Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind...

The article presents an algorithm to compute a C[t]-module basis G for a given subalgebra A over a polynomial ring R = C[x] with a Euclidean domain C as the domain of coefficients and t a given element of A. The reduction modulo G allows a subalgebra membership test. The algorithm also works for more general rings R, in particular for a ring R ⊂ C((q)) with the property that f ∈ R is zero if an...

In this paper, starting with a simply laced root system, we define a triangulated category which we call the m-cluster category, and we show that it encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. For Φ any root sys...

A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty subsets of G where Y=G−X and every member of X precedes every member of Y. A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has a least member, but not both; if every Dedekind cut of G is a continuous cut, G is said to be (Dedekind) continuous. The ordered abelian group R of real numbers is, ...

let $g$ be a group with identity $e$. let $r$ be a $g$-graded commutative ring with a non-zero identity and $m$ be a graded $r$-module. in this article, we introduce the concept of graded almost semiprime submodules. also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.

Abstract. We consider the series Φ(α) = ∑∞ m=1 1 m2 sin(2πm α) cot(πmα), a twist of the famous continuous but almost nowhere differentiable sine series defined by Riemann. In a slightly different but equivalent form, this series appeared in the first author’s paper [On the distribution of multiple of real numbers, Monatsh. Math 164.3 (2011), 325–360]. We pursue here the study of Φ, which is alm...

The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and its applications to arithmetical functions involving Dedekind sums and Kloosterman sums. As was discovered by Andre Weil, automorphic forms of non-integral weight correspond to invariant funtions on Metaplectic groups. We thus give an explicit description of Meptaplectic groups corresponding to ...

In general, a product has a different theory to that of the structures in the set Mi. For example a product of fields is just a ring. Note also that if φ is an atomic formula then ∏ iMi |= φ((ai )i, · · · , (ai )i) iff {i ∈ I : Mi |= φ(ai , · · · , ai )} = I. In particular this shows that if J ⊆ I then the restiction function (ai)i∈I → (ai)i∈J is a surjective homomorphism from ∏ i∈IMi to ∏ i∈JM...

here, we propose an add-drop and a channel drop filter based on two-dimensional photonic crystal all circular ring resonators. these structures are made of a square lattice of silicon rods with the refractive index n1=3.464 surrounded by air (with refractive index n2=1). the broadest photonic band gap occurs at the filling ratio of r/a = 0.17. two linear defect w1 waveguides couple to the ring....