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1.
John S. Caughman 《Graphs and Combinatorics》1998,14(4):321-343
Let Y=(X,{R
i
}0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A
0, A
1,…, A
D
of the associate matrices, and Q-polynomial with respect to the ordering E
0, E
1,…,E
D
of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv).
(i)
(ii) D is even, and
(iii) θ*
0>θ0, and
(iv) θ*
0>θ0, D is odd, and
Received: February 13, 1996 / Revised: October 16, 1996 相似文献
2.
Rostom Getsadze 《Journal d'Analyse Mathématique》2007,102(1):209-223
Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E
0 ⊂ I
2 is any Lebesgue measurable set such that μ2
E
0 > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L1(I
2) and a set E
0
′
, ⊂ E
0, μ2
E
0
′
> 0, such that
and the sequence of double Cesàro means of Fourier series of f with respect to the system {ϕn(x)ϕm(y): n,m = 1, 2,...} is unbounded in the sense of Pringsheim (by rectangles) on E
0
′
. This statement gives critical integrability conditions for the Cesàro summability a.e. of Fourier series in the class of
all complete orthonormal systems of the type {ϕ n(x)ϕm(y): n,m = 1, 2,...}. 相似文献
3.
Etsuko Bannai 《Journal of Algebraic Combinatorics》2006,24(4):391-414
Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ
n
. For an integer t, a finite subset X of ℝ
n
given together with a weight function w is a Euclidean t-design if holds for any polynomial f(x) of deg(f)≤ t, where {S
i
, 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X
i
= X∩ S
i
, and w:X→ ℝ> 0. (The case of X⊂ S
n−1 with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres. 相似文献
4.
Brian Curtin 《Graphs and Combinatorics》1999,15(2):143-158
Let Γ=(X,E) denote a bipartite distance-regular graph with diameter D≥4, and fix a vertex x of Γ. The Terwilliger algebra
T=T(x) is the subalgebra of Mat
X(C) generated by A, E
*
0, E
*
1,…,E
*
D, where A denotes the adjacency matrix for Γ and E
*
i denotes the projection onto the i
TH subconstituent of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimE
*
i
W≤1 for 0≤i≤Di. The endpoint of W is min{i|E
*
i
W≠0}.
We determine the structure of the (unique) irreducible T-module of endpoint 0 in terms of the intersection numbers of Γ. We show that up to isomorphism there is a unique irreducible
T-module of endpoint 1 and it is thin. We determine its structure in terms of the intersection numbers of Γ. We determine the
structure of each thin irreducible T-module W of endpoint 2 in terms of the intersection numbers of Γ and an additional real parameter ψ=ψ(W), which we refer to as the type of W.
We now assume each irreducible T-module of endpoint 2 is thin and obtain the following two-fold result. First, we show that the intersection numbers of Γ
are determined by the diameter D of Γ and the set of ordered pairs
where Φ2 denotes the set of distinct types of irreducible T-modules with endpoint 2, and where mult(ψ) denotes the multiplicity with which the module of type ψ appears in the standard
module. Secondly, we show that the set of ordered pairs {(ψ,mult(ψ)) |ψ∈Φ2} is determined by the intersection numbers k, b
2, b
3 of Γ and the spectrum of the graph , where
and where ∂ denotes the distance function in Γ. Combining the above two results, we conclude that if every irreducible T-module of endpoint 2 is thin, then the intersection numbers of Γ are determined by the diameter D of Γ, the intersection numbers k, b
2, b
3 of Γ, and the spectrum of Γ2
2.
Received: November 13, 1995 / Revised: March 31, 1997 相似文献
5.
Suppose D is an NTA domain, E
D is any closed set, and P
x
0(E) is the projection with respect to a point x
0D of the set E onto the boundary of D. The projection P
x
0 satisfies certain geometric properties so that it is a generalization of the notion of radial projection with respect to a point x
0 onto a boundary of a domain. It is shown that the harmonic measure of E with respect to the domain DE evaluated at the point x
0 is bounded below by a constant times the harmonic measure of the set P
x
0(E) with respect to the domain D evaluated at the point x
0. The constant is independent of the set E but it may depend upon x
0. 相似文献
6.
Let f(x, y) be a periodic function defined on the region D
with period 2π for each variable. If f(x, y) ∈ C
p (D), i.e., f(x, y) has continuous partial derivatives of order p on D, then we denote by ω
α,β(ρ) the modulus of continuity of the function
and write
For p = 0, we write simply C(D) and ω(ρ) instead of C
0(D) and ω
0(ρ).
Let T(x,y) be a trigonometrical polynomial written in the complex form
We consider R = max(m
2 + n
2)1/2 as the degree of T(x, y), and write T
R(x, y) for the trigonometrical polynomial of degree ⩾ R.
Our main purpose is to find the trigonometrical polynomial T
R(x, y) for a given f(x, y) of a certain class of functions such that
attains the same order of accuracy as the best approximation of f(x, y).
Let the Fourier series of f(x, y) ∈ C(D) be
and let
Our results are as follows
Theorem 1 Let f(x, y) ∈ C
p(D (p = 0, 1) and
Then
holds uniformly on D.
If we consider the circular mean of the Riesz sum S
R
δ
(x, y) ≡ S
R
δ
(x, y; f):
then we have the following
Theorem 2 If f(x, y) ∈ C
p (D) and ω
p(ρ) = O(ρ
α (0 < α ⩾ 1; p = 0, 1), then
holds uniformly on D, where λ
0
is a positive root of the Bessel function J
0(x)
It should be noted that either
or
implies that f(x, y) ≡ const.
Now we consider the following trigonometrical polynomial
Then we have
Theorem 3 If f(x, y) ∈ C
p(D), then uniformly on D,
Theorems 1 and 2 include the results of Chandrasekharan and Minakshisundarm, and Theorem 3 is a generalization of a theorem
of Zygmund, which can be extended to the multiple case as follows
Theorem 3′ Let f(x
1, ..., x
n) ≡ f(P) ∈ C
p
and let
where
and
being the Fourier coefficients of f(P). Then
holds uniformly.
__________
Translated from Acta Scientiarum Naturalium Universitatis Pekinensis, 1956, (4): 411–428 by PENG Lizhong. 相似文献
7.
Vladimir Ivanovich Danchenko Roman Vasil’evich Rubay 《Journal of Mathematical Sciences》2010,171(1):34-45
In this paper, properties of solutions of the convolution-type integral equation
( 1 + w(x) )P(x) = ( m*P )(x) + Cm(x) \left( {1 + w(x)} \right)P(x) = \left( {m*P} \right)(x) + Cm(x) 相似文献
8.
V. N. Margaryan H. G. Ghazaryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2018,53(1):6-15
A linear differential operator P(x, D) = P(x1,... x n , D1,..., D n ) = ∑αγα(x)Dα with coefficients γα(x) defined in E n is called formally almost hypoelliptic in E n if all the derivatives DνξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in En. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions. 相似文献
9.
Results on the existence and non-existence of nontrivial
\mathbb L1{\mathbb L^1}-solutions of the refinement equation
|