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1.
For \(k,l\in \mathbf {N}\), let
$$\begin{aligned}&P_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k-1} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }\\&\quad \text{ and }\quad Q_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }. \end{aligned}$$
We prove that the inequality
$$\begin{aligned} \frac{1}{4}\le P_{k,l} \end{aligned}$$
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
$$\begin{aligned} P_{k,l}<\frac{1}{2} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$
An immediate corollary is that
$$\begin{aligned} \frac{1}{4}\le P_{k,l}<\frac{1}{2} <Q_{k,l}\le \frac{3}{4} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$
The constant bounds are sharp.
  相似文献   

2.
For \(n \ge 1\) let
$$\begin{aligned} {\mathcal {A}}_n := \bigg \{ P: P(z) = \sum \limits _{j=1}^n{z^{k_j}}: 0 \le k_1 < k_2 < \cdots < k_n, k_j \in {\mathbb {Z}} \bigg \}, \end{aligned}$$
that is, \({\mathcal {A}}_n\) is the collection of all sums of \(n\) distinct monomials. These polynomials are also called Newman polynomials. Let
$$\begin{aligned} M_{p}(Q) := \left( \int _{0}^{1}{\left| Q(e^{i2\pi t}) \right| ^p\,dt} \right) ^{1/p}, \qquad p > 0. \end{aligned}$$
We define
$$\begin{aligned} S_{n,p} := \sup _{Q \in {\mathcal {A}}_n}{\frac{M_p(Q)}{\sqrt{n}}} \qquad \text{ and } \qquad S_p := \liminf _{n \rightarrow \infty }{S_{n,p}} \le \Sigma _p := \limsup _{n \rightarrow \infty }{S_{n,p}}. \end{aligned}$$
We show that
$$\begin{aligned} \Sigma _p \ge \Gamma (1+p/2)^{1/p}, \qquad p \in (0,2). \end{aligned}$$
The special case \(p=1\) recaptures a recent result of Aistleitner [1], the best known lower bound for \(\Sigma _1\).
  相似文献   

3.
We present the conditions under which every nonoscillator solution x(t) of the forced fractional differential equation
$$\begin{aligned} ^{\mathrm{C}}D_{\mathrm{c}}^{\alpha } y ( t ) = e ( t ) +f ( {t, x ( t )} ), c > 1,\alpha \in ( {0,1} ), \quad \mathrm{{and}} \,\, \delta \ge 1, \end{aligned}$$
where \(y(t)= ( {a(t) ( {{x}'(t)} )^{\delta }})^{\prime },c_0 =\frac{y(c)}{\Gamma (1)}= y(c)\), is a real constant which satisfies
$$\begin{aligned} |x(t)|=O\left( {t^{1/\delta }e^{t}\int _{\mathrm{c}}^{t} {a^{-1/\delta }} (s)\mathrm{d}s} \right) , \quad t \rightarrow \infty \end{aligned}$$
It is shown that the technique can be applied to some related fractional differential equations. Examples are inserted to illustrate the relevance of the obtained results.
  相似文献   

4.
In this paper, we study the existence of solutions for the boundary value problems of fractional perturbation differential equations
$$\begin{aligned} D^{\alpha }\left( \frac{x(t)}{f(t,x(t))}\right) =g(t,x(t)),\;\;a.e.\;t\in J=[0,1], \end{aligned}$$
or
$$\begin{aligned} D^{\alpha }\left( x(t)-f(t,x(t))\right) =g(t,x(t)),\;\;a.e.\;t\in J, \end{aligned}$$
subject to
$$\begin{aligned} x(0)=y(x),\;\;x(1)=m, \end{aligned}$$
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.
  相似文献   

5.
For the singularly perturbed system
$$\begin{aligned} \varDelta u_{i,\beta }=\beta u_{i,\beta }\sum _{j\ne i}u_{j,\beta }^2, \quad 1\le i\le N, \end{aligned}$$
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,
$$\begin{aligned} \sum _iu_{i,\beta }\ge c\beta ^{-1/4}. \end{aligned}$$
.
  相似文献   

6.
The paper is devoted to sharp weak type \((\infty ,\infty )\) estimates for \({\mathcal {H}}^{\mathbb {T}}\) and \({\mathcal {H}}^{\mathbb {R}}\), the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that
$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {T}}f\right\| _{W({\mathbb {T}})}\le \Vert f\Vert _{L^\infty ({\mathbb {T}})} \end{aligned}$$
and
$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {R}}f\right\| _{W({\mathbb {R}})}\le \Vert f\Vert _{L^\infty ({\mathbb {R}})}, \end{aligned}$$
where \(W({\mathbb {T}})\) and \(W({\mathbb {R}})\) stand for the weak-\(L^\infty \) spaces introduced by Bennett, DeVore and Sharpley. In both estimates, the constant \(1\) on the right is shown to be the best possible.
  相似文献   

7.
We consider \(\text {pod}_3(n)\), the number of 3-regular partitions with odd parts distinct, whose generating function is
$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3(n)q^n=\frac{(-q;q^2)_\infty (q^6;q^6)_\infty }{(q^2;q^2)_\infty (-q^3;q^3)_\infty }=\frac{\psi (-q^3)}{\psi (-q)}, \end{aligned}$$
where
$$\begin{aligned} \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}=\sum _{-\infty }^\infty q^{2n^2+n}. \end{aligned}$$
For each \(\alpha >0\), we obtain the generating function for
$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3\left( 3^{\alpha }n+\delta _\alpha \right) q^n, \end{aligned}$$
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.
We show that the sequence {\(\text {pod}_3(n)\)} satisfies the internal congruences
$$\begin{aligned} \text {pod}_3(9n+2)\equiv \text {pod}_3(n)\pmod 9, \end{aligned}$$
(0.1)
$$\begin{aligned} \text {pod}_3(27n+20)\equiv \text {pod}_3(3n+2)\pmod {27} \end{aligned}$$
(0.2)
and
$$\begin{aligned} \text {pod}_3(243n+182)\equiv \text {pod}_3(27n+20)\pmod {81}. \end{aligned}$$
(0.3)
  相似文献   

8.
In this paper we study a Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on \(L^2(\partial \Omega )\) given by \(\varphi \mapsto \partial _{\nu }u\) where u is a weak solution of
$$\begin{aligned} \left\{ \begin{aligned}&-\mathrm{div}\, (a\nabla u) +b\cdot \nabla u -\mathrm{div}\, (cu)+du =\lambda u \ \ \text {on}\ \Omega ,\\&u|_{\partial \Omega } =\varphi . \end{aligned} \right. \end{aligned}$$
Under suitable assumptions on the matrix-valued function a, on the vector fields b and c, and on the function d, we investigate positivity, sub-Markovianity, irreducibility and domination properties of the associated Dirichlet-to-Neumann semigroups.
  相似文献   

9.
We prove that, for all integers \(n\ge 1\),
$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$
and
$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$
with the best possible constants
$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$
  相似文献   

10.
Abdulkadir Dogan 《Positivity》2018,22(5):1387-1402
This paper deals with the existence of positive solutions of nonlinear differential equation
$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$
subject to the boundary conditions
$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$
where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality
$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$
  相似文献   

11.
We prove several numerical radius inequalities for certain 2 × 2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then
$$w\left( \left[\begin{array}{cc} X &; Y \\ Z &; W \end{array} \right] \right) \geq \max \left(w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right) $$
and
$$w\left( \left[\begin{array}{cc}X &; Y \\ Z &; W\end{array} \right] \right) \leq \max \left( w(X), w(W)\right)+\frac{w(Y+Z)+w(Y-Z)}{2}. $$
As an application of a special case of the second inequality, it is shown that
$$\frac{\left\Vert X\right\Vert }{2}+\frac{\left\vert \left\Vert\operatorname{Re}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4}+\frac{ \left\vert \left\Vert \operatorname{Im}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4} \leq w(X), $$
which is a considerable improvement of the classical inequality \({\frac{ \left\Vert X\right\Vert }{2}\leq w(X)}\) . Here w(·) and || · || are the numerical radius and the usual operator norm, respectively.
  相似文献   

12.
D. D. Hai 《Positivity》2018,22(5):1269-1279
We prove the existence of positive solutions for the boundary value problem
$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$
for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.
  相似文献   

13.
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
$$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$
This is similar to Wolstenholme’s classical congruences
$$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$
for any prime \(p>3\).
  相似文献   

14.
Here we consider the q-series coming from the Hall-Littlewood polynomials,
$$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$
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
$$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$
  相似文献   

15.
In this paper the inequality
$$\begin{aligned} \bigg ( \int _0^{\infty } \bigg ( \int _x^{\infty } \bigg ( \int _t^{\infty } h \bigg )^q w(t)\,dt \bigg )^{r / q} u(x)\,{ ds} \bigg )^{1/r}\le C \,\int _0^{\infty } h v, \quad h \in {\mathfrak {M}}^+(0,\infty ) \end{aligned}$$
is characterized. Here \(0< q ,\, r < \infty \) and \(u,\,v,\,w\) are weight functions on \((0,\infty )\).
  相似文献   

16.
In this paper, the prescribed \(\sigma \)-curvature problem
$$\begin{aligned} P_{\sigma }^{g_0} u={\tilde{K}}(x)u^{\frac{N+2\sigma }{N-2\sigma }}, x\in {\mathbb {S}}^N,u>0 \end{aligned}$$
is considered. When \({\tilde{K}}(x)\) is some axis symmetric function on \({\mathbb {S}}^N\), by using singular perturbation method, it is proved that this problem possesses infinitely many non-radial solutions for \(0<\sigma \le 1\) and \(N> 2\sigma +2\).
  相似文献   

17.
We study the Eisenstein series and constant term functors in the framework of geometric theory of automorphic functions. Our main result says that for a parabolic \(P\subset G\) with Levi quotient M, the !-constant term functor
$$\begin{aligned}{\text {CT}}_!:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M)\end{aligned}$$
is canonically isomorphic to the *-constant term functor
$$\begin{aligned} {\text {CT}}^-_*:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M), \end{aligned}$$
taken with respect to the opposite parabolic \(P^-\).
  相似文献   

18.
This paper is devoted to a substantial generalization of previous work on the analytic hypoellipticity of sums of squares \(P=\sum _1^4X^2_j\) of real vector fields with real analytic coefficient in three variables. For p(xy) quasi-homogeneous in (xy), consider the vector fields
$$\begin{aligned} X_1 = \frac{\partial }{\partial x}, \quad X_2=-\frac{\partial }{\partial y} + p(x,y)\frac{\partial }{\partial t}, \quad X_3=x^{n_1}\frac{\partial }{\partial t}, \quad X_4=y^{n_2}\frac{\partial }{\partial t}, \end{aligned}$$
\( n_1, n_2 \ne 0\). We show that the operator
$$\begin{aligned} P=\sum _1^4 X_j^2, \end{aligned}$$
well known to be \(C^\infty \)-hypoelliptic, is actually analytic hypoelliptic near the origin in \({\mathcal {R}}^3\).
  相似文献   

19.
In this paper, we investigate the existence results for fractional differential equations of the form
$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T)\left( 0<T\le \infty \right) , \quad q \in (1,2),\\ x(0)=a_{0},\quad x^{'}(0)=a_{1}, \end{array}\right. } \end{aligned}$$
(0.1)
and
$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T), \quad q \in (0,1),\\ x(0)=a_{0}, \end{array}\right. } \end{aligned}$$
(0.2)
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 \([0,+\infty )\).
  相似文献   

20.
We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations
$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$
(0.1)
where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by
$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$
(0.2)
Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that
$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$
and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).
  相似文献   

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