Diophantine equations between polynomials obeying second order recurrences

Periodica Mathematica Hungarica -     

On uniform distribution of double sequences

In this paper a new notion of uniform distribution of double sequences is introduced. In the most elementary case a double sequence (x_mn)_{m, nN} is called uniformly distributed in this sense if all finite matrices of equal size appear with the same frequency as submatrices of the infinite matrix (x_mn). Some special double sequences with such a distribution property are studied and metric theorems are proved.     

Operatorenkalkül über freien Monoiden III: Lagrangeinversion und Sheffersysteme

In this part the relations between the theory of binomial systems for noncommuting variables (of part II) and the results ofRota, Cigler et al. for the commutative case are studied in some detail. Afterwards for binomial systems of diagonal type there are given generalizations of the Rodrigues formula and the theorem ofLagrange-Good. A short development of Sheffer sequences follows. Finally the results are extended to binomial systems the structure of which is determined by any partial order of linear type.     

Zur diskrepanz von 0, 1-Folgen

In this paper theorems concerning the distribution of sequences on finite sets are proved. Generalizations of some results due to Meijer, Niederreiter, and Tijdeman are obtained.     

The Sum-of-Digits Function for Complex Bases

We consider digital expansions with respect to complex integerbases. We derive precise information about the length of theseexpansions and the corresponding sum-of-digits function. Furthermorewe give an asymptotic formula for the sum-of-digits functionin large circles and prove that this function is uniformly distributedwith respect to the argument. Finally the summatory functionof the sum-of-digits function along the real axis is analyzed.     

Multidimensional digital searching—alternative data structures

We analyze the average cost of partial match queries in M-dimensional Digital Search Trees and Patricia Tries. Our results allow a precise comparison of the average behavior of these data structures with the M-dimensional Tries studied by Flajolet and Puech [J. Assoc. Comput. Mach., 33 , 371–407 (1986)]. It turns out that Patricia is superior to Digital Search Tres, the latter ones being superior to Tries. © 1994 John Wiley & Sons, Inc.     

Operatorenkalkül über freien Monoiden I: Strukturen

In part I algebraic structures (esp. rings) on the sets of polynomials and formal power series on an at most countable alphabetA are considered. Given a partial order onA the words ofA ^* are mixed together in consistence with it. It is shown that the structures derived are associative iff the given partial order is of linear type. The coefficients appearing at these operations are identified as generalizations of the ordinary binomial coefficients and a number of relations involving them are listed up.(Part II will bring a generalization ofRota's theory of polynomial sequences of binomial type to the structures studied in I.In Part III the theory of special binomial systems will be continued until the analogue of Lagrange inversion and a short development of generalized Sheffer polynomials will be given).     

On the average shape of simply generated families of trees

This paper deals with the limiting distribution of the level number of the jth leaf of a planted plane tree (where leaves are enumerated from left to right) for the so-called “simply generated families” introduced by Meir and Moon. The mathematical apparatus is determined by the idea of an asymptotic analysis of a given sequence of numbers by studying the location and nature of the singularities of appropriate generating functions.     

Hc2-Anisotropy in TaN single crystals

Experiments on the anisotropy of H_c2 in a tantalum single crystal doped with nitrogen in order to change its behaviour from type-I to type-II superconductivity are reported. The results show a disappearance of the anisotropy coefficient a₆ and are compared to the anisotropy in niobium single crystals.     

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα _s (j), which is of orderj ^1/2. Thej-th ”s-hyperoscillation”α ₁(j)−α _s+1(j) is given by 1/2α ₁(s)+O(j ^−1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s ^1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod.