共查询到20条相似文献,搜索用时 15 毫秒
1.
We generalize the classical Voronoi formula for and as an application, we derive a sharp bound for the shifted convolution sum convolving the Fourier coefficients of holomorphic cusp forms with those of theta series.
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$r_{l}(n) = \#\{ (n_{1}, \ldots , n_{l}) \in \mathbf{Z}^{l}, n_{1}^{2} + \cdots + n_{l}^{2} = n \},$
2.
Let \(\mathbb {F}_{q}\) be the finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. For a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and let \({\mathrm {Tr}}_{m}\) be the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}\setminus \{(0,0,\ldots ,0)\} : {\mathrm {Tr}}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\},\) we define a p-ary linear code \(\mathcal {C}_{D}\) by where We shall present the complete weight enumerators of the linear codes \(\mathcal {C}_{D}\) and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016).
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$$\begin{aligned} \mathcal {C}_{D}=\{\mathbf {c}(a_{1},a_{2},\ldots ,a_{t}) : (a_{1},a_{2},\ldots ,a_{t})\in \mathbb {F}^{t}_{q}\}, \end{aligned}$$
$$\begin{aligned} \mathbf {c}(a_{1},a_{2},\ldots ,a_{t})=({\mathrm {Tr}}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2}+\cdots +a_{t}x^{2}_{t}))_{(x_{1},x_{2},\ldots ,x_{t}) \in D}. \end{aligned}$$
3.
For any prime \(p>3,\) we prove that where \(E_{0},E_{1},E_{2},\ldots \) are Euler numbers and \(\left( \frac{\cdot }{p}\right) \) is the Legendre symbol. This result confirms a conjecture of Z.-W. Sun. We also re-prove that for any odd prime \(p,\) using WZ method.
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$$\begin{aligned} \sum _{k=0}^{p-1}\frac{3k+1}{(-8)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-1}{p}\right) +p^3E_{p-3}\pmod {p^4}, \end{aligned}$$
$$\begin{aligned} \sum _{k=0}^{\frac{p-1}{2}}\frac{6k+1}{(-512)^k}{2k\atopwithdelims ()k}^3\equiv p\left( \frac{-2}{p}\right) \pmod {p^2} \end{aligned}$$
4.
Let ω ≥ 0 be a given number and let I be a subinterval of \({{\mathbb Z}}\). We say that a sequence \({(f_k)_{k \in I}}\) is ω -strongly quasiconvex, ω-strongly quasiconcave, ω-strongly quasiaffine ifWe characterize ω-strongly quasiconvex, ω-strongly quasiconcave and ω-strongly quasiaffine sequences. We also show that these notions lead naturally to analogous notions for functions defined on subintervals of \({{\mathbb R}}\).
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$\begin{array}{lll}f_k \leq \max(f_{k-1},f_{k+1})-\omega\quad\quad{\rm for}\,\,\,k:k-1, k, k+1 \in I;\\ f_k \geq \max(f_{k-1},f_{k+1})-\omega\quad\quad{\rm for}\,\,\,k:k-1, k, k+1 \in I;\\ f_k = \max(f_{k-1},f_{k+1})-\omega\quad\quad{\rm for}\,\,\,k:k-1, k, k+1 \in I.\end{array}$
5.
We use bounds of mixed character sums modulo a square-free integer q of a special structure to estimate the density of integer points on the hypersurface for some polynomials \(f_i \in {\mathbb {Z}}[X]\) and nonzero integers a and \(k_i\), \(i=1, \ldots , n\). In the case of the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.
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$$\begin{aligned} f_1(x_1) + \cdots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} \end{aligned}$$
$$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^2\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$
$$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^n\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$
6.
Here we consider the q-series coming from the Hall-Littlewood polynomials,These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence
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$$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$
$$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$
7.
In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if or where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).
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$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$
$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$
8.
For \(n \ge 1\) let that is, \({\mathcal {A}}_n\) is the collection of all sums of \(n\) distinct monomials. These polynomials are also called Newman polynomials. Let We define We show that The special case \(p=1\) recaptures a recent result of Aistleitner [1], the best known lower bound for \(\Sigma _1\).
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$$\begin{aligned} {\mathcal {A}}_n := \bigg \{ P: P(z) = \sum \limits _{j=1}^n{z^{k_j}}: 0 \le k_1 < k_2 < \cdots < k_n, k_j \in {\mathbb {Z}} \bigg \}, \end{aligned}$$
$$\begin{aligned} M_{p}(Q) := \left( \int _{0}^{1}{\left| Q(e^{i2\pi t}) \right| ^p\,dt} \right) ^{1/p}, \qquad p > 0. \end{aligned}$$
$$\begin{aligned} S_{n,p} := \sup _{Q \in {\mathcal {A}}_n}{\frac{M_p(Q)}{\sqrt{n}}} \qquad \text{ and } \qquad S_p := \liminf _{n \rightarrow \infty }{S_{n,p}} \le \Sigma _p := \limsup _{n \rightarrow \infty }{S_{n,p}}. \end{aligned}$$
$$\begin{aligned} \Sigma _p \ge \Gamma (1+p/2)^{1/p}, \qquad p \in (0,2). \end{aligned}$$
9.
In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying for some \(C \ge 1\) and \(\gamma >1\). We first prove the Lebesgue-type inequalities for such series Moreover, we obtain necessary and sufficient conditions for the sum of such series to belong to the generalized Lipschitz, Nikolskii, and Zygmund spaces. We also prove similar results for trigonometric series with weak monotone coefficients, i.e., satisfying for some \(C \ge 1\) and \(\gamma >1\). Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem–Hardy type.
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$$\begin{aligned} \sum \limits _{k=n}^{2n} |a_k - a_{k+1}| \le C \sum \limits _{k=[{n}/{\gamma }]}^{[\gamma n]} \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$
$$\begin{aligned} n|a_n|\le C \omega (f,1/n). \end{aligned}$$
$$\begin{aligned} |a_n | \le C \sum \limits _{k=[{n}/{\gamma }]}^{\infty } \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$
10.
Liuquan Wang 《The Ramanujan Journal》2017,44(2):343-358
Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that and
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$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$
$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$
11.
In this work, we consider the second-order discontinuous equation in the real line,with \({k > 0}\) and \({f : \mathbb{R}^{3} \rightarrow \mathbb{R}}\) an \({L^{1}}\)-Carathéodory function. The existence of homoclinic solutions in presence of not necessarily ordered lower and upper solutions is proved, without periodicity assumptions or asymptotic conditions. Some applications to Duffing-like equations are presented in last section.
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$$u^{\prime \prime}(t)-ku(t) = f( t, u(t), u^{\prime}(t)), \quad a.e.t \in \mathbb {R},$$
12.
G. Cupini Bernard Dacorogna O. Kneuss 《Calculus of Variations and Partial Differential Equations》2009,36(2):251-283
We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfyingwhere k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfieswith no sign hypothesis on f.
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$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$
$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$
13.
For \(k,l\in \mathbf {N}\), let We prove that the inequality is valid for all natural numbers k and l. The sign of equality holds if and only if \(k=l=1\). This complements a result of Vietoris, who showed that An immediate corollary is that The constant bounds are sharp.
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$$\begin{aligned}&P_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k-1} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }\\&\quad \text{ and }\quad Q_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }. \end{aligned}$$
$$\begin{aligned} \frac{1}{4}\le P_{k,l} \end{aligned}$$
$$\begin{aligned} P_{k,l}<\frac{1}{2} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$
$$\begin{aligned} \frac{1}{4}\le P_{k,l}<\frac{1}{2} <Q_{k,l}\le \frac{3}{4} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$
14.
In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem where in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.
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$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$
$$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$
15.
D. T. Luyen 《Mathematical Notes》2017,101(5-6):815-823
In this paper, we study the existence of multiple solutions for the boundary-value problem where Ω is a bounded domain with smooth boundary in R N (N ≥ 2) and Δ γ is the subelliptic operator of the type We use the three critical point theorem.
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$${\Delta _\gamma }u + f\left( {x,u} \right) = 0in\Omega ,u = 0on\partial \Omega ,$$
$${\Delta _\gamma }u = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}u} \right)} ,{\partial _{{x_j}}}u = \frac{{\partial u}}{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \ldots ,{\gamma _N}} \right).$$
16.
Guo-Shuai Mao 《The Ramanujan Journal》2018,45(2):319-330
In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.
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$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$
$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$
17.
Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition implies the uniform Kreiss resolvent condition We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T m for each integer m ? 2.
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$\parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M}{{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots ,$
$\left\| {\sum\limits_{k = 0}^n {\frac{{T^k }}{{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L}{{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots .$
18.
Zhi-Wei Sun 《The Ramanujan Journal》2016,40(3):511-533
Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have This is similar to Wolstenholme’s classical congruences for any prime \(p>3\).
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$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$
$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$
19.
We prove that, for all integers \(n\ge 1\), and with the best possible constants
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$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$
$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$
$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$
20.
In this paper, we study the uniform Hölder continuity of the generalized Riemann function \({R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}\)) defined by using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of \({R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}\) tends to infinity.
相似文献
$$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$