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1.
P. Erdös 《Israel Journal of Mathematics》1964,2(3):183-190
Anr-graph is a graph whose basic elements are its vertices and r-tuples. It is proved that to everyl andr there is anε(l, r) so that forn>n
0 everyr-graph ofn vertices andn
r−ε(l, r) r-tuples containsr. l verticesx
(j), 1≦j≦r, 1≦i≦l, so that all ther-tuples
occur in ther-graph. 相似文献
2.
Emin Özçag 《Proceedings Mathematical Sciences》1999,109(1):87-94
The distributionF(x
+, −r) Inx+ andF(x
−, −s) corresponding to the functionsx
+
−r lnx+ andx
−
−s respectively are defined by the equations
(1) and
(2) whereH(x) denotes the Heaviside function. In this paper, using the concept of the neutrix limit due to J G van der Corput [1], we evaluate
the non-commutative neutrix product of distributionsF(x
+, −r) lnx+ andF(x
−, −s). The formulae for the neutrix productsF(x
+, −r) lnx
+ ox
−
−s, x+
−r lnx+ ox
−
−s andx
−
−s o F(x+, −r) lnx+ are also given forr, s = 1, 2, ... 相似文献
3.
Dr. Herbert E. Salzer 《Numerische Mathematik》1971,18(2):144-153
We considern-point Lagrange-Hermite extrapolation forf(x), x>1, based uponf(x
i
),i=1(1)n, –1x
i
1, including non-distinct pointsx
i
in confluent formulas involving derivatives. The problem is to find the pointsx
i
that minimize the factor
in the remainderP
n
(x)f
(n)()/n, –1<<x subject to the condition|P
n
(x)|M, –1x1,2–n+1M2
n
. The solution is significant only when a single set of pointsx
i
suffices for everyx>1. The problem is here completely solved forn=1(1)4. Forn>4 it may be conjectured that there is a single minimal
, 0 rn, whererr(M) is a non-decreasing function ofM, P
n
(–1)=(–1)
n
M, and for 0rn–2, thej-th extremumP
n
(x
e, j
)=(–1)
n–j
M,j=1(1)n–r–1 (except forM=M
r
,r=1(1)n–1, whenj=1(1)n–r). 相似文献
4.
K. F. Cheng 《Annals of the Institute of Statistical Mathematics》1982,34(1):479-489
Summary Letf
n
(p)
be a recursive kernel estimate off
(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of
and show that the rate of almost sure convergence of
to zero isO(n
−α), α<(r−p)/(2r+1), iff
(r),r>p≧0, is a continuousL
2(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of
to zero under different conditions onf.
This work was supported in part by the Research Foundation of SUNY. 相似文献
5.
M. Sh. Shabozov 《Mathematical Notes》1996,59(1):104-111
We find the exact value of the expression $$\varepsilon ^{(l,q)} {\mathbf{ }}(W^{(r,s)} ){\mathbf{ }}H^{w_1 ,w_2 } (G)) = \sup \{ ||f^{(l,q)} ( \cdot {\mathbf{ }}, \cdot ) - S_{1,1}^{(l,q)} (f;{\mathbf{ }} \cdot {\mathbf{ }}, \cdot )||_{C(G)} :f \in W^{(r,{\mathbf{ }}s)} H^{w_1 ,w_2 } (G)\} ,$$ , where? (l,q) (x,y)=? 1+q ?/?x l ?y q (l, q=0, 1, 1≤l+q≤2) andS 1,1(f; x, y) is a bilinear spline interpolatingf(x, y) in the nodes of the grid Δ mn =Δ m x ×Δ n y with Δ m x :x i =i/m (i=0, ..., m) and Δ n y :y j =j/n (j=0, ..., n). Here $(W^{(r,s)} ){\mathbf{ }}H^{w_1 ,w_2 } (G)$ is the class of functionsf(x, y) with continuous derivativesf (r,s)(x, y) (r, s=0, 1, 1≤r+s≤2) on the squareG=[0, 1]×[0, 1] and with the modulus of continuity satisfying the inequalityω(f (r,s);t, τ)≤ω 1 (t)+ω 2 (τ), whereω 1 (τ) andω 2 (τ) are the given moduli of continuity. 相似文献
6.
7.
Uniform estimates for positive solutions to quasi-linear differential equations of even order 总被引:1,自引:0,他引:1
I. V. Astashova 《Journal of Mathematical Sciences》2006,135(1):2616-2624
The existence of uniform estimates for positive solutions with the same domain to the even-order differential equation
with k > 1 is proved. The estimates for solutions depend on those for the continuous coefficients p(x) > 0 and a
i
(x), not on the coefficients themselves.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 21–34, 2005. 相似文献
8.
Akira Hiraki 《Graphs and Combinatorics》2009,25(1):65-79
Let Γ be a distance-regular graph of diameter d ≥ 3 with c
2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
In [12] we have shown that the condition (SC)
m
holds if and only if both of the conditions (BB)
i
and (CA)
i
hold for i = 1,...,m. In this paper we show that if a
1 = 0 < a
2 and the condition (BB)
i
holds for i = 1,...,m, then the condition (CA)
i
holds for i = 1,...,m. In particular, the condition (SC)
m
holds. Applying this result we prove that a distance-regular graph with classical parameters (d, b, α, β) such that c
2 > 1 and a
1 = 0 < a
2 satisfies the condition (SC)
i
for i = 1,...,d − 1. In particular, either (b, α, β) = (− 2, −3, −1 − (−2)
d
) or holds. 相似文献
(SC) m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. | |
(BB) m : Let (x, y, z) be a triple of vertices with ∂Γ(x, y) = 1 and ∂Γ(x, z) = ∂Γ(y, z) = m. Then B(x, z) = B(y, z). | |
(CA) m : Let (x, y, z) be a triple of vertices with and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z). |
9.
We investigate the convergence of simultaneous Hermite-Padé approximants to then-tuple of power series $$f_i (z) = \sum\limits_{k = 0}^\infty {C_k^{(i)} z^k ,} i = 1,2,...,n,$$ where $$C_0^{(i)} = 1;C_k^{(i)} = \prod\limits_{p = 0}^{k - 1} {\frac{1}{{(C - q^{\gamma i + p} )}},} k \ge 1.$$ HereC, q∈?, γ i ∈?,i=1, 2,...,n. For |C|≠1, ifq=eiθ, θ∈(0, 2π) and θ/2π is irrational, eachf i (z),i=1,...,n, has a natural boundary on its circle of convergence. We show that “close-to-diagonal” and other sequences of Hermite-Padé approximants converge in capacity to (f 1(z),..., fn (z)) inside the common circle of convergence of eachf i (z),i=1,...,n. 相似文献
10.
This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr
1 andr
2 be rank functions of two matroids defined on the same setE. For everyS ⊂E, letr
12(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r
12(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2
E
→Z
+ which maximizes
and
, subject toy≧0, ∀j∈E,
. 相似文献
11.
Herbert E. Salzer 《Numerische Mathematik》1962,4(1):381-392
Equally-weighted formulas for numerical differentiation at a fixed pointx=a, which may be chosen to be 0 without loss in generality, are derived for (1)
whereR
2n
=0 whenf(x) is any (2n)th degree polynomial. Equation (1) is equivalent to (2)
,r=1,2,..., 2n. By choosingf(x)=1/(z–x),x
i
fori=1,..., n andx
i
fori=n+1,..., 2n are shown to be roots ofg
n
(z) andh
n
(z) respectively, satisfying (3)
. It is convenient to normalize withk=(m–1)!. LetP
s
(z) denotez
s
· numerator of the (s+1)th diagonal member of the Padé table fore
x
, frx=1/z, that numerator being a constant factor times the general Laguerre polynomialL
s
–2s–1
(x), and letP
s
(X
i
)=0, i=1, ...,s. Then for anym, solutions to (1) are had, for2n=2ms, forx
i
, i=1, ...,ms, andx
i
, i=ms+1,..., 2ms, equal to all them
th rootsX
i
1/m
and (–X
i
)1/m
respectively, and they give {(2s+1)m–1}th degree accuracy. For2sm2n(2s+1)m–1, these (2sm)-point solutions are proven to be the only ones giving (2n)th degree accuracy. Thex
i
's in (1) always include complex values, except whenm=1, 2n=2. For2sm<2n(2s+1)m–1,g
n
(z) andh
n
(z) are (n–sm)-parameter families of polynomials whose roots include those ofg
ms
(z) andh
ms
(z) respectively, and whose remainingn–ms roots are the same forg
n
(z) andh
n
(z). Form>1, and either 2n<2m or(2s+1)m–1<2n<(2s+2)m, it is proven that there are no non-trivial solutions to (1), real or complex. Form=1(1)6, tables ofx
i
are given to 15D, fori=1(1)2n, where 2n=2ms ands=1(1) [12/m], so that they are sufficient for attaining at least 24th degree accuracy in (1).Presented at the Twelfth International Congress of Mathematicians, Stockholm, Sweden, August 15–22, 1962.General Dynamics/Astronautics. A Division of General Dynamics Corporation. 相似文献
12.
Owe Axelsson 《BIT Numerical Mathematics》1964,4(2):69-86
For the numerical solution of the initial value problemy=f(x,y), –1x1;y(–1)=y
0 a global integration method is derived and studied. The method goes as follows.At first the system of nonlinear equations is solved. The matrix (A
i,k
(n)
) of quadrature coefficients is nearly lower left triangular and the pointsx
k,n
,k=1,2,...,n are the zeros ofP
n
–P
n–2, whereP
n
is the Legendre polynomial of degreen. It is showed that the errors From the valuesf(x
i,n
,y
i,n
),i=1,2,...,n an approximation polynomial is constructed. The approximation is Chebyshevlike and the error at the end of the interval of integration is particularly small. 相似文献
13.
We study some properties of sets of differences of dense sets in ℤ2 and ℤ3 and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results.
(i) | If E ⊂ ℤ2, $
\bar d
$
\bar d
(E) > 0 and p
i
, q
i
∈ ℤ[x], i = 1, ..., m satisfy p
i
(0) = q
i
(0) = 0, then there exists B ⊂ ℤ such that $
\bar d
$
\bar d
(B) > 0 and
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