alpha _n^\nu< \infty ,$$   相似文献   

3.
Integral operators of the type $$(Tf)(x) = \int_0^1 {\frac{{x^\beta y^\gamma }}{{(x + y)^\alpha }}} f(y)dy,$$ the kernels of which have a singularity at a single point, are discussed. H. Widom's method and some of his results are used to show that, if α>0, β, γ>?1/2, ρ=β+γ?α+1>0, then we have for the distribution function of the singular numbers of the operator, $$\mathop {\lim }\limits_{\varepsilon \to 0} N(\varepsilon ,T)ln^{ - 2} {\textstyle{1 \over \varepsilon }} = {\textstyle{1 \over {2\pi ^2 \varepsilon }}}.$$   相似文献   

4.
Let \(T(x) = \sum\limits_{ord(G) \leqq x} {t(G),} \) , wheret(G) define the number of direct factors of a finite Abelian group.E. Krätzel ([5]) defined a remainderΔ 1(x) in the asymptotic ofT(x) and proved $$\Delta _1 (x)<< x^{{5 \mathord{\left/ {\vphantom {5 {12}}} \right. \kern-\nulldelimiterspace} {12}}} \log ^4 x.$$ Using two different methods to estimate a special three-dimensional exponential sum we get the better results $$\Delta _1 (x)<< x^{{{282} \mathord{\left/ {\vphantom {{282} {683}}} \right. \kern-\nulldelimiterspace} {683}}} \log ^4 x$$ and $$\Delta _1 (x)<< x^{{{45} \mathord{\left/ {\vphantom {{45} {109}}} \right. \kern-\nulldelimiterspace} {109}} + \varepsilon } (\varepsilon > 0).$$   相似文献   

5.
ПустьM m - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$   相似文献   

6.
Chebyshev determined $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n + a_1 x^{n - 1} + \cdots + a_n |$$ as 21?n , which is attained when the polynomial is 21?n T n(x), whereT n(x) = cos(n arc cosx). Zolotarev's First Problem is to determine $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n - n\sigma x^{n - 1} + a_2 x^{n - 2} + \cdots + a_n |$$ as a function ofn and the parameter σ and to find the extremal polynomials. He solved this in 1878. Another discussion was given by Achieser in 1928, and another by Erdös and Szegö in 1942. The case when 0≤|σ|≤ tan2(π/2n) is quite simple, but that for |σ|> tan2(π/2n) is quite different and very complicated. We give two new versions of the proof and discuss the change in character of the solution. Both make use of the Equal Ripple Theorem.  相似文献   

7.
В стАтьЕ пОлУЧЕНО УсИ лЕНИЕ НЕскОлькИх РЕж УльтАтОВ О РАцИОНАльНОИ АппРОк сИМАцИИx α НА [0,1]. ДОкАжАНО, ЧтО НАИ лУЧшИЕ пРИБлИжЕНИьr n (x α) ФУНкцИИx α НА [0,1] РАцИОНА льНыМИ ДРОБьМИ пОРьДкАn Дль лУБОгО НЕцЕлОгО пОлО жИтЕльНОгО А УДОВлЕтВОРьУт сООт НОшЕНИУ $$\mathop {\lim }\limits_{n \to \infty } r_n^{1/\sqrt n } (x^\alpha ) = \exp ( - 2\pi \sqrt \alpha ).$$ гИпОтЕжА О спРАВЕДлИ ВОстИ ЁтОИ ОцЕНкИ Был А ВыскАжАНА А. А. гОНЧАРОМ В 1974 г. кРОМЕ тОгО, тОЧНАь ОцЕ НкА, пОлУЧЕННАь Н. с. Вь ЧЕслАВОВНы Дль слУЧАь А=1/2 РАс-пРОс тРАНьЕтсь В РАБОтЕ НА пРОИжВОль НыЕ пОлОжИтЕльНыЕ РА цИОНАльНыЕ ЧИслА α: $$b_\alpha |\sin \pi \alpha |< r_n (x^\alpha )\exp (2\pi \sqrt {\alpha n} )< B_{p,q} ,$$ жДЕсь α=p/q.  相似文献   

8.
9.
We prove the following inequalities involving Euler’s beta function. (i) Let α and β be real numbers. The inequalities $\left( {\frac{{y^{z - x} }} {{x^{z - y} z^{y - x} }}} \right)^\alpha \leqslant \frac{{B(x,x)^{z - y} B(z,z)^{y - x} }} {{B(y,y)^{z - x} }} \leqslant \left( {\frac{{y^{z - x} }} {{x^{z - y} z^{y - x} }}} \right)^\beta $ hold for all x, y, z with 0 < xyz if and only if α ≤ 1/2 and β ≥ 1. (ii) Let a and b be non-negative real numbers. For all positive real numbers x and y we have $\delta (a,b) \leqslant \frac{{x^a B(x + b,y) + y^a B(x,y + b)}} {{(x + y)^a B(x,y)}} \leqslant \Delta (a,b) $ with the best possible bounds $\delta (a,b) = \min \{ 2^{ - a} ,2^{1 - a - b} \} and\Delta (a,b) = \max \{ 1,2^{1 - a - b} \} . $ .  相似文献   

10.
We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.  相似文献   

11.
LetA be an operator of the calculus of variations of order 2m onW m,p (Ω) andj a normal convex integrand. ForfL p (Ω), the equation $$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$ may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and $$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$ by the study of the subdifferential of the function $$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$   相似文献   

12.
Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μj+1j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pvj(x)} последовател ьности {рn(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).  相似文献   

13.
On the basis of an analysis on the adelic group (Tate's formula) a regularization is proposed for the divergent infinite product ofp-adic Γ functions: $$\Gamma _p (\alpha ) = \frac{{1 - p^{\alpha - 1} }}{{1 - p^{ - \alpha } }}, p = 2,3,5...,$$ and the adelic formula $$reg\prod\limits_{p = 2}^\infty {\Gamma p(\alpha ) = \frac{{(\zeta \alpha )}}{{\zeta (1 - \alpha )}},} $$ where ζ(α) is the Riemann ζ function, is proved.  相似文献   

14.
LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW 2 (x) are finite. Let {p n (W 2;x)} 0 denote the sequence of orthonormal polynomials with respect to the weightW 2, and let {α n } 1 and {β n } 1 denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec n 1 is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.  相似文献   

15.
Let(?)=B_ηu:2(q-(?))+(⊿((?)-2q))+(2q_x+(?)_x))η=0,2(r-(?)+(⊿(2(?)-r)+(r_x+2(?)_x))η=0,u=(q,r)~Tbe the Backlund transformation (BT) of the hierarchy of AKNS equations,where η is a parameterand Δ=integral from -∞ to x (qr-(?))dx′.It is shown in this paper the infinitesimal BT B_(η+ε)B_η~(-1) admits thefollowing expansionB_(η+ε)B_η~(-1)u=u+εsum from n=0 to ∞ β_n(JL~(n+1)u)η~n,β_n=1+(-1)~n2~(-n-1),where L is the recurrence operator of the hierarchy and ε is an infinitesimal parameter.Thisexpansion implies the equivalence between the permutabiliy of BTs and the involution in pairs ofconserved densities.  相似文献   

16.
LetG be an arbitrary domain in \(\bar C\) ,f a function meromorphic inG, $$M_f \mathop = \limits^{def} \mathop {\lim \sup }\limits_{G \mathrel\backepsilon z \to \partial G} \left| {f(z)} \right|< \infty ,$$ andR the sum of the principal parts in the Laurent expansions off with respect to all its poles inG. We set $$f_G (z) = R(z) - \alpha ,{\mathbf{ }}where{\mathbf{ }}\alpha = \mathop {\lim }\limits_{z \to \infty } (f(z) - R(z))$$ in case ∞?G, andα=0 in case ∞?G. It is proved that $$\left\| {f_G } \right\|_{C(\partial G)} \leqq 50(\deg f_G )M_f ,{\mathbf{ }}\left\| {f'_G } \right\|_{L_1 (\partial G)} \leqq 50(\deg f_G )V(\partial G)M_f ,$$ where $$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$   相似文献   

17.
18.
For anyx ∈ r put $$c(x) = \overline {\mathop {\lim }\limits_{t \to \infty } } \mathop {\min }\limits_{(p,q\mathop {) \in Z}\limits_{q \leqslant t} \times N} t\left| {qx - p} \right|.$$ . Let [x0; x1,..., xn, ...] be an expansion of x into a continued fraction and let \(M = \{ x \in J,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n< \infty \}\) .ForxM put D(x)=c(x)/(1?c(x)). The structure of the set \(\mathfrak{D} = \{ D(x),x \in M\}\) is studied. It is shown that $$\mathfrak{D} \cap (3 + \sqrt 3 ,(5 + 3\sqrt 3 )/2) = \{ D(x^{(n,3} )\} _{n = 0}^\infty \nearrow (5 + 3\sqrt 3 )/2,$$ where \(x^{(n,3)} = [\overline {3;(1,2)_n ,1} ].\) This yields for \(\mu = \inf \{ z,\mathfrak{D} \supset (z, + \infty )\}\) (“origin of the ray”) the following lower bound: μ?(5+3√3)/2=5.0n>(5 + 3/3)/2=5.098.... Suppose a∈n. Put \(M(a) = \{ x \in M,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n = a\}\) , \(\mathfrak{D}(a) = \{ D(x),x \in M(a)\}\) . The smallest limit point of \(\mathfrak{D}(a)(a \geqslant 2)\) is found. The structure of (a) is studied completely up to the smallest limit point and elucidated to the right of it.  相似文献   

19.
LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that $$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that \(\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )\) for every ε>0. In this paper, we prove the following results on the assumption of R.H.: (1) $$\frac{1}{x}\int\limits_1^x {\Delta (t)dt} = O(x^{\tfrac{1}{{12}} + \varepsilon } ),$$ (2) $$\int\limits_1^x {\frac{{\Delta (t)}}{t}\log } ^{v - 1} \left( {\frac{x}{t}} \right) = O(x^{\tfrac{1}{{12}} + \varepsilon } )$$ for any integer ν≥1. In fact, we prove some general results and deduce the above from them. On the basis of (1) and (2) above, we conjecture that \(\Delta (x) = O(x^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}} + \varepsilon } )\) under the assumption of R.H.  相似文献   

20.
ПустьΦN-функция Юнг а со свойствами $$\Phi (x)x^{ - 1} \downarrow 0, \exists \alpha > 1 \Phi (x)x^{ - \alpha } \uparrow (x \downarrow 0),$$ илиΦ(х)=х, {λk} — положи тельная, неубывающая последовательность и $$S_\Phi \{ \lambda \} = \left\{ {f:\left\| {\sum\limits_{k = 0}^\infty \Phi (\lambda _k |f - s_k |)} \right\|_\infty< \infty } \right\}.$$ В работе найдены необ ходимые и достаточны е условия для вложений $$S_\Phi \{ \lambda \} \subset W^r F(r \geqq 0),$$ , гдеF=C, L , Lip α (0<α≦1). С этой то чки зрения рассматриваются и др угие классы (например, \(W^r H^\omega ,\tilde W^r F\) ).  相似文献   

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