How does truncation of the mask affect a refinable function?

If the mask of a refinable function has infinitely many coefficients, or if the coefficients are irrational, then it is often replaced by a finite mask with coefficients with terminating decimal expansions when it comes to applications. This note studies how such truncation affects the refinable function.Communicated by Charles A. Micchelli     

How many graphs are unions of k‐cliques?

We study the number of n‐vertex graphs that can be written as the edge‐union of k‐vertex cliques. We obtain reasonably tight estimates for in the cases (i) k = n ? o(n) and (ii) k = o(n) but . We also show that exhibits a phase transition around . We leave open several potentially interesting cases, and raise some other questions of a similar nature. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 87–107, 2006     

How many bits of information does an independent variable yield in a multiple regression?

A simple measure from statistical information theory is proposed to measure the contribution of each independent variable in a multiple regression.     

Sublimation‐like behavior of cardiac dynamics in heart failure: A malignant phase transition?

An abrupt transition from sinus rhythm to atrial fibrillation (AF) is common in patients with chronic heart failure (CHF). We propose a conceptual framework for viewing this malignant transition in terms of a type of sublimation marked by the switch from highly periodic sinus interbeat interval dynamics characteristic of CHF to a state of random disorganization with AF. Sublimation of physical substances involves an increase in entropy via heat transfer. In contrast, the disease‐related sublimation‐like behavior involves a loss of information content, associated decreases in cardiac bioenergetic capacity and in multiscale entropy. © 2016 Wiley Periodicals, Inc. Complexity 21: 24–32, 2016     

How many T‐tessellations on k lines? Existence of associated Gibbs measures on bounded convex domains

The paper bounds the number of tessellations with T‐shaped vertices on a fixed set of k lines: tessellations are efficiently encoded, and algorithms retrieve them, proving injectivity. This yields existence of a completely random T‐tessellation, as defined by Kiêu et al. (Spat Stat 6 (2013) 118–138), and of its Gibbsian modifications. The combinatorial bound is sharp, but likely pessimistic in typical cases. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 561–587, 2015     

How f ar is $${\left( 1 + \fr ac{ a}{n}\right) ^n} $$1+ ann from $$ e^ a $$e a when a is an element of a B an ach algebr a?

We present effective upper and lower bounds for the distance from \(\displaystyle \left( 1 + \frac{a}{n}\right) ^n \) to \(\displaystyle e^a \) for an element a of a complex unital Banach algebra and positive integer n. Specifically

$$\begin{aligned} \frac{1}{2n} \sup \left\{ \left| \mathfrak {R}(z^2) \right| e^{\mathfrak {R}(z)} : z \in \sigma (a) \right\} \lesssim _{(2)} \left\| e^a - \left( 1 + \frac{a}{n}\right) ^n \right\| \le \frac{ \left\| a \right\| ^2}{2n} \ e^{ \left\| a \right\| }, \end{aligned}$$

where \(\sigma (a)\) is the spectrum of a. The symbol \(\lesssim _{(p)}\) means “less than or equal to, up to a term of order \(n^{-p}\)”as discussed below. Following some technical preliminaries (Sect. 1) we treat the real case (Sect.  2), extend to the complex case (Sect.  3), and then generalise to the case of a norm-unital Banach algebra (Sect.  4).