共查询到20条相似文献,搜索用时 125 毫秒
1.
Takao Komatsu 《Monatshefte für Mathematik》2005,145(1):47-60
The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e1/a (a 1), ae1/a and (1/a)e1/a are included as special cases. The corresponding Tasoev continued fractions are also derived. 相似文献
2.
Edward B. Burger 《Monatshefte für Mathematik》1999,128(3):201-209
Here we provide a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to
insure that they are elements of the same quadratic number field over ℚ. Such a condition has implications to simultaneous
diophantine approximation. In particular, we deduce an improvement to Dirichlet’s Theorem in this context which, as an immediate
consequence, shows the Littlewood Conjecture to hold for all numbers α and β both from . Specifically, for all such pairs we have .
(Received 10 August 1998; in revised form 23 November 1998) 相似文献
3.
In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.This work was supported by Yonsei University Research Fund of 2003. 相似文献
4.
J. Schoissengeier 《Monatshefte für Mathematik》2000,129(2):147-151
Let x be a real number, α an irrational number with continued fraction expansion and convergents a positive integer, be chosen such that the fractional part of x and J. Beck [1] proved that We give a shorter proof, thereby using Dedekind sums. We may (and do) assume w.l.o.g. that . There are non-negative integers such that and (the so-called Ostrowski-expansion of N with respect to α). Let us put for and, for . Then . Let us finally put and, for and . (Received 1 March 1999) 相似文献
5.
The aim of this paper is to study the moduli space of solutions of the Dirichlet problem associated to the equation of Monge-Ampeère
type, det , on an exterior planar domain. We prove that this moduli space is either empty or a differentiable manifold of dimension
five.
(Received 23 April 1999) 相似文献
6.
Summary. In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau
equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann
equation in the spirit of [21,23]. We show that the kernel modes that define the spectral method have the correct grazing
collision limit providing a consistent spectral method for the limiting Fokker-Planck-Landau equation. In particular, for
small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cut-off Boltzmann equation
which, similarly to the Fokker-Planck-Landau case, can be computed with a fast algorithm. The uniform spectral accuracy of
the method with respect to the grazing collision parameter is also proved.
Received July 10, 2001 / Revised version received October 12, 2001 / Published online January 30, 2002 相似文献
7.
8.
Wendelin Werner 《Probability Theory and Related Fields》1997,108(1):131-152
Summary. We study the asymptotic behaviour of disconnection and non-intersection exponents for planar Brownian motionwhen the number
of considered paths tends to infinity. In particular, if η
n
(respectively ξ (n, p)) denotes the disconnection exponent for n paths (respectively the non-intersection exponent for n paths versus p paths), then we show that lim
n →∞
η
n
/n = 1 2 and that for a > 0 and b > 0,lim
n →∞
ξ ([na],[nb])/n = (√ a + √ b)
2
/2.
Received: 28 February 1996 / In revised form: 3 September 1996 相似文献
9.
Ryan I. Fernandes 《Numerische Mathematik》1997,77(2):223-241
Summary. Piecewise Hermite bicubic orthogonal spline collocation Laplace-modified and alternating-direction schemes for the approximate
solution of linear second order hyperbolic problems on rectangles are analyzed. The schemes are shown to be unconditionally
stable and of optimal order accuracy in the and discrete maximum norms for space and time, respectively. Implementations of the schemes are discussed and numerical results
presented which demonstrate the accuracy and rate of convergence using various norms.
Received November 7, 1994 / Revised version received April 29, 1996 相似文献
10.
When trains of impulse controls are present on the right-hand side of a system of ordinary differential equations, the solution
is no longer smooth and contains jumps which can accumulate at several points in the time interval. In technological and physical
systems the sum of the absolute value of all the impulses is finite and hence the total variation of the solution is finite.
So the solution at best belongs to the space BV of vector functions with bounded variation. Unless variable node methods are
used, the loss of smoothness of the solution would a priori make higher-order methods over a fixed mesh inactractive. Indeed
in general the order of -convergence is and the nodal rate is . However in the linear case -convergence rate remains but the nodal rate can go up to by using one-step or multistep scheme with a nodal rate up to when the solution belongs to . Proofs are given of error estimates and several numerical experiments confirm the optimality of the estimates.
Received March 15, 1996 / Revised version received January 3, 1997 相似文献
11.
P. Dutt 《Numerische Mathematik》1999,81(3):323-344
Summary. In this paper we consider hyperbolic initial boundary value problems with nonsmooth data. We show that if we extend the time
domain to minus infinity, replace the initial condition by a growth condition at minus infinity and then solve the problem
using a filtered version of the data by the Galerkin-Collocation method using Laguerre polynomials in time and Legendre polynomials
in space, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth.
For this we have to perform a local smoothing of the computed solution.
Received August 1, 1995 / Revised version received August 19, 1997 相似文献
12.
Julien Bernat 《Discrete Mathematics》2006,306(22):2828-2850
Let φ be the golden ratio. We define and study a continued φ-fraction algorithm, inspired by Euclid's algorithm. We show that any non-negative element of Q(φ) has a finite continued φ-fraction. 相似文献
13.
Algebroid functions, Wirsing's theorem and their relations 总被引:1,自引:0,他引:1
Min Ru 《Mathematische Zeitschrift》2000,233(1):137-148
In this paper, we first point out a relationship between the Second Main Theorem for algebriod functions in Nevanlinna theory
and Wirsing's theorem in Diophantine approximation. This motivates a unified proof for both theorems. The second part of this
paper deals with “moving targets” problem for holomorphic maps to Riemann surfaces. Its counterpart in Diophantine approximation
follows from a recent work of Thomas J. Tucker. In this paper, we point out Tucker's result in the special case of the approximation
by rational points could be obtained by doing a “translation” and applying the corresponding result with fixed target. However,
we could not completely recover Tucker's result concerning the approximation by algebraic points. In the last part of this
paper, cases in higher dimensions are studied. Some partial results in higher dimensions are obtained and some conjectures
are raised.
Received August 26, 1997; in final form June 30, 1998 相似文献
14.
We prove that the Osgood-Thue Theorem, about Diophantine Approximation in function fields, holds under a more general condition
when the ground field is finite.
Received 28 January 1998 相似文献
15.
We study the metrical theory of fibred systems, in particular, in the case of continued fraction mixing systems. We get the
limit distribution of the largest value of a continued fraction mixing stationary stochastic process with infinite expectation
and some related results. These are analogous to J. Galambos, W. Philipp, and H. G. Diamond–J. D. Vaaler theorems for the
regular continued fractions. As an application, we see that these theorems hold for Jacobi-Perron algorithm.
Received September 30, 2001; in revised form January 8, 2002 相似文献
16.
Summary. For evolution equations with a strongly monotone operator we derive unconditional stability and discretization error estimates valid for all . For the -method, with , we prove an error estimate , if , where is the maximal integration step for an arbitrary choice of sequence of steps and with no assumptions about the existence
of the Jacobian as well as other derivatives of the operator , and an optimal estimate under some additional relation between neighboring steps. The first result is an improvement over the implicit midpoint method
, for which an order reduction to sometimes may occur for infinitely stiff problems. Numerical tests illustrate the results.
Received March 10, 1999 / Revised version received April 3, 2000 / Published online February 5, 2001 相似文献
17.
Giovanni Panti 《Monatshefte für Mathematik》2008,154(3):247-264
The Minkowski question mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates
the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the M?nkemeyer continued fraction algorithm with an appropriate
tent map.
Author’s address: Department of Mathematics, University of Udine, via delle Scienze 208, 33100 Udine, Italy 相似文献
18.
Igor Moret 《Numerische Mathematik》1994,68(3):341-353
Summary. Certain types
of singular solutions of nonlinear parameter-dependent
operator equations were characterized by
Griewank and Reddien [5, 6] as regular solutions of
suitable augmented systems. For their numerical
approximation an approach based on the use of
Krylov subspaces is here presented. The application
to boundary value problems is illustrated by
numerical examples.
Received March 8, 1993 / Revised version received December 13,
1993 相似文献
19.
Summary.
The ``L--curve' is a plot (in ordinary or doubly--logarithmic scale) of the
norm of (Tikhonov--) regularized solutions of an ill--posed
problem versus the norm of the residuals. We show that the popular criterion
of choosing the parameter corresponding to the point with maximal
curvature of the L--curve does not yield
a convergent regularization strategy to solve the ill--posed problem.
Nevertheless, the L--curve can be used to compute the
regularization
parameters produced by Morozov's discrepancy principle and by an
order--optimal variant of the
discrepancy principle proposed by Engl and Gfrerer in an alternate way.
Received June 29, 1993 /
Revised version received February 2, 1994 相似文献
20.
Summary. In the study of the choice of the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems, Scherzer, Engl and Kunisch proposed an a posteriori strategy in 1993. To prove the optimality of the strategy, they imposed many very restrictive conditions on the problem under consideration. Their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid. In this paper we give a further study on this strategy, and show that Tikhonov regularization is order optimal for each with the regularization parameter chosen according to this strategy under some simple and easy-checking assumptions. This paper weakens the conditions needed in the existing results, and provides a theoretical guidance to numerical experiments. Received August 8, 1997 / Revised version received January 26, 1998 相似文献