9,475 research outputs found
Davenport constant with weights
For the cyclic group and any non-empty
. We define the Davenport constant of with weight ,
denoted by , to be the least natural number such that for any
sequence with , there exists a non-empty
subsequence and such that
. Similarly, we define the constant to be
the least such that for all sequences with
, there exist indices , and with . In the present paper, we show that
. This solve the problem raised by Adhikari and Rath
\cite{ar06}, Adhikari and Chen \cite{ac08}, Thangadurai \cite{th07} and
Griffiths \cite{gr08}.Comment: 6page
Three topics in additive prime number theory
This is an expository article to accompany my two lectures at the CDM
conference. I have used this an excuse to make public two sets of notes I had
lying around, and also to put together a short reader's guide to some recent
joint work with T.Tao. Contents: 1. An exposition, without much detail, of the
work of Goldston, Pintz and Yildirim on gaps between primes; 2. A detailed
discussion of the work of Mauduit and Rivat establishing that 50 percent of the
primes have odd digit sum when written in base 2; 3. A reader's guide to recent
work of T.Tao and the author on linear equations in primes.
The sections can be read independently.Comment: 40 pages, notes to accompany my lectures at the Current Developments
in Mathematics Conference, Harvard, 16th-17th November 200
Arithmetic-Progression-Weighted Subsequence Sums
Let be an abelian group, let be a sequence of terms
not all contained in a coset of a proper subgroup of
, and let be a sequence of consecutive integers. Let
which is a particular kind of weighted restricted sumset. We show that , that if , and also
characterize all sequences of length with . This
result then allows us to characterize when a linear equation
where are
given, has a solution modulo with all
distinct modulo . As a second simple corollary, we also show that there are
maximal length minimal zero-sum sequences over a rank 2 finite abelian group
(where and ) having
distinct terms, for any . Indeed, apart from
a few simple restrictions, any pattern of multiplicities is realizable for such
a maximal length minimal zero-sum sequence
Davenport constant for semigroups II
Let be a finite commutative semigroup. The Davenport constant
of , denoted , is defined to be the least
positive integer such that every sequence of elements in
of length at least contains a proper subsequence
() with the sum of all terms from equaling the sum of all terms
from . Let be a prime power, and let \F_q[x] be the ring of
polynomials over the finite field \F_q. Let be a quotient ring of
\F_q[x] with 0\neq R\neq \F_q[x]. We prove that where denotes
the multiplicative semigroup of the ring , and denotes
the group of units in .Comment: In press in Journal of Number Theory. arXiv admin note: text overlap
with arXiv:1409.1313 by other author
On weighted zero-sum sequences
Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A
be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest
positive integer , denoted by s_A(G), such that any sequence {c_i}_{i=1}^m
with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which
there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0.
When G is a p-group, A contains no multiples of p and any two distinct
elements of A are incongruent mod p, we show that s_A(G) is at most if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is
the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is
essentially best possible.
In the case A={1,-1}, we determine the asymptotic behavior of s_{{1,-1}}(G)
when exp(G) is even, showing that, for finite abelian groups of even exponent
and fixed rank, s_{{1,-1}}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends
to the infinity. Combined with a lower bound of
, where with 1<n_1|... |n_r, this determines s_{{1,-1}}(G), for even exponent
groups, up to a small order error term. Our method makes use of the theory of
L-intersecting set systems.
Some additional more specific values and results related to s_{{1,-1}}(G) are
also computed.Comment: 24 pages. Accepted version for publication in Adv. in Appl. Mat
Representation of Finite Abelian Group Elements by Subsequence Sums
Let be a finite and nontrivial
abelian group with . A conjecture of Hamidoune says that if
is a sequence of integers, all but at most one relatively prime
to , and is a sequence over with ,
the maximum multiplicity of at most , and ,
then there exists a nontrivial subgroup such that every element
can be represented as a weighted subsequence sum of the form
, with a subsequence of . We give two
examples showing this does not hold in general, and characterize the
counterexamples for large .
A theorem of Gao, generalizing an older result of Olson, says that if is
a finite abelian group, and is a sequence over with , then either every element of can be represented as a
-term subsequence sum from , or there exists a coset such that
all but at most terms of are from . 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 can be relaxed to , 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 of the are
relatively prime to
Childhood Mortality & Nutritional Status as Indicators of Standard of Living: Evidence from World War I Recruits in the United States
This paper examines variations in stature and the Body Mass Index (BMI) across space for the United States in 1917/18, using published data on the measurement of approximately 890,000 recruits for the American Army for World War I. It also connects those anthropometric measurements with an index of childhood mortality estimated from the censuses of 1900 and 1910. This index is taken to be an indicator of early childhood environment for these recruits. Aggregated data were published for states and groups of counties by the Surgeon General after the war. These data are related to regional data taken primarily from the censuses of 1900 and 1910. The results indicate that early childhood mortality was a good (negative) predictor of height and the body mass index, while it is also possible to predict early childhood experience from terminal adult height. Urbanization was important, although the importance declined over time. Income apparently had little effect on health in this period.
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