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1.
Let M be a closed extremal hypersurface in Sn+1 with the same mean curvature of the Willmore torus Wm, n-m. We proved that if Specp(M) = Specp(Wm, n-m) for p = 0,1,2, then M is Wm, m. 相似文献
2.
Zhen Guo 《数学学报(英文版)》2009,25(1):77-84
Let x : Mn^n→ R^n+1 be an n(≥2)-dimensional hypersurface immersed in Euclidean space Rn+1. Let σi(0≤ i≤ n) be the ith mean curvature and Qn = ∑i=0^n(-1)^i+1 (n^i)σ1^n-iσi. Recently, the author showed that Wn(x) = ∫M QndM is a conformal invariant under conformal group of R^n+1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper. 相似文献
3.
Assuming m − 1 < kp < m, we prove that the space C
∞(M, N) of smooth mappings between compact Riemannian manifolds M, N (m = dim M) is dense in the Sobolev space W
k,p
(M, N) if and only if π
m−1(N) = {0}. If π
m−1(N) ≠ {0}, then every mapping in W
k,p
(M, N) can still be approximated by mappings M → N which are smooth except in finitely many points. 相似文献
4.
Let A be a function with derivatives of order m and D γ A ∈■β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ∞ (R n ) × L s (S n 1 ) (s ≥ n/(n β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ A Ω and its variation μ A Ω are bounded from L p (R n ) to L q (R n ), where 1 < p < n/β and 1/q = 1/p β/n. The authors also consider the boundedness of μ A Ω and its variation μ A Ω on Hardy spaces. 相似文献
5.
Susana Elena Trione 《Studies in Applied Mathematics》1988,79(2):127-141
Let t = (t1,…,tn) be a point of ?n. We shall write . We put, by the definition, Wα(u, m) = (m?2u)(α ? n)/4[π(n ? 2)/22(α + n ? 2)/2Г(α/2)]J(α ? n)/2(m2u)1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. Wα(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m2}Wα + 2(u, m) = Wα(u, m), where {□ + m2} is the ultrahyperbolic operator. Then we express Wα(u, m) as a linear combination of Rα(u) of differntial orders; Rα(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W?2k(u, m) = {□ + m2}kδ, k = 0, 1,…; W0(u, m) = δ; and {□ + m2}kW2k(u, m) = δ. Finally we prove that Wα(u, m = 0) = Rα(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method. 相似文献
6.
For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative
on the Grassmann bundle ^ := Gm(T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rn this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on
the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ Gm(TpN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving
) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel
second fundamental form can be obtained in this way.
Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21. 相似文献
7.
Vito Lampret 《Central European Journal of Mathematics》2012,10(2):775-787
An asymptotic approximation of Wallis’ sequence W(n) = Π
k=1
n
4k
2/(4k
2 − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates
of Wallis’ ratios w(n) = Π
k=1
n
(2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example,
W(n) ·(an + bn ) < p < W(n) ·(an + b¢n )W(n) \cdot (a_n + b_n ) < \pi < W(n) \cdot (a_n + b'_n ) 相似文献
8.
B. Bojarski 《Proceedings of the Steklov Institute of Mathematics》2006,255(1):65-81
We prove that a function f is in the Sobolev class W
loc
m,p
(ℝ
n
) or W
m,p
(Q) for some cube Q ⊂ ℝ
n
if and only if the formal (m − 1)-Taylor remainder R
m−1
f(x,y) of f satisfies the pointwise inequality |R
m−1
f(x,y)| ≤ |x − y|
m
[a(x) + a(y)] for some a ε L
p
(Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth
functions on closed subsets of ℝ
n
.
Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday
The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference
and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005. 相似文献
9.
In this paper, we first consider a delay difference equation of neutral type of the form:
Δ(y
n
+ py
n−k
+ q
n
y
n−l
= 0 for n∈ℤ+(0) (1*)
and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form:
Δ1(u
n,m
+ pu
n−k,m
) + q
n,m
u
n−l,m
= a
2Δ2
2
u
n
+1,
m−1
for (n,m) ∈ℤ+ (0) ×Ω, (2*)
study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses.
Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000 相似文献
10.
Let m and n be nonnegative integers. Denote by P(m,n) the set of all triangle-free graphs G such that for any independent m-subset M and any n-subset N of V(G) with M ∩ N = Ø, there exists a unique vertex of G that is adjacent to each vertex in M and nonadjacent to any vertex in N. We prove that if m ? 2 and n ? 1, then P(m,n) = Ø whenever m ? n, and P(m,n) = {Km,n+1} whenever m > n. We also have P(1,1) = {C5} and P(1,n) = Ø for n ? 2. In the degenerate cases, the class P(0,n) is completely determined, whereas the class P(m,0), which is most interesting, being rich in graphs, is partially determined. 相似文献
11.
We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝ
n
, A∈ℝ
n×n
, b∈ℝ
n
, C∈ℝ
m×n
, d∈ℝ
m
, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc
(p), respectively. It is proved that if the point-to-set mapping M
loc
(·) is lower semicontinuous at p then M
loc
(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
= is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.
Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000 相似文献
12.
For a distribution ?? over labeled bipartite (multi) graphs G = (W, M, E), |W| = |M| = n, let L(n) denote the size of the largest planar matching of G (here W and M are posets drawn on the plane as two ordered rows of nodes and edges are drawn as straight lines). We study the asymptotic (in n) behavior of L(n) for different distributions ??. Two interesting instances of this problem are Ulam's longest increasing subsequence problem and the longest common subsequence problem. We focus on the case where ?? is the uniform distribution over the k‐regular bipartite graphs on W and M. For k = o(n1/4), we establish that $L(n) \slash \sqrt{kn}$ tends to 2 in probability when n → ∞. Convergence in mean is also studied. Furthermore, we show that if each of the n2 possible edges between W and M are chosen independently with probability 0 < p < 1, then L(n)/n tends to a constant γp in probability and in mean when n → ∞. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 162–181, 2002 相似文献
13.
O. M. Iksanov 《Ukrainian Mathematical Journal》2006,58(3):368-387
Let M
n
, n = 1, 2, ..., be a supercritical branching random walk in which the number of direct descendants of an individual may be infinite
with positive probability. Assume that the standard martingale W
n
related to M
n
is regular and W is a limit random variable. Let a(x) be a nonnegative function regularly varying at infinity with index greater than −1. We present sufficient conditions for
the almost-sure convergence of the series
. We also establish criteria for the finiteness of EW ln+
Wa(ln+
W) and E ln+|Z
∞|a(ln+|Z
∞|), where
and (M
n
, Q
n
) are independent identically distributed random vectors not necessarily related to M
n
.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 326–342, March, 2006. 相似文献
14.
Hei-Chi Chan 《数学学报(英文版)》2011,27(4):625-634
In this paper, we study a certain partition function a(n) defined by Σ
n≥0
a(n)q
n
:= Π
n=1(1 − q
n
)−1(1 − q
2n
)−1. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m
j
). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n). 相似文献
15.
Klaus Pinn 《Complexity》1999,4(3):41-46
A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/nα, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of xm = R(n) − R(n − m) is given by pm(xm) = λm p*(xm/λm), with λm → √2 for large m. © 1999 John Wiley & Sons, Inc. 相似文献
16.
Laurent-Padé (Chebyshev) rational approximantsP
m
(w, w
−1)/Q
n
(w, w
−1) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP
m
/Q
n
matches that of a given functionf(w, w
−1) up to terms of orderw
±(m+n)
, based only on knowledge of the Laurent series coefficients off up to terms inw
±(m+n)
. This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent
series ofP
m
matches that ofQ
n
f up to terms of orderw
±(m+n
), but based on knowledge of the series coefficients off up to terms inw
±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and
corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of
the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations
are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably
similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the
use of either. 相似文献
17.
Yu-Chung Chang 《Monatshefte für Mathematik》2009,4(4):1-22
In this paper we consider a compact oriented hypersurface M
n
with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S
n+1. Denote by fij{\phi_{ij}} the trace free part of the second fundamental form of M
n
, and Φ be the square of the length of fij{\phi_{ij}} . We obtain two integral formulas by using Φ and the polynomial
PH,m(x)=x2+ \fracn(n-2m)?{nm(n-m)}H x -n(1+H2){P_{H,m}(x)=x^{2}+ \frac{n(n-2m)}{\sqrt{nm(n-m)}}H x -n(1+H^{2})} . Assume that B
H,m
is the square of the positive root of P
H,m
(x) = 0. We show that if M
n
is a compact oriented hypersurface immersed in the sphere S
n+1 with constant mean curvatures H having two distinct principal curvatures λ and μ then either F = BH,m{\Phi=B_{H,m}} or F = BH,n-m{\Phi=B_{H,n-m}} . In particular, M
n
is the hypersurface Sn-m(r)×Sm(?{1-r2}){S^{n-m}(r)\times S^{m}(\sqrt{1-r^{2}})} . 相似文献
18.
James R. Holub 《Israel Journal of Mathematics》1985,52(3):231-238
LetW(D) denote the set of functionsf(z)=Σ
n=0
∞
A
n
Z
n
a
nzn for which Σn=0
∞|a
n
|<+∞. Given any finite set lcub;f
i
(z)rcub;
i=1
n
inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f
1(z)z
kn
,f
2(z)z
kn+1, …,f
n
(z)z
(k+1)n−1rcub;
k=0
∞
is a basis forW(D) which is equivalent to the basis lcub;z
m
rcub;
m=0
∞
. (ii) The generalized shift sequence is complete inW(D), (iii) The function
has no zero in |z|≦1, wherew=e
2πiti
/n. 相似文献
19.
For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by
20.
We consider generalized Morrey type spaces Mp( ·),q( ·),w( ·)( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets
W ì \mathbbRn \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with
standard kernel. We prove a Sobolev–Adams type embedding theorem Mp( ·),q1( ·),w1( ·)( W) ? Mq( ·),q2( ·),w2( ·)( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I
α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles. 相似文献
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