4,519 research outputs found
On constant-multiple-free sets contained in a random set of integers
For a rational number , a set of positive integers is called an
-multiple-free set if does not contain any solution of the equation .
The extremal problem on estimating the maximum possible size of
-multiple-free sets contained in has been studied for its
own interest in combinatorial number theory and application to coding theory.
Let , be positive integers such that and the greatest common
divisor of and is 1. Wakeham and Wood showed that the maximum size of
-multiple-free sets contained in is .
In this paper we generalize this result as follows. For a real number , let be a set of integers obtained by choosing each element
randomly and independently with probability . We show that the
maximum possible size of -multiple-free sets contained in is
with probability that goes to
1 as .Comment: 9 pages, 1 figure, Abstract was modifie
A unique pure mechanical system revealing dipole repulsion
We study multiple elastic collisions of a block and a ball against a rigid
wall in one dimension. The complete trajectory of the block is solved as an
analytic function of time. Near the turning point of the block the force
carried by the ball is proportional to 1/x^3, where x is the distance between
the wall and the block, in the limit that the block is sufficiently heavier
than the ball. This is a unique pure mechanical system that reveals dipole-like
repulsion.Comment: 20 pages, 3 figures, 1 table, version published in Am. J. Phy
Magic mass ratios of complete energy-momentum transfer in one-dimensional elastic three-body collisions
We consider the one-dimensional scattering of two identical blocks of mass
that exchange energy and momentum via elastic collisions with an
intermediary ball of mass . Initially, one block is incident upon
the ball with the other block at rest. For , the three objects will
make multiple collisions with one another. In our analysis, we construct a
Euclidean vector whose components are proportional to the
velocities of the objects. Energy-momentum conservation then requires a
covariant recurrence relation for that transforms like a pure
rotation in three dimensions. The analytic solutions of the terminal velocities
result in a remarkable prediction for values of , in cases where the
initial energy and momentum of the incident block are completely transferred to
the scattered block. We call these values for "magic mass ratios."Comment: 32 pages, 6 figures, 2 table
The independence number of non-uniform uncrowded hypergraphs and an anti-Ramsey type result
We prove the following: Fix an integer , and let be a real
number with . Let \cH=(V,\cE_2\cup \cE_3\cup\dots\cup\cE_k) be a
non-uniform hypergraph with the vertex set and the set \cE_i of edges of
size . Suppose that \cH has no -cycles (regardless of
sizes of edges), and neither contains -cycles nor -cycles consisting of
-element edges. If the average degrees t_i^{i-1} := i |\cE_i|/ |V| satisfy
that for ,
then there exists a constant , depending only on , such that
\alpha(\cH)\geq C_k \frac{|V|}{T} (\ln T)^{\frac{1}{k-1}}, where
\alpha(\cH) denotes the independence number of \cH. This extends results of
Ajtai, Koml\'os, Pintz, Spencer and Szemer\'edi and Duke, R\"odl and the second
author for uniform hypergraphs.
As an application, we consider an anti-Ramsey type problem on non-uniform
hypergraphs. Let \cH=\cH(n;2,\ldots,\ell) be the hypergraph on the -vertex
set in which, for , each -subset of is a hyperedge
of \cH. Let be an edge-coloring of \cH satisfying the following:
(a) two hyperedges sharing a vertex have different colors; (b) two hyperedges
with distinct size have different colors; (c) a color used for a hyperedge of
size appears at most times. For such a coloring , let
be the maximum size of a subset of
such that each hyperedge of \cH[U] has a distinct color, and let
We
determine up to a multiplicative logarithm factor.Comment: 17 page
Higher order operator splitting Fourier spectral methods for the Allen-Cahn equation
The Allen-Cahn equation is solved numerically by operator splitting Fourier
spectral methods. The basic idea of the operator splitting method is to
decompose the original problem into sub-equations and compose the approximate
solution of the original equation using the solutions of the subproblems.
Unlike the first and the second order methods, each of the heat and the
free-energy evolution operators has at least one backward evaluation in higher
order methods. We investigate the effect of negative time steps on a general
form of third order schemes and suggest three third order methods for better
stability and accuracy. Two fourth order methods are also presented. The
traveling wave solution and a spinodal decomposition problem are used to
demonstrate numerical properties and the order of convergence of the proposed
methods
Towards extending the Ahlswede-Khachatrian theorem to cross t-intersecting families
Ahlswede and Khachatrian's diametric theorem is a weighted version of their
complete intersection theorem, itself an extension of the -intersecting
Erd\H{o}s-Ko-Rado theorem. Their intersection theorem says that the maximum
size of a family of subsets of , every pair of which
intersects in at least elements, is the size of certain trivially
intersecting families proposed by Frankl. We address a cross intersecting
version of their diametric theorem.
Two families and of subsets of are {\em
cross -intersecting} if for every and , and intersect in at least elements. The -weight of
a element subset of is , and the weight of a
family is the sum of the weights of its sets. The weight of a
pair of families is the product of the weights of the families.
The maximum -weight of a -intersecting family depends on the value of
. Ahlswede and Khachatrian showed that for in the range , the maximum -weight of a -intersecting
family is that of the family consisting of all subsets of
containing at least elements of the set .
In a previous paper we showed a cross -intersecting version of this for
large in the case that . In this paper, we do the same in the case
that . We show that for in the range the maximum -weight of a cross -intersecting pair of families, for
, is achieved when both families are . Further, we
show that except at the endpoints of this range, this is, up to isomorphism,
the only pair of -intersecting families achieving this weight.Comment: 22 pages (18 plus appendix), 3 figure
On the Heterogeneous Distributions in Paper Citations
Academic papers have been the protagonists in disseminating expertise.
Naturally, paper citation pattern analysis is an efficient and essential way of
investigating the knowledge structure of science and technology. For decades,
it has been observed that citation of scientific literature follows a
heterogeneous and heavy-tailed distribution, and many of them suggest a
power-law distribution, log-normal distribution, and related distributions.
However, many studies are limited to small-scale approaches; therefore, it is
hard to generalize. To overcome this problem, we investigate 21 years of
citation evolution through a systematic analysis of the entire citation history
of 42,423,644 scientific literatures published from 1996 to 2016 and contained
in SCOPUS. We tested six candidate distributions for the scientific literature
in three distinct levels of Scimago Journal & Country Rank (SJR) classification
scheme. First, we observe that the raw number of annual citation acquisitions
tends to follow the log-normal distribution for all disciplines, except for the
first year of the publication. We also find significant disparity between the
yearly acquired citation number among the journals, which suggests that it is
essential to remove the citation surplus inherited from the prestige of the
journals. Our simple method for separating the citation preference of an
individual article from the inherited citation of the journals reveals an
unexpected regularity in the normalized annual acquisitions of citations across
the entire field of science. Specifically, the normalized annual citation
acquisitions have power-law probability distributions with an exponential
cut-off of the exponents around 2.3, regardless of its publication and citation
year. Our results imply that journal reputation has a substantial long-term
impact on the citation.Comment: 8 pages, 7 figure
On the total variation distance between the binomial random graph and the random intersection graph
When each vertex is assigned a set, the intersection graph generated by the
sets is the graph in which two distinct vertices are joined by an edge if and
only if their assigned sets have a nonempty intersection. An interval graph is
an intersection graph generated by intervals in the real line. A chordal graph
can be considered as an intersection graph generated by subtrees of a tree. In
1999, Karo\'nski, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999),
131--159] introduced a random intersection graph by taking randomly assigned
sets. The random intersection graph has vertices and sets
assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set
of size where each element of belongs to each random subset with
probability , independently of all other elements in . Fill, Scheinerman
and Singer-Cohen [Random Struct Algorithms 16 (2000), 156--176] showed that the
total variation distance between the random graph and the
Erd\"os-R\'enyi graph tends to for any if , , where is chosen so that the
expected numbers of edges in the two graphs are the same. In this paper, it is
proved that the total variation distance still tends to for any whenever .Comment: revised version of the 1st draft "On a phase transition of the random
intersection graph: Supercritical region
Dynamic coloring of graphs having no minor
We prove that every simple connected graph with no minor admits a
proper 4-coloring such that the neighborhood of each vertex having more
than one neighbor is not monochromatic, unless the graph is isomorphic to the
cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and
W.-J. Park on planar graphs.Comment: Rewriting with a major change (14 pages, 1 figure
A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions
We present a new integral equation method for the calculation of
two-dimensional scattering from periodic structures involving triple-points
(multiple materials meeting at a single point). The combination of a robust and
high-order accurate integral representation and a fast direct solver permits
the efficient simulation of scattering from fixed structures at multiple angles
of incidence. We demonstrate the performance of the scheme with several
numerical examples.Comment: 19 Pages. 8 Figure
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