Remarks on the plus-minus weighted Davenport constant

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsf{D}_{\pm}(G)$, is the smallest $\ell$ such that each sequence $g_1 ... g_{\ell}$ over $G$ has a weighted zero-subsum with weights +1 and -1, i.e., there is a non-empty subset $I \subset \{1,..., \ell\}$ such that $\sum_{i \in I} a_i g_i =0$ for $a_i \in \{+1,-1\}$. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of groups

Representation of Finite Abelian Group Elements by Subsequence Sums

Let $G\cong C_{n_1}\oplus ... \oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1... w_n$ is a sequence of integers, all but at most one relatively prime to $|G|$, and $S$ is a sequence over $G$ with $|S|\geq |W|+|G|-1\geq |G|+1$, the maximum multiplicity of $S$ at most $|W|$, and $\sigma(W)\equiv 0\mod |G|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\sum_{i=1}^{n}w_is_i$, with $s_1... s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples for large $|W|\geq {1/2}|G|$. 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 $|S|\geq |G|+D(G)-1$, 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 $|S|\geq |G|+D(G)-1$ can be relaxed to $|S|\geq |G|+d^*(G)$, where d^*(G)=\Sum_{i=1}^{r}(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 $|G|$

Highlights in toxicology

Every year our cooperating journal, the Archives of Toxicology, publishes and analyzes its most cited articles. In 2009/2010 the most popular articles focussed on ethanol-induced liver damage, tea polyphenols as anti-carcinogens and concepts of dose-response modelling. To keep our readers informed about recent developments in toxicology we reproduce a table summarizing the take home messages of the most cited articles (Table; from: Bolt and Hengstler, 2011)

Inverse results for weighted Harborth constants

For a finite abelian group $(G,+)$ the Harborth constant is defined as the smallest integer $\ell$ such that each squarefree sequence over $G$ of length $\ell$ has a subsequence of length equal to the exponent of $G$ whose terms sum to $0$. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling $0$ is required, that is, when forming the sum one can chose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constant for certain groups, in particular for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erd\H{o}s--Ginzburg--Ziv constant

ATPase Domain of Heat Shock protein 70—isoform 2—(Hsp70-2) and their role in activating the adaptive immune response: An in silico approach

A lot of molecules play fundamental roles in neoplasic processes and cancer. Heat shock proteins may enhance this severe pathology and favor the protumoral milieu. However, a look at the literature tells us that these molecules intervene in both to promote or attack cancer cells. In the case of breast cancer is known that Hsp70 (isoform 2) improve this establishment and progression in the patient, and is possible that the ATPase domain of Hsp70-2 favors this disease. Thus, is relevant to know if this molecular region has immunogenic activity as well as which epitopes are essential to stimulate immune cells, and whether could induce the attack of the tumor mass. In this aim, the immunogenicity of ATPase domain of Hsp70-2 was studied in silico. The results suggest that the majority of the molecule had immunogenic epitopes that boosts the immune response through activation of B cells and T cells. However, in vitro synthesis and in vivo experimental studies to evaluate the efficacy of this therapeutic candidate are required to ensure efficacy and safety in people.