let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

We study thermodynamic properties and phase structures of topological black holes in Einstein theory with a Gauss-Bonnet term and a negative cosmological constant. The event horizon of these topological black holes can be a hypersurface with positive, zero or negative constant curvature. When the horizon is a zero curvature hypersurface, the thermodynamic properties of black holes are completel...

We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime hypersurface is called null, if its induced metric degenerate. The speed of spacelike surfaces projection codimension-two vector onto hypersurface. impose fairly mild conditions on Then for an outer un-trapped initial surface, condition which resembles mean-convexity surface Euclidean space, we prove that exist...

In the context of effective Friedmann equation we classify the cosmologies in multiscalar models with an arbitrary scalar potential V according to their geometric properties. It is shown that all flat cosmologies are geodesics with respect to a conformally rescaled metric on the ‘augmented’ target space. Non-flat cosmologies with V = 0 are also investigated. It is shown that geodesics in a ‘dou...

If X is a smooth hypersurface in complex projective space, the Fano variety of lines on stratified by splitting type normal bundle line. We show that for general hypersurfaces, these strata have expected dimension and, this case, compute class closure Chow ring Grassmannian space. For certain types, we also provide upper bounds hold all X.

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spin...