A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweil?s theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweilʼs theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

Invariance principles for Diophantine approximation of formal Laurent series over a finite base field

In a recent paper, the first and third author proved a central limit theorem for the number of coprime solutions of the Diophantine approximation problem for formal Laurent series in the setting of the classical theorem of Khintchine. In this note, we consider a more general setting and show that even an invariance principle holds, thereby improving upon earlier work of the second author. Our result yields two consequences: (i) the functional central limit theorem and (ii) the functional law of the iterated logarithm. The latter is a refinement of Khintchine's theorem for formal Laurent series. Despite a lot of research efforts, the corresponding results for Diophantine approximation of real numbers have not been established yet.     

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).     

An analogue of a theorem of Szüsz for formal Laurent series over finite fields

About 40 years ago, Szüsz proved an extension of the well-known Gauss-Kuzmin theorem. This result played a crucial role in several subsequent papers (for instance, papers due to Szüsz, Philipp, and the author). In this note, we provide an analogue in the field of formal Laurent series and outline applications to the metric theory of continued fractions and to the metric theory of diophantine approximation.     

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.     

A note on badly approximabe sets in projective space

Recently, Ghosh and Haynes (J Reine Angew Math 712:39–50, 2016) proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarník-type result also holds for ‘badly approximable’ points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine’s theorem for convergence relating to ‘intrinsic’ approximation of points in these compact sets.     

Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI(T-ξ,K) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399–406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307–317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321–337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55–71].     

Note on Diophantine inequality and Linear Artin Approximation over a local ring

G. Rond [G. Rond, Approximation diophantienne dans les corps de séries en plusieurs variables, Ann. Institut Fourier 56 (2) (2006) 299–308. [10]] has proved Linear version of Artin Approximation theorem (LAA) and Diophantine inequality for a single homogeneous polynomial equation in two unknowns with coefficients in a formal (or convergent) power series ring over a field. M. Hickel and H. Ito, S. Izumi have generalized Rond's result to certain good local domains, independently, in 2008. This is a complementary Note to theirs. The most important point is that we can delete the equicharacteristic assumption in both papers. To cite this article: M. Hickel et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p

The aim of this paper is to give a geometric interpretation of the continued fraction expansion in the field of formal Laurent series in X ^–1 over , in terms of the action of the modular group on the Bruhat–Tits tree of , and to deduce from it some corollaries for the diophantine approximation of formal Laurent series in X ^–1 by rational fractions in X.     

An uncertainty inequality for Fourier-Dunkl series

An uncertainty inequality for the Fourier-Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939-2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.     

Formal Laurent series in several variables

We explain the construction of fields of formal infinite series in several variables, generalizing the classical notion of formal Laurent series in one variable. Our discussion addresses the field operations for these series (addition, multiplication, and division), the composition, and includes an implicit function theorem.     

Selections and countably-approachable points

The present paper improves a result of Gutev [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by characterizing the countably-approachable points in sense of [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by a natural extreme-like condition in the spirit of [V. Gutev, T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. 129 (2001) 2809–2815; V. Gutev, T. Nogura, Selection pointwise-maximal spaces, Topology Appl. 146–147 (2005) 397–408]. This demonstrates the natural relationship between different extreme-like points with respect to continuous selections for the Vietoris hyperspace of nonempty closed subsets.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

Invariant approximations in CAT(0) spaces

Some common fixed point and invariant approximation results for CAT(0) spaces are obtained. Our results improve and extend some results of Shahzad and Markin [N. Shahzad, J. Markin, Invariant approximation for commuting mappings in hyperconvex and CAT(0) spaces, J. Math. Anal. Appl. 337 (2008) 1457–1464] and Dhompongsa, Kaewkhao, and Panyanak [S. Dhompongsa, A. Kaewkhao, B. Panyanak, Lim’s theorem for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl. 312 (2005) 478–487].     

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).     

Factorization of multivariate positive Laurent polynomials

Recently Dritschel proved that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Fejér–Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting.     

Weak subdifferential for set-valued mappings and its applications

In this paper, the existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (2) (1998) 187–200] and Theorem 4.1 in [J.W. Peng, H.W.J. Lee, W.D. Rong, X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005) 281–297], respectively, are proved by virtue of a Hahn–Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.     

Denisov's theorem on recurrence coefficients

Recently Denisov (aka Dennisov) (Proc. Amer. Math. Soc.) has proved the following remarkable extension of Rakhmanov's theorem (Math. USSR-Sb. 46 (1983) 105; Russian Original, Mat. Sb. 118 (1982) 104) (see also (Mate et al., Constr. Approx., 1 (1985) 63; Nevai, J. Approx. Theory 65 (1991) 322)) which was conjectured in (Nevai, in: Approximation Theory IV, Vol. II, Academic Press, New York, 1989, pp. 449–489, Conjecture 2.7, p. 453).     

Metric properties and exceptional sets of β-expansions over formal Laurent series

This paper is concerned with the metric properties of β-expansions over the field of formal Laurent series. We will see that there are essential differences between β-expansions of the formal Laurent series case and the classical real case. Also the Hausdorff dimensions of some exceptional sets, with respect to the Haar measure, are determined.