266,896 research outputs found
Decompositions of edge-colored infinite complete graphs into monochromatic paths
An -edge coloring of a graph or hypergraph is a map . Extending results of Rado and answering questions of Rado,
Gy\'arf\'as and S\'ark\"ozy we prove that
(1.) the vertex set of every -edge colored countably infinite complete
-uniform hypergraph can be partitioned into monochromatic tight paths
with distinct colors (a tight path in a -uniform hypergraph is a sequence of
distinct vertices such that every set of consecutive vertices forms an
edge),
(2.) for all natural numbers and there is a natural number such
that the vertex set of every -edge colored countably infinite complete graph
can be partitioned into monochromatic powers of paths apart from a
finite set (a power of a path is a sequence of
distinct vertices such that implies that is an
edge),
(3.) the vertex set of every -edge colored countably infinite complete
graph can be partitioned into monochromatic squares of paths, but not
necessarily into ,
(4.) the vertex set of every -edge colored complete graph on
can be partitioned into monochromatic paths with distinct colors
The adjacency matrix of one type of graph and the Fibonacci numbers
Recently there is huge interest in graph theory and intensive study on
computing integer powers of matrices. In this paper, we investigate
relationships between one type of graph and well-known Fibonacci sequence. In
this content, we consider the adjacency matrix of one type of graph with 2k
(k=1,2,...) vertices. It is also known that for any positive integer r, the
(i,j)th entry of A^{r} (A is the adjacency matrix of the graph) is just the
number of walks from vertex i to vertex j, that use exactly k edges
Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence
A bijection is presented between (1): partitions with conditions
and , where is the frequency of the
part in the partition, and (2): sets of ordered partitions such that
and ,
where is the number of parts in . This bijection entails an
elementary and constructive proof of the Andrews multiple-sum enumerating
partitions with frequency conditions. A very natural relation between the
ordered partitions and restricted paths is also presented, which reveals our
bijection to be a modification of Bressoud's version of the Burge
correspondence.Comment: 12 pages; minor corrections, version to appear in Discrete Mat
Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata
Define a certain gambler's ruin process \mathbf{X}_{j}, \mbox{ \ }j\ge 0,
such that the increments
take values and satisfy ,
all , where if , and if .
Here denote persistence parameters and with
. The process starts at and terminates when
. Denote by , , and ,
respectively, the numbers of runs, long runs, and steps in the meander portion
of the gambler's ruin process. Define and let for some . We show exists in an explicit form. We obtain a
companion theorem for the last visit portion of the gambler's ruin.Comment: Presented at 8th International Conference on Lattice Path
Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been
streamlined, with references added, including reference to a companion
document with details of calculations via Mathematica. The 3rd version has 2
new figures and improved presentatio
Single hole dynamics in the t-J model on a square lattice
We present quantum Monte Carlo (QMC) simulations for a single hole in a t-J
model from J=0.4t to J=4t on square lattices with up to 24 x 24 sites. The
lower edge of the spectrum is directly extracted from the imaginary time
Green's function. In agreement with earlier calculations, we find flat bands
around , and the minimum of the dispersion at
. For small J both self-consistent Born approximation and
series expansions give a bandwidth for the lower edge of the spectrum in
agreement with the simulations, whereas for J/t > 1, only series expansions
agree quantitatively with our QMC results. This band corresponds to a coherent
quasiparticle. This is shown by a finite size scaling of the quasiparticle
weight that leads to a finite result in the thermodynamic limit for
the considered values of . The spectral function is
obtained from the imaginary time Green's function via the maximum entropy
method. Resonances above the lowest edge of the spectrum are identified, whose
J-dependence is quantitatively described by string excitations up to J/t=2
Statistics of Partial Minima
Motivated by multi-objective optimization, we study extrema of a set of N
points independently distributed inside the d-dimensional hypercube. A point in
this set is k-dominated by another point when at least k of its coordinates are
larger, and is a k-minimum if it is not k-dominated by any other point. We
obtain statistical properties of these partial minima using exact probabilistic
methods and heuristic scaling techniques. The average number of partial minima,
A, decays algebraically with the total number of points, A ~ N^{-(d-k)/k}, when
1<=k<d. Interestingly, there are k-1 distinct scaling laws characterizing the
largest coordinates as the distribution P(y_j) of the jth largest coordinate,
y_j, decays algebraically, P(y_j) ~ (y_j)^{-alpha_j-1}, with
alpha_j=j(d-k)/(k-j) for 1<=j<=k-1. The average number of partial minima grows
logarithmically, A ~ [1/(d-1)!](ln N)^{d-1}, when k=d. The full distribution of
the number of minima is obtained in closed form in two-dimensions.Comment: 6 pages, 1 figur
On the uniform generation of modular diagrams
In this paper we present an algorithm that generates -noncrossing,
-modular diagrams with uniform probability. A diagram is a labeled
graph of degree over vertices drawn in a horizontal line with arcs
in the upper half-plane. A -crossing in a diagram is a set of
distinct arcs with the property . A diagram without any
-crossings is called a -noncrossing diagram and a stack of length
is a maximal sequence
. A diagram is
-modular if any arc is contained in a stack of length at least
. Our algorithm generates after preprocessing time,
-noncrossing, -modular diagrams in time and space
complexity.Comment: 21 pages, 7 figure
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