Topological Data Analysis with Bregman Divergences

Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory

Flat sheet metal girders with very thin metal web. Part II : sheet metal girders with spars resistant to bending - oblique uprights - stiffness

Noting that the stiffness of the girder increases very rapidly as Beta increases, the result can be summed up as follows: When the cross stress preponderates in one direction and when the web plate is to be given the dimensions commensurate to its stresses, it is advisable (regardless of any ensuing structural difficulties) to set the uprights at about Beta = 120 degrees, thereby lowering the weight of the plate wall 15 percent (in contrast to Beta = 90 degrees), and raising the stiffness 55 percent. But, when the cross stresses alternate and are approximately of the same intensity in both directions, or, if the web plate thickness is determined by other structural reasons, then Beta = 90 degrees should be chosen

Flat sheet metal girders with very thin metal web. Part III : sheet metal girders with spars resistant to bending - the stress in uprights - diagonal tension fields

This report continues the work presented in NACA-TM 605 and expands the scope to include the change in specific number of wrinkles from direction x to z, so that b and f become variable in direction z. Moreover, it seems likely that b and f increase from the edge toward the center if the sheet is infinitely thin

The Analysis of Aircraft Structures as Space Frameworks. Method Based on the Forces in the Longitudinal Members

The following examples do not take up the discussion of viewpoints to be heeded in determining the design of a framework for given external conditions. Rather they are methods for determining the forces in airplane fuselages and wings, though similar considerations are applied to certain simple cases of a different kind. The object of this treatise is to summarize and amplify these considerations from definite viewpoints

Remarks on airplane struts and girders under compressive and bending stresses : index values

In this paper the behavior of straight, centrally loaded compression struts is discussed and values computed

Flat sheet metal girders with very thin metal web. Part I : general theories and assumptions

The object of this report was to develop the structural method of sheet metal girders and should for that reason be considered solely from this standpoint. The ensuing methods were based on the assumption of the infinitely low stiffness in bending of the metal web. This simplifies the basis of calculations to such an extent that many questions of great practical importance can be examined which otherwise cannot be included in any analysis of the bending stiffness of the buckled plate. This report refers to such points as the safety in buckling of uprights to the effect of bending flexibility of spars, to spars not set parallel, etc

Torsion and buckling of open sections

In this paper is a discussion of the general principles for open sections of any shape. In what follows the torsion will be computed and on the basis of the results it will be possible to obtain a proper design of section in each case. The torsion of buckling members for the case where they are centrally loaded, leads to a problem in pure stability and is similar to that of stressed beams

Landing of seaplanes

This report begins with a brief resume of a theoretical consideration and its results. Then a description is presented of the processes at the moment a keeled bottom strikes the water and the formulas and solutions are given. Lastly a discussion of the application of these results to practical problems is given