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1.
2.
The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ2. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the
discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.The first author is supported by the Australian Research Council under its Center of Excellence Program.The second author is supported by the Austrian Research Foundation (FWF), Project S 8305 and Project P17022-N12. 相似文献
3.
Ligia L. Cristea Josef Dick Gunther Leobacher Friedrich Pillichshammer 《Numerische Mathematik》2007,105(3):413-455
In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives
are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t,m,s)-nets over which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the
root mean square worst-case error converges with order for any ɛ > 0, where 2
m
is the number of points. A similar result for lattice rules has previously been shown by Hickernell.
Ligia L. Cristea is supported by the Austrian Research Fund (FWF), Project P 17022-N 12 and Project S 9609.
Josef Dick is supported by the Australian Research Council under its Center of Excellence Program.
Gunther Leobacher is supported by the Austrian Research Fund (FWF), Project S 8305.
Friedrich Pillichshammer is supported by the Austrian Research Fund (FWF), Project P 17022-N 12, Project S 8305 and Project
S 9609. 相似文献
4.
Steven J. Altschuler Lang F. Wu 《Calculus of Variations and Partial Differential Equations》1994,2(1):101-111
In this work, we study surfaces over convex regions in 2 which are evolving by the mean curvature flow. Here, we specify the angle of contact of the surface to the boundary cylinder. We prove that solutions converge to ones moving only by translation.Partially supported by the NSF grant no: DMS-9100383Partially supported by the NSF grant no: DMS 9108269.A01 相似文献
5.
Gerhard Larcher Harald Niederreiter Wolfgang Ch. Schmid 《Monatshefte für Mathematik》1996,121(3):231-253
A detailed study of digital (t, m, s)-nets and digital (T,s)-sequences constructed over finite rings is carried out. We present general existence theorems for digital nets and sequences and also explicit constructions. Special attention is devoted to the case where the finite ring is a residue class ring of the integers. This study is motivated by the problem of numerical integration of multivariate Walsh series by quasi-Monte Carlo methods, for which we also provide a general error bound.The third author was supported by the CEI Project PACT, WP5.1.2.1.3. 相似文献
6.
Roswitha Hofer Gerhard Larcher Friedrich Pillichshammer 《Monatshefte für Mathematik》2008,154(3):199-230
We introduce a generalized weighted digit-block-counting function on the nonnegative integers, which is a generalization of
many digit-depending functions as, for example, the well known sum-of-digits function. A formula for the first moment of the
sum-of-digits function has been given by Delange in 1972. In the first part of this paper we provide a compact formula for
the first moment of the generalized weighted digit-block-counting function and show that a (weak) Delange type formula holds
if the sequence of weights converges. The question, whether the converse is true as well, can only be answered partially at
the moment.
In the second part of this paper we study distribution properties of generalized weighted digit-block-counting sequences and
their d-dimensional analogues. We give an if and only if condition under which such sequences are uniformly distributed modulo one.
Roswitha Hofer, Recipient of a DOC-FFORTE-fellowship of the Austrian Academy of Sciences at the Institute of Financial Mathematics
at the University of Linz (Austria).
Friedrich Pillichshammer, Supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian
National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.
Dedicated to Prof. Robert F. Tichy on the occasion of his 50th birthday
Authors’ address: Roswitha Hofer, Gerhard Larcher and Friedrich Pillichshammer, Institut für Finanzmathematik, Universit?t
Linz, Altenbergerstra?e 69, A-4040 Linz, Austria 相似文献
7.
On the distribution of irreducible algebraic integers 总被引:1,自引:0,他引:1
Jerzy Kaczorowski 《Monatshefte für Mathematik》2009,156(1):47-71
We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic
number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate
for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this
is always the case when h = 2.
Author’s address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań,
Poland
The author was supported in part by KBN Grant # N N201 1482 33. 相似文献
8.
We present a method to estimate the L
2-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method
to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov.
(Received 14 September 2000) 相似文献
9.
Jirří Matousšek 《Monatshefte für Mathematik》1999,127(4):325-336
We consider so-called Tusnády’s problem in dimension d: Given an n-point set P in R
d
, color the points of P red or blue in such a way that for any d-dimensional interval B, the number of red points in differs from the number of blue points in by at most Δ, where should be as small as possible. We slightly improve previous results of Beck, Bohus, and Srinivasan by showing that , with a simple proof. The same asymptotic bound is shown for an analogous problem where B is allowed to be any translated and scaled copy of a fixed convex polytope A in R
d
. Here the constant of proportionality depends on A and we give an explicit estimate. The same asymptotic bounds also follow for the Lebesgue-measure discrepancy, which improves
and simplifies results of Beck and of Károlyi.
Received 17 November 1997; in revised form 30 July 1998 相似文献
10.
R. Nair 《Monatshefte für Mathematik》2001,132(4):341-348
Let ? be a class of real valued integrable functions on [0,1). We will call a strictly increasing sequence of natural numbers
an sequence if for every f in ? we have
almost everywhere with respect to Lebesgue measure. Here, for a real number y we have used to denote the fractional part of y. For a finite set A we use to denote its cardinality. In this paper we show that for strictly increasing sequences of natural numbers and , both of which are sequences for all , if there exists such that
then the sequence of products of pairs of elements in a and b once ordered by size is also an sequence.
(Received 2 March 2000; in revised form 3 January 2001) 相似文献
11.
G. S. Weiss 《Calculus of Variations and Partial Differential Equations》2003,17(3):311-340
The equation where converges to the Dirac measure concentrated at with mass has been used as a model for the propagation of flames with high activation energy. For initial data that are bounded in
and have a uniformly bounded support, we study non-negative solutions of the Cauchy problem in as We show that each limit of is a solution of the free boundary problem in on (in the sense of domain variations and in a more precise sense). For a.e. time t the graph of u(t) has a unique tangent cone at -a.e. The free boundary is up to a set of vanishing measure the sum of a countably n-1-rectifiable set and of the set on which vanishes in the mean. The non-degenerate singular set is for a.e. time a countably n-1-rectifiable set. As key tools we introduce a monotonicity formula and, on the singular set, an estimate for the parabolic
mean frequency.
Received: 8 August 2001 / Accepted: 8 May 2002 / Published online: 5 September 2002
RID="a"
ID="a" Partially supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Japan. 相似文献
12.
We estimate the linear complexity profile of m-ary sequences in terms of their correlation measure, which was introduced by Mauduit and Sárközy. For prime m this is a direct extension of a result of Brandstätter and the second author. For composite m, we define a new correlation measure for m-ary sequences, relate it to the linear complexity profile and estimate it in terms of the original correlation measure. We apply our results to sequences of discrete logarithms modulo m and to quaternary sequences derived from two Legendre sequences. 相似文献
13.
Elliptic curve analogue of Legendre sequences 总被引:1,自引:0,他引:1
Zhixiong Chen 《Monatshefte für Mathematik》2008,154(1):1-10
The Legendre symbol is applied to the rational points over an elliptic curve to output a family of binary sequences with strong
pseudorandom properties. That is, both the well-distribution measure and the correlation measure of order k, which are evaluated by using estimation of certain character sums along elliptic curves, of the resulting binary sequences
are “small”. A lower bound on the linear complexity profile of these sequences is also presented. Our results indicate that
the behavior of such sequences is very similar to that of the Legendre sequences.
Research partially supported by the Science and Technology Foundation of Putian City (No. 2005S04), the Open Funds of Key
Lab of Fujian Province University Network Security and Cryptology (No. 07B005) and the Foundation of the Education Department
of Fujian Province (No. JA07164).
Author’s addresses: Department of Mathematics, Putian University, Putian, Fujian 351100, China; and Key Lab of Network Security
and Cryptology, Fujian Normal University, Fuzhou, Fujian 350007, China 相似文献
14.
By a well known result of Philipp (1975), the discrepancy D
N
(ω) of the sequence (n
k
ω)
k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n
k + 1/n
k
≥ q > 1 (k = 1, 2, …). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions
on (n
k
), for subexpenentially growing (n
k
), but in general the behavior of (n
k
ω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties
of the sequence (n
k
), in this paper we prove a law of the iterated logarithm for the discrepancy D
N
(ω) for subexponentially growing (n
k
) without number theoretic assumptions.
C. Aistleitner, Research supported by FWF grant S9603-N13.
I. Berkes, Research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961.
Authors’ addresses: C. Aistleitner, Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz,
Austria; I. Berkes, Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010 Graz, Austria 相似文献
15.
An explicit estimate for the lattice point discrepancy of ellipsoids of rotation. For the lattice point discrepancy
(i.e., the number of integer points minus the volume) of the ellipsoid (u
1
2 + u
2
2)/a + a
2
u
3
2 ≤ x (a, x > 0), this paper provides an estimate of the form
terms of smaller order in x.
Die Autoren danken dem ?sterreichischen Fonds zur F?rderung der wissenschaftlichen Forschung (FWF) für finanzielle Unterstützung
unter der Projekt-Nr. P18079-N12. 相似文献
16.
Karl Entacher 《Monatshefte für Mathematik》2000,130(2):99-108
We apply the Haar function system to estimate the star-discrepancy of special digital (t,m,s)-nets in dimension . We use a basic technique based on discretization combined with an exact calculation of the discrete star-discrepancy.
(Received 30 August 1999; in revised form 17 January 2000) 相似文献
17.
In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation
for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small .
Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002
RID="*"
ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China 相似文献
18.
Jun Wu 《Monatshefte für Mathematik》2008,19(1):83-87
For an irrational number x and n ≥ 1, we denote by k
n
(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x. G. Lochs proved that for almost all x, with respect to the Lebesgue measure
In this paper, we prove that an iterated logarithm law for {k
n
(x): n ≥ 1}, more precisely, for almost all x,
for some constant σ > 0.
Author’s address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China 相似文献
19.
We consider the function μ(G), introduced by W. Narkiewicz, which associates to an abelian group G the maximal cardinality of a half-factorial subset of it. In this article, we start a systematic study of this function in
the case where G is a finite cyclic group and prove several results on its behaviour. In particular, we show that the order of magnitude of
this function on cyclic groups is the same as the one of the number of divisors of its cardinality.
This work was supported by the Austrian Science Fund FWF (Project P16770-N12) and by the Austrian-French Program ``Amadeus
2003–2004'. 相似文献
20.
Wen-Bin Zhang 《Monatshefte für Mathematik》2003,138(4):319-353
We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory.
Received October 18, 2001; in final form April 11, 2002 相似文献