The notion of generalized locally bounded $I$-topological vectorspaces is introduced. Some of their important properties arestudied. The relationship between this kind of spaces and thelocally bounded $I$-topological vector spaces introduced by Wu andFang [Boundedness and locally bounded fuzzy topological vectorspaces, Fuzzy Math. 5 (4) (1985) 87$-$94] is discussed. Moreover, wealso use the fam...

the notion of generalized locally bounded $i$-topological vectorspaces is introduced. some of their important properties arestudied. the relationship between this kind of spaces and thelocally bounded $i$-topological vector spaces introduced by wu andfang [boundedness and locally bounded fuzzy topological vectorspaces, fuzzy math. 5 (4) (1985) 87$-$94] is discussed. moreover, wealso use the fam...

Thermodynamic quantities and correlation functions (CFs) of the classical antiferromagnet on the checkerboard lattice are studied for the exactly solvable infinite-component spin-vector model, D → ∞. In contrast to conventional two-dimensional magnets with continuous symmetry showing extended short-range order at distances smaller than the correlation length, r ∼ ξc ∝ exp(T /T ), correlations i...

Abstract. The security of lattice-based cryptosystems such as NTRU, GGH and Ajtai-Dwork essentially relies upon the intractability of computing a shortest non-zero lattice vector and a closest lattice vector to a given target vector in high dimensions. The best algorithms for these tasks are due to Kannan, and, though remarkably simple, their complexity estimates have not been improved since ov...

Abstract. The security of lattice-based cryptosystems such as NTRU, GGH and Ajtai-Dwork essentially relies upon the intractability of computing a shortest non-zero lattice vector and a closest lattice vector to a given target vector in high dimensions. The best algorithms for these tasks are due to Kannan, and, though remarkably simple, their complexity estimates have not been improved since ov...

Lattice control can unify nonlinear control systems where the basic vector and signal superpositions or transformations are based on the lattice supremum and infimum. In this paper we introduce a special case of lattice control that can model fuzzy dynamical systems in state space. Vector and signal transformations are represented as lattice dilations or erosions. The state and output responses...

We study lattice bases where the angle between any basis vector and the linear subspace spanned by the other basis vectors is at least π 3 radians; we denote such bases as “nearly orthogonal.” We show that a nearly orthogonal lattice basis always contains a shortest lattice vector. Moreover, we prove that if the basis vector lengths are “nearly equal,” then the basis is the unique nearly orthog...

All the identities and integral theorems of vector calculus are contained in the calculus of differential forms. The analogy between the exterior calculus of forms and the homology theory of a cell complex yields discrete lattice models for an array of interesting physical phenomena. These models, based on arbitrary combinations of coupled scalar and polar or axial vector field quantities, can ...

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...