Volume 13, Issue 4

Let 𝑅 be a commutative Noetherian ring and let 𝑀 be a finitely generated 𝑅-module. Let 𝑋 = Spec 𝑅 (𝑀) be the topological space with Zariski topology. Our main goal in this paper is to describe the connectedness dimension of 𝑅 in terms of Krull dimension of some quotient of 𝑀 and prove that 𝑐(Spec 𝑅 (𝑀)) = 𝑐(Supp(𝑀)).

The concept of a triangulated homotopy category can be canonically lifted to the level of a groupoid-enriched category. This way two natural axioms on track triangles replace the four somewhat obscure axioms of a triangulated category.

Using two natural axioms on track triangles in triangulated track category we show that the mapping cone construction is functorial, i.e., principal maps between mapping cones in a triangulated category can be identified with the homotopy classes of homotopy pairs. This result is applied to obtain an algebraic model of the category of principal maps between 2-stage spectra.

On the singular complex of a space, a local system of acyclic spaces is constructed leading, for Serre fibrations, to a bigraded differential model for the chain complex of the total space.

We classify all lacunary modular forms corresponding to the two-eta-products η 𝑟 (𝑧) η 𝑠 (𝑚𝑧) for 𝑚 = 3, 4, 5, where 𝑟 + 𝑠 is even and 𝑟𝑠 ≠ 0. We show that there are no lacunary non-cusp forms corresponding to the eta-product η 𝑟 (𝑧) η 𝑠 (𝑚𝑧), 𝑚 ≥ 4.

Motivated by two results of Ramanujan, we give a family of 15 results and 4 related ones. Several have interesting interpretations in terms of the number of representations of an integer by a quadratic form , where λ 1 + . . . + λ 𝑚 = 2, 4 or 8. We also give a new and simple combinatorial proof of the modular equation of order seven.

Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ (𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ (𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑 0 𝑑 1 , if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ (𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

Using quadratic maps we define a category Niq with objects class two nilpotent groups. Hom's in this category are groups and we investigate functoriality properties of this group. The classification problem in Niq is reduced to a certain linear problem.

The basis of the space of cusp forms is constructed for any integer 𝑘 and character mod 12.

The Eisenstein series corresponding to quadratic forms of type (𝑓/2, 4𝑁, χ ) (𝑁 is a square-free natural number) are constructed using the bases of the spaces of Eisenstein series given in [Kachakhidze, Georgian Math. J. 13: 55–78, 2006].

We complete a certain diagram (the operadic butterfly) of the categories of algebras involving Com , As , and Lie by constructing a type of algebras which have 4 generating operations and 16 relations. The associated operad is self-dual for Koszul duality.

This is a survey of the main results obtained by the mathematicians of Georgia on the representation of natural numbers by integral quadratic forms before 2000.

The number of representations of an integer as a sum of 𝑛 odd squares for 𝑛 ≡ 0 mod 4 is expressed by the number of representations as a sum of 𝑛 squares of integers and the number of representations by the lattice .

Let 𝑕, 𝑘 be integers such that 0 < 𝑕 < 𝑘 and (𝑕, 𝑘) = 1. If 1 ≤ 𝑖 ≤ 𝑘 – 1, let 𝑟 𝑖 be the least positive residue (mod 𝑘) of 𝑕𝑖. Let the permutation For 1 ≤ 𝑖 < 𝑗 ≤ 𝑘 – 1, if 𝑟 𝑖 > 𝑟 𝑗 , this is called an inversion of σ 𝑕, 𝑘 . Let 𝐼(𝑕, 𝑘) denote the total number of inversions of σ 𝑕, 𝑘 . In this note, we prove several identities concerning 𝐼(𝑕, 𝑘).

The modular properties of general theta-functions with characteristics are used to build a cusp form of weight 9/2.

An arithmetic identity is used to prove a relation satisfied by the double series . As an application an explicit formula is given for the number of representations of the positive integer 𝑛 by the form .

In this paper we classify the finite groups satisfying the following property 𝑃 3 : every three distinct orders of elements are setwise relatively prime.