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1.
A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following: 1. , for 1a5, a≠2. 2. A8. 3. PSL(3,4)e, for 1e10. 4. A5×PSL(3,4)e, for 1e10. Based on this result, we virtually show that if G is an ah-group with π( G) 2,3,5,7 , then F( G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S3, then the non-abelian socle of G is either trivial or isomorphic to one of the following: 1. , for 3a5. 2. PSL(3,4)e, for 1e10. Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z2Sn. These results are of independent interest and are located in appendices for greater autonomy. 相似文献
2.
Let k1 be an integer and G be a graph of order n3 k satisfying the condition that σ2( G) n+ k-1. Let v1,…, vk be k independent vertices of G, and suppose that G has k vertex-disjoint triangles C1,…, Ck with viV( Ci) for all 1 ik.Then G has k vertex-disjoint cycles such that - (i) for all 1ik.
- (ii) , and
- (iii) At least k-1 of the k cycles are triangles.
The condition of degree sum σ2( G) n+ k-1 is sharp.
Keywords: Degree sum condition; Independent vertices; Vertex-disjoint cycles 相似文献
3.
Let I be a finite interval, , and 1 p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x(r)Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in Lq for s> r+1 and ( r, p, q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈ n-2. 相似文献
4.
In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2 nu1 (HSOLSSOM(2 nu1)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3 u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3 u/2+1 or (2) n263 and n2( u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs ( n, u), the largest of which is (53,28). 相似文献
5.
Let 1< p<∞, and k, m be positive integers such that 0( k−2 m) pn. Suppose Ω Rn is an open set, and Δ is the Laplacian operator. We will show that there is a sequence of positive constants cj such that for every f in the Sobolev space Wk,p(Ω), for all xΩ except on a set whose Bessel capacity Bk−2m,p is zero. 相似文献
6.
We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition Δ kfLp1, where p=1 or 2 and Δ is the Laplace operator. 相似文献
7.
Given an integrable function f, we are concerned with the construction of a spline Hn( f) of degree n with prescribed knots that satisfies the histopolation conditionsfor some fixed . Additionally, the resulting spline operator should be local and reproduce all polynomials of degree n. Our approach of generating such a histospline is based on a local spline interpolation operator that is exact for all polynomials of degree n. 相似文献
8.
We discuss the existence of a diffeomorphism such that | where are closed differential forms and 2kn. Our main results (the case k=n having been handled by Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] and Dacorogna and Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) are that- – when n is even and k=2, under some natural non-degeneracy condition, we can prove the existence of such diffeomorphism satisfying Dirichlet data on the boundary of a bounded open set and the natural Hölder regularity; at the same time we get Darboux theorem with optimal regularity;
- – we are also able to handle the degenerate cases when k=2 (in particular when n is odd), k=n−1 and some cases where 3kn−2.
Résumé
Nous montrons l'existence d'un difféomorphisme
satisfaisant
où
sont des formes différentielles fermées et 2
kn. Nos résultats principaux (le cas
k=
n a été discuté notamment dans Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] et Dacorogna et Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) sont les suivants.
- – Si n est pair, k=2 et sous des conditions naturelles de non dégénérescence, nous montrons l'existence et la régularité dans les espaces de Hölder d'un tel difféomorphisme satisfaisant de plus une condition de Dirichlet. On obtient aussi le théorème de Darboux avec la régularité optimale.
- – Par ailleurs quand k=2 et n est impair ou k=n−1, ainsi que quelques cas particuliers où 3kn−2, nous montrons l'existence locale d'un tel difféomorphisme satisfaisant, en outre, des conditions de Cauchy.
Keywords: Darboux theorem; Symplectic forms; Pullback; Hölder regularity
相似文献
9.
Suppose that
G is a graph with
n vertices and
m edges, and let
μ be the spectral radius of its adjacency matrix.Recently we showed that if
G has no 4-cycle, then
μ2-
μn-1, with equality if and only if
G is the friendship graph.Here we prove that if
m9 and
G has no 4-cycle, then
μ2m, with equality if
G is a star. For 4
m8 this assertion fails.
相似文献
10.
Let
Lq (1
q<∞) be the space of functions
f measurable on
I=[−1,1] and integrable to the power
q, with norm
L∞ is the space of functions measurable on
I with normWe denote by
AC the set of all functions absolutely continuous on
I. For
nN,
q[1,∞] we set
Wn,q={
f:
f(n−1)AC,
f(n)Lq}.In this paper, we consider the problem of accuracy of constants
A,
B in the inequalities
(1)||
f(m)||
qA||
f||
p+
B||
f(m+k+1)||
r,
mN,
kW;
p,
q,
r[1,∞],
fWm+k+1,r.
相似文献
11.
A graph
G is (
m,
n)-
linked if for any two disjoint subsets
R,
BV(
G) with |
R|
m and |
B|
n,
G has two disjoint connected subgraphs containing
R and
B, respectively. We shall prove that a planar graph with at least six vertices is (3,3)-linked if and only if
G is 4-connected and maximal.
相似文献
12.
We develop a general context for the computation of the determinant of a Hankel matrix
Hn = (
αi+j)0i,jn, assuming some suitable conditions for the exponential (or ordinary) generating function of the sequence (
αn)
n0. Several well-known particular cases are thus derived in a unified way.
相似文献
13.
Let
Γ denote a distance-regular graph with diameter
D3. Let
θ denote a nontrivial eigenvalue of
Γ and let
denote the corresponding dual eigenvalue sequence. In this paper we prove that
Γ is
Q-polynomial with respect to
θ if and only if the following (i)–(iii) hold:
- (i) There exist such that
- (ii) There exist such that the intersection numbers ai satisfyfor 0iD, where and are the scalars which satisfy Eq. (1) for i=0, i=D, respectively.
- (iii) for 1iD.
Keywords: Distance-regular graph;
Q-polynomial; Association scheme
相似文献
15.
Let μ be a real measure on the line such that its Poisson integral
M(
z) converges and satisfies|
M(
x+
iy)|
Ae−cyα,
y→+∞,for some constants
A,
c>0 and 0<α1. We show that for 1/2<α1 the measure μ must have many sign changes on both positive and negative rays. For 0<α1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.
相似文献
16.
New pointwise inversion formulae are obtained for the
d-dimensional totally geodesic Radon transform on the
n-dimensional real hyperbolic space, 1
dn−1, in terms of polynomials of the Laplace–Beltrami operator and intertwining fractional integrals. Similar results are established for hyperbolic cosine and sine transforms.
相似文献
17.
Let
m and
n be positive integers with
n2 and 1
mn−1. We study rearrangement-invariant quasinorms
R and
D on functions
f: (0, 1)→
such that to each bounded domain
Ω in
n, with Lebesgue measure |
Ω|, there corresponds
C=
C(|
Ω|)>0 for which one has the Sobolev imbedding inequality
R(
u*(|
Ω|
t))
CD(|
mu|* (|
Ω|
t)),
uCm0(
Ω), involving the nonincreasing rearrangements of
u and a certain
mth order gradient of
u. When
m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which
D need not be rearrangement-invariant,
R(
u*(|
Ω|
t))
CD((d/d
t) ∫
{x
n : |u(x)|>u*(|Ω| t)} |(
u)(
x)| d
x),
uC10(
Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that
R cannot be replaced by an essentially larger quasinorm and
D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger.
相似文献
18.
For a compact convex set
the well-known general Markov inequality holds asserting that a polynomial
p of degree
n must have
p′c(
K)
n2p. On the other hand for polynomials in general,
p′ can be arbitrarily small as compared to
p.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set
K. This was first investigated by Turán, who showed the lower bounds
p′(
n/2)
p for the unit disk
D and
for the unit interval
I[-1,1]. Although partial results provided general lower estimates of order
, as well as certain classes of domains with lower bounds of order
n, it was not clear what order of magnitude the general convex domains may admit here.Here we show that for all bounded and convex domains
K with nonempty interior and polynomials
p with all their zeroes lying in
K p′c(
K)
np holds true, while
p′C(
K)
np occurs for any
K. Actually, we determine
c(
K) and
C(
K) within a factor of absolute numerical constant.
相似文献
19.
The continuity conditions at the endpoints of interpolation theorems,
TaBjMj aAj for
j=0, 1, can be written with the help of the approximation functional:
E(
t,
Ta;
B1,
B0)
L∞M0 aA0 and
E(
t,
Ta;
B0,
B1)
L∞M1 aA1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the
L∞ norms can be replaced by BMO(
+) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition
fL0, i.e.,
f*
L∞, where
f*(
γ)=
μ{|
f|>
γ} is the distribution function of
f, can be replaced in interpolation with
L(
p,
q) spaces by the weaker
f*BMO(
+).
相似文献
20.
Let
K be a convex body in
d (
d2), and denote by
Bn(
K) the set of all polynomials
pn in
d of total degree
n such that |
pn|1 on
K. In this paper we consider the following question: does there exist a
p*
nBn(
K) which majorates
every element of
Bn(
K)
outside of
K? In other words can we find a minimal
γ1 and
p*
nBn(
K) so that |
pn(
x)|
γ |
p*
n(
x)| for every
pnBn(
K) and
x
d\
K? We discuss the magnitude of
γ and construct the universal majorants
p*
n for
evenn. It is shown that
γ can be 1 only on
ellipsoids. Moreover,
γ=
O(1) on
polytopes and has at most polynomial growth with respect to
n, in general, for every convex body
K.
相似文献