共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators. 相似文献
2.
Albertas Zinevi?ius 《Lithuanian Mathematical Journal》2012,52(4):477-487
Let p(y) = p m y m + p m?1 y m?1 + ?+ p 0 ?? $ \mathbb{Z} $ [y] be a polynomial of degree m > 0 in an integer variable. We estimate the number of times it equals some homogeneous polynomial in two variables with integer coefficients, degree at most n, and Euclidean norm at most N evaluated at a pair of small coprime integers (we count this number with the occurring multiplicities). For pairs of coprime integers of absolute value at most $ H<N/\sqrt{n} $ , this estimate is ?? n,p (H)N n+1/m + O(N n+1/m?1 H 3 + N n H 2), where ?? n,p (H) does not depend on N. 相似文献
3.
We consider, for odd primes p, the function N(p, m, α) which equals the number of subsets S?{1,…,p ? 1} with the property that Σ∞∈Sxm ≡ α (mod p). We obtain a closed form expression for N(p, m, α). We give simple explicit formulas for N(p, 2, α) (which in some cases involve class numbers and fundamental units), and show that for a fixed m, the difference between N(p, m, α) and its average value p?12p?1 is of the order of or less. Finally, we obtain the curious result that if p ? 1 does not divide m, then N(p, m, 0) > N(p, m, α) for all α ? 0 (mod p). 相似文献
4.
Josué Meléndez 《Bulletin of the Brazilian Mathematical Society》2014,45(3):385-404
Let M n be a compact oriented hypersurface of a unit sphere \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H. Given an integer k between 2 and n ? 1, we introduce a tensor ? related to H and to the second fundamental form A of M, and show that if |?|2 ≤ B H,k and tr(? 3) ≤ C n,k |?|3, where B H,k and C n,k are numbers depending only on H, n and k, then either |?|2 ≡ 0 or |?|2 ≡ B H,k . We characterize all M n with |?|2 ≡ B H,k . We also prove that if \(\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}\) and tr(? 3) ≤ C n,k |?|3 then |A|2 is constant and characterize all M n with |A|2 in the interval \(\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] \) . We also study the behavior of |?|2, with the condition additional tr(? 3) ≤ C n,k |?|3, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup M |?|2 = B H,k and this supremum is attained in M n then M n is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n ? k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ? 3 is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n ? 1 and \(H \geqslant 1/\sqrt {2n - 1} \) there is a complete hypersurface M n in \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H such that sup M |?|2 = B H,k , and this supremum is attained in M n , and which is not a product of spheres. 相似文献
5.
We obtain asymptotic formulae for the number of primes p ≤ x for which the reduction modulo p of the elliptic curve $$ E_{a,b} :Y^2 = X^3 + aX + b $$ satisfies certain “natural” properties, on average over integers a and b such that |a| ? A and |b| ? B, where A and B are small relative to x. More precisely, we investigate behavior with respect to the Sato-Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer m. 相似文献
6.
Kai Wang 《Journal of Number Theory》1984,18(3):306-312
For a positive integer m, let and let n = |A|. For an integer x, let R(x) be the least positive residue of x modulo m and if (x, m) = 1, let x′ be the inverse of x modulo m. If m is odd, then |R(ab′)|a,b∈A = ?21?n(∏χ(Σa = 1m ? 1aχ(a))), where χ runs over all the odd Dirichlet characters modulo m. 相似文献
7.
Let p_n(z)=∑_(k-0)~n a_kz~k be a polynomial of degree n such that |p_n(z)|≤M for |z|≤1. It is well.known that for 0≤u相似文献
8.
The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C 1 and C 2 of length N = qn + 1 such that |C 1 ? C 2| = k · |P i |/p for each k ∈ {0,..., p · K ? 2, p · K}, where q = p r , p is prime, r ≥ 1, $n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}$ , m ≥ 2, |P i | = p nr(q?2)+n , and K = p n(2r?1)?r(m?1). We show also that there exist two q-ary perfect codes of length N which are intersected by p nr(q?3)+n codewords. 相似文献
9.
In this paper we continue our investigation on “Extremal problems under dimension constraint” introduced in [2]. Let E(n, k) be the set of (0,1)-vectors in ? n with k one's. Given 1 ≤ m, w ≤ n let X ? E(n, m) satisfy span (X) ∩ E(n, w) = ?. How big can |X| be? This is the main problem studied in this paper. We solve this problem for all parameters 1 ≤ m, w ≤ n and n > n 0(m, w). 相似文献
10.
William T. Sledd 《Monatshefte für Mathematik》1988,106(1):65-73
The following result is proved: Letp>0,a>?1. Suppose thatG is a measurable subset ofB, the unit ball in ? N , for which there exists a positive constantA 1, so that $$\int\limits_B {\left( {1 - \left| x \right|} \right)^a \left| {f(x)} \right|^p dm \leqslant A_1 } \int\limits_G {\left( {1 - \left| x \right|} \right)^a \left| {f(x)} \right|^p dm}$$ for each function that is harmonic inB and for which the left-hand side of the above inequality is finite. Then there is a positive constantA 2 so that for each ballK with center on ?B, $$m\left( {K \cap B} \right) \leqslant A_2 m\left( {K \cap G} \right).$$ Herem denotes Lebesgue measure in ? N . This result answers a question left open byDan Luecking [2]. 相似文献
11.
The paper deals with variational problems of the form $$\mathop {\inf }\limits_{u \in W^{1,p} (\Omega )} \int\limits_\Omega {a(\varepsilon ^{ - 1} x)(\left| {\nabla u} \right|^p + \left| {u - g} \right|^p )} dx,$$ where Ω is a bounded Lipschitzian domain in ? N , g∈Lp(Ω). The function a(x) is assumed to satisfy the following conditions:
- a(x) is periodic and lower semicontinuous;
- 0≤a(x)≤1 and the set {∈? N , a(x)>0} is connected in ? N Under these conditions, basic properties of homogenization (convergence of energies and generalized solutions) and properties of Г-convergence type are proved. Bibliography: 3 titles.
12.
The following theorem is proved: Suppose that H = (X; E1, E2, …, Em) is a hypergraph without odd cycles with n vertices and p components, such that any two edges have at most k vertices in common. If for any cycle C in H, there exist two vertices of C contained in at least two common edges of H, then Σi=1m (|Ei| ? k) ≤ n ? pk. 相似文献
13.
P.Sivaramakrishna Das 《Journal of Mathematical Analysis and Applications》1981,84(1):264-269
Let ? be an algebraic integer in a quadratic number field whose minimum polynomial is x2 + p1 + p0. Then all the elements of the ring [?] can be written uniquely in the base ? as Σkm=0akπk, where 0 ? ak < |p0|, if and only if p0 ? 2 and ?1 ? p1 ? p0. 相似文献
14.
The Hermite series estimate of a density f?Lp, p > 1, convergessin the mean square to f (x) for almost all x? |R, ifN (n) → ∞ and N (n) / n2 → ) as n → ∞, where N is the number of the Hermite functions in the estimate while n is the number of observations. Moreover, the mean square and weak consistency are equivalent. For m times differentiable densities, the mean squares convergence rate is O(n?(2m?1)/2m). Results for complete convergence are also given. 相似文献
15.
H.L Abbott 《Journal of Number Theory》1983,17(2):135-143
Denote by k = k(N) the least integer for which there exists integers b1, b2, …, bk satisfying 0 ≤ b1 ≤ b2 ≤ … ≤ bk ≤ N such that every integer in |1, N| can be written in the form i2 + bj. It is shown that for all sufficiently large N, k ≥ (1.147)√N. 相似文献
16.
John A.R. Holbrook 《Linear algebra and its applications》1977,18(3):229-256
For complex Hilbert space H of d dimensions and for any number K ? 1, we may define m(K, d) as the least number with the following property: if 6p(T)6 ? K for all polynomials p mapping the complex unit disk into itself, then the operator T may be made a contraction by changing to a new norm |·|, derived from an inner product, such that It is a long-standing open question whether m(K, d) has a finite bound independent of d. The present paper studies this and related questions and provides, in particular, an explicit estimate for m(K,d)—which, however, grows with d. 相似文献
17.
M. Z. Garaev 《Mathematical Notes》2010,88(3-4):330-337
Suppose that p is a large prime. In this paper, we prove that, for any natural number N < p the following estimate holds: $$ \left. {\mathop {\max }\limits_{\left( {a,p} \right) = 1} } \right|\left. {\sum\limits_{q \leqslant N} {e^{{{2\pi iaq*} \mathord{\left/ {\vphantom {{2\pi iaq*} p}} \right. \kern-\nulldelimiterspace} p}} } } \right| \leqslant \left( {N^{{{15} \mathord{\left/ {\vphantom {{15} {16}}} \right. \kern-\nulldelimiterspace} {16}}} + N^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} p^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} } \right)p^{0\left( 1 \right)} , $$ where q is a prime and q* is the least natural number satisfying the congruence qq* ≡ 1 (modp). This estimate implies the following statement: if p > N > p 16/17+? , where ? > 0, and if we have λ ? 0 (modp), then the number J of solutions of the congruence $$ q_1 \left( {q_2 + q_3 } \right) \equiv \lambda \left( {\bmod p} \right) $$ for the primes q 1, q 2, q 3 ≤ N can be expressed as $$ J = \frac{{\pi \left( N \right)^3 }} {p}\left( {1 + O\left( {p^{ - \delta } } \right)} \right), \delta = \delta \left( \varepsilon \right) > 0. $$ This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition p > N > p 38/39+? was required. 相似文献
18.
Jörgen Boo 《Arkiv f?r Matematik》1997,35(2):225-251
TheH p corona problem is the following: Letg 1, ...,g m be bounded holomorphic functions with 0<δ≤Σ‖g i ‖. Can we, for anyH p function ?, findH p functionsu 1, ...,u m such that Σg i u i =?? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. TheH p estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc. 相似文献
19.
P Frankl 《Journal of Combinatorial Theory, Series A》1977,22(2):249-251
The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2n. Then there is a family , || = m, such that F ∈ , G ? X, | G | > | F | implies G ∈ and minimizes the number of pairs (F1, F2), F1, F2 ∈ F1 ∩ F2 = ? over all families consisting of m subsets of X. 相似文献
20.
Let G = (V, E) be a graph. A mapping f: E(G) → {0, l} m is called a mod 2 coding of G, if the induced mapping g: V(G) → {0, l} m , defined as \(g(v) = \sum\limits_{u \in V,uv \in E} {f(uv)}\) , assigns different vectors to the vertices of G. Note that all summations are mod 2. Let m(G) be the smallest number m for which a mod 2 coding of G is possible. Trivially, m(G) ≥ ?Log2 |V|?. Recently, Aigner and Triesch proved that m(G) ≤ ?Log2 |V|? + 4. In this paper, we determine m(G). More specifically, we prove that if each component of G has at least three vertices, then $$mG = \left\{ {\begin{array}{*{20}c} {k,} & {if \left| V \right| \ne 2^k - 2} \\ {k + 1,} & {else} \\ \end{array} ,} \right.$$ where k = ?Log2 |V|?. 相似文献