We examine the Landau constants defined by $$G_n:=\sum_{m\,=0}^{n}\frac{1}{2^{4 m}}\,\binom{2 m}{m}^2\qquad(n=0, 1, 2, \cdots)$$ by making use of the celebrated Ramanujan formula expressing $G_n$ in terms of the Clausenian ${}_3F_2$ hypergeometric series. It is shown that it could be used to deduce other, mostly new, Ramanujan type formulas for the Landau constants involving the terminating and non-terminating hypergeometric series. In addition, by this approach we derive once again, in a simple and unified manner, almost all of the known results and also establish several new results for $G_n$. These new results include (for example) the generating function and asymptotic expansions and estimates for $G_n$.

The purpose of this paper is to characterize in terms of scalar preinvexity (resp. prequasiinvexity) the set-valued maps which are K-preinvex (resp. K-prequasiinvex) with respect to a closed convex cone K. Moreover, as applications of our results some conditions under which a local solution of set-valued scalar optimization for (VP) is a global supper efficient solution for (VP) are given.

In this paper, we present some new random coincidence point and random fixed point theorems for multifunctions in separable complete metric spaces, which improve some existing results in the literature (even some results in the non-random case).

The nonlinear matrix equations $X\pm A^{*}X^{-1}A=I$ are investigated, where $A$ is an $n\times n$ nonsingular matrix and $I$ is an $n\times n$ identity matrix. Some new perturbation bounds for Hermitian positive definite solutions of these equations are derived by using elementary calculus techniques developed in[Sun J G , BIT, 31(1991), pp.341-352] and [Barrlund A, BIT, 31(1991), pp.358-363]. The new results are illustrated by numerical examples.

In the present paper submanifolds of generalized Sasakian-spaceforms are studied. We focus on almost semi-invariant submanifolds, these generalize invariant, anti-invariant, and slant submanifolds. Sectional curvatures, Ricci tensor and scalar curvature are also studied. The paper finishes with some results about totally umbilical submanifolds.

Under the Continuum Hypothesis, it is proved that any nonzero $\sigma$-finite metrically transitive invariant measure on a group of cardinality continuum admits a nonseparable invariant extension. An application of this result to the left Haar measure on a $\sigma$-compact locally compact topological group of the same cardinality is also presented.

We prove a new characterization of arbitrary (including nonlinear) open maps between pseudometrizable topological vector spaces. We use it to give a new proof that nonconstant analytic functions are open.

In this paper we consider the following nonlinear third order two-point boundary value problem \begin{align*} & x'''(t)+f(t,x(t))=0,\quad a \lt t \lt b,\\ & x(a)=x''(a)=x(b)=0. \end{align*} By using the Leggett-Williams and Krasnosel'skii fixed-point theorems, we offer criteria for the existence of three positive solutions to the boundary value problem. Examples are also included to illustrate the results obtained.

We prove that the $g$-function operator $g_\phi$, where $\phi(x)=h(|x|)\Omega(x)$ with $\Omega(x)=\Omega(x')\in H^1(S^{n-1})$ and $h(s)$ satisfing certain continuity hypothesis, is bounded on Triebel-Lizorkin space $F^{\alpha,q}_p(R^n)$ when $0 \lt \alpha \lt 1$ and $1 \lt p,q \lt \infty$. In particular, we get that the Marcinkiewicz integral operator $\mu_\Omega$ with $H^1$-kernel is bounded on $F^{\alpha,q}_p$

Let $p\in(0,1]$, $q\in(1,\infty)$, $\alpha\in[n(1-1/q),\infty)$ and $w_1,\,w_2\in A_1$. The author proves that the norms in weighted Herz-type Hardy spaces $HK^{\alpha,\,p}_q(w_1,\,w_2)$ and $HK^{\alpha,\,p}_q(w_1,\,w_2)$ can be achieved by finite central atomic decompositions in some dense subspaces of them. As an application, the author proves that if $T$ is a sublinear operator and maps all central $(\alpha,\, q,\,s;\,w_1,\,w_2)_0$-atoms (resp. central $(\alpha,\, q,\,s;\,w_1,\,w_2)$-atoms of restrict type) into uniformly bounded elements of certain quasi-Banach space $\cal B$ for certain nonnegative integer $s$ no less than the integer part of $\alpha-n(1-1/q)$, then $T$ uniquely extends to a bounded operator from $H\dot K^{\alpha,\,p}_q(w_1,\,w_2)$ (resp. $HK^{\alpha,\, p}_q(w_1,\,w_2)$) to $\cal B$.

It shall be shown that there exist real symmetric differential expressions of every even order with a nonempty essential spectrum which are not in the limit-point case.

Let $C_n$ be the cyclic group of order $n$, $n\geq 20$, and let $S=\prod_{i=1}^k g_i$ be a minimal zero-sum sequence of elements in $C_n$. We say that $S$ is insplitable if for any $g_i\in S$ and any two elements $x,y\in C_n$ satisfying $x+y=g_i$, $Sg_i^{-1}xy$ is not a minimal zero-sum sequence any more. We define $\mbox{Index}(S)=\min_{(m,n)=1} \{\sum_{i=1}^k|mg_i|\}$, where $|x|$ denotes the least positive inverse image under homomorphism from the additive group of integers $\mathbb{Z}$ onto $C_n$. In this paper we prove that for an insplitable minimal zero-sum sequence $S$, if $\mbox{Index}(S)=2n$, then $|S|\leq \lfloor\frac{n}{2}\rfloor+1$.

The structure of left filial algebras over fields is studied. Roughly speaking, these are algebras in which the relation ”being a left ideal” is transitive. It is shown that semiprime algebras are left filial if and only if they are strongly regular and that prime radical algebras are left filial if and only if they are H-algebras, i.e., algebras in which all subalgebras are ideals. In the general case structure theorems describing left filial algebras are obtained. They make it possible to get a complete classification of finite dimensional left filial algebras over some fields.

An element $u\in R$ is pseudo invertible if there exist $v,w\in R$ such that $R(1-uv)R(1-wu)R$ is a nilpotent ideal. A ring $R$ is a $QB_{\infty}$ ring provided whenever $aR+bR=R$ with $a,b\in R$, there exists $y\in R$ such that $a+by$ is pseudo invertible. We prove, in this paper, that an exchange ring $R$ is a $QB_{\infty}$-ring if and only if whenever $x=xyx$, there exists a pseudo invertible $u\in R$ such that $x=xyu=uyx$ if and only if whenever $x=xyx$, there exists $a\in R$ such that $y+a$ is pseudo invertible and $1+xa$ is invertible. Also we characterize exchange $QB_{\infty}$-rings by virtue of pseudo unit-regularity. These generalize the main results of Wei (2004, Theorem 3, Theorem 7; 2005, Theorem 2.2, Theorem 2.4 and Theorem 3.6).

The main purpose of this paper is to prove the boundedness of commutators of fractional integral operators associated to a measure on metric space satisfying just a mild growth condition.

We study the Fourier-Laplace series on the unit sphere of higher dimensional Euclidean spaces and obtain a condition for convergence of Fourier- Laplace series on the unit sphere. The result generalizes Carleson’s Theorem to higher dimensional unit spheres.

Under suitable assumptions on the Fourier transform of the delay operator $F$, we give necessary and sufficient conditions for the inhomogeneous abstract delay equations: $ u'(t)=Au(t)+Fu_{t}+f(t), \ (t\in \mathbf T)$ to have maximal regularity in Besov spaces $B_{p,q}^s(\mathbf T, X)$ and Triebel-Lizorkin spaces $F_{p,q}^s(\mathbf T, X)$. .

We establish explicit necessary and sufficient conditions for the existence of nonnegative solutions of the $p$-Laplacian boundary blow-up problem \begin{equation*} \left\{ \begin{array}{l} \left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }=\lambda f(u(x)), \, 0 \lt x \lt 1, \\ \lim\limits_{x\rightarrow 0^{+}}u(x)=\infty =\lim\limits_{x\rightarrow 1^{-}}u(x), \end{array} \right. \end{equation*} where $p>1$, $\varphi _{p}\left( y\right) =\left\vert y\right\vert ^{p-2}y$ and $\left( \varphi _{p}(u^{\prime })\right) ^{\prime }$ is the one-dimensional $p$-Laplacian, $\lambda $ is a positive bifurcation parameter and $f$ is a locally Lipschitz continuous function on $[0,\infty)$. The gap is extremely small between the explicit necessary condition and the explicit sufficient condition for the existence of nonnegative solutions. Our results improve and extend some main results of Anuradha, Brown and Shivaji [2] and of Wang [30] from $p=2$ to any $p\gt 1$.

The existence of multiple nontrivial solutions for two types of double eigenvalue problems involving the p-Laplacian is derived. To prove the existence of at least two nontrivial solutions we use a Ricceri-type three critical point result for non-smooth functions of S. Marano and D. Motreanu [12]. The existence of at least three nontrivial solutions is shown by combining a result of B. Ricceri [17] and a Pucci-Serrin mountain pass type theorem of S. Marano and D. Motreanu [12].