The Kissing Triangles: The Aesthetics of Mathematical Discovery

Papert's (1978) appeal to reconsider the power and possibilities of the aesthetic in mathematics learning is often ignored in mathematics education research. This paper begins with the premise, put forth by Dewey (1934), that the aesthetic structures many dimensions of inquiry and experience. In the same way that using particular paintings, musical compositions, or even everyday experiences has been instrumental to attempts by philosophers to understand the aesthetic dimensions of meaning and experience in artistic domains, I propose that analysing a particular encounter with mathematics may help reveal the nature and role of the often nebulous responses of elegance, beauty, and `fit' to which mathematicians lay claim in their mathematical activity. To achieve this, I draw on and adapt the defining features of the aesthetic character of experience set forth by the aesthetician Beardsley (1982). This, in turn, sheds light on the role thataesthetics can play in mathematical inquiry and experience, and provides initial categories and conjectures that can be used to investigate the potential roles of aesthetics in mathematics learning contexts.This revised version was published online in September 2005 with corrections to the Cover Date.     

Kolmogorov Width of Classes of Smooth Functions on the Sphere

Let ^d−1{(x₁,…,x_d) ^d:x²₁+···+x²_d=1} be the unit sphere of the d-dimensional Euclidean space ^d. For r>0, we denote by B^r_p (1p∞) the class of functions f on ^d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on ^d−1, and P^λ_k(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of B^r_p in L^q( ^d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces X_N of L^q( ^d−1).The main result in this paper is that for r2(d−1)²,where A_NB_N means that there exists a positive constant C, independent of N, such that C⁻¹A_NB_NCA_N.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.     

On removable even circuits in graphs

Let G be a connected graph with minimum degree at least 3. We prove that there exists an even circuit C in G such that G−E(C) is either connected or contains precisely two components one of which is isomorphic to a 1-bond. We further prove sufficient conditions for there to exist an even circuit C in a 2-connected simple graph G such that G−E(C) is 2-connected. As a consequence of this, we obtain sufficient conditions for there to exist an even circuit C in a 2-connected graph G for which G−E(C) is 2-connected.     

Upon the removal of the edges of a 1-factor from an even circuit in a 2-connected graph

Let G be a 2-connected graph with minimum degree at least 3. We prove that there exists an even circuit C in G with factorization F={F₁,F₂} such that G−E(F₁) is 2-connected.     

Investigation of the chemical state of ultrathin Hf-Al-O films during high temperature annealing

Al₂O₃ incorporated HfO₂ films grown by atomic layer deposition (ALD) were investigated by high-resolution X-ray photoelectron spectroscopy (HRXPS). The core level energy state of a 15 Å thick film showed a shift to higher binding energy, as the result of a silicate formation and Al₂O₃ incorporation. The incorporation of Al₂O₃ into the HfO₂ film had no effect on silicate formation at the interface between the film and Si, while the ionic bonding characteristics and hybridization effects were enhanced compared to a pure HfO₂ film. The dissociation of the film in an ultrahigh vacuum (UHV) is effectively suppressed compared to a pure HfO₂ film, indicating an enhanced thermal stability of Hf-Al-O. Any dissociated Al₂O₃ on the film surface was completely removed into the vacuum by vacuum annealing treatment over 850 °C, while HfO₂ contributed to Hf silicide formation on the film surface.     

Predicting the Mechanical Properties of Binary Blends of Immiscible Polymers

We couple a morphological study of an immiscible binary AB mixture with a micromechanical simulation to determine how the spatial distribution of the A and B domains and the interfacial region (interphase) affects the mechanical behavior of the blend. The morphological studies are conducted through a three-dimensional Cahn-Hilliard (CH) simulation. Through the CH calculations, we obtain the size and structure of the domains for different blend compositions. The output of the CH model serves as the input to the Lattice Spring Model (LSM), which consists of a three-dimensional network of springs. In particular, the location of the different phases is mapped onto the LSM lattice and the appropriate force constants are assigned to the LSM sites. A stress is applied to the LSM lattice and we calculate the elastic response of the material. We find that the local stress and strain fields are highly dependent on the morphology of the system. By integrating the morphological and mechanical models, we can isolate how modifications in the composition of the mixture affect the macroscopic behavior. Thus, we can establish how choices made in the components affect the ultimate performance of the material.