121 research outputs found
A good leaf order on simplicial trees
Using the existence of a good leaf in every simplicial tree, we order the
facets of a simplicial tree in order to find combinatorial information about
the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire
splitting of the ideal, as well as a refinement of a recursive formula of H\`a
and Van Tuyl for computing the graded Betti numbers of simplicial trees.Comment: 17 pages, to appear; Connections Between Algebra and Geometry,
Birkhauser volume (2013
Equation-regular sets and the Fox–Kleitman conjecture
Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero
integer b such that the 2k-variable linear Diophantine equation
∑k
i=1
(xi − yi) = b
is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all
b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for
all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of
the corresponding equation for all b ≥ 1. In particular, this independently confirms the
conjecture for k = 3. We also briefly discuss the case k = 4
On the degree of regularity of a certain quadratic Diophantine equation
We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic
Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic
progressions
Numerical semigroups of Szemerédi type
Given any length k ≥ 3 and density 0 < δ ≤ 1, we introduce and study the set Sz(k, δ)
consisting of all positive integers n such that every subset of {1, 2, . . . , n} of density at
least δ contains an arithmetic progression of length k. A famous theorem of Szemerédi
guarantees that this set is not empty. We show that Sz(k, δ)∪{0} is a numerical semigroup
and we determine it for (k, δ) = (4, 1/2) and for more than thirty pairs (3, δ) with δ > 1/5
Weak Schur numbers and the search for G.W. Walker’s lost partitions
AbstractA set A of integers is weakly sum-free if it contains no three distinct elements x,y,z such that x+y=z. Given k≥1, let WS(k) denote the largest integer n for which {1,…,n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5)=196, without proof. Here we show WS(5)≥196, by constructing a partition of {1,…,196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k=6, we obtain WS(6)≥572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver
On the finiteness of some n-color Rado numbers
For integers k, n, c with k, n ≥ 1, the n-color Rado number Rk(n, c) is defined to be
the least integer N if any, or infinity otherwise, such that for every n-coloring of the set
{1, 2, . . . , N}, there exists a monochromatic solution in that set to the linear equation
x1 + x2 + · · · + xk + c = xk+1.
A recent conjecture of ours states that Rk(n, c) should be finite if and only if every divisor
d ≤ n of k−1 also divides c. In this paper, we complete the verification of this conjecture for
all k ≤ 7. As a key tool, we first prove a general result concerning the degree of regularity
over subsets of Z of some linear Diophantine equations
On the ground states of the Bernasconi model
The ground states of the Bernasconi model are binary +1/-1 sequences of
length N with low autocorrelations. We introduce the notion of perfect
sequences, binary sequences with one-valued off-peak correlations of minimum
amount. If they exist, they are ground states. Using results from the
mathematical theory of cyclic difference sets, we specify all values of N for
which perfect sequences do exist and how to construct them. For other values of
N, we investigate almost perfect sequences, i.e. sequences with two-valued
off-peak correlations of minimum amount. Numerical and analytical results
support the conjecture that almost perfect sequences do exist for all values of
N, but that they are not always ground states. We present a construction for
low-energy configurations that works if N is the product of two odd primes.Comment: 12 pages, LaTeX2e; extended content, added references; submitted to
J.Phys.
Travelling waves in pipe flow
A family of three-dimensional travelling waves for flow through a pipe of
circular cross section is identified. The travelling waves are dominated by
pairs of downstream vortices and streaks. They originate in saddle-node
bifurcations at Reynolds numbers as low as 1250. All states are immediately
unstable. Their dynamical significance is that they provide a skeleton for the
formation of a chaotic saddle that can explain the intermittent transition to
turbulence and the sensitive dependence on initial conditions in this shear
flow.Comment: 4 pages, 5 figure
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