Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

Meromorphic Functions That Share One Finite Value CM or IM with Their k-th Derivative

In this paper we shall prove that if a non-constant meromorphic f and its k-th derivative f ^(k) (k ≥ 2) share the value ${a\not= 0,\infty\; CM}$ (IM) and if ${\bar{N}(r,\frac{1}{f})=S(r,f)\;\left(\bar{N}\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{f^{(k)}}\right)=S(r,f)\right)}$ , then ${f \equiv f^{(k)}}$ . These results extend the results in Al-Khaladi (J Al-Anbar Univ Pure Sci 3:69–73, 2009).     

Symplectic mean curvature flow in CP 2

Let Σ be an immersed symplectic surface in CP ² with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP ². In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ .     

Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

Well-posedness of a semilinear heat equation with weak initial data

This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation $$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$ with initial data in $\dot L_{r,p} $ is studied. We prove the well-posedness when $$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$ and construct non-unique solutions for $$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$ In the second part the well-posedness of the avove IVP for k=2 with μ₀?H ^s (? ⁿ ) is proved if $$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$ and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u₀, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].     

On the approximation of conjugate functions from Holder classes by Fejér means

Получены асимптотич еские равенства для в еличин гдеr≧0 — целое, ω(t) — выпу клый модуль непрерыв ности и $$\bar \sigma _n (f;x) = - \frac{1}{\pi } \mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2}ctg\frac{t}{2} - \frac{1}{{4(n + 1)}}\frac{{\sin (n + 1)t}}{{\sin ^2 \tfrac{1}{2}t}}} \right)dt$$ сумма Фейера функцииf(х), сопряженной сf(x).     

Many Neighborly Polytopes and Oriented Matroids

In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of ${\big (( r+d ) ^{( \frac{r}{2}+\frac{d}{2} )^{2}}\big )}\big /{\big ({r}^{{(\frac{r}{2})}^{2}} {d}^{{(\frac{d}{2})}^{2}}{\mathrm{e}^{3\frac{r}{2}\frac{d}{2}}}\big )}$ for the number of combinatorial types of vertex-labeled neighborly polytopes in even dimension d with $r+d+1$ vertices. This improves current bounds on the number of combinatorial types of polytopes. The previous best lower bounds for the number of neighborly polytopes were found by Shemer in 1982 using a technique called the Sewing Construction. We provide a new simple proof that sewing works, and generalize it to oriented matroids in two ways: to Extended Sewing and to Gale Sewing. Our lower bound is obtained by estimating the number of polytopes that can be constructed via Gale Sewing. Combining both new techniques, we are also able to construct many non-realizable neighborly oriented matroids.     

Closed geodesics on Finsler spheres

In this paper, we prove that for every Finsler n-sphere (S ⁿ ,?F) all of whose prime closed geodesics are non-degenerate with reversibility λ and flag curvature K satisfying ${\left(\frac{\lambda}{\lambda+1}\right)^2 < K \le 1,}$ there exist ${2[\frac{n+1}{2}]-1}$ prime closed geodesics; moreover, there exist ${2[\frac{n}{2}]-1}$ non-hyperbolic prime closed geodesics provided the number of prime closed geodesics is finite.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

A Note on the Largest Size of Families of Sets with a Forbidden Poset

Consider families of subsets of [n]:?=?{1,2,...,n} that do no contain a given poset P as a subposet. Let La(n, P) denote the largest size of such families and h(P) denote the height of P. The best known general upper bound for La(n, P) is $\left(\frac{1}{2}(|P|+h(P))-1\right)\left( \begin{array}{l}\,\,\,n \\ \lfloor \frac{n}{2} \rfloor\end{array}\right)$ , due to Bursi and Nagy (2012). This paper provides an improved upper bound $\frac{1}{m+1} \left(|P|+\frac{1}{2}(m^2+3m-2)(h(P)-1)-1\right) \left( \begin{array}{l} \,\,\,n \\ \lfloor \frac{n}{2} \rfloor\end{array}\right) $ , where m can be any positive integer less than $\lceil \frac{n}{2}\rceil$ .     

On Certain Integral Operators of Fractional Type

In this paper we study integral operators of the form $$T\,f\left( x \right) = \int {k_1 \left( {x - a_1 y} \right)k_2 \left( {x - a_2 y} \right)...k_m \left( {x - a_m y} \right)f\left( y \right)dy} ,$$ $$k_i \left( y \right) = \sum\limits_{j \in Z} {2^{\frac{{jn}}{{q_i }}} } \varphi _{i,j} \left( {2^j y} \right),\,1 \leqq q_i < \infty ,\frac{1}{{q_1 }} + \frac{1}{{q_2 }} + ... + \frac{1}{{q_m }} = 1 - r,$$ $0 \leqq r < 1$ , and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:L^p \left( {R^n } \right) \to T:L^q \left( {R^n } \right)$ for $1 < p < \frac{1}{r}$ and $\frac{1}{q} = \frac{1}{p} - r$ .     

Sign-Changing Multi-Peak Solutions for Nonlinear Schrödinger Equations with Compactly Supported Potential

In this paper, we aim to prove the existence and concentration of sign-changing H ¹(? ^N ) solutions to the following nonlinear Schrödinger equations $$-\varepsilon^2\Delta u_{\varepsilon}+V(x)u_{\varepsilon}=K(x)|u_{\varepsilon}|^{p-1}u_{\varepsilon} $$ with N≥3, $1<p<\frac{N+2}{N-2}$ , and ε>0. When the potential function V(x) has compact support, $V(x)\not\equiv 0$ and V(x)≥0, K(x) is permitted to be unbounded under some necessary restrictions, we will show that one sign-changing H ¹(? ^N ) solution exists and exhibits concentration profile around local minimum points of the ground energy function $G(\xi)\equiv V^{\theta}(\xi) K^{-\frac{2}{p-1}}(\xi)$ with $\theta=\frac{p+1}{p-1}-\frac{N}{2}$ in the semiclassical limit ε→0.     

Asymptotics of the approximation of continuous and differentiable functions by the singular integrals of de la Vallée Poussin

The complete asymptotic developments in powers of 1/n are derived for quantities characterizing approximation by singular integrals of de la Vallée Poussin $$V_n (f:x) = \frac{1}{{\Delta _n }}\int_{ - \pi }^\pi {f(x + t)} \cos ^{2n} \frac{t}{2}dt;\Delta _n = \int_{ - \pi }^\pi {\cos ^{2n} \frac{t}{2}dt}$$ of the function classes Lipa, 0w ^(r), r?1 an integer.     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.     

Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the fractional diffusion-wave equation

In this paper, we consider the numerical approximation for the fractional diffusion-wave equation. The main purpose of this paper is to solve and analyze this problem by introducing an implicit fully discrete local discontinuous Galerkin method. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully we prove that our scheme is unconditionally stable and get L ₂ error estimates of \(O(h^{k+1}+(\Delta t)^{2}+(\Delta t)^{\frac {\alpha }{2}}h^{k+1})\) . Finally numerical examples are performed to illustrate the efficiency and the accuracy of the method.     

Sums of cubes of primes in short intervals

In this paper, we are able to sharpen Hua??s result by proving that almost all integers satisfying some necessary congruence conditions can be represented as $$N=p_1^3+ \cdots+p_s^3 \quad \mbox{with } \biggl \vert p_j-\sqrt[3]{\frac {N}{s}}\biggr \vert \leqslant U, j=1,\ldots, s, $$ where p _j are primes and $U=N^{\frac{1}{3}-\delta_{s}+\varepsilon }$ with $\delta_{s}=\frac{s-4}{6s+72}$ , where s=5,6,7,8.     

First moment of Rankin–Selberg central L-values and subconvexity in the level aspect

Let 1≤N<M with N and M coprime and square-free. Through classical analytic methods we estimate the first moment of central L-values $L(\frac{1}{2},f\times g)$ where $f\in\mathcal{S}^{*}_{k}(N)$ runs over primitive holomorphic forms of level N and trivial nebentypus and g is a given form of level M. As a result, we recover the bound $L(\frac{1}{2},f\times g) \ll_{\varepsilon}(N + \sqrt{M}) N^{\varepsilon}M^{\varepsilon}$ when g is dihedral. The first moment method also applies to the special derivative $L'(\frac{1}{2},f\times g)$ under the assumption that it is non-negative for all $f\in\mathcal{S}^{*}_{k}(N)$ .     

Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥?3, we show that for every ε?>?0, there are $\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon)}$ polynomials $f \in \mathbb{F}_{q}[x]$ with $\deg f=L$ , for which the class group of the quadratic extension $\mathbb{F}_{q}(x, \sqrt{f})$ has an element of order g. This sharpens the previous lower bound $q^{L(\frac{1}{2}+\frac{1}{g})}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.