On convex triangle functions

We prove that the strongest (largest convex) solution of the functional inequality $$\tau \left( {\frac{{F + G}}{2},\frac{{H + K}}{2}} \right) \le \frac{{\tau (F,H) + \tau (G,K)}}{2},$$ whereF, G, H andK are arbitrary distribution functions, is the triangle function τ(F, G)(x) = Max(F(x) +G(x) ? 1, 0).     

Triangles III: Complex triangle functions

Summary This paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In this paper, we look at the triangle coordinates of the special points of a triangle, and show that they are functions of its shape. We then show how these functions can be used to prove theorems about triangles, and to gain some insight into what makes a special point of a triangle a centre.     

Triangles II: Complex triangle coordinates

Summary This paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we coordinatize the Euclidean plane using cross ratios, and use these triangle coordinates to prove theorems about triangles. We develop a complex version of Ceva's theorem, and apply it to proofs of several new theorems. The remaining paper of this series will deal with complex triangle functions.     

On Hecke groups,Schwarzian triangle functions and a class of hyper-elliptic functions

Let m be a positive integer \(\ge \)3 and \(\lambda =2\cos \frac{\pi }{m}\). The Hecke group \(\mathfrak {G}(\lambda )\) is generated by the fractional linear transformations \(\tau + \lambda \) and \(-\frac{1}{\tau }\) for \(\tau \) in the upper half plane \(\mathbb H\) of the complex plane \(\mathbb C\). We consider a set of functions \(\mathfrak {f}_0, \mathfrak {f}_i\) and \(\mathfrak {f}_{\infty }\) automorphic with respect to \(\mathfrak {G}(\lambda )\), constructed from the conformal mapping of the fundamental domain of \(\mathfrak {G}(\lambda )\) to the upper half plane \(\mathbb H\), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of \(\lambda =1\) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of \(\pi \) in terms of the hypergeometric series.     

A Catalan triangle

We develop an arithmetic triangle similar to Pascal's triangle. The entries are interpreted in terms of numbers of pairs of nonintersecting paths in the first quadrant. The main applications are results about the Catalan numbers and various random walk problems.     

A note on the triangle conjecture

We prove some particular cases of the following conjecture of Perrin and Schützenberger, known as “the triangle conjecture.” Let A = {a, b} be a two-letter alphabet, d a positive integer and let B_d = {aⁱba^j| 0 ? i + j ? d}. If X ? B_d is a code, then |X| ? d + 1.     

Characterization of intermediate values of the triangle inequality II

In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C _n that satisfy 0 ≤ C _n ≤ Σ _j=1 ⁿ ‖x _j ‖ ? ‖Σ _j=1 ⁿ x _j ‖, x ₁,...,x _n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.     

On one-relator products induced by generalized triangle groups II

This is the second of two papers in which we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz.     

A counterexample to the triangle conjecture

The triangle conjecture sets a bound on the cardinality of a code formed by words of the form aⁱba^j. A counterexample exceeding that bound is given. This also disproves a stronger conjecture that every code is commutatively equivalent to a prefix code.     

A note about the Catalan triangle

A new identity is obtained for the Catalan triangle introduced by Shapiro.     

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.     

A subprime lending market primer

Opening a copy of TheMathematical Intelligencer you may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.     

A primer of simple theories

 We present a self-contained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah's 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay. Received: 12 November 1999 / Revised version: 1 June 2001 / Published online: 12 July 2002