In this paper, we give an overview of new results that define and explicitly construct Ramanujan Cayley biregular bipartite graphs. We also study the extremal combinatorial properties of these graphs. This parallels the work of Lubotzky, Phillips and Sarnak on regular Ramanujan Cayley graphs, with several interesting differences. Furthermore, this work also proposes a stronger definition of Ramanujan graphs than has been used in the past, which opens the door to future studies.
We introduce conditions weaker than Gauduchon K & auml;hler-like and use them to derive rigidity results. These results extend those obtained in [15,16] under the stronger Gauduchon K & auml;hler-like assumption.
Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map f : M -> Sn from a closed connected Riemannian spin manifold M with scalar curvature scal >= n(n-1) to the standard sphere Sn is an isometry if the degree off is nonzero. We investigate if one can replace the condition deg(f) /= 0 by the weaker condition that f is surjective. The answer turns out to be "no" for n >= 3 but "yes" for n = 2. If we replace the scalar curvature by Ricci curvature, the answer is "yes"in all dimensions.
In this article we classify all simple modules over a noncommutative and noncocommutative bialgebra M ( p , q ) assuming q is a root of unity.
