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1.
The Hamilton-Waterloo problem asks for a 2-factorisation of K
v
in which r of the 2-factors consist of cycles of lengths a
1, a
2,…, a
t
and the remaining s 2-factors consist of cycles of lengths b
1, b
2,…, b
u
(where necessarily ∑
i=1
t
a
i
=∑
j=1
u
b
j
=v). In this paper we consider the Hamilton-Waterloo problem in the case a
i
= m, 1≤ i≤ t and b
j
= n, 1≤ j≤ u. We obtain some general constructions, and apply these to obtain results for ( m, n)∈{(4,6),(4,8),(4,16),(8,16),(3,5),(3,15),(5,15)}.
Received: July 5, 2000 相似文献
2.
We prove that a function F of the Selberg class ℐ is a b-th power in ℐ, i.e., F=H
b for some Hσ ℐ, if and only if b divides the order of every zero of F and of every p-component F
p. This implies that the equation F
a=G b with ( a, b)=1 has the unique solution F=H
b and G=H
a in ℐ. As a consequence, we prove that if F and G are distinct primitive elements of ℐ, then the transcendence degree of ℂ[ F,G] over ℂ is two. 相似文献
3.
Let K be a function field in one variable over ℂ and a
1,..., a
m
, b non-zero elements of K, such that b is linearly independent from a
1,..., a
m
over ℂ. We show that for n sufficiently large, the equation ∑
i=1
m
a
i
x
i
n
has no non-constant solutions in K. 相似文献
4.
In this paper, we findall metacyclic groups (a,b:a m=e,b s=e,b -1ab=a r), where m=10,14,15,20,21,22, such that the cusp forms associated with all elements of these groups by an exact representation are multiplicative-products. We also consider the correspondence between multiplicative -products and elements of finite order in SL(5,C) by the adjoint representation. 相似文献
5.
In this paper, we study some properties of semigroups with presentation 〈 a, b ; a
p
= b
r
, a
q
= b
s
〉. We will also study their potential as platforms for the Diffie-Hellman key exchange protocol. 相似文献
6.
For all integers m3 and all natural numbers a1, a2,…, am−1, let n= R( a1, a2,…, am−1) represent the least integer such that for every 2-coloring of the set {1,2,…, n} there exists a monochromatic solution to | Let t=min{a1,a2,…,am−1} and b=a1+a2++am−1−t. In this paper it is shown that whenever t=2, R(a1,a2,…,am−1)=2b2+9b+8.
It is also shown that for all values of t, R(a1,a2,…,am−1)tb2+(2t2+1)b+t3.
相似文献
7.
A survey of solvability conditions for the embedding problem of number fields, in which the kernel is a non-Abelian group
of order p
4, is completed. As a kernel, the two 2-groups with two generators a, b and with the following relations are considered: a
8
=
1, b
2
=
1, [a,b]=a
−2
in the first group, and a
8
=
1, b
2
=a
4
, [a,b]=a
−2
in the second. Bibliography: 7 titles.
Translated from
Zapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 127–132.
Translated by V. V. Ishkhanov.
相似文献
8.
Let
G be a finite solvable group with {1,
a,
b,
c,
ab,
ac} as the character degree set, where
a ,
b, and
c are pairwise coprime integers greater than 1. We show that the derived length of
G is at most 4. This verifies that the Taketa inequality, dl(
G) ≤ |cd(
G)|, is valid for solvable groups with {1,
a,
b,
c,
ab,
ac} as the character degree set. Also, as a corollary, we conclude that if
a,
b,
c, and
d are pairwise coprime integers greater than 1 and
G is a solvable group such that cd(
G) = {1,
a,
b,
c,
d,
ac,
ad,
bc,
bd}, then dl(
G) ≤ 5. Finally, we construct a family of solvable groups whose derived lengths are 4 and character degree sets are in the
form {1,
p,
b,
pb,
q
p
,
pq
p
}, where
p is a prime,
q is a prime power of an odd prime, and
b > 1 is integer such that
p,
q, and
b are pairwise coprime. Hence, the bound 4 is the best bound for the derived length of solvable groups whose character degree
set is in the form {1,
a,
b,
c,
ab,
ac} for some pairwise coprime integers
a,
b, and
c.
相似文献
9.
In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2+b^2=c^r,a〉b,a≡3 (mod4),b≡2 (mod4) and c-1 is not a square, thena a^x+b^y=c^z has only the positive integer solution (x, y, z) = (2, 2, r).
Let m and r be positive integers with 2|m and 2 r, define the integers Ur, Vr by (m +√-1)^r=Vr+Ur√-1. If a = |Ur|,b=|Vr|,c = m^2+1 with m ≡ 2 (mod 4),a ≡ 3 (mod 4), and if r 〈 m/√1.5log3(m^2+1)-1, then a^x + b^y = c^z has only the positive integer solution (x,y, z) = (2, 2, r). The argument here is elementary.
相似文献
11.
In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup
from a semilattice. Finally, we give two examples of the closure of
B
( −∞, ∞ )1, a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation
a
∨
b=sup{
a,
b}.
The author would like to thank to the referee for useful suggestions.
相似文献
12.
We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium
N*=1/(
a+∑
i=0mbi) of the following differential equation with piecewise constant arguments:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, ∑
i=0mbi>0,
bi0,
i=0,1,2,…,
m, and
a+∑
i=0mbi>0. These new conditions depend on
a,
b0 and ∑
i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case
m=0 and
r(
t)≡
r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, 1−
ax−
g(
x,
x,…,
x)=0 has a unique solution
x*>0 and
g(
x0,
x1,…,
xm)
C1[(0,+∞)×(0,+∞)××(0,+∞)].
相似文献
13.
For integers
a, b and
c, the group
F
a,b,−c is defined to be the group 〈
R, S : R
2=
RS
aRS
bRS
−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of order
p
n (for certain
p and
n) as groups of type
F
a,b,−c for certain (not unique) choices of
a, b and
c.
相似文献
14.
Let
a,
b,
m, and
t be integers such that 1≤
a<
b and 1≤
t≤⌉(
b−
m+1)/
a⌉. Suppose that
G is a graph of order |
G| and
H is any subgraph of
G with the size |
E(
H)|=
m. Then we prove that
G has an [
a,
b]-factor containing all the edges of
H if the minimum degree is at least
a, |
G|>((
a+
b)(
t(
a+
b−1)−1)+2
m)/
b, and |
N
G
(
x
1)∪⋯ ∪
N
G
(
x
t
)|≥(
a|
G|+2
m)/(
a+
b) for every independent set {
x
1,…,
x
t
}⊆
V(
G). This result is best possible in some sense and it is an extension of the result of H. Matsuda (A neighborhood condition
for graphs to have [
a,
b]-factors, Discrete Mathematics
224 (2000) 289–292).
Received: October, 2001 Final version received: September 17, 2002
RID="*"
ID="*" This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement
of Young Scientists, 13740084, 2001
相似文献
15.
A new generalized Radon transform
R
α, β
on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform,
the generalized biaxially symmetric potential operator Δ
α, β
, and the Jacobi polynomials
Pk(b, a)(
t)P_{k}^{(\beta,\,\alpha)}(t). The transform
R
α, β
and its dual
Ra, b*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for
R
α, β
for functions in
La, bp(\mathbb
R2+)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform
R
α, β
is used to represent the solutions of the partial differential equations
Lu:=?
j=1majD
a, bju=
fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients
a
j
and the Cauchy problem for the generalized wave equation associated with the operator Δ
α, β
. Another application is that, by an invariant property of
R
α, β
, a new product formula for the Jacobi polynomials of the type
Pk(b, a)(
s)
C2ka+b+1(
t)=
còò
Pk(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.
相似文献
16.
Let
a,
b,
c be relatively prime positive integers such that
a
p
+
b
q
=
c
r
for fixed integers
p,
q,
r ≥ 2. Terai conjectured that the equation
a
x
+
b
y
=
c
z
in positive integers has only the solution (
x,
y,
z) = (
p,
q,
r) except for specific cases. In this paper, we consider the case
q =
r = 2 and give some results related to exceptional cases.
相似文献
17.
Let
P
n
be a set of
n=2
m points that are the vertices of a convex polygon, and let ℳ
m
be the graph having as vertices all the perfect matchings in the point set
P
n
whose edges are straight line segments and do not cross, and edges joining two perfect matchings
M
1 and
M
2 if
M
2=
M
1−(
a,
b)−(
c,
d)+(
a,
d)+(
b,
c) for some points
a,
b,
c,
d of
P
n
. We prove the following results about ℳ
m
: its diameter is
m−1; it is bipartite for every
m; the connectivity is equal to
m−1; it has no Hamilton path for
m odd,
m>3; and finally it has a Hamilton cycle for every
m even,
m≥4.
Received: October 10, 2000 Final version received: January 17, 2002
RID="*"
ID="*" Partially supported by Proyecto DGES-MEC-PB98-0933
Acknowledgments. We are grateful to the referees for comments that helped to improve the presentation of the paper.
相似文献
18.
Summary. A parametric curve
f
L
2
(m)
([
a,
b]ℝ
d
) is a ``near-interpolant' to prescribed data
z
ij
ℝ
d
at data sites
t
i
[
a,
b] within tolerances 0<ɛ
ij
≤∞ if |
f
(j−1)
(
t
i
)−
z
ij
|≤ɛ
ij
for
i=1:
n and
j=1:
m, and a ``best near-interpolant' if it also minimizes ∫
a
b
|
f
(m)
|
2. In this paper optimality conditions are derived for these best near-interpolants. Based on these conditions it is shown
that the near-interpolants are actually smoothing splines with weights that appear as Lagrange multipliers corresponding to
the constraints. The optimality conditions are applied to the computation of near-interpolants in the last sections of the
paper.
Received September 4, 2001 / Revised version received July 22, 2002 /
Published online October 29, 2002
Mathematics Subject Classification (1991): 41A05, 41A15, 41A29
相似文献
19.
Soit
M(Ω, η, ξ,
g) une variété à (2
m+1)-dimensions presque cosymplectique (i. e. Ω∈Λ
2
M est de rang 2
m et Ω
m
Λη≠0). On définit
M comme étant une variété semi-cosymplectique si en termes de
d
ω-cohomologie la paire (Ω, η) satisfait à
dη=0,
d
−cη Ω=Ψ∈Λ
3
M,
c=constant. Dans ce cas le champ vectoriel de structure ξ=
b
−1(η) est un champ conforme horizontal et si
M est une forme-espace elle est nécessairement du type hyperbolique. Différentes propriétés de cette structure sont étudiés
et le cas où
M est une variété para Sasakienne dans le sens large est discuté.
相似文献
20.
Let
M be a monoid presented by <Σ;
u=
v> where
u and
v are words on the finite alphabet Σ./ Deciding the world problem for
M is still an open question, though it seems decidable and partial results exist. The remaining cases to solve are presentations
of the form <
a, b; bva=
aua>,
u, v∈{
a, b}
⋆ The word problem is then closely related to the left-divisibility problem, as shown by S.I. Adjan who introduced a procedure
that “almost” allows to decide the problem. The main contribution, due to Adjan and Oganesjan, states that if
bva is an unbordered factor of
aua then the word problem is deciable. We restrict Adjan's procedure to study the case when
bva is unbordered, which allows us to extend Adjan and Oganesjan's theorem. More specifically, we show (Proposition 4.24) that
the word problem is decidable for presentations <
a, b; bva=
aua> with the only following condition: In
bva, the leftmost train of
b is strictly longer than the others. The following corollary naturally holds: The word problem is decidable for presentations
of the form <
a, b; b
m
a
n
=
aua>,
u∈{
a, b}
⋆,
m, n>0
相似文献