Nodal intersections for random waves against a segment on the 3-dimensional torus

We consider random Gaussian eigenfunctions of the Laplacian on the three-dimensional flat torus, and investigate the number of nodal intersections against a straight line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on the arithmetic properties of the straight line. The considerations made establish a close relation between this problem and the theory of lattice points on spheres.Comment: 40 page

Random waves on T3\mathbb{T}^3: nodal area variance and lattice point correlations

We consider the ensemble of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or `energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.Comment: To appear in IMR

Fashioning Desire at B. Altman & Co.: Ethics and Consumer Culture in Early Department Stores

We live in an age of fast fashion. Clothing is produced in greater volumes than ever before and the lifecycle of each garment keeps getting shorter and shorter. Many items are manufactured to be worn only one time and then thrown away—as disposable as a cup of coffee. There is much to be learned about our current fashion ecosystem by looking into the past. Beyond the garments themselves we must understand the larger historical and sociological context in which these articles of clothing were produced. How does the shopping environment shape the buying habits and fashion trends of an era? How does that system inform the worn identities of the individuals operating within it? The experiential quality of department stores has been eclipsed by consumer demands for faster, cheaper, and more convenient products, but e-commerce has yet to find a way to deliver the delicious and tactile experience of shopping. Did the mass culture of the early 20th century prefigure the fashion industry as it exists today? Can ethical business practices co-exist with modern fashion

On self-duality and unigraphicity for 33-polytopes

Recent literature posed the problem of characterising the graph degree sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we characterise the sequences with exactly one self-dual $3$-polytopal realisation. An algorithm in the literature constructs a self-dual $3$-polytope for any admissible degree sequence. To do so, it performs operations on the radial graph, so that the corresponding $3$-polytope and its dual are modified in exactly the same way. To settle our question and construct the relevant graphs, we apply this algorithm, we introduce some modifications of it, and we also devise new ones. The speed of these algorithms is linear in the graph order