Inverse zero-sum problems and algebraic invariants

In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of rank two.Comment: 17 pages, to appear in Acta Arithmetic

On a combinatorial problem of Erdos, Kleitman and Lemke

In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Abelian groups. Building among others on earlier works by Alon and Dubiner and by the author, our main theorem gives a new upper bound for this invariant in the general case, and provides its right order of magnitude.Comment: 15 page

On the existence of zero-sum subsequences of distinct lengths

In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland and Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao's conjecture is solved using Alon's Combinatorial Nullstellensatz. To conclude, we show that, assuming every integer satisfies Property B, this conjecture holds in the case of finite Abelian groups of rank two.Comment: 10 pages, to appear in Rocky Mountain Journal of Mathematic

Inverse zero-sum problems in finite Abelian p-groups

In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, the method that we use here enables us to show that, if we denote by exp(G) the exponent of the finite Abelian p-group G which is considered, then a zero-sumfree sequence S with maximal possible length in G must contain at least exp(G)-1 elements of maximal order, which improves a previous result of W. Gao and A. Geroldinger.Comment: 13 pages, submitte

Local optical field variation in the neighborhood of a semiconductor micrograting

The local optical field of a semiconductor micrograting (GaAs, 10x10 micro m) is recorded in the middle field region using an optical scanning probe in collection mode at constant height. The recorded image shows the micro-grating with high contrast and a displaced diffraction image. The finite penetration depth of the light leads to a reduced edge resolution in the direction to the illuminating beam direction while the edge contrast in perpendicular direction remains high (~100nm). We use the discrete dipole model to calculate the local optical field to show how the displacement of the diffraction image increases with increasing distance from the surface.Comment: 12 pages, 3 figure

On additive bases in infinite abelian semigroups

27 pagesIn this paper, building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in a class of infinite abelian semigroups, which we term \em translatable \em semigroups. These include all numerical semigroups as well as all infinite abelian groups. We show that, for every such semigroup $T$, the number of essential subsets of any additive basis is finite, and also that the number $E_T(h, k)$ of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon ${\mathbf{N}}$. Also, using invariant means, we address a classical problem, initiated by Erd\H{o}s and Graham and then generalized by Nash and Nathanson both in the case of ${\mathbf{N}}$, of estimating the maximal order $X_T(h, k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that, whenever $T$ is a translatable semigroup, $X_T(h, k)$ is $O(h^{2k+1})$ for every integer $k \ge 1$. This result is new even in the case where $k = 1$ and $T$ is an infinite abelian group. Besides the maximal order $X_T(h, k)$, the typical order $S_T(h, k)$ is also studied

Law in other contexts: stand bravely brothers! a report from the law wars

This essay argues against the two pillars of current research on law and globalisation, from the perspective of legal theory and political philosophy: first, the distinction between ‘well-ordered’ and ‘not so well-ordered’ societies; second, the sociological model of the subject as pacified, fearful and isolated (to sum up, in harmony). It is argued that mainstream legal theory and political philosophy merely reflects the actual rules of the game of competition, dispute and conflict. In contrast, this essay takes sides with the anthropological and philosophical tradition that conceives the subject as antagonistic and in state of lack, profoundly concerned with the other, whom she imitates and whose standpoint she must be able to share if she is to make sense of the world. Furthermore, it is argued that transitivity or imitation lies at the very origin of conflict and dispute; lack and antagonism remain thus at the core of society, in spite of the surface appearance of harmony that characterises post-modern societies. Because of this, any general theory of law and society that wishes to be relevant at the time of globalisation must make the experience of antagonism and violence, motivated by imitation and envy, and its containment, its object of study. To do this, it must abandon the dualist conception of subjects and societies expressed in the distinction between ‘well-ordered’ (more violent) and ‘not-so-well-ordered’ (less violent) societies that has informed its investigation to this day, in order to declare in the most general terms a critique of violence from the standpoint of the victim, as of a piece with its demand for global social and political justice. Description from publisher website at: http://journals.cambridge.org/action/displayIssue?jid=IJC&volumeId=4&issueId=02&iid=243936

A nullstellensatz for sequences over F_p

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.Comment: 23 page

k-Sums in abelian groups

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.Comment: 15 page