On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function φ(q): φ(e^{−7π√7}) = 7^{−3/4}φ(e^{−π√7}){1 + ( )^{2/7} + ( )^{2/7} + ( )^{2/7}}. We give the three missing terms. In addition, we calculate the class invariant G_{343} and further special values of φ(e^{−nπ}), for n = 7, 21, 35, and 49.

The density ridge framework for estimating principal curves and surfaces has in a number of recent works been shown to capture manifold structure in data in an intuitive and effective manner. However, to date there exists no efficient way to traverse these manifolds as defined by density ridges. This is unfortunate, as manifold traversal is an important problem for example for shape estimation in ...

Students of physics in Norway typically spend a lot of time on mandatory problemsolving exercises that they must get a passing grade on to be allowed to take the exam. This is often a primarily summative assessment, which can limit the learning potential. <p> <p>In this contribution, we present an intervention in which traditional mandatory assignments are modified to put more emphasis on ...

We use the recent frame-indifferent theory of diffusive momentum transport to identify internal barriers in wall-bounded turbulence. Formed by the invariant manifolds of the Laplacian of the velocity field, the barriers block the viscous part of the instantaneous momentum flux in the flow. We employ the level sets of single-trajectory Lagrangian diagnostic tools, the trajectory rotation average ...

We consider a q-analogue of abstract simplicial complexes, called q-complexes, and discuss the notion of shellability for such complexes. It is shown that q-complexes formed by independent subspaces of a q-matroid are shellable. Further, we explicitly determine the homology of q-complexes corresponding to uniform q-matroids. We also outline some partial results concerning the determination of homology ...

The arithmetic mean/geometric mean inequality (AM/GM inequality) facilitates classes of nonnegativity certificates and of relaxation techniques for polynomials and, more generally, for exponential sums. Here, we present a first systematic study of the AM/GM-based techniques in the presence of symmetries under the linear action of a finite group. We prove a symmetry-adapted representation theorem and ...

Statistiske metoder som regresjonsanalyse og kausale design basert på naturlige eksperi-menter har etter hvert blitt hyllevare innen de fleste fag som behandler store datamengder,ikke minst innen epidemiologi, meteorologi og klimavitenskap.1 Et eksempel er prinsipalkomponentanalyse (PCA), som brukes til å finne karakteristiske mønstre i geofysiske felt.Mønstrene, eller prinsipalkomponentene, ...

The global atmospheric circulation is fundamental for the local weather and climate by redistributing energy and moisture. To the present day, there is a knowledge gap at which spatial scales the energy and its components are transported. Therefore, we separate the meridional atmospheric energy transport in the ERA5 reanalysis by the spatial scales, the quasi-stationary and transient flow patterns, ...

A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of ...

Lattice conditional independence models [Andersson and Perlman, Lattice models for conditional independence in a multivariate normal distribution, Ann. Statist. 21 (1993), 1318–1358] are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via ...