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981.
We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the ai are integers coprime to p and the numbers are distinct . 相似文献
982.
Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem
governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation.
Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive
finite element approximation schemes for the control problem.
Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002
RID="*"
ID="*" Supported by EPSRC research grant GR/R31980 相似文献
983.
We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer
k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,
this last inequality also holding if z ⩾ 2. We then use these inequalities to obtain probabilistic results and we state a conjecture. Finally, using (*), we show
that the probability that the abc conjecture does not hold is 0.
Research supported in part by a grant from NSERC.
Re?u le 17 décembre 2001; en forme révisée le 23 mars 2002
Publié en ligne le 11 octobre 2002 相似文献
984.
Effects of uncertainties in the domain on the solution of Dirichlet boundary value problems 总被引:1,自引:0,他引:1
Summary. A domain with possibly non-Lipschitz boundary is defined as a limit of monotonically expanding or shrinking domains with
Lipschitz boundary. A uniquely solvable Dirichlet boundary value problem (DBVP) is defined on each of the Lipschitz domains
and the limit of these solutions is investigated. The limit function also solves a DBVP on the limit domain but the problem
can depend on the sequences of domains if the limit domain is unstable with respect to the DBVP. The core of the paper consists
in estimates of the difference between the respective solutions of the DBVP on two close domains, one of which is Lipschitz
and the other can be unstable. Estimates for starshaped as well as rather general domains are derived. Their numerical evaluation
is possible and can be done in different ways.
Received October 16, 2001 / Revised version received January 16, 2002 / Published online: April 17, 2002
RID="*"
ID="*" The research was funded partially by the National Science Foundation under the grants NSF–Czech Rep. INT-9724783 and
NSF DMS-9802367
RID="**"
ID="**" Support for Jan Chleboun coming from the Grant Agency of the Czech Republic through grant 201/98/0528 is appreciated 相似文献
985.
986.
987.
In this paper we present an inertia result for Stein equations with
an indefinite right hand side. This result is applied to establish connnections
between the inertia of invertible hermitian block Toeplitz matrices and associated
orthogonal polynomials. 相似文献
988.
Ronald G.?DouglasEmail author Gadadhar?Misra 《Integral Equations and Operator Theory》2003,47(4):435-456
The notion of a quasi-free Hilbert module over a function algebra
$\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m
space is introduced. It is shown that quasi-free Hilbert modules correspond to
the completion of the direct sum of a certain number of copies of the algebra
$\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there
exists a module map from a quasi-free module with dense range (respectively,
onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a
constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and
$\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported
then it is weakly regular. The paper identifies several other classes of Hilbert
modules which are weakly regular. In addition, this result is extended to yield
topologically exact resolutions of such modules by quasi-free ones. 相似文献
989.
We introduce a class of function algebras, that we call unimodular,
and study Toeplitz operators on the Hardy spaces associated to representing
measures on these algebras.We show that our class of function algebras is very
extensive and that a number of important results for Toeplitz operators and
their associated C*-algebras extend to the very general setting we consider.
Submitted: July 1. 2001. 相似文献
990.
Given an admissible measure μ on where is an open set, we define a realization of the Laplacian in with general Robin boundary conditions and we show that generates a holomorphic C
0
-semigroup on which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity
assumption, the generator of each sandwiched semigroup is of the form . We also show that if contains smooth functions, then μ is of the form (where σ is the (n-1)-dimensional Hausdorff measure and β a positive measurable bounded function on ); i.e. we have the classical Robin boundary conditions.
RID="h1"
ID="h1"Dédié à Philippe Bénilan
RID="*"
ID="h1"This work is part of the DGF-Project: "Regularit?t und Asymptotik für elliptische und parabolische Probleme". 相似文献