形式Laurent级数域上交错Oppenheim展开的研究

该文介绍了形式Laurent级数域上交错Oppenheim展开的算法,得到了该展开中数字的强(弱)大数定理、中心极限定理和重对数率,并且研究了这些级数部分和的逼近的度.     

一类Oppenheim展式的例外关系集

本文研究了Oppenheim展式中一类例外关系集的Hausdorff维数,作为其应用,我们得到了Lüroth级数展式中一些集合的Hausdorff维数的确切值,并给出了这些确切值的一个估计式     

Approximation orders of formal Laurent series by Oppenheim rational functions

We study formal Laurent series which are better approximated by their Oppenheim convergents. We calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations are faster than the approximation of typical Laurent series (with respect to the Haar measure).     

Cantor sets determined by partial quotients of continued fractions of Laurent series

In this paper, two types of general sets determined by partial quotients of continued fractions over the field of formal Laurent series with coefficients from a given finite field are studied. The Hausdorff dimensions of and are determined completely, where A_n(x) denotes the partial quotients in the continued fraction expansion (in case of Laurent series) of x and (n) is a positive valued function defined on natural numbers N.     

On sums of degrees of the partial quotients in continued fraction expansions of Laurent series

For any formal Laurent series with coefficients cn lying in some given finite field, let x=[a₀(x);a₁(x),a₂(x),…] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.     

Inhomogeneous Diophantine approximation over the field of formal Laurent series

De Mathan [B. de Mathan, Approximations diophantiennes dans un corps local, Bull. Soc. Math. France, Suppl. Mém. 21 (1970)] proved that Khintchine's theorem on homogeneous Diophantine approximation has an analogue in the field of formal Laurent series. Kristensen [S. Kristensen, On the well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 135 (2003) 255–268] extended this metric theorem to systems of linear forms and gave the exact Hausdorff dimension of the corresponding exceptional sets. In this paper, we study the inhomogeneous Diophantine approximation over a field of formal Laurent series, the analogue Khintchine's theorem and Jarnik–Besicovitch theorem are proved.     

罗朗级数环的主拟Baer性

称环 R为右主拟 Baer环（简称为右p·q.Baer环）,如果 R的任意主右理想的右零化子可由幂等元生成.本文证明了,若环 R满足条件Sl（R）（?）C（R）,则罗朗级数环R[[x,x-1]]是右p.q.Baer环当且仅当R是右p.q.Baer环且R的任意可数多个幂等元在I（R）中有广义join.同时还证明了,R是右p.q.Baer环当且仅当R[x,x-1]是右P.q.Baer环.     

Laurent Series Ring over Semiperfect Ring Can Not be Semiperfect

One of the main results of the article [2 Sonin , K. I. ( 1996 ). Semiprime and semiperfect rings of Laurent series . Mathematical Notes 60 : 222 – 226 .[Crossref], [Web of Science ®] , [Google Scholar]] says that, if a ring R is semiperfect and ? is an authomorphism of R, then the skew Laurent series ring R((x, ?)) is semiperfect. We will show that the above statement is not true. More precisely, we will show that, if the Laurent series ring R((x)) is semilocal, then R is semiperfect with nil Jacobson radical.     

Approximation orders of formal Laurent series by β-expansions

We prove that almost all (with respect to Haar measure) formal Laurent series are approximated with the linear order −(degβ)n by their β-expansions convergents. Hausdorff dimensions of sets of Laurent series which are approximated by all other orders, are determined. In contrast, the corresponding theory of real case has not been established.     

Hausdorff dimension of some kind of alternating Oppenheim series expansion

For Oppenheim series epansions, the authors of [7] discussed the exceptional sets Bm={x∈（0,1]：1〈dj（x）/h（j-1）（d（j-1）（x））≤m for any j ≥2} In this paper, we investigate the Hausdorff dimension of a kind of exceptional sets occurring in alternating Oppenheim series expansion. As an application, we get the exact Hausdorff dimension of the-set in Luroth series expansion, also we give an estimate of such dimensional number.     

Independence of continued fractions in the field of Laurent series

Criteria for independence, both algebraic and linear, are derived for continued fraction expansions of elements in the field of Laurent series. These criteria are then applied to examples involving elements recently discovered to have explicit series and continued fraction expansions. Communicated by Attila Pethő     

Double Series Identity and Some Laurent Type Hypergeometric Generating Relations

The main aim of this article is to obtain certain Laurent type hypergeometric generating relations. Using a general double series identity, Laurent type generating functions(in terms of Kampéde Fériet double hypergeometric function) are derived. Some known results obtained by the method of Lie groups and Lie algebras, are also modified here as special cases.     

On Metric Diophantine Approximation in the Field of Formal Laurent Series

B. deMathan (1970, Bull. Soc. Math. France Supl. Mem.21) proved that Khintchine’s Theorem has an analogue in the field of formal Laurent series. First, we show that in case of only one inequality this result can also be obtained by continued fraction theory. Then, we are interested in the number of solutions and show under special assumptions that one gets a central limit theorem, a law of iterated logarithm and an asymptotic formula. This is an analogue of a result due to W. J. LeVeque (1958, Trans. Amer. Math. Soc.87, 237–260). The proof is based on probabilistic results for formal Laurent series due to H. Niederreiter (1988, in Lecture Notes in Computer Science, Vol. 330, pp. 191–209, Springer-Verlag, New York/Berlin).     

Multiple Laurent series and fundamental solutions of linear difference equations

Using the notion of fundamental solution, we obtain a solution to the Cauchy problem for a multidimensional homogeneous linear difference equation with constant coefficients.