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1.
Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature   总被引:3,自引:0,他引:3  
The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere Sn–1(1). The hypersurfaceSk(c1)xSnk(c2) in a unit sphere Sn+1(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sn+1(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sn+1(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S1xSn–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sn+1(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and   相似文献   

2.
On the Optimum Criterion of Polynomial Stability   总被引:1,自引:0,他引:1  
The purpose of this note is to answer the question raised byNie & Xie (1987). Let f(x)=a0xn+a1xn–1+...+an be apositive-coefficient polynomial. The numbers 1=ai-1ai+2/aiai+1(i=1, ..., n–2) are called determining coefficients. Theoptimum criterion problem was posed as follows: for n3, findthe maximal number (n) such that the polynomial f(x) is stableif i < (n) (1in–2). For n6, we show that (n)=ß,where ß is the unique real root of the equation x(x+1)2=1.  相似文献   

3.
Let M denote a connected complete Riemannian manifold (possiblywith a convex boundary), the Riemannian distance function froma fixed point and V C2 (M) such that dµV eV d xis a probability measure. For any K 0, we prove that K/2 isthe infimum over all > 0 such that RicM – HessVKand imply the log-Sobolevinequality for the Dirichlet form µV(| f |2).  相似文献   

4.
On the Discreteness and Convergence in n-Dimensional Mobius Groups   总被引:5,自引:0,他引:5  
Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rn), the Clifford algebraCn–1, the Clifford matrix group SL(2, n), the Cliffordnorm of ||A||=(|a|2+|b|2+|c|2+|d|2) (1) and the Clifford metric of SL(2, n) or of the Möbius groupM(Rn) d(A1,A2)=||A1A2||(|a1a2|2+|b1b2|2+|c1c2|2+|d1d2|2)(2) where |·| is the norm of a Clifford number and represents fi M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.  相似文献   

5.
The low-dimensional projective irreducible representations incross characteristics of the projective special linear groupPSLn(q) are investigated. If n 3 and (n, q) (3,2), (3,4), (4,2), (4,3), all such representationsof the first degree (which is (qnq)/(q – 1) – with = 0 or 1) and the second degree (which is (qn –1)/(q – 1) come from Weil representations. We show thatthe gap between the second and the third degree is roughly q2n-4.1991 Mathematics Subject Classification: 20C20, 20C33.  相似文献   

6.
Let =(n)n1 be a log concave sequence such that lim infn+n/nc>0for some c>0 and ((log n)/n)n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||Tn||=O(n)as n tends to + with n11/(n log n)=+, then T is unitary. Onthe other hand, if n11/(n log n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||Tn||=O(n) as n tends to +, andlim supn+||Tn||=+.  相似文献   

7.
The paper examines blow-up phenomena for the inequality utLu–|u|q–1utL–||q–1 (*) in the half-space x Rn, n 1, where L is a linear second-order partial differential operatorin divergence form. The paper studies weak solutions of (*) that belong only locallyto the corresponding Sobolev spaces in the half-space x Rn. It also requires no conditionsfor the behavior of solutions of (*) on the hyperplane t = 0. The existence of critical blow-up exponents is obtained forsolutions of (*) as a special case of a comparison principlefor the corresponding solutions of (*). For example, the well-knownFujita result is a consequence of the comparison principle. The approach developed in the paper is directly applicable tothe study of analogous problems involving nonlinear differentialoperators. Its elliptic analogue has been recently developedby the authors.  相似文献   

8.
Spaces of Harmonic Functions   总被引:1,自引:0,他引:1  
It is important and interesting to study harmonic functionson a Riemannian manifold. In an earlier work of Li and Tam [21]it was demonstrated that the dimensions of various spaces ofbounded and positive harmonic functions are closely relatedto the number of ends of a manifold. For the linear space consistingof all harmonic functions of polynomial growth of degree atmost d on a complete Riemannian manifold Mn of dimension n,denoted by Hd(Mn), it was proved by Li and Tam [20] that thedimension of the space H1(M) always satisfies dimH1(M) dimH1(Rn)when M has non-negative Ricci curvature. They went on to askas a refinement of a conjecture of Yau [32] whether in generaldim Hd(Mn) dimHd(Rn)for all d. Colding and Minicozzi made animportant contribution to this question in a sequence of papers[5–11] by showing among other things that dimHd(M) isfinite when M has non-negative Ricci curvature. On the otherhand, in a very remarkable paper [16], Li produced an elegantand powerful argument to prove the following. Recall that Msatisfies a weak volume growth condition if, for some constantA and , (1.1) for all x M and r R, where Vx(r) is the volume of the geodesicball Bx(r) in M; M has mean value property if there exists aconstant B such that, for any non-negative subharmonic functionf on M, (1.2) for all p M and r > 0.  相似文献   

9.
One of the most famous theorems in number theory states thatthere are infinitely many positive prime numbers (namely p =2 and the primes p 1 mod4) that can be represented in the formx21+x22, where x1 and x2 are positive integers. In a recentpaper, Fouvry and Iwaniec [2] have shown that this statementremains valid even if one of the variables, say x2, is restrictedto prime values only. In the sequel, the letter p, possiblywith an index, is reserved to denote a positive prime number.As p21=p22 = p is even for p1, p2 > 2, it is reasonable toconjecture that the equation p21=p22 = 2p has an infinity ofsolutions. However, a proof of this statement currently seemsfar beyond reach. As an intermediate step in this direction,one may quantify the problem by asking what can be said aboutlower bounds for the greatest prime divisor, say P(N), of thenumbers p21=p22, where p1, p2 N, as a function of the realparameter N 1. The well-known Chebychev–Hooley methodcombined with the Barban–Davenport–Halberstam theoremalmost immediately leads to the bound P(N) N1–, if N No(); here, denotes some arbitrarily small fixed positivereal number. The first estimate going beyond the exponent 1has been achieved recently by Dartyge [1, Théorème1], who showed that P(N) N10/9–. Note that Dartyge'sproof provides the more general result that for any irreduciblebinary form f of degree d 2 with integer coefficients the greatestprime divisor of the numbers |f(p1, p2)|, p1, p2 N, exceedsNd, where d = 2 – 8/(d = 7). We in particular wantto point out that Dartyge does not make use of the specificfeatures provided by the form x21+x22. By taking advantage ofsome special properties of this binary form, we are able toimprove upon the exponent 2 = 10/9 considerably.  相似文献   

10.
Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let ph(x) = inf||–1(Ricx(,)–2Hess(hx(,)).Here Ricx denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if hph<0. Here h =:+2Lh is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.  相似文献   

11.
Shapiro's cyclic sum is defined by , If K is the cone in Rn of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T2, where T is the non-linear operatordefined by (Tx)i = xni+1/(xni+2 + xni+3)2for i = 1,2,...,n. It is conjectured that Tx = Skx, where Sis the shift operator in Rn, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, ai(x) = ani+1–k(x) fori = 1,2,...,n.  相似文献   

12.
This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation ut = (um)– up in Rn x (0,) with (1 – 2/n)+ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of ut = (um) with an additional logarithmicanomalous decay exponent in time as t .  相似文献   

13.
Every compact, connected PL manifold Mn, with MnØ, collapsesto a codimension-one subpolyhedron Qn–1, called a spineof Mn. The purpose of this paper is to prove that, if Qn–1is appropriately chosen, one can reconstruct Mn from Qn–1,after taking the Cartesian product with an interval I=[0, 1].  相似文献   

14.
Let Fn be the free group of rank n with basis x1, x2, ..., xn,and let d(G) denote the minimal number of generators of thefinitely generated group G. Suppose that n d(G). There existsan exact sequence and wemay view the free abelian group as a right ZG-module by defining (rR')g = rg–1R' for allg G, where g–1 is any preimage of g under , and = (g–1)–1 r(g–1),the conjugate of r by g–1. We call the relation module of G associated with the presentation(1), and say that has ambient rank n. Furthermore, we call the group Fn/R' the free abelianizedextension of G associated with (1). 1991 Mathematics SubjectClassification 20F05, 20C07.  相似文献   

15.
This paper examines the norms of the projections from C[–1,1] to PM[–1, 1] which are obtained by truncating Chebyshevand ultraspherical expansions after n terms. The norms are evaluatednumerically for n = 1, 2, 3 , ... , 10 and –0.1 5 wherethe ultraspherical weight function is (1 – x2)x.In particular the data show that the norm of the Chebyshev projectionis not the smallest in this range for 1 n 10.  相似文献   

16.
In this paper we consider the modified successive overrelaxation(MSOR)methodto appropriate the solution of the linear system D-1/2 Ax =D-1/2b, where A is a symmetric, positive definite and consistentlyordered matrix and D is a diagonal matrix with the diagonalidentical to that of A. The main purpose of this paper is to obtain some theoreticalresults, namely a bound for the norm of n = v –vn in termsof the norms nvn-1, n+1 –vn and their inner product,where v =D-1/2 x and vn is the nth iteration vector, obtainedusing the (MSOR)method.  相似文献   

17.
It is shown that the asymptotic behaviour of the coefficientsan at high order n and at large wave steepness ak is determinedmainly by the limiting form of the wave crest. In a lower rangeof n, an, decreases like n, corresponding to the Stokes120° corner flow. In an upper range, an, decreases exponentiallywith n. The transition occurs when n3 is O(1) where is relatedto the steepness ak of the waves by 2 = 2.0[(ak)maxak].  相似文献   

18.
Let (an)n0 be a sequence of complex numbers, and, for n0, let A number of results are proved relating the growth of the sequences(bn) and (cn) to that of (an). For example, given p0, if bn= O(np and for all > 0,then an=0 for all n > p. Also, given 0 < p < 1, then for all > 0 if and onlyif . It is further shown that, given rß > 1, if bn,cn=O(rßn), then an=O(n),where , thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredientsof the proogs are a Phragmén-Lindelöf theorem forentire functions of exponential type zero, and an estimate forthe expected value of e(X), where X is a Poisson random variable.2000 Mathematics Subject Classification 05A10 (primary), 30D15,46H05, 60E15 (secondary).  相似文献   

19.
Hormander's Hp Multiplier Theorem for the Heisenberg Group   总被引:1,自引:0,他引:1  
Let Hn denote the (2n+1)-dimensional Heisenberg group. Givenan operator-valued function M, define the operator TM with ‘^’ denoting theFourier transform. Hörmander-type sufficient conditionsare determined on M for the Hp-boundedness, p 1, of the operatorTM on Hn.  相似文献   

20.
Let A be a commutative ring. A graded A-algebra U = n0 Un isa standard A-algebra if U0 = A and U = A[U1] is generated asan A-algebra by the elements of U1. A graded U-module F = n0Fnis a standard U-module if F is generated as a U-module by theelements of F0, that is, Fn = UnF0 for all n 0. In particular,Fn = U1Fn–1 for all n 1. Given I, J, two ideals of A,we consider the following standard algebras: the Rees algebraof I, R(I) = n0Intn = A[It] A[t], and the multi-Rees algebraof I and J, R(I, J) = n0(p+q=nIpJqupvq) = A[Iu, Jv] A[u, v].Consider the associated graded ring of I, G(I) = R(I) A/I =n0In/In+1, and the multi-associated graded ring of I and J,G(I, J) = R(I, J) A/(I+J) = n0(p+q=nIpJq/(I+J)IpJq). We canalways consider the tensor product of two standard A-algebrasU = p0Up and V = q0Vq as a standard A-algebra with the naturalgrading U V = n0(p+q=nUp Vq). If M is an A-module, we havethe standard modules: the Rees module of I with respect to M,R(I; M) = n0InMtn = M[It] M[t] (a standard R(I)-module), andthe multi-Rees module of I and J with respect to M, R(I, J;M) = n0(p+q=nIpJqMupvq) = M[Iu, Jv] M[u, v] (a standard R(I,J)-module). Consider the associated graded module of M withrespect to I, G(I; M) = R(I; M) A/I = n0InM/In+1M (a standardG(I)-module), and the multi-associated graded module of M withrespect to I and J, G(I, J; M) = R(I, J; M) A/(I+J) = n0(p+q=nIpJqM/(I+J)IpJqM)(a standard G(I, J)-module). If U, V are two standard A-algebras,F is a standard U-module and G is a standard V-module, thenF G = n0(p+q=nFp Gq) is a standard U V-module. Denote by :R(I) R(J; M) R(I, J; M) and :R(I, J; M) R(I+J;M) the natural surjective graded morphisms of standard RI) R(J)-modules. Let :R(I) R(J; M) R(I+J; M) be . Denote by :G(I) G(J; M) G(I, J; M) and :G(I, J; M) G(I+J; M) the tensor productof and by A/(I+J); these are two natural surjective gradedmorphisms of standard G(I) G(J)-modules. Let :G(I) G(J; M) G(I+J; M) be . The first purpose of this paper is to prove the following theorem.  相似文献   

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