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1.
We investigate the correlation between the constants K(ℝn) and
, where
is the exact constant in a Kolmogorov-type inequality, ℝ is the real straight line,
, L
l
p, p
(G
n) is the set of functions ƒ ∈ L
p
(G
n
) such that the partial derivative
belongs to L
p
(G
n
),
, 1 ≤ p ≤ ∞, l ∈ ℕn, α ∈ ℕ
0
n
= (ℕ ∪ 〈0〉)n, D
α
f is the mixed derivative of a function ƒ, 0 < μi < 1,
, and ∑
i=0
n
. If G
n
= ℝ, then μ0=1−∑
i=0
n
(α
i
/l
i
), μi = αi/l
i
,
if
, then μ0=1−∑
i=0
n
(α
i
/l
i
) − ∑
i=0
n
(λ/l
i
), μi = αi/ l
i
+ λ/l
i
,
, λ ≥ 0. We prove that, for λ = 0, the equality
is true.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 597–606, May, 2006. 相似文献
2.
Sun Zhonghua Qi Wenfeng 《高校应用数学学报(英文版)》2007,22(4):469-477
Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed. 相似文献
3.
A general result on precise asymptotics for linear processes of positively associated sequences 总被引:2,自引:0,他引:2
Let {εt; t ∈ Z^+} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2+1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^+} is a sequence of real numbers satisfying ∑j=0^∞|aj|〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E|ε1|^2+δ′〈 for some δ′〉0 and μ(n)=O(n^-ρ) for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}. 相似文献
4.
Let
be a nondecreasing sequence of positive numbers and let l
1,α be the space of real sequences
for which
. We associate every sequence ξ from l
1,α with a sequence
, where ϕ(·) is a permutation of the natural series such that
, j ∈ ℕ. If p is a bounded seminorm on l
1,α and
, then
Using this equality, we obtain several known statements.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 1002–1006, July, 2005. 相似文献
5.
Achiya Dax 《BIT Numerical Mathematics》1997,37(3):600-622
This paper presents a proximal point algorithm for solving discretel
∞ approximation problems of the form minimize ∥Ax−b∥∞. Let ε∞ be a preassigned positive constant and let ε
l
,l = 0,1,2,... be a sequence of positive real numbers such that 0 < ε
l
< ε∞. Then, starting from an arbitrary pointz
0, the proposed method generates a sequence of points z
l
,l= 0,1,2,..., via the rule
. One feature that characterizes this algorithm is its finite termination property. That is, a solution is reached within
a finite number of iterations. The smaller are the numbers ε
l
the smaller is the number of iterations. In fact, if ε
0
is sufficiently small then z1 solves the original minimax problem.
The practical value of the proposed iteration depends on the availability of an efficient code for solving a regularized minimax
problem of the form minimize
where ∈ is a given positive constant. It is shown that the dual of this problem has the form maximize
, and ify solves the dual thenx=A
T
y solves the primal. The simple structure of the dual enables us to apply a wide range of methods. In this paper we design
and analyze a row relaxation method which is suitable for solving large sparse problems. Numerical experiments illustrate
the feasibility of our ideas. 相似文献
6.
Shan-tao Liao 《Frontiers of Mathematics in China》2006,1(1):1-52
Let M
n
be an n-dimensional compact C
∞-differentiable manifold, n ≥ 2, and let S be a C
1-differential system on M
n
. The system induces a one-parameter C
1 transformation group φ
t
(−∞ < t < ∞) over M
n
and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M
n
. The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group.
Among various results established in the paper, we mention here only the following, which might describe quite well the nature
of our study.
(A) Let M be the set of regular points in M
n
of the differential system S. With respect to a given C
∞ Riemannian metric of M
n
, we consider the bundle
of all (n−2) spheres Q
x
n−2, x∈M, where Q
x
n−2 for each x consists of all unit tangent vectors of M
n
orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ
t
#
(−∞<t<∞) of
. For an l-frame α = (u
1, u
2,⋯, u
l
) of M
n
at a point x in M, 1 ≥ l ≥ n−1, each u
i
being in
, we shall denote the volume of the parallelotope in the tangent space of M
n
at x with edges u
1, u
2,⋯, u
l
by υ(α), and let
. This is a continuous real function of t. Let
α is said to be positively linearly independent of the mean if I
+
*(α) > 0. Similarly, α is said to be negatively linearly independent of the mean if I
−
*(α) > 0.
A point x of M is said to possess positive generic index κ = κ
+
*(x) if, at x, there is a κ-frame
,
, of M
n
having the property of being positively linearly independent in the mean, but at x, every l-frame
, of M
n
with l >
κ does not have the same property. Similarly, we define the negative generic index κ
−
*(x) of x. For a nonempty closed subset F of M
n
consisting of regular points of S, invariant under φ
t
(−∞ < t < ∞), let the (positive and negative) generic indices of F be defined by
Theorem
κ
+
*(F)=κ
−
*(F).
(B) We consider a nonempty compact metric space x and a one-parameter transformation group ϕ
t
(−∞ < t < ∞) over X. For a given positive integer l ≥ 2, we assume that, to each x∈X, there are associated l-positive real continuous functions
of −∞ < t < ∞. Assume further that these functions possess the following properties, namely, for each of k = 1, 2,⋯, l,
for each x∈X, each −∞ < s < ∞, and each −∞ < t < ∞.
Theorem
With X, etc., given above, let μ
be a normal measure of X that is ergodic and invariant under ϕ
t
(−∞
< t < ∞). Then, for a certain permutation k→p(k) of k= 1, 2,⋯, l, the set W of points x of X such that all the inequalities
(I
k
)
(II
k
)
(k=2, 3,⋯, l) hold is invariant under ϕ
t
(−∞
< t < ∞) and is μ-measurable with μ-measure1.
In practice, the functions h
xk
(t) will be taken as length functions of certain tangent vectors of M
n
. This theory, established such as in this paper, is expected to be used in the study of structurally stable differential
systems on M
n
.
Translated from Qualitative Theory of Differentiable Dynamical Systems, Beijing, China: Science Press, 1996, by Dr. SUN Wen-xiang, School of Mathematical Sciences, Peking University, Beijing 100871,
China. The Chinese version of this paper was published in Acta Scientiarum Naturalium Universitatis Pekinensis, 1963, 9: 241–265, 309–326 相似文献
(i*) | h k (x, t) = h xk (t) is a continuous function of the Cartesian product X×(−∞, ∞). |
(ii*) |
7.
Xi Mei WU Qin YUE 《数学学报(英文版)》2007,23(11):2061-2068
Let F = Q(√-p1p2) be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol (p1/p2) = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold: (1) The quartic residue symbols (p1/p2)4 = (p2/p1)4 = 1; (2) Either both
p1 and p2 are represented by the form a^2 + 32b^2 over Z and p^h2+(2p1)/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 + 32b^2 over Z and p^h2+(2p1)/4=ε(2x^2-p1y^2),x,y∈Z,ε∈{±1},where h+(2p1) is the narrow class number of Q(√2p1),Moreover, we also generalize these results. 相似文献
8.
9.
Let r ∈ N, α, t ∈ R, x ∈ R 2, f: R 2 → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R 2, R + 2 }, and ψ n ∈ L 1(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R + such that A ? E, f ∈ L p (G), α ∈ [0, 2π] when G =R 2 and α ∈ [0, π/2] when G = R + 2 Denote Δ n, k = ∫ A t k ψ n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ m, r > 0, Δ m, r + 1 < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K Let r ∈ N, α, t ∈ R, x ∈ R
2, f: R
2 → C, and denote
In this paper, we investigate the relation between the behavior of the quantity
as n → ∞ (here, E ⊂ R, G ∈ {R
2, R
+2}, and ψ
n
∈ L
1(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity:
Here is one of the results obtained.
Theorem 1. Let E and A be intervals in
R
+
such that A ⊂ E, f ∈ L
p
(G), α ∈ [0, 2π] when G =R
2
and α ∈ [0, π/2] when G = R
+2
Denote Δ
n, k
= ∫
A
t
k
ψ
n
(t)dt. If there exists an r ∈ N
such that, for any m ∈ N, we have Δ
m, r
> 0, Δ
m, r + 1 < ∞, and
then the relations
are equivalent. Particular methods of approximation are considered. We establish
Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and
Then the relations and are equivalent.
Original Russian Text ? N.Yu. Dodonov, V.V. Zhuk, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1.
Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 23–33. 相似文献
10.
E.M.E.ZAYED 《数学学报(英文版)》2003,19(4):679-694
The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained. 相似文献
11.
E.M.E.ZAYED 《数学学报(英文版)》2004,20(2):209-222
The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|. 相似文献
12.
V. V. Makeev 《Journal of Mathematical Sciences》2007,140(4):558-563
Let ℝn be the n-dimensional Euclidean space, and let { · } be a norm in Rn. Two lines ℓ1 and ℓ2 in ℝn are said to be { · }-orthogonal if their { · }-unit direction vectors e
1 and e
2 satisfy {e
1 + e
2} = {e
1 − e
2}. It is proved that for any two norms { · } and { · }′ in ℝn there are n lines ℓ1, ..., ℓn that are { · }-and { · }′-orthogonal simultaneously. Let
be a continuous function on the unit sphere
with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in
, and such that all sums of values of f at the vertices of (n − 3)-faces of C are pairwise equal. If the function f is even,
then there exists an n-cube with the same properties. Furthermore, there exists an orthonormal basis e
1, ..., e
n such that for 1 ≤ i ≤ j ≤ n we have
. Bibliography: 8 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117. 相似文献
13.
Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices 总被引:1,自引:0,他引:1
Hua WANG Shang Bin CUI Dong Gao DENG 《数学学报(英文版)》2007,23(8):1435-1446
We first prove that the Cauchy problem of the Kawahara equation, δtu + uδxu +βδx^3u+γδx^5u = 0, is locally solvable if the initial data belong to H^r(R) and r〉 r≥-7/5, thus improving the known local well-posedness result of this equation. Next we use this local result and the method of "almost conservation law" to prove that global solutions exist if the initial data belong to H^r(R) and r〉-1/2. 相似文献
14.
J. A. López Molina M. E. Puerta M. J. Rivera 《Bulletin of the Brazilian Mathematical Society》2006,37(2):191-216
Let
, be a family of compatible couples of Lp-spaces. We show that, given a countably incomplete ultrafilter
in
, the ultraproduct
of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type
, an intermediate K?the space between
and
being a purely atomic measure space, and a K?the function space K(Ω3) defined on some purely non atomic measure space (Ω3, ν3) in such a way that Ω2 ∪ Ω3 ≠∅.
The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group
03/050. 相似文献
15.
Saharon Shelah 《Israel Journal of Mathematics》2008,166(1):61-96
We show that, consistently, there is an ultrafilter on ω such that if N
nℓ = (P
nℓ ∪ Q
nℓ, P
nℓ, Q
nℓ, R
nℓ) (for ℓ = 1, 2, n < ω), P
nℓ ∪ Q
nℓ ⊆ ω, and are models of the canonical theory t
ind of the strong independence property, then every isomorphism from onto is a product isomorphism.
The first version of this work done in 93; First typed: May 1993.
This research was partially supported by the United States-Israel Binational Science Foundation. Publication 509 相似文献
16.
Bin Heng SONG Huai Yu JIAN 《数学学报(英文版)》2005,21(5):1183-1190
We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1(u^mi)xixi in R^n × (O,∞), where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n(2 + ∑i^n=1 mi). 相似文献
17.
We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of
in the limitN→∞. The resulting Hopf algebra Rep
is a tensor product of its Hopf subalgebras Repa
,a ∈ ℂ×/q2ℤ. Whenq is generic (resp.,q
2 is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep
a
and the algebra of regular functions on the prounipotent proalgebraic group
(resp.,
). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep
a
spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the
coefficients of the determinant of an element of
considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations.
In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic
quiver withl vertices) on Rep
a
and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra. 相似文献
18.
R. Nair 《Israel Journal of Mathematics》2009,171(1):197-219
We consider a system of “generalised linear forms” defined at a point x = (x
(i, j)) in a subset of R
d
by
for k ≥ 1. Here d = d
1 + ⋯ + d
l
and for each pair of integers (i, j) ∈ D, where D = {(i, j): 1 ≤ i ≤ l, 1 ≤ j ≤ d
i
} the sequence of functions (g
(i, j), k
(x))
k=1∞ are differentiable on an interval X
ij
contained in R. We study the distribution of the sequence on the l-torus defined by the fractional parts X
k
(x) = ({ L
1(x)(k)}, ..., {L
l
(x)(k)}) ∈ T
l
, for typical x in the Cartesian product . More precisely, let R = I
1 × ⋯ × I
l
be a rectangle in T
l
and for each N ≥ 1 define a pair correlation function
and a discrepancy , where the supremum is over all rectangles in T
l
and χ
R
is the characteristic function of the set R. We give conditions on (g
(i, j), k
(x))
k=1∞ to ensure that given ε > 0, for almost every x ∈ T
l
we have Δ
N
(x) = o(N(log N)
l+∈). Under related conditions on(g
(i, j), k
(x))
k =1∞ we calculate for appropriate β ∈ (0, 1) the Hausdorff dimension of the set {x : lim sup
N→∞
N
β Δ
N
(x > 0)}. Our results complement those of Rudnick and Sarnak and Berkes, Philipp, and Tichy in one dimension and M. Pollicott
and the author in higher dimensions. 相似文献
19.
Zhao-hui Huo 《应用数学学报(英文版)》2005,21(3):441-454
The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu+αδx^5u+βδx^3u+rδxu+μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s(s≥-3/8) by the Fourier restriction norm method. Moreover, the global well-posedness for L^2 data follows from the local well-posedness and the conserved quantity. For data in H^s(s〉0), the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L^2 norm of initial data. 相似文献
20.
E. M. E. Zayed 《数学学报(英文版)》2000,16(4):627-636
Abstract
Small-time asymptotics of the trace of the heat semigroup
where {μ
ν
} are the eigenvalues of the negative Laplacian
in the (x
1, x
2)-plane, is studied for a general bunded domain Ω with a smooth boundary ∂Ω, where a finite number of Dirichlet, Neumann and
Robin boundary conditions, on the piecewise smooth parts Γ
i
(i = 1, ..., n) of ∂Ω such that
, are considered. Some geometrical properties associated with Ω are determined. 相似文献