共查询到20条相似文献,搜索用时 78 毫秒
1.
For the singularly perturbed system we prove that flat segregated interfaces are uniformly Lipschitz as \(\beta \rightarrow +\infty \). As a byproduct of the proof we also obtain the optimal lower bound near flat interfaces, .
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$$\begin{aligned} \varDelta u_{i,\beta }=\beta u_{i,\beta }\sum _{j\ne i}u_{j,\beta }^2, \quad 1\le i\le N, \end{aligned}$$
$$\begin{aligned} \sum _iu_{i,\beta }\ge c\beta ^{-1/4}. \end{aligned}$$
2.
Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\), and
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$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$
$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$
3.
Let \((x_i)_{i=1}^{+\infty }\) be the digits sequence in the unique terminating dyadic expansion of \(x\in [0,1)\). The run-length function \(l_n(x)\) is defined by Erdös and Rényi proved that In this note, we show that for each pair of numbers \(\alpha ,\beta \in [0,+\infty ]\) with \(\alpha \le \beta \), the following exceptional set has Hausdorff dimension one.
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$$\begin{aligned} l_n(x):=\max \left\{ j:x_{i+1}=x_{i+2}=\cdots =x_{i+j}=1\ \text {for some}\ 0\le i\le n-j\right\} . \end{aligned}$$
$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=1, \text {a.e.}\ x\in [0,1). \end{aligned}$$
$$\begin{aligned} E_{\alpha ,\beta }=\left\{ x\in [0,1):\liminf _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\alpha ,\ \limsup _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\beta \right\} \end{aligned}$$
4.
Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example, where \(\alpha , \beta \ge 0\).
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$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$
5.
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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$$\begin{aligned} {\mathcal {A}}_n := \bigg \{ P: P(z) = \sum \limits _{j=1}^n{z^{k_j}}: 0 \le k_1 < k_2 < \cdots < k_n, k_j \in {\mathbb {Z}} \bigg \}, \end{aligned}$$
$$\begin{aligned} M_{p}(Q) := \left( \int _{0}^{1}{\left| Q(e^{i2\pi t}) \right| ^p\,dt} \right) ^{1/p}, \qquad p > 0. \end{aligned}$$
$$\begin{aligned} S_{n,p} := \sup _{Q \in {\mathcal {A}}_n}{\frac{M_p(Q)}{\sqrt{n}}} \qquad \text{ and } \qquad S_p := \liminf _{n \rightarrow \infty }{S_{n,p}} \le \Sigma _p := \limsup _{n \rightarrow \infty }{S_{n,p}}. \end{aligned}$$
$$\begin{aligned} \Sigma _p \ge \Gamma (1+p/2)^{1/p}, \qquad p \in (0,2). \end{aligned}$$
6.
We consider \(\text {pod}_3(n)\), the number of 3-regular partitions with odd parts distinct, whose generating function is where For each \(\alpha >0\), we obtain the generating function for where \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha }}\) if \(\alpha \) is even, \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha +1}}\) if \(\alpha \) is odd.
$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3(n)q^n=\frac{(-q;q^2)_\infty (q^6;q^6)_\infty }{(q^2;q^2)_\infty (-q^3;q^3)_\infty }=\frac{\psi (-q^3)}{\psi (-q)}, \end{aligned}$$
$$\begin{aligned} \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}=\sum _{-\infty }^\infty q^{2n^2+n}. \end{aligned}$$
$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3\left( 3^{\alpha }n+\delta _\alpha \right) q^n, \end{aligned}$$
We show that the sequence {\(\text {pod}_3(n)\)} satisfies the internal congruences and
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$$\begin{aligned} \text {pod}_3(9n+2)\equiv \text {pod}_3(n)\pmod 9, \end{aligned}$$
(0.1)
$$\begin{aligned} \text {pod}_3(27n+20)\equiv \text {pod}_3(3n+2)\pmod {27} \end{aligned}$$
(0.2)
$$\begin{aligned} \text {pod}_3(243n+182)\equiv \text {pod}_3(27n+20)\pmod {81}. \end{aligned}$$
(0.3)
7.
Donna Q. J. Dou 《The Ramanujan Journal》2016,40(3):535-540
In this note we investigate the function \(B_{k,\ell }(n)\), which counts the number of \((k,\ell )\)-regular bipartitions of n. We shall prove an infinite family of congruences modulo 11: for \(\alpha \ge 2\) and \(n\ge 0\),
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$$\begin{aligned} B_{3,11}\left( 3^{\alpha }n+\frac{5\cdot 3^{\alpha -1}-1}{2}\right) \equiv 0\ (\mathrm{mod\ }11). \end{aligned}$$
8.
We prove that, for all integers \(n\ge 1\), and with the best possible constants
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$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$
$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$
$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$
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.
Ri-An Yan Shu-Rong Sun Dian-Wu Yang 《Journal of Applied Mathematics and Computing》2015,48(1-2):187-203
In this paper, we study the existence of solutions for the boundary value problems of fractional perturbation differential equations or subject to where \(1<\alpha <2,\,D^{\alpha }\) is the standard Caputo fractional derivatives. Using some fixed point theorems, we prove the existence of solutions to the two types. For each type we give an example to illustrate our results.
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$$\begin{aligned} D^{\alpha }\left( \frac{x(t)}{f(t,x(t))}\right) =g(t,x(t)),\;\;a.e.\;t\in J=[0,1], \end{aligned}$$
$$\begin{aligned} D^{\alpha }\left( x(t)-f(t,x(t))\right) =g(t,x(t)),\;\;a.e.\;t\in J, \end{aligned}$$
$$\begin{aligned} x(0)=y(x),\;\;x(1)=m, \end{aligned}$$
11.
Qingfeng Sun 《The Ramanujan Journal》2017,44(1):13-36
Let f be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus and denote by \(\lambda _f(n)\) its nth Hecke eigenvalue. Let In this paper, we study the shifted convolution sum and establish uniform bounds with respect to the shift h for \(\mathcal {S}_h(X)\).
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$$\begin{aligned} r(n)=\#\left\{ (n_1,n_2)\in \mathbb {Z}^2:n_1^2+n_2^2=n\right\} . \end{aligned}$$
$$\begin{aligned} \mathcal {S}_h(X)=\sum _{n\le X}\lambda _f(n+h)r(n), \qquad 1\le h\le X, \end{aligned}$$
12.
Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies and where \(\delta _{p}\) designates Dirac mass at the point \(p\).
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$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$
$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$
$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$
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.
István Mező 《The Ramanujan Journal》2018,46(1):161-171
In this paper, we study the sequences where \(\mathrm {Li}_n\) is the nth polylogarithm function. Among others, we determine their generating functions, asymptotic behaviour and their connection to the well-known log-sine integrals With the help of the explicit forms of \(I_n\) and \(J_n\), we deduce closed-form evaluations for a number of polylog-trigonometric definite integrals.
相似文献
$$\begin{aligned} I_n=\int _0^1\mathrm {Li}_n(\sin \pi x)\mathrm {d}x\quad \text{ and }\quad J_n=\int _0^1\mathrm {Li}_n(\cos \pi x)\mathrm {d}x, \end{aligned}$$
$$\begin{aligned} S_n=(-1)^n\int _0^1\log ^n(\sin \pi x)\mathrm {d}x. \end{aligned}$$
15.
In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families \(\mathcal T:=\{T_\varepsilon \}_{\varepsilon >0}\) of truncated singular integrals and \(\mathcal M:=\{M_t\}_{t>0}\) of averaging operators with rough kernels, which are defined respectively by and where the kernel \(\Omega \) belongs to \(L\log ^+\!\!L(\mathbf S^{n-1})\) or \(H^1(\mathbf S^{n-1})\) or \(\mathcal {G}_\alpha (\mathbf S^{n-1})\) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.
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$$\begin{aligned} T_\varepsilon f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy \end{aligned}$$
$$\begin{aligned} M_t f(x)=\frac{1}{t^n}\int _{|y|<t}\Omega (y')f(x-y)dy, \end{aligned}$$
16.
Esteban Andruchow 《Complex Analysis and Operator Theory》2016,10(6):1383-1409
An idempotent operator E in a Hilbert space \({\mathcal {H}}\) \((E^2=1)\) is written as a \(2\times 2\) matrix in terms of the orthogonal decomposition (R(E) is the range of E) as We study the sets of idempotents that one obtains when \(E_{1,2}:R(E)^\perp \rightarrow R(E)\) is a special type of operator: compact, Fredholm and injective with dense range, among others.
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$$\begin{aligned} {\mathcal {H}}=R(E)\oplus R(E)^\perp \end{aligned}$$
$$\begin{aligned} E=\left( \begin{array}{l@{\quad }l} 1_{R(E)} &{} E_{1,2} \\ 0 &{} 0 \end{array} \right) . \end{aligned}$$
17.
18.
In this paper, we investigate the existence results for fractional differential equations of the form and where \(D_{c}^{q}\) is the Caputo fractional derivative. We prove the above equations have solutions in C[0, T). Particularly, we present the existence and uniqueness results for the above equations on \([0,+\infty )\).
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$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T)\left( 0<T\le \infty \right) , \quad q \in (1,2),\\ x(0)=a_{0},\quad x^{'}(0)=a_{1}, \end{array}\right. } \end{aligned}$$
(0.1)
$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T), \quad q \in (0,1),\\ x(0)=a_{0}, \end{array}\right. } \end{aligned}$$
(0.2)
19.
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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$$\begin{aligned} f_1(x_1) + \cdots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} \end{aligned}$$
$$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^2\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$
$$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^n\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$
20.
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
相似文献
$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$
$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$