In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours. 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
We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator. 相似文献
The expansion of f ∈ Lp(0, 1) Fourier series of Bessel functions of order converges to f in Lp whenever Let be the space of p-integrable functions with respect to the measure t dt and where {sn}, n = 1, 2, …, is the set of positive zeros of Jv. Then, the expansion of in a Fourier series of functions ψn, ?1 < ν < ?½, converges to in whenever 相似文献
For each 0 < s < 1, define where , denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with . In this paper we give a nontrivial estimation of c(s), namely, for each 0 < s < 1, where . As an application, we obtain a lower density theorem for Hausdorff measures. 相似文献
We consider random PATRICIA trees constructed from n i.i.d. sequences of independent equiprobable bits. We study the height Hn (the maximal distance between the root and a leaf), and the minimal fill-up level Fn (the minimum distance between the root and a leaf). We give probabilistic proofs of . 相似文献
In this paper a finite element approximation scheme for the system curl is considered. The use of pointwise approximation of the boundary condition leads to a nonconforming method. The error estimate is proved and numerically tested. 相似文献
We consider the half‐linear boundary value problem where and the weight function q is assumed to change sign. We prove the existence of two sequences , of eigenvalues and derive asymptotic estimates for as . 相似文献
Let be an arbitrary integer base and let be the number of different prime factors of with , . Further let be the set of points on the unit circle with finite –adic expansions of their coordinates and let be the set of angles of the points . Then is an additive group which is the direct sum of infinite cyclic groups and of the finite cyclic group . If in case of the points of are arranged according to the number of digits of their coordinates, then the arising sequence is uniformly distributed on the unit circle. On the other hand, in case of the only points in are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions of . 相似文献
We prove convergence laws for logics of the form , where is a properly chosen collection of generalized quantifiers, on very sparse finite random structures. We also study probabilistic collapsing of the logics , where is a collection of generalized quantifiers and k ∈ ℕ+, under arbitrary probability measures of finite structures. 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 study the derived functors of Hom that are computed by using flat resolutions of Hom. These are denoted n. We compare these with the usual Extn's and show that 1⊂ Ext1and indicate (using MacLane's terminology) why the class of associated short exact sequences is a proper class. When the ring is a Dedekind domain we classify the N such that n(–, N) = 0 and show that unlike the situation for other classically defined right derived functors of Hom, Hom is not balanced relative to the two classes of modules that make 1 vanish. 相似文献
Let (ω)(ℝ) denote the non–quasianalytic class of Beurling type on ℝ. For μ, ν ∈ ′(ω)(ℝ) we give necessary conditions for the inclusion Tν( (ω)(ℝ)) ⊂ Tμ( (ω)(ℝ)), thus extending previous work of Malgrange and Ehrenpreis . 相似文献