In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set K in a normed space K , a subset K in a Hausdorff topological vector space K , and a closed convex cone K in K . Let K , K be two multifunctions and K be a single-valued mapping. Find a point K such that\begin {gather}(\hat x,\hat y)\in\Gamma (\hat x)\times\Phi (\hat x),\,\,{\rm and}\,\,\{f (\hat x,\hat y, z): z\in\Gamma (\hat x)\}\cap (-{\rm Int} C)=\emptyset.\notag\end {gather} We prove some existence theorems for the problem in which K can be discontinuous and K can be unbounded.

In this paper, we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems by taking adiabatic limits. As an application, we give a Kastler-Kalau-Walze type theorem for foliations.

We present a generalized learning with errors (LWE) problem, which is essentially a simple and direct extension of the original LWE problem to the case of matrices. Then we use this new version of LWE problem, which we call matrix LWE problem to build new cryptographic schemes, which include a new scalable key distribution scheme, a new key exchanges scheme and a new simple identity-based encryption scheme.

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