Multidimensional continued fractions and a Minkowski function

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     

Learning from dependent observations

In most papers establishing consistency for learning algorithms it is assumed that the observations used for training are realizations of an i.i.d. process. In this paper we go far beyond this classical framework by showing that support vector machines (SVMs) only require that the data-generating process satisfies a certain law of large numbers. We then consider the learnability of SVMs for α-mixing (not necessarily stationary) processes for both classification and regression, where for the latter we explicitly allow unbounded noise.     

Invariant (α, β)-metrics on homogeneous manifolds

In this paper we study invariant (α, β)-metrics on homogeneous spaces. We first give a method to construct invariant (α, β)-metrics on homogeneous spaces. Then we obtain some conditions for some special type of (α, β)-metrics to be of Berwald type and Douglas type. At last, we give a rigidity result concerning the Randers metrics and Matsumoto metrics of Berwald type on homogeneous spaces. Members of LPMC and supported by NSFC (no. 10671096) and NCET of China. Second author was corresponding author. Authors’ address: Huihui An and Shaoqiang Deng, School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China     

Normality of orbit closures for Dynkin quivers¶of type ?n

It is known that the orbit closures for the representations of the equioriented Dynkin quivers ? _n are normal and Cohen–Macaulay varieties with rational singularities. In the paper we prove the same for the Dynkin quivers ? _n with arbitrary orientation. Received: 25 October 2000 / Revised version: 28 February 2001     

Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains

Consider the nonstationary Stokes equations in exterior domains with the compact boundary . We show first that the solution decays like for all as . This decay rate is optimal in the sense that for some as occurs if and only if the net force exerted by the fluid on is zero. Received: 15 June 2000 / Published online: 18 June 2001     

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, I

In this paper, we consider mixed problems with a spacelike boundary derivative condition for semilinear wave equations with exponential nonlinearities in a quarter plane. Results similar to those obtained earlier by Caffarelli-Friedman for Cauchy problems and power nonlinearities are proved in the present situation, namely we show that solutions either are global or blow up on a spacelike curve. Weaker results are also obtained if the boundary vector field is tangent to the characteristic which leaves the domain in the future. Received January 7, 2000 / Accepted July 17, 2000 /Published online December 8, 2000     

Infinite paths with bounded or recurrent partial sums

We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or −1, with probability 1 −p. We ask whether an infinite path π exists, with the property that the partial sums of the ±1s along π are uniformly bounded, and whether there exists an infinite path π' with the property that the partial sums along π' are equal to zero infinitely often. The answers to these question depend on the type of path one allows, the value of p and the uniform bound specified. We show that phase transitions occur for these phenomena. Moreover, we make a surprising connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each vertex at most finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson process in the plane. For the recurrence problem, we also show that the probability of finding such a path is monotone in p, for p≥?. Received: 10 January 2000 / Revised version: 14 August 2000 / Published online: 9 March 2001     

Some results on supersolvability of finite groups

We get new characterizations of finite supersolvable groups based on the structure of the generalized Fitting subgroup. Then we extend our results to the saturated formations containing the class of all supersolvable groups. Authors’ addresses: M. Asaad, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt; Piroska Cs?rg?, Department of Algebra and Number Theory, E?tv?s University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary     

The $H^{\infty}-$calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an calculus. We then apply this to the joint functional calculus of two sectorial operators when one has an calculus. Using this we prove theorem of Dore-Venni type on sums of sectorial operators and apply our results to the problem of maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of spaces and spaces, to obtain some more detailed results in this setting. Received: 3 July 2000 / Revised version: 31 January 2001 / Published online: 23 July 2001