This paper generalizes properties of mixtilinear incircles. Let (S) be any circle in the plane of triangle ABC. Suppose there are circles (Sa), (Sb), and (Sc) each tangent internally to (S); and (Sa) is inscribed in angle BAC (similarly for (Sb) and (Sc)). Let the points of tangency of (Sa), (Sb), and (Sc) with (S) be X, Y , and Z, respectively. Then it is shown that the lines AX, BY , and CZ m...

The article below is the first in what we hope will be a series—which we will call CrossWires. In this series, articles will be published that ‘invite’ response or commentary. We will be happy to receive such replies and, subject to review, publish them as articles in WIREs Cognitive Science. We will, on a case-by-case basis, solicit replies, but in all cases, we will be open to submitted repli...

Let N denote the set of all nonnegative integers. Let k ≥ 3 be an integer and A0 = {a1, . . . , at} (a1 < . . . < at) be a nonnegative set which does not contain an arithmetic progression of length k. We denote A = {a1, a2, . . . } defined by the following greedy algorithm: if l ≥ t and a1, . . . , al have already been defined, then al+1 is the smallest integer a > al such that {a1, . . . , al}...

This paper examines the extremal problem of how many 1-entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P . For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also settles the conjecture of Stanley and Wilf on the number of n-permutations avoiding a fixed permuta...

Article history: Received 1 September 2013 Available online xxxx

IN HER brief portrait of Mrs. Catherine Gladstone in 1956, Mrs. Georgina Battiscombe wrote that Mrs. Gladstone's efforts 'to be of some use in the [Crimean] war troubles' brought her into 'a most unfortunate collision' with Florence Nightingale when, under the influence of Mrs. Elizabeth Herbert, Mrs. Gladstone 'threw herself heart and soul ... into the good work of providing nurses for the mil...