Non-linear dose response relationships pose statistical challenges for their discovery. Even when an initial linear approximation is followed by other approaches, the results may be misleading and, possibly, preclude altogether the discovery of the nonlinear relationship under investigation. We review a variety of straightforward statistical approaches for detecting nonlinear relationships and ...

Children's ability to place fractions on a number line strongly correlates with math achievement. But does the number line play a causal role in fraction learning or does it simply index more advanced fraction knowledge? The number line may be a particularly effective representation for fraction learning because its properties align with the desired mental representation and take advantage of p...

In this paper, we give a combinatorial interpretation of the r-Whitney-Eulerian numbers by means of coloured signed permutations. This sequence is a generalization of the well-known Eulerian numbers and it is connected to r-Whitney numbers of the second kind. Using generating functions, we provide some combinatorial identities and the log-concavity property. Finally, we show some basic congruen...

This paper describes treeClust, an R package that produces dissimilarities useful for clustering. These dissimilarities arise from a set of classification or regression trees, one with each variable in the data acting in turn as a the response, and all others as predictors. This use of trees produces dissimilarities that are insensitive to scaling, benefit from automatic variable selection, and...

This project found improved lower bounds for van der Waerden numbers. The number W (k, r) is the length of the shortest sequence of colors with r different colors that guarantees an evenly-spaced subsequence (or arithmetic progression) of the same color of length k. We applied existing techniques: Rabung’s using primitive roots of prime numbers and Herwig et al’s Cyclic Zipper Method as improve...

In 14] and 11] we have studied compact embeddings of weighted function spaces on R n , p 2 (R n), s1 > s2, 1 < p1 p2 < 1, s1 ? n=p1 > s2 ? n=p2, and w(x) of the type w(x) = (1 + jxj) (log(2 + jxj)) , 0, 2 R. We have determined the asymptotic behaviour of the corresponding entropy numbers e k (idH). Now we are interested in the limiting case s1 ?n=p1 = s2 ?n=p2. of the so modiied embedding idH;a...

in the following work we present a proof for the strong law of large numbers for pairwise negatively dependent random variables which relaxes the usual assumption of pairwise independence. let be a double sequence of pairwise negatively dependent random variables. if for all non-negative real numbers t and , for 1 < p < 2, then we prove that (1). in addition, it also converges to 0 in . the res...

The Catalan numbers C n = (2n)!/n! (n + 1)! are are well-known integers that arise in many combinatorial problems. The numbers 6(2n)!/n! (n + 2)!, 60(2n)!/n! (n + 3)!, and more generally (2r + 1)!/r! · (2n)!/n! (n + r + 1)! are also integers for all n. We study the properties of these numbers and of some analogous numbers that generalize the ballot numbers, which we call super ballot numbers.

2. Fuzzy stochastic processes First we give some mathematical notations regarding fuzzy numbers. Let (Ω,M, P ) be a probability space, where M is a σ-field and P is a non-atomic probability measure. R denotes the set of all real numbers, and let C(R) be the set of all non-empty bounded closed intervals. A ‘fuzzy number’ is denoted by its membership function ã : R → [0, 1] which is normal, upper...

In this article we introduce and investigate the notion of ∆(r)-null, ∆(r)-convergent and ∆(r)-bounded sequences of fuzzy numbers which generalize the notion of null, convergent and bounded sequence of fuzzy numbers. 2000 Mathematics Subject Classification: 40A05, 40D25.