The first author conjectured in 1990 (see [18]) that for any simply-connected elliptic space, the total dimension of the rational homotopy does not exceed that of its rational cohomology. Our main purpose in this paper is to investigate the following: does the Hilali conjecture holds for the configuration spaces of a rationally elliptic and simply connected topological space when it already holds for the space itself. We will prove that this statement is true for closed manifolds.

We prove non-existence of proper biharmonic hypersurfaces of zero scalar curvature in Euclidean space $E^5$.

In this article, we establish some existence results for solutions of a initial value problem of a nonlinear fractional differential system on half line involving the sequential Riemann-Liouville fractional derivatives. Our analysis relies on the Schauder fixed point theorem. An efficiency example is presented to illustrate the main theorem. As far as the author knows, the present work is perhaps the first one that deals with such kind of initial value problems for fractional differential systems on half line.

In this paper, we establish several new Schläfli-type mixed modular equations of composite degrees. These equations are analogous to those recorded by Ramanujan in his second notebook. As an application, we establish several new explicit values for the Ramanujan-Weber class invariant $G_{n}$ for $n=12, 48, 51, 57, 3/4, 3/16, 3/17$ and $3/19$.

In this paper we obtain some operator inequalities for functions defined by power series with complex coefficients and, more specifically, with nonnegative coefficients. In order to obtain these inequalities the classical Wielandt and some reverses of the Schwarz inequality for vectors in inner product spaces are utilized. Natural applications for some elementary functions of interest are also provided.

A hypermodule is a multivalued algebraic system satisfying the module like axioms. In this paper, we construct quotient hypermodule. Let $M$ be a hypermodule, $N$ be a subhypermodule of $M$ and $I$ be a hyperideal of $R$. Then, $[M:N^{\ast}]$ is $R$-hypermodule and $[R:I^{\ast}]$-hypermodule, and prove that when $N$ is normal subhypemodule, $[M:N^{\ast}]$ is a $[R:I^{\ast}]$-module. Hence, the quotient hypermodules considered by Anvarieh and Davvaz are modules.

We prove that the mean value (for some measure $\mu =\chi dx$ with $\chi \geq 0,dx=$ Riemannian measure) of the squared norm of the gradient of the unitary direction of a Jacobi field along an eigenmapping $v$ (associated to an eigenvalue $\lambda \geq 0$) of the tension field, for mappings from a compact Riemannian manifold $(M,g)$ into a symmetric Riemannian manifold $(N,h)$ of positive sectional curvature, is smaller than $c\lambda $, where $c>0$ depends only on the diameter and upper and lower curvature bounds of $(N,h)$. For negative $\lambda $, we prove that there is no nonvanishing Jacobi field along the eigenmappings, under the same assumptions on $(M,g)$ and $(N,h)$.