... equality holds in (1), then for any independent set I of cardinality α(Γ
2
)
in Γ
2
, φ
−1
(I) is an independent set of cardinality α(Γ
1
) in Γ
1
.
2 Intersecting families in the alternating group
Throughout, ... maximal size in S
n
are the cosets of point
stabilizers. We show that, under mild restrictions, analogous results hold for the
alternating group and...
... concerning the exclusive use of Chinese concerned the
building and maintenance of a Sinophone identity. The exclusive use of Chinese was
regarded as sign of a pure Sinophone and mixing-codes ... minority groups in the People’s Republic of China (Hu and
Alsagoff 2010; Zhou and Sun 2004) and Taiwan (Chen and Hsin 2011); in post-colonial
contexts such as Sin...
... proof of Theorem 2.
2.2 Proof of Theorem 3
Here we assume that s
2
≥
2
n,thecases
1
≥
2
nis similar. Denote by P
r
the product of
the primes not exceeding r. The rough outline of the proof will ... denote the greatest common divisor and [a, b] the least common multiple of integers
a, b. Consider the coprime graph on the integers. This is the graph whose ver...
... only at 0 and 1. Furthermore, if ρ =0,then
φ is the characteristic function of the interval (0, 1].
(ii) Let X
n
be a random variable equal to the sum of the fixed points in an element of
G
n
\D
n
chosen ... n} whose fixed points sum to r, shows a sharp discontinuity in the neigh-
borhood of r = n. We explain this discontinuity and study the possible existence
of o...
... consequence of Lemmas 4.1, 4 .2 , 4.3, 4.4 while the second line
follows fr om Lemma 4.5.
Lemma 4.6 proves item (vi) of Theorem 2.1 thereby completing the proof of the whole
theorem.
Given the great ... 2}.
The first string of +3’s must exhaust the 0-congruence class and therefore we reach
up to the vertex 3k + 3. The second string of −3’s will describe another cong...
... constraint in
the structure of trees or indeed all higher plants.
If we seek meaning in the list as a whole there
remains the task of “seeing the wood for the trees”,
selecting and arranging them into ... section of each type and
the attachment and relaying will be terminal. If on the
other hand the axis is indeterminate there will often be
several sections o...
... k
1
+ ···+ k
m
= n.
In order to prepare the proof of Theorem 5, let introduce the notation of rays in
generalized Pascal pyramids. As usual, a lattice ray in the domain D
p
of the function
p is ... his/her helpful
comments.
the electronic journal of combinatorics 18 (2011), #P79 8
2 Results and proofs
Using the notation above, the principal result of this work...