We introduce a new concept of strong vanishing property and investigate this property for minimally elliptic singularities. Acomparison theorem about strong vanishing property is also obtained. We use Zariski decomposition to give a formula of plurigenera for Gorenstein singularities and use it to test whether a divisor is f-nef.

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek <i>et al.</i> in Taiwan. J.Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo <i>et al.</i> (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J.Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J.Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive

Let X be an equivariant embedding of a connected reductive group X over an algebraically closed field X of positive characteristic. Let X denote a Borel subgroup of X . A X -Schubert variety in X is a subvariety of the form $\diag (G)\cdot V $, where X is a X -orbit closure in X . In the case where X is the wonderful compactification of a group of adjoint type, the X -Schubert varieties are the closures of Lusztig's X -stable pieces. We prove that X admits a Frobenius splitting which is compatible with all X -Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any X -Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that X -Schubert varieties have nice singularities we present an example of a non-normal X -Schubert variety in the wonderful compactification of a group of type X . Finally we also extend the Frobenius splitting results to the more general class of X -Schubert varieties.

We classify the Ext-quivers of hearts in the bounded derived category D(A_n) and the finite-dimensional derived category D(\Gamma_N A_n) of the Calabi-Yau-N Ginzburg algebra D(\Gamma_N A_n). This provides the classification for Buan-Thomas' colored quiver for higher clusters of A-type. We also give explicit combinatorial constructions from a binary tree with n+2 leaves to a torsion pair in mod k\overrightarrow{A_n} and a cluster tilting set in the corresponding cluster category, for the straight oriented A-type quiver \overrightarrow{A_n}. As an application, we show that the orientation of the n-dimensional ssociahedron induced by poset structure of binary trees coincides with the orientation induced by poset structure of torsion pairs in mod k\overrightarrow{A_n} (under the correspondence above).

LWE/RLWE-based cryptosystems require sampling error term from discrete Gaussian distribution. However, some existing samplers are somehow slow under certain circumstances therefore efficiency of such schemes is restricted. In this paper, we introduce a more efficient discretized Gaussian sampler based on ziggurat sampling algorithm. We also analyze statistical quality of our sampler to prove that it can be adopted in LWE/RLWE-based cryptosystems. Compared with ziggurat-based sampler by Buchmann et al. related samplers by Peikert, Ducas et al. and Knuth-Yao, our sampler achieves more than 2x speedup when standard deviation is large. This can benefit constructions rely on noise flooding (e.g., homomorphic encryption). We also present two applications: First, we use our sampler to optimize the RLWE-based authenticated key exchange (AKE) protocol by Zhang et al. We achieve 1.14x speedup on total runtime of this protocol over major parameter choices. Second, we give practical post-quantum Transport Layer Security (TLS) ciphersuite. Our ciphersuite inherits advantages from TLS and the optimized AKE protocol. Performance of our proof-of-concept implementation is close to TLS v1.2 ciphersuites and one post-quantum TLS construction.