Showing papers on "Torsion-free abelian group published in 2007"

Invariant decomposition of functions with respect to commuting invertible transformations

Abstract: Consider a 1 ,..., , an ∈ R arbitrary elements. We characterize those functions f: R→ R that decompose into the sum of a j -periodic functions, i.e., f=f 1 +---+f n with Δ aJ f(x):= f(x+aj)-f(x) = 0. We showthat f has such a decomposition if and only if for all partitions Β 1 UB 2 U...US N = {a 1 ,..., a n ) } with Bj consisting of commensurable elements with least common multiples b j one has Δ b1 ... Δb N f = 0. Actually, we prove a more general result for periodic decompositions of functions f: A→ R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f: A → R with respect to commuting, invertible self-mappings of some abstract set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

Invariant decomposition of functions with respect to commuting invertible transformations

Abstract: Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real functions f that decompose into the sum of a_j-periodic functions, i.e., f=f_1+...+f_n with D_{a_j}f(x):=f(x+a_j)-f(x)=0. We show that f has such a decomposition if and only if for all partitions to B_1, B_2,... B_N of {a_1,a_2,...,a_n} with B_j consisting of commensurable elements with least common multiples b_j, one has D_{b_1}... D_{b_N}f=0. Actually, we prove a more general result for periodic decompositions of real functions f defined on an Abelian group A, and, in fact, we even consider invariant decompositions of functions f defined on some abstract set A, with respect to commuting, invertible self-mappings of the set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real valued periodic decomposition of an integer valued function implies the existence of an integer valued periodic decomposition with the same periods.

The p-rank of Extℤ(G, ℤ) in certain models of ZFC

Abstract: We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal µ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π0 and Π1, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2µ = µ+ such that the p-rank of Ext ℤ(G, ℤ) equals 2µ = µ+ for every p ∈ Π0 and 0 otherwise, that is, for p ∈ Π1.

Enumerations and Torsion Free Abelian Groups

Abstract: We study possible spectrums of torsion free Abelian groups. We code families of finite sets into group and set up the correspondence between their algorithmic complexities.

Ruelle type L-functions versus determinants of Laplacians for torsion free abelian groups

Abstract: We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $ \ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $ =2,4$ and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus $\bR^{ }/\G$ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.

Invariant decomposition of functions with respect to commuting invertible

Abstract: Consider a1, . . . , an 2 R arbitrary elements. We characterize those functions f : R ! R that decompose into the sum of aj-periodic functions, i.e., f = f1+···+fn withaj f(x) := f(x+aj) f(x) = 0. We show that f has such a decomposition if and only if for all partitions B1(B2(···(BN = {a1, . . . , an} with Bj consisting of commensurable elements with least common multiples bj one hasb1 . . .�bN f = 0. Actually, we prove a more general result for periodic decompositions of functions f : A ! R defined on an Abelian group A, in fact, we even consider invariant decompositions of functions f : A ! R with respect to commuting, invertible self-mappings of some abstract set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real valued periodic decomposition of an integer valued function implies the existence of an integer valued periodic decomposition with the same periods.