... norm on R
d
.
In recent years, there has been an increasing interest in the study of the asymptotic behavior of the
solutions of both convolution and non-convolution-type linear and nonlinear Volterra ... aim in this section is to obtain sufficient condition for the boundedness of the solution of (1.1) under
the initial condition (1.2), but in the linea...
... +1)
6
49n
6
.
7. The number of ascendants of a given node in a LBST
As in the case of the number of ascendants in a random BST, computing the probability that
the j
th
node in a random LBST has m ascendants ... other random variables: the number of descendants D
n
and the number of ascendants A
n
of a randomly chosen internal node...
... n)k
−1
φ(k).
In either case, the inequality in part 1 of the lemma holds.
On the number of perfect matchings and Hamilton
cycles in -regular non-bipartite graphs
Alan Frieze
∗
Department of Mathematical ... of combinatorics 7 (2000), #R57 4
3 Hamilton Cycles
A Hamilton cycle is the union of two perfect matchings and so h(G) ≤
1
2
m(G)
2
a...
... in the case when m is prime.
Whether a permutation of
n
admits an m-th root can be read on the partition
of n determined by the lengths of the permutation’s cycles, because the class of such
the ... is the exponential generating function (EGF, formal series) of the
m-thpowersinthegroups
n
. This means that the number of m-thpowersin
n
is
p
n
(m)ìn!foreachn....
... we are not in one of these cases, then there is exactly one edge e attached to v which is
below l.Theedgee is on a loop of the configuration. Let us start with e and follow the
part of the loop ... which is based on a variation of the above reasoning, see
Section 5. Finally, for the proof of the more specific assertions in Conjectures 1.1 and 1.2
on the i...
... x and y to be row and column sums of a matrix
is that they have the same weight.
Clearly, the row sums of a member of M are at most n.Conversely,ifx =(x
1
, ,x
m
)
and 0 x
i
n,letR = R(x)bethem ... (5)
the 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 e...
... (2.6) and (2.8), the number of copies of K
k
in E is
(1 + o(1))
|E|
k
k!
q
−
(
k
2
)
.
.
This implies that the number of the number of k-tuples of k mutually orthogonal
vectors in E is also
(1 ... o(1))
|E|
k
k!
q
−
(
k
2
)
,
completing the proof of Theorem 1.2.
Acknowledgments
The research is performed during the authors visit at the Erwin Schrăodinger I...
... such that the tree together with
the circle it is drawn on can be embedded in a surface of genus one, but not of genus zero.
Hough [3] observed that the number of genus one labeled circle trees ... offspring of C by definition.
Proposition 4. The genus of the reduced form T
3
of a u-c-tree T is one if and only if the
genus of T is one.
Proof. L...
... N
i
denote the number of Eulerian orientations of G
0
which have i valid cross over
boxes, that is N
is the number of Eulerian orientations of G. The number of Eulerian
orientations of G
k
is
EO(G
k
) ... bound for the number of almost alternating Eulerian orientations
yields a lower bound for the number of 2 -orientations of G
k,
.
For the...
... get the numerator of the right-hand side of (4.10). Divide
this number of choices by the number of permutations of {1, . . . , rn} to deduce the lemma.
✷
Using the methods in the proof of Theorem ... :=
r
i=1
A
i
. Let m ∈ min(p, q). Then perm
m
A is the number
of m -matchings of G := ∨
r
i=1
G
i
, which is equal to the number of m -matchings o...
... .
2 The number of matchings in a tree
In t his section, we turn to the number of matchings in a graph. This is a lso known as the
Hosoya index, or the Z-index in mathematical chemistry. For a rooted ... An interesting paper of Wagner [5] looks at the number of independent sets modulo m.
Wagner showed that the pr oportion of trees on n vertices w...
... subsequences with a given sum
in a nite abelian group
Gerard Jennhwa Chang,
123
Sheng-Hua Chen,
13
Yongke Qu,
4
Guoqing Wang,
5Đ
and Haiyan Zha ng
6ả
1
Department of Mathematics, National Taiwan University, ... Wang, A quantitative aspect of non-unique
factorizations: the Narkiewicz constants, International Journal of Number Theory,
to appear.
[13] W.D. Gao and J.T...
... Introduction
Despite great strides in our understanding of the genetic
regulation of germ cell determination in recent years [1],
the size of the founding germ cell population in humans
remains ... supported by the observation
that alkaline phosphatase is present not only in germ cells
of the mouse, but also in somatic cells that surround these
germ...