1,321 research outputs found
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 , the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality contained in an additive basis of order at most can be bounded in terms of and alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon . 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 , of estimating the maximal order that a basis of cocardinality contained in an additive basis of order at most can have. Among other results, we prove that, whenever is a translatable semigroup, is for every integer . This result is new even in the case where and is an infinite abelian group. Besides the maximal order , the typical order 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 âwell-orderedâ and ânot so well-orderedâ 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 âwell-orderedâ (more violent) and ânot-so-well-orderedâ (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
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