... G
∗
∼
=
P
D+1
and for some k ∈ [2, D − 1], the vertices u
∗
k−1
and u
∗
k+1
of G
∗
are both not of type (1). Then u
∗
k−1
and u
∗
k+1
are both of type (N).
Proof. Let x and y be twins of u
k−1
and u
k+1
respectively. ... of
type (1) are adjacent, and one of them is a leaf of type (K), then the other is also of type
(K).
Proof. As illustrated in Figure 12, le t x a...
... from which Corollary 1 follows. ✷
Discrepancy of Matrices of
Zeros and Ones
Richard A. Brualdi
∗
and Jian Shen
†
Department of Mathematics
University of Wisconsin
Madison, Wisconsin 53706
brualdi@math.wisc.edu ... each i =1, 2, and
2. The intersection of each row of A with region i is connected for each i =1, 2,
and define, for each A ∈A(R, S), the discrepancy d(A)ofA t...
... properties of nonnegative matrices and their permanents, Soviet
Math. Dokl. 14:945-949 (1973).
[4] R. A. Brualdi, J. L. Goldwasser and T. S. Michael, Maximum permanents of matrices
of zeroes and ones, ... moments, (6).
Maximising the permanent of (0,1) -matrices and
the number of extensions of Latin rectangles
B. D. McKay and I. M. Wanless
Department of Computer...
... ∈ X
d
,sof is a bijection. For all x ∈ X
d
\ D the f–orbit
O
f
(x)ofx has order 2 and consists of x and f(x). Further we have
{x
i
|i ∈ [d]} = {f(x)
i
|i ∈ [d]},
the electronic journal of combinat ... fixed d.
Our proof of the lower bound is a variation of the Fourier transform method (in the
literature also called circle–method). The novelty of our proof is the application of...
... polynomial in x and y of degree ≤ deg f +1.
By induction and (4) it follows that the coefficients of G are rational numbers. This
proves the lemma.
5 End of the proof
Since G is a polynomial of degree ... electronic journal of combinatorics 13 (2006), #N8 5
3 Start of the proof
Let N = {0, 1, }. Define the weight of a matrix N to be the sum of its entries, and
write |N| for...
... point
of X either red or blue, in such a way that any of the sets of F has roughly the same
number of red points and blue points. The maximum deviation from an even splitting,
over all sets of F, ... elements of X as
x
1
,x
2
, ,x
n
and the sets of F as S
1
,S
2
, ,S
m
in some arbitrary order. The incidence
matrix of (X, F)isthem × n matrix A, with columns corresponding to...
... used in the proofs of Theorem 8 and Theorem 10
are different.
References
[1] J. Beck and V. T. S´os, Discrepancy theory, in R. Graham, M. Gr¨otschel, and
L. Lov´asz, Editors, Handbook of Combinatorics, ... respect to χ.
Proof of Lemma 7. The proof is based on an argument from Ramsey theory. First we
verify the statement of Lemma 7 for a fixed simplex. Then, by induction over the n...
... recall the definition of the
density matrices of a graph and define the tensor product of three graphs, reconsider the
tripartite entanglement properties of the density matrices of graphs introduced ... for separability of density matrices of nearest point graphs and perfect
the electronic journal of combinatorics 14 (2007), #R40 1
where I
n
and J
n
is the n ×n ident...
... then fo llows from Theorem 3 and ess sup
0x1
|g(x)| = D
∗
N
(P).
3.5 The proof of Theorem 5
Proof. We go back to our definition of D
P
N
as the expected value of the random variable
# {x
i
: ... The proof of Theorem 4
Proof. We start with the lower bound. Taking a random interval could be simulated as
follows: we start by t aking any random point α in [0, 1] and then take...
... analysis, and writing of
the manuscript. AW and SJ participated in the conception and design of the study and
revised the manuscript for important intellectual content. All authors read and
approved ... Watcharapasorn
1,2
, and Sukanda Jiansirisomboon*
1,2
1
Department of Physics and Materials Science, Faculty of Science, Chiang Mai
University, Chiang Mai, 50200,...