In this paper, we introduce the notion of L-topological spaces based on a complete bounded integral residuated lattice and discuss some properties of interior and left (right) closure operators.

Let {gt} be a nonquasiunipotent one-parameter subgroup of a connected semisimple Lie group G without compact factors; we prove that the set of points in a homogeneous spaceG/Γ (Γ an irreducible lattice inG) with bounded {gt}-trajectories has full Hausdorff dimension. Using this we give necessary and sufficient conditions for this property to hold for any Lie group G and any lattice Γ in G.

We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank l is bounded above by the effective central charge c̃. We show that lattice vertex operator algebras may be characterized by the equalities c̃ = l = c, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator a...

Fix an integer n > 0. For a multivariate function defined on a (not necessarily rectangular) lattice, an extension is constructed to have, ∀k ≤ n, derivatives of total degree k that are bounded by the function’s tensor product divided differences of total degree k times a constant independent of the lattice and the function. The extension is locally constructed, can have any prescribed smoothne...

In this paper we will discuss a few ways of thinking about Minkowski’s lattice point theorem. The Minkowski’s Lattice Point Theorem essentially states that bounded, convex, centrally symmetric sets in the n-dimensional Euclidean space Rn will always contain a nontrivial lattice point of Zn, provided that their volume is big enough. This simple and natural result was proved by Hermann Minkowski ...

This paper presents the study of modular lattice under rough set environment and referred to herein a concept is called rough modular lattice. We study the properties of lattice in an approximation space and defined as rough lattice, rough sublattice and bounded rough lattice. We consider the approximation space by means of an equivalence relation and also we present the rough set as pair of se...

Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time. We demonstrate that each such rule fills the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density can be computed exactly, and others in ...

We present a probabilistic polynomial-time reduction from the lattice Bounded Distance Decoding (BDD) problem with parameter 1/( √ 2 · γ) to the unique Shortest Vector Problem (uSVP) with parameter γ for any γ > 1 that is polynomial in the lattice dimension n. It improves the BDD to uSVP reductions of [Lyubashevsky and Micciancio, CRYPTO, 2009] and [Liu, Wang, Xu and Zheng, Inf. Process. Lett.,...

1.1 Basic Definitions Throughout these notes, V will be a finite dimensional Q-vector space. By a lattice in V we mean a finitely generated sybmodule that generates V as a Q-vector space. A subset X ⊆ V is said to be bounded if X is contained in a lattice. For a given function f on V and a vector ` ∈ V we say ` is a period of f if f(x + `) = f(x) for all x ∈ V . The set of all periods of f will...