8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

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.     

A Nontrivial Product of Filtration s ＋ 5 in the Stable Homotopy of Spheres

In this paper, some groups Ext A^s.t （Zp, Zp） with specialized s and t are first computed by the May spectrM sequence. Then we make use of the Adams spectral sequence to prove the existence of a new nontrivial family of filtration s＋5 in the stable homotopy groups of spheres πpnq＋（s＋3）pq＋（s＋1）q-5S which is represented （up to a nonzero scalar） by β＋2bohh∈ExtA^s＋5,P^nq＋（n＋3）pq＋（n＋1）q＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 5 is a prime number, n ≥3, 0≤ s 〈 p - 3, q = 2（p - 1）.     

On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.     

Hua＇s Theorem on Five Almost Equal Prime Squares

In this paper we sharpen Hua＇s result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N=p1^2＋p2^2＋p3^2＋p4^2＋p5^2,with │pj-√N/5│≤U=N^1/2-1/28＋ε,where pj are primes.     

The Average Widths of Sobolev-Wiener Classes and Besov-Wiener Classes

This paper concerns the problem of average σ-width of Sobolev-Wiener classes W^rpq(R^d),W^rpq(M,R^d),and Besov-Wiener classes S^rpqθb(R^d).S^rpqθB(R^d),S^rpqθb(M,R^d),S^rpqθB(R^d)in the metric Lq(R^d) for 1≤q≤p≤∞.The weak asymptotic results concerning the average linear widths,the average Bernstein widths and the infinite-dimensional Gel‘fand widths are obtained,respectively.     

Waring-Goldbach Problem for Unlike Powers

In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due to C. Bauer.     

Reproducing Spaces and Localization Operators

This paper, by using of windowed Fourier transform (WFT), gives a family of embedding operators Tn:L^2(R)→L^2(C,e^-|z|^2/2dzd-↑z/4πi), s.t. TnL^2(R) lontain in L^2(C,e^-|z|^2/2dzd-↑z/4πi) are reproducing subspaces (n=0, Bargmann Space); and gives a reproducing kernel and an orthonormal basis (ONB) of TnL^2(R), Furthermore, it shows the orthogonal spaces decomposition of L^2(C,e^-|z|^2/2dzd-↑z/4πi). Finally, by using the preceding results, it shows the eigenvalues and eigenfunctions of a class of localization operators associated with WFT, which extends the result of Daubechies in [1] and [6].     

Boundedness of Some Marcinkiewicz Integral Operators Related to Higher Order Commutators on Hardy Spaces

In this paper, the authors study the boundedness properties of μΩ↑m,b generated by the function b ∈Lipβ（R^n）（0 〈β≤ 1/m） and the Marcinkiewicz integrals operator μΩ. The boundednesses are established on the Hardy type spaces Hb^m^p,n（R^n） and the Herz Hardy type spaces Hbm Kq^α,p（R^b）.     

Upper Bounds on Character Sums with Rational Function Entries

We obtain formulac and estimates for character sums of the type S(x,f,p^m)=∑x=1^p^mx(f(f)),where p^m is a prime power with m≥2,x is a multiplicative character(mod p^m),and f=f1/f2 is a rational function over Z.In particular,if p is odd,d=deg(f1) deg(f2) and d^*=max(deg(f1),deg(f2)) then we obtain |S(x,f,p^m)|≤(d-1)p^m(1-1/d^*) for any non-constant f (mod p) and primitive character x.For p=2 an extra factor of 2√2 is needed.     

Universal Similarity Factorization Equalities over Generalized Clifford Algebras

For any element a in a generalized 2^n-dimensional Clifford algebra Lln （F） over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln（F） such that Pn^-1DnPn= φ（α）∈F^2n×2n, where φ（a） is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate.     

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p (\mathbb{S}^{{d - 1}} ) (0 < p ≤ 1)

Let be a unit sphere of the d–dimensional Euclidean space ℝ ^d and let (0 < p ≤ 1) denote the real Hardy space on For 0 < p ≤ 1 and let E _j (f,H ^p ) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space Given a distribution f on its Cesàro mean of order δ > –1 is denoted by For 0 < p ≤ 1, it is known that is the critical index for the uniform summability of in the metric H ^p . In this paper, the following result is proved: Theorem Let 0<p<1 and Then for

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).     

Rigidity Theorem of Hypersurfaces with Constant Scalar Curvature in a Unit Sphere

In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is an n-dimensional complete hypersurface in Sn＋1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations     

Wavelets and Geometric Structure for Function Spaces

Abstract With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel–Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L ¹ are studied. Supported by NNSF of China (Grant No. 10001027)     

On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Let G be a non-empty closed(resp．bounded closed)boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X．Let K(X)denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance．Moreover,let KG(X)denote the closure of the set {A∈K(x)：A∩G=0}．We prove that the set of all A∈KG(X)(resp．A∈K(X)),such that the minimization (resp．maximization)problem min(A,G)(resp．max(A,G))is well posed,contains a dense Gδ-subset of KG(X)(resp．K(X))．thus extending the recent results due to Blasi,Myjak and Papini and Li．     

A Note on Certain Block Spaces on the Unit Sphere

In this note, we clarify a relation between block spaces and the Hardy space. We obtain Bq^0.v belong to H^1（S^n-1）＋L（ln＋L）^1＋v（s^n-1）,v〉-1,q〉1,Furthermore,if v≥ 0, q 〉 1. we verify that block spaces Rq^0.v（S^n-1）are proper subspaces of H1 （S^n- 1）,     

Sums of Five Almost Equal Prime Squares

Let Pi, 1≤i≤5, be prime numbers. It is proved that every integer N that satisfies N=5 （mod 24） can be written as N=p1^2＋p2^2＋P3^2＋p4^2 ＋p5^2, where │√N5-Pi│≤N^1/2-19/850＋∈.     

The Asymptotics of the Heat Semigroup for a General Bounded Domain with Mixed Boundary Conditions

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.