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Let G ∼ = Cn 1 ⊕ · · · ⊕ Cn r be a finite and nontrivial abelian group with n 1 |n 2 |. .. |nr. A conjecture of Hamidoune says that if W = w 1 · · · wn is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S| ≥ |W | + |G| − 1 ≥ |G| + 1, the maximum multiplicity of S at most |W |, and σ(W) ≡ 0 mod |G|, then there exists a nontrivial subgroup H such that every element g ∈ H can be represented as a weighted subsequence sum of the form g = n P i=1 w i s i , with s 1 · · · sn a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W | ≥ 1 2 |G|. A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S| ≥ |G| + D(G) − 1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g + H such that all but at most |G/H| − 2 terms of S are from g + H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S| ≥ |G| + D(G) − 1 can be relaxed to |S| ≥ |G| + d * (G), where d * (G) = r P i=1 (n i − 1). We also use this method to derive a variation on Hamidoune's conjecture valid when at least d * (G) of the w i are relatively prime to |G|. 1. Notation We follow the conventions of [9] for notation concerning sequences over an abelian group. For real numbers a, b ∈ R, we set [a, b] = {x ∈ Z | a ≤ x ≤ b}. Throughout, all abelian groups will be written additively. Let G be an abelian group, and let A, B ⊆ G be nonempty subsets. Then A + B = {a + b | a ∈ A, b ∈ B} denotes their sumset. The stabilizer of A is defined as H(A) = {g ∈ G | g + A = A}, and A is called periodic if H(A) = {0}, and aperiodic otherwise. If A is a union of H-cosets (i.e., H ≤ H(A)), then we say A is H-periodic. The order of an element g ∈ G is denoted ord(g), and we use φ H : G → G/H to denote the natural homomorphism. We use gcd(a, b) to denote the greatest common divisor of a, b ∈ Z. 2000 Mathematics Subject Classification. 11B75 (20K01).

Abstract. A subset S of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of S is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, p-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary p-groups of rank at most 2, paralleling and building on recent results on this problem for the Olson constant.

Let G be an abelian group of order n and let μ be a sequence of elements of G with length 2n−k+1 taking k distinct values. Assuming that no value occurs n−k+3 times, we prove that the sums of the n-subsequences of μ must include a non-null subgroup. As a corollary we show that if G is cyclic then μ has an n-subsequence summing to 0. This last result, conjectured by Bialostocki, reduces to the Erdos–Ginzburg–Ziv theorem for k=2.

minimal correctionInternational audienceThe Harborth constant of a finite abelian group is the smallest integer $\ell$ such that each subset of $G$ of cardinality $\ell$ has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this constant is defined in the same way except that instead of considering the sum of all elements of the subset one can choose to add either the element or its inverse. We determine these constants for certain groups, mainly groups that are the direct sum of a cyclic group and a group of order $2$. Moreover, we contrast these results with existing results and conjectures on these problems

Let G be a finite abelian group and let A ⊆ Z be nonempty. Let DA(G) denote the minimal integer such that any sequence over G of length DA(G) must contain a nontrivial subsequence s1 · · · sr such that r P i=1 wisi = 0 for some wi ∈ A. Let EA(G) denote the minimal integer such that any sequence over G of length EA(G) must contain a subsequence of length |G|, s1 · · · s |G| , such that |G| P i=1 wisi = 0 for some wi ∈ A. In this paper, we show thatconfirming a conjecture of Thangadurai and the expectations of Adhikari, et al. The case A = {1} is an older result of Gao, and our result extends much partial work done by Adhikari, Rath, Chen, David, Urroz, Xia, Yuan, Zeng and Thangadurai. Moreover, under a suitable multiplicity restriction, we show that not only can zero be represented in this manner, but an entire nontrivial subgroup, and if this subgroup is not the full group G, we obtain structural information for the sequence generalizing another non-weighted result of Gao. Our full theorem is valid for more general n-sums with n ≥ |G|, in addition to the case n = |G|.2000 Mathematics Subject Classification. 11B75 (20K01).

Abstract-Product Line Architecture (PLA) is the main tangible element shared by all products of a Software Product Line (SPL); it covers common functionality and the required variability of SPL products. Responding to industrial practice, this paper proposes a reactive refactoring bottom-up process to build a PLA from existing similar software product architectures of a domain, expressed by UML logical views. An architecture is represented by a connected graph or valid architectural configuration (P, R), where P and R represent components and connectors of the product. This process constructs a graph (RG) for each product, organized by levels, containing intermediate valid configurations or connected induced sub-graphs of (P, R). A candidate PLA is automatically constructed followed by an optimization process to obtain the PLA using the domain quality model. The refactoring process is applied to a case study in the robotics industry domain. Automatic parts of the process are tool supported.

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