1 The Initial Results This note has its source in [3], in a different setting. Consider the system of recurrences, with a 0 = b 0 = 1, a n = a n−1 + kb n−1 , (1) b n = a n−1 + b n−1. (2) Later on we will generalize and show that the ratio an bn is related to conver-gents to the square root of rationals. We postpone the proof by induction of the following formulas (" summation formulas ")

We investigate extremal functions exe(F, n) and exi(F, n) counting maximum numbers of edges and maximum numbers of vertex-edge incidences in simple hypergraphs H which have n vertices and do not contain a fixed hypergraph F ; the containment respects linear orderings of vertices. We determine both functions exactly if F has only distinct singleton edges or if F is one of the 55 hypergraphs with...

We resolve a conjecture proposed by D.E. Knuth concerning a recurrence arising in the satisfiability problem. Knuth’s recurrence resembles recurrences arising in the analysis of tries, in particular PATRICIA tries, and asymmetric leader election. We solve Knuth’s recurrence exactly and asymptotically, using analytic techniques such as the Mellin transform and analytic depoissonization.

A 321-k-gon-avoiding permutation π avoids 321 and the following four patterns: k(k + 2)(k + 3) · · · (2k − 1)1(2k)23 · · · (k − 1)(k + 1), k(k + 2)(k + 3) · · · (2k − 1)(2k)12 · · · (k − 1)(k + 1), (k + 1)(k + 2)(k + 3) · · · (2k − 1)1(2k)23 · · · k, (k + 1)(k + 2)(k + 3) · · · (2k − 1)(2k)123 · · · k. The 321-4-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW]...

This paper introduces logarithmic curvature and torsion graphs for analyzing planar and space curves. We present a method for drawing these graphs from any differentiable parametric curves and clarify the characteristics of these graphs. We show several examples of theses graphs drawn from planar and 3D Bézier curves. From the graphs, we can see some interesting properties of curves that cannot...

We study the asymptotic behavior of the terms in sequences satisfying recurrences of the form an = an−1 + n−d k=d f(n, k)akan−k where, very roughly speaking, f(n, k) behaves like a product of reciprocals of binomial coefficients. Some examples of such sequences from map enumerations, Airy constants, and Painlevé I equations are discussed in detail. 1 Main results There are many examples in the ...

were examined arthroscopically The intraarticular disease was recorded on a special documentation sheet (containing 67 descriptive items). The patients were divided into 5 subgroups: first-time dislocation (n = 91; first or second recurrence (n = 12); 3 to 5 recurrences (n = 23); 6 or more recurrences (n = 32); and chronic subluxafions (n = 15). A// data were examined statistically. Each lesion...

In the learning models considered, the events that occur on successive trials act additively. New derivations are given of a criterion for recurrence or transience of response probability, and a formula for the asymptotic response frequency in the recurrent case. It is shown that there are stationary probability distributions in the recurrent case, and that they are approximately normal with sm...

We present explicit solutions of a class of recurrences related to the Quickselect algorithm. Thus we are immediately able to solve recurrences arising at the partial sorting problem, which are contained in this class. We show how the partial sorting problem is connected to the Multiple Quickselect algorithm and present a method for the calculation of solutions for a class of recurrences relate...

Muscle injuries with hematomas are one of the most common events occurring in sport traumatology and require careful clinical and instrumental evaluation and timely treatment in order to restore a good functional outcome. The consequences of a failed treatment can be very serious, postponing an athlete's return to sports for weeks or months because of possible recurrences and complications (Gab...