Let X be a projective algebraic manifold of dimension n (over C), CH1(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A1(X) ? CH1(X) the subgroup of cycles algebraically equivalent to zero. We say that A1(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH1(Γ × X) such that z*:A1(Γ)-A1(X) is surjective. There is the well known Abel-Jacobi map λ1:A1(X)-J(X), where J(X) is the lth Lieberman Jacobian. It is easy to show that A1(X)→J(X) A1(X) finite dimensional. Now set with corresponding map A*(X)→J(X). Also define Level . In a recent book by the author, there was stated the following conjecture: where it was also shown that (?) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A*(X) finite dimensional ?? Level (H*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k-planes on X, where ([…] = greatest integer function) and d = deg X; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given. 相似文献
Let F(θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave number k and direction a by a sound-soft cylinder of cross section D. Suppose F has the Fourier expansion where an = an(k, . Then if ?2 is a Dirichlet eigenvalue for D, sufficient conditions are given on D for the existence of a nontrivial sequence |bn| where the bn are independent of such that for all directions Domains for which this is true are called generalized Herglotz domains. The conditions for a domain to be a generalized Herglotz domain are given either in terms of the Schwarz function for the analytic boundary ?D or in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these conditions. 相似文献
We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??3 with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L2, ?µ(ΩT), initial velocity v0∈H(Ω), µ∈?+\? there exist velocity v∈H(ΩT) and the pressure p, ?p∈L2, ?µ(ΩT) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:
We study the following initial and boundary value problem: In section 1, with u0 in L2(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L2(0, T; H ∩ H2) ∩ L∞(0, T; H) if we assume u0 in H and f in C1(?,?). Numerical results are given for these two cases. 相似文献
The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor-valued function on G satisfying certain restrictions, and let {n} be a sequence of vector-valued functions on G where the L2-norms of {n}, {curl n}, and {div(β n)} are bounded, and where all n either satisfy x n = 0 or (β Fn) = 0 at the boundary ?G of G ( = normal to ?G): then {n} has a L2-convergent subsequence. The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ?G if β is interpreted as electric dielectricity ? or as magnetic permeability μ, respectively. These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations. 相似文献
The d-dimensional Hardy spaces Hp ( T × … × T ) (d = d1 + … + dkand a general summability method of Fourier series and Fourier transforms are introduced with the help of integrable functions θj having integrable Fourier transforms. Under some conditions on θj we show that the maximal operator of the θ-means of a distribution is bounded from Hp ( T × … × T ) to Lp ( T d) where p0 < p < ∞ and p0 < 1 is depending only on the functions θj. By an interpolation theorem we get that the maximal operator is also of weak type ( L1) (i = 1, …, k) where the Hardy space is defined by a hybrid maximal function and if k = 1. As a consequence we obtain that the θ-means of a function (log L)k–1 converge a.e. to the function in question. If k = 1 then we get this convergence result for all f ∈ L1. Moreover, we prove that the θ-means are uniformly bounded on the spaces Hp ( T × … × T ) whenever p0 <p < ∞, thus the θ-means converge to f in ( T × … × T ) norm. The same results are proved for the conjugate θ-means and for d-dimensional Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations. 相似文献
Let t = (t1, …, tn) be a point of ?n. We shall write . We put by definition Rα(u) = u(α?n)/2/Kn(α); here α is a complex parameter, n the dimension of the space, and Kn(α) is a constant. First we evaluate □Rα(u) = Rα(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R?2k(u) = □kδ; R0(u) = δ; and □kR2k(u) = δ, k = 0, 1, …. The first result is the n-dimensional ultrahyperbolic correlative of the well-known one-dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously. 相似文献