Strain induced strengthening of soft thermoplastic polyurethanes under cyclic deformation

We investigate the cyclic mechanical behavior in uniaxial tension of three different commercial thermoplastic polyurethane elastomers (TPU) often considered as a sustainable replacement for common filled elastomers. All TPU have similar hard segment contents and linear moduli but sensibly different large strain properties as shown by X-ray analysis. Despite these differences, we found a stiffening effect after conditioning in step cyclic loading which greatly differs from the common softening (also referred as Mullins effect) observed in chemically crosslinked filled rubbers. We propose that this self-reinforcement is related to the fragmentation of hard domains, naturally present in TPU, in smaller but more numerous sub-units that may act as new physical crosslinking points. The proposed stiffening mechanism is not dissimilar to the strain-induced crystallization observed in stretched natural rubber, but it presents a persistent nature. In particular, it may cause a local reinforcement where an inhomogeneous strain field is present, as is the case of a crack propagating in cyclic fatigue, providing a potential explanation for the well-known toughness and wear resistance of TPU.     

Linear Quadratic Gaussian Homing for Markov Processes with Regime Switching and Applications to Controlled Population Growth/Decay

Methodology and Computing in Applied Probability - The problem of optimally controlling one-dimensional diffusion processes until they enter a given stopping set is extended to include Markov...     

Effect of absorbed pump power on the quality of output beam from monolithic microchip lasers

The dependence of the beam propagation factor (M ² parameter) with the absorbed pump power in the case of monolithic microchip laser under face-cooled configuration is extensively studied. Our investigations show that the M ₂ parameter is related to the absorbed pump power through two parameters (α and β) whose values depend on the laser material properties and laser configuration. We have shown that one parameter arises due to the oscillation of higher order modes in the microchip cavity and the other parameter accounts for the spherical aberration associated with the thermal lens induced by the pump beam. Such dependency of M ² parameter with the absorbed pump power is experimentally verified for a face-cooled monolithic microchip laser based on Nd³⁺ -doped GdVO₄ crystal and the values of α and β parameters were estimated from the experimentally measured data points.     

The Moore Bound for Irregular Graphs

 What is the largest number of edges in a graph of order n and girth g? For d-regular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem ([4] p. 163, problem 10). Received: June 27, 2000 Final version received: July 3, 2001     

Metric Embeddings—Beyond One-Dimensional Distortion

The extensive study of metric spaces and their embeddings has so far focused on embeddings that preserve pairwise distances. A very intriguing concept introduced by Feige allows us to quantify the extent to which larger structures are preserved by a given embedding. We investigate this concept, focusing on several major graph families such as paths, trees, cubes, and expanders. We find some similarities to the regular (pairwise) distortion, as well as some striking differences.     

The structure of students' emotions experienced during a mathematical achievement test

Several models have been developed in order to categorize the numerous expressions that people use in order to describe their emotional experiences. The focus of the present study is on one of these theoretical classifications proposed by Pekrun (1992) specifically concerning emotions which are directly related to learning and achievement in mathematics. In his model, emotions are classified according to their valence (positive vs. negative) and their level of activation. In testing the assumptions of this model, we investigated students' enjoyment, anxiety, anger and boredom experienced before, during, and after the completion of a math test. Correspondence analyses which were used to generate a graphical illustration of structural interrelationships between these emotions provide empirical support for the theoretical classification.     

Visualisation and proof: a brief survey of philosophical perspectives

The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification.     

Teaching proof in the context of physics

This paper describes an application of statics to geometrical proofs in the classroom. The aim of the study was to find out whether the use of concepts and arguments from statics can help students understand and produce proofs of geometrical theorems. The two theorems studied were (1) that the medians in a triangle meet at a single point which is the centre of gravity of the triangle, and (2) the Varignon theorem, that the lines joining the midpoints of successive sides of a quadrilateral form a parallelogram. The classroom experiment showed that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems. These findings lend support to the claim that the introduction of statics helps students produce proofs and grasp their meaning.