304 research outputs found
The inverse problem for representation functions for general linear forms
The inverse problem for representation functions takes as input a triple
(X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a
function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A
\subseteq X such that there are f(x) solutions (counted appropriately) to
L(x_1,...,x_h) = x for every x \in X, or a proof that no such subset exists.
This paper represents the first systematic study of this problem for
arbitrary linear forms when X = Z, the setting which in many respects is the
most natural one. Having first settled on the "right" way to count
representations, we prove that every primitive form has a unique representation
basis, i.e.: a set A which represents the function f \equiv 1. We also prove
that a partition regular form (i.e.: one for which no non-empty subset of the
coefficients sums to zero) represents any function f for which {f^{-1}(0)} has
zero asymptotic density. These two results answer questions recently posed by
Nathanson.
The inverse problem for partition irregular forms seems to be more
complicated. The simplest example of such a form is x_1 - x_2, and for this
form we provide some partial results. Several remaining open problems are
discussed.Comment: 15 pages, no figure
Limit points in the range of the commuting probability function on finite groups
If G is a finite group, then Pr(G) denotes the fraction of ordered pairs of
elements of G which commute. We show that, if l \in (2/9,1] is a limit point of
the function Pr on finite groups, then l \in \Q and there exists an e = e_l > 0
such that Pr(G) \not\in (l - e_l, l) for any finite group G. These results lend
support to some old conjectures of Keith Joseph.Comment: 11 pages, no figure
Answers to two questions posed by Farhi concerning additive bases
Let A be an asymptotic basis for N and X a finite subset of A such that A\X
is still an asymptotic basis. Farhi recently proved a new batch of upper bounds
for the order of A\X in terms of the order of A and a variety of parameters
related to the set X. He posed two questions concerning possible improvements
to his bounds. In this note, we answer both questions.Comment: 7 pages, no figures. This is v3 : I found a gap in the proof of Lemma
3.2 of v2. This has now been corrected and the same result is Lemma 3.3 in
this versio
Some explicit constructions of sets with more sums than differences
We present a variety of new results on finite sets A of integers for which
the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums
than differences) sets. First we show that there is, up to affine
transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting
from some examples of size 9, we present several new constructions of infinite
families of MSTD sets. Thirdly we show that for every fixed ordered pair of
non-negative integers (j,k), as n -> \infty a positive proportion of the
subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.Comment: 21 pages, no figures. Section 4 has been rewritten and Theorem 8 is a
strengthening of Theorem 9 in previous version. Reference list updated, plus
some other cosmetic change
On the notion of balance in social network analysis
The notion of "balance" is fundamental for sociologists who study social
networks. In formal mathematical terms, it concerns the distribution of triad
configurations in actual networks compared to random networks of the same edge
density. On reading Charles Kadushin's recent book "Understanding Social
Networks", we were struck by the amount of confusion in the presentation of
this concept in the early sections of the book. This confusion seems to lie
behind his flawed analysis of a classical empirical data set, namely the karate
club graph of Zachary. Our goal here is twofold. Firstly, we present the notion
of balance in terms which are logically consistent, but also consistent with
the way sociologists use the term. The main message is that the notion can only
be meaningfully applied to undirected graphs. Secondly, we correct the analysis
of triads in the karate club graph. This results in the interesting observation
that the graph is, in a precise sense, quite "unbalanced". We show that this
lack of balance is characteristic of a wide class of starlike-graphs, and
discuss possible sociological interpretations of this fact, which may be useful
in many other situations.Comment: Version 2: 23 pages, 4 figures. An extra section has been added
towards the end, to help clarify some things. Some other minor change
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