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1.
In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality is solvable, where p 1,···,p 5 are primes.  相似文献   

2.
Let 1<c<11/10. In the present paper it is proved that there exists a numberN(c)>0 such that for each real numberN>N(c) the inequality is solvable in prime numbersp 1,p 2,p 3, wherec 1 is some absolute positive constant. Project supported by the National Natural Science Foundation of China (grant: 19801021) and by MCSEC  相似文献   

3.
Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.  相似文献   

4.
LetR n be n-dimensional Euclidean space with n>-3. Demote by Ω n the unit sphere inR n. ForfɛL n ) we denote by σ N δ its Cesàro means of order σ for spherical harmonic expansions. The special value l = \tfracn - 22\lambda = \tfrac{{n - 2}}{2} of σ is known as the critical one. For 0<σ≤λ, we set p0 = \tfrac2ld+ lp_0 = \tfrac{{2\lambda }}{{\delta + \lambda }} . This paper proves that
limN ? ¥ || sNd (f) - f ||p0 = 0\mathop {\lim }\limits_{N \to \infty } \left\| {\sigma _N^\delta (f) - f} \right\|p_0 = 0  相似文献   

5.
We prove that if q = p h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q k and 1 < k < n, determining N directions where
\fracqk - 1p - 1 < N £ \fracq+32 q k-1+ qk-2 +...+q2 + q \frac{{q^k} - 1}{p - 1} < N \le \frac{q+3}{2} q ^{k-1}+ q^{k-2} +\dots+q{^2} + q  相似文献   

6.
We prove that the inequality ||g (·/ n ) ||L1[-1,1] ||Pn+k||L1[-1,1] £ 2 ||gPn+k||L1[-1,1]\vert\vert g (\cdot / n ) \vert\vert_{L_{1}[-1,1]} \vert\vert P_{n+k}\vert\vert_{L_{1}[-1,1]} \leq 2 \vert\vert gP_{n+k}\vert\vert_{L_{1}[-1,1]}, where g : [-1, 1]→ℝ is a monotone odd function and P n+k is an algebraic polynomial of degree not higher than n + k, is true for all natural n for k = 0 and all natural n ≥ 2 for k = 1. We also propose some other new pairs (n, k) for which this inequality holds. Some conditions on the polynomial P n+k under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.  相似文献   

7.
This paper is concerned mainly with the logarithmic Bloch space ℬlog  which consists of those functions f which are analytic in the unit disc \mathbbD{\mathbb{D}} and satisfy sup|z| < 1(1-|z|)log\frac11-|z||f(z)| < ¥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:
•  A bounded univalent function in $\bigcup_{p>1}B^{p}$\bigcup_{p>1}B^{p} but not in the logarithmic Bloch space.  相似文献   

8.
Let A, B, and X be operators on a complex separable Hilbert space such that A and B are positive, and let 0 ≤ v ≤ 1. The Heinz inequalities assert that for every unitarily invariant norm | | | ·| | | ,{\left\vert \left\vert \left\vert \cdot \right\vert \right\vert \right\vert ,}
2| | | A1/2XB1/2| | | £ | | | AvXB1-v+A1-vXBv| | | £ | | | AX+XB| | |.2\left\vert \left\vert \left\vert A^{1/2}XB^{1/2}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert A^{v}XB^{1-v}+A^{1-v}XB^{v}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert AX+XB\right\vert \right\vert \right\vert.  相似文献   

9.
We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation rt=div(|?rm |p-2 ?(rm)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRn{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.  相似文献   

10.
A polynomial P(X) with coefficients {ǃ} of odd degree N - 1 is cyclotomic if and only if¶¶P(X) = ±Fp1X)Fp2Xp1) ?FprXp1 p2 ?pr-1) P(X) = \pm \Phi_{p1} (\pm X)\Phi_{p2}(\pm X^{p1}) \cdots \Phi_{p_r}(\pm X^{p1 p2 \cdots p_r-1}) ¶where N = p1 p2 · · · pr and the pi are primes, not necessarily distinct, and where Fp(X) : = (Xp - 1) / (X - 1) \Phi_{p}(X) := (X^{p} - 1) / (X - 1) is the p-th cyclotomic polynomial. This is a conjecture of Borwein and Choi [1]. We prove this conjecture for a class of polynomials of degree N - 1 = 2r pl - 1 N - 1 = 2^{r} p^{\ell} - 1 for any odd prime p and for integers r, l\geqq 1 r, \ell \geqq 1 .  相似文献   

11.
D'après [6] et [7] l'anneau des entiers du corps quadratique Q(?d), d \not = -3{\bf Q}(\sqrt {d}), d \not = -3, possède une extension cyclique cubique monogène (de discriminant 1) si, et seulement si, l'équation diophantienne¶¶ 4m3 = y2d + 274m^3 = y^2d + 27 a une solution avec d \not o 21d \not \equiv 21 (mod 36) et m \not o 3m \not \equiv 3 (mod 9).¶¶ On démontre ici que pour qu'une telle extension existe il faut que 3 divise h (d) et, lorsque d o 1d \equiv 1 (mod 8), d'où (2) = \frak p1\frak p2(2) = \frak p_1\frak p_2 où \frak p1\frak p_1 et \frak p2\frak p_2 sont deux idéaux premiers distincts de Ad, que la classe [\frak p1][\frak p_1] de \frak p1\frak p_1 dans le groupe de classes de Q(?d){\bf Q}(\sqrt {d}) ne soit pas un cube. Pour |d||d| < 100'000 cela élimine 68,37 % des valeurs restantes, les valeurs éliminées passent ainsi de 90 à 97 %.¶ De plus d ne doit pas être de la forme pq ou -3 pq pour lesquels le symbole d'Aigner T(p *q)T(p \star q) vale -1. L'article comporte aussi deux corrections, des résultats complétant [6] et [7], parus dans une thèse, et d'autres (en particulier l'indépendance des critères et des résultats numériques) parus ailleurs.  相似文献   

12.
We show that if λ 1,λ 2,λ 3,λ 4 are nonzero real numbers, not all of the same sign, η is real, and at least one of the ratios λ 1/λ j (j=2,3,4) is irrational, then given any real number ω>0, there are infinitely many ordered quadruples of primes (p 1,p 2,p 3,p 4) for which
|l1 p1+l2 p22+l3 p23+l4p24+h| < (maxpj)-\frac128+w.\bigl|\lambda_1 p_1+\lambda_2 p^2_2+\lambda_3 p^2_3+\lambda_4p^2_4+\eta \bigr|<(\max p_j)^{-\frac{1}{28}+\omega}.  相似文献   

13.
Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant H 3 X [7/24]+7dH\ge X^{\,{7\over24}+7\delta} we have |E(X,H) | << H1-d/600\vert E(X,H) \vert \ll H^{1-\delta/600} .  相似文献   

14.
Let H be an atomic monoid. For k ? \Bbb Nk \in {\Bbb N} let Vk (H){\cal V}_k (H) denote the set of all m ? \Bbb Nm \in {\Bbb N} with the following property: There exist atoms (irreducible elements) u 1, …, u k , v 1, …, v m H with u 1· … · u k = v 1 · … · v m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets Vk (H){\cal V}_k (H) are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every k ? \Bbb Nk \in {\Bbb N}, max V2k+1 (H) = k |G|+ 1{\cal V}_{2k+1} (H) = k \vert G\vert + 1 which settles Problem 38 in [4].  相似文献   

15.
Let B denote the unit ball in [(?)\tilde] \widetilde{\nabla\hskip-4pt}\hskip4pt denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces Dg{{\cal D}_{\gamma}} , $\gamma > (n-1)$\gamma > (n-1) , and weighted Bergman spaces Apa{A^p_{\alpha}} , 0 < p < ¥0 < p < \infty , $\alpha > n$\alpha > n , of holomorphic functions f on B for which Dgf)D_{\gamma}(\,f) and || f||Apa\Vert\, f\Vert_{A^p_{\alpha}} respectively are finite, where Dgf)=òB (1-|z|2)g|[(?)\tilde]  f(z)|2dt(z),D_{\gamma}(\,f)=\int_B (1-\vert z\vert^2)^{\gamma}\vert\widetilde{\nabla\hskip-4pt}\hskip4pt f(z)\vert^2d\tau(z), and || f||pApaB(1-|z|2)af(z)|pdt(z).\Vert\, f\Vert^p_{A^p_{\alpha}}=\int_B(1-\vert z\vert^2)^{\alpha}\vert\, f(z)\vert^pd\tau(z). The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and $\alpha > n$\alpha > n .  相似文献   

16.
Consider an irreducible, admissible representation π of GL(2,F) whose restriction to GL(2,F) +  breaks up as a sum of two irreducible representations π  +  + π −. If π = r θ , the Weil representation of GL(2,F) attached to a character θ of K * does not factor through the norm map from K to F, then c ? [^(K*)]\chi\in \widehat{K^*} with (c. q-1)| F * =w K/F(\chi . \theta ^{-1})\vert _{ F^{ * }}=\omega _{ {K/F}} occurs in r θ  +  if and only if e(qc-1,y0)=e([`(q)]c-1,y0)=1\epsilon(\theta\chi^{-1},\psi_0)=\epsilon(\overline \theta\chi^{-1},\psi_0)=1 and in r θ − if and only if both the epsilon factors are − 1. But given a conductor n, can we say precisely how many such χ will appear in π? We calculate the number of such characters at each given conductor n in this work.  相似文献   

17.
We show that the derivative of an arbitrary rational function R of degree n that increases on the segment [−1, 1] satisfies the following equality for all 0 < ε < 1 and p, q > 1:
|| R ||Lp[ - 1 + \upvarepsilon ,1 - \upvarepsilon ]C ·9n( 1 - 1 / p )\upvarepsilon 1 / p - 1 / q - 1|| R ||Lq[ - 1,1 ], {\left\| {R^{\prime}} \right\|_{{L_p}\left[ { - 1 + {\upvarepsilon },1 - {\upvarepsilon }} \right]}} \leq C \cdot {9^{n\left( {1 - {{1} \left/ {p} \right.}} \right)}}{{\upvarepsilon }^{{{1} \left/ {p} \right.} - {{1} \left/ {q} \right.} - 1}}{\left\| {R} \right\|_{{L_q}\left[ { - 1,1} \right]}},  相似文献   

18.
The large time behaviour of the Lq L^q -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ ut - Du + ?i=1m |uxi|pi = 0      in   \mathbbR+×\mathbbRN,u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,¶¶is studied for q=1 q=1 and q=¥ q=\infty , where m ? {1,?,N} m\in\{1,\ldots,N\} and pi ? [1,+¥) p_i\in [1,+\infty) for i ? {1,?,m} i\in\{1,\ldots,m\} . The limit of the L1 L^1 -norm is identified, and temporal decay estimates for the L L^\infty -norm are obtained, according to the values of the pi p_i 's. The main tool in our approach is the derivation of L L^\infty -decay estimates for ?(ua ), a ? (0,1] \nabla\left(u^\alpha \right), \alpha\in (0,1] , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.  相似文献   

19.
A k-dimensional box is a Cartesian product R 1 × · · · × R k where each R i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some a ? \mathbbN 3 2{\alpha\in\mathbb{N}_{\geq 2}}, then box(G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree D < ?\fracn(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some a ? \mathbbN 3 2{\alpha \in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. We also demonstrate a graph having box(G) > α but with D = n\frac(a-1)2a+ \fracn2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if D < ?\fracn(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some a ? \mathbbN 3 2{\alpha\in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box(G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box(G) ≤ 2. We also show that both these bounds are tight.  相似文献   

20.
We integrate the Lifting cocycles Y2n+1, Y2n+3, Y2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Difn of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H·Lie(\frak g\frak lfin(Difn);\Bbb C) = ù·(Y2n+1, Y2n+3, Y2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .  相似文献   

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