in this paper, our purpose is twofold. firstly, the tensor andresiduum operations on $l-$nested systems are introduced under thecondition of complete residuated lattice. then we show that$l-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. thus the new representation theorem of$l-$subsets on complete re...

Let L = {1 b1 , . . . , as bs }, where for every i ∈ [s], ai bi ∈ [0, 1) is an irreducible fraction. Let F = {A1, . . . , Am} be a family of subsets of [n]. We say F is a fractional L-intersecting family if for every distinct i, j ∈ [m], there exists an ab ∈ L such that |Ai ∩Aj | ∈ {ab |Ai|, ab |Aj |}. In this paper, we introduce and study the notion of fractional L-intersecting families.

We discuss the idea of a “family of L-functions” and describe various methods which have been used to make predictions about L-function families. The methods involve a mixture of random matrix theory and heuristics from number theory. Particular attention is paid to families of elliptic curve L-functions. We describe two random matrix models for elliptic curve families: the Independent Model an...

Let R be the ring of integers in a finite extension K of Qp, let k be its residue field and let χ : π1(X) → R× = GL1(R) be a ”geometric” rank one representation of the arithmetic fundamental group of a smooth affine k-scheme X. We show that the locally Kanalytic characters κ : R× → Cp are the Cp-valued points of a K-rigid space W and that L(κ ◦ χ, T ) = ∏ x∈X 1 1− (κ ◦ χ)(Frobx)T deg(x) , viewe...

Evolutionary computation has proven its utility in automating the process of engineering design. However, little attention has been paid to the scalability of generated designs, which is an important issue. This paper addresses this issue and proves the viability of evolving families of designs using parameterized L-Systems as a representation. The rest of the paper is organized as follows: fir...

For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square L-functions of...