In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227251] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the FischerBurmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Yamashita, M. Fukushima, New NCP functions and

We prove the existence of global sections trivializing the Hodge bundles on the Hodge metric completion space of the Torelli space of CalabiYau manifolds, a global splitting property of these Hodge bundles. We also prove that a compact CalabiYau manifold can not be deformed to its complex conjugate. These results answer certain open questions in the subject. A general result about certain period map to be bi-holomorphic from the Hodge metric completion space of the Torelli space of CalabiYau type manifolds to their period domains is proved and applied to the cases of K3 surfaces, cubic fourfolds, and hyperkhler manifolds.

In this paper we present a new variant of the Zhuang-Zi algorithm, which solves multivariate polynomial equations over a finite field by converting it into a single variable problem over a large extension field. The improvement is based on the newly developed concept of mutant in solving multivariate equations.

In this paper, we prove that the degree of regularity of the family of Square systems, an HFE type of systems, over a prime finite field of odd characteristics q is exactly q, and therefore prove that \begin{itemize} \item inverting Square systems algebraically is exponential, when q=O(n), where n is the number of variables of the system. \end{itemize}

We provide a non conformal generalization of the Compère-Song-Strominger (CSS) boundary conditions for AdS3 gravity that breaks the \hat{u}(1) Kac-Moody-Virasoro symmetry to two u(1)s. The holographic dual specified by the new boundary conditions can be understood as an irrelevant deformation of a warped conformal field theory (WCFT). Upon consistent reduction to two dimensions, AdS3 gravity results in a deformed Jackiw-Teitelboim dilaton gravity model coupled to a Maxwell field. We show that near extremality the boundary conditions inherited from generalized CSS boundary conditions in three dimensions give rise to an effective action exhibiting the same symmetry breaking pattern as the complex Sachdev-Ye-Kitaev models. Besides the Schwarzian term reflecting the breaking of conformal symmetry, the effective action contains an additional term that captures the breaking of the \hat{u}(1) Kac-Moody symmetry.