On generalized Dedekind sums attached to Dirichlet characters

Generalized reciprocity formulas and Dedekind-Petersson-Knopp-type formulas are given to generalized Dedekind sums attached to Dirichlet characters, defined on a certain congruence subgroup of SL₂($\text{Z}$). In addition, these formulas are respectively construed as transformational and eigen properties of those sums redefined on a certain set of cusps.     

A footnote on residually finite groups

We give an easy proof that a finitely generated group which is residually (finite and soluble of bounded rank) is nilpotent by quasi-linear. This can be used to shorten the proofs of some recent theorems about residually finite groups.     

A Menon-type identity using Klee’s function

Czechoslovak Mathematical Journal - Menon’s identity is a classical identity involving gcd sums and the Euler totient function φ. A natural generalization of φ is the Klee’s...     

Rings with all modules residually finite

Define a ringA to be RRF (resp. LRF) if every right (resp. left) A-module is residually finite. Refer to A as an RF ring if it is simultaneously RRF and LRF. The present paper is devoted to the study of the structure of RRF (resp. LRF) rings. We show that all finite rings are RF. IfA is semiprimary, we show thatA is RRF ⇔A is finite ⇔A is LRF. We prove that being RRF (resp. LRF) is a Morita invariant property. All boolean rings are RF. There are other infinite strongly regular rings which are RF. IfA/J(A) is of bounded index andA does not contain any infinite family of orthogonal idempotents we prove:

Commutators in residually finite groups

The following result is proved. Let n be a positive integer and G a residually finite group in which every product of at most 68 commutators has order dividing n. Then G′ is locally finite.     

Centralizers in residually finite torsion groups

Let be a residually finite torsion group. We show that, if has a finite 2-subgroup whose centralizer is finite, then is locally finite. We also show that, if has no -torsion, and is a finite 2-group acting on in such a way that the centralizer is soluble, or of finite exponent, then is locally finite.

     

A residually finite associative algebra with an undecidable word problem

For any constructive commutative ring k with unity, we furnish an example of a residually finite, finitely generated, recursively defined associative k-algebra with unity whose word problem is undecidable. This answers a question of Bokut’ in [3]. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 441–451, July–August, 2000.     

Multilinear commutators in residually finite groups

The following result is proved. Let w be a multilinear commutator and n a positive integer. Suppose that G is a residually finite group in which every product of at most 896 w-values has order dividing n. Then the verbal subgroup w(G) is locally finite.     

Dedekind sums with a parameter in finite fields

We first introduce the multiple Dedekind–Rademacher sum with a parameter in finite fields and establish its reciprocity law. We then construct an analog of the higher-dimensional Apostol–Dedekind sums, and establish their reciprocity laws using the parameterized Dedekind sum.     

Linear core conditions in residually finite groups

LetG be a residually finite or pro-finite group. We say thatG satisfies the linear core condition with constantc if all finite index (open) subgroups ofG contain a subgroup of index at mostc which is normal inG. Answering a question of L. Pyber we give a complete characterisation of finitely generated residually finite and pro-finite groups satisfying a linear core generated residually finite and pro-finite groups satisfying a linear core condition. In the case of infinitely generated groups we prove that such groups are abelian-by-finite. Research supported by the Hungarian National Research Foundation (OTKA), grant no. 16432 and F023436.     

Distribution problems in Dedekind domains and submeasures

This paper is a continuation of[PAS], where uniformly distributed sequences in Dedekind domains were investigated. In the present article the concepts of well-distribution and complete distribution are studied. Furthermore general matrix summation methods are considered. Both authors are supported by the exchange program of the Austrian Academy of Sciences. The second author is supported by the Austrian Science Foundation project 10223-PHY.     

Sums with convolutions of Dirichlet characters

We bound short sums of the form ${\sum_{n\le X}(\chi_1{*}\chi_2)(n)}$ , where χ ₁*χ ₂ is the convolution of two primitive Dirichlet characters χ ₁ and χ ₂ with conductors q ₁ and q ₂, respectively.     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

A Lie algebra with a single defining relation need not be residually finite

Translated from Matematicheskie Zametki, Vol. 51, No. 4, pp. 3–7, April, 1992.