This is an expositiry article on collapsing theory written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applications of collapsing theory to Riemannian geometry are not discussed in this survey, due to page limit...

Nonequilibrium systems that are driven or drive themselves towards a critical point have been studied for almost three decades. Here we present a minimalist example of such a system, motivated by experiments on collapsing active elastic networks. Our model of an unstable elastic network exhibits a collapse towards a critical point from any macroscopically connected initial configuration. Taking...

INTRODUCTION Fresnel lenses are a type of lens that eliminates the bulk of a conventional lens by collapsing the lens thickness while retaining the curvature necessary for focusing. Figure 1 shows the Fresnel lens equivalent of a conventional lens. For a rotationally symmetric lens, the discontinuities or grooves formed when collapsing the lens are annular. This produces a much thinner lens, pa...

Two relativistic models for collapsing spheres at different stages of evolution, which include pre-relaxation processes, are presented. The influence of relaxation time on the outcome of evolution in both cases is exhibited and established. It is shown that relaxation processes can drastically change the final state of the collapsing system. In particular, there are cases in which the value of ...

The neutrino-nucleon scattering in a collapsing star envelope with a strong magnetic field is investigated. The transferred by neutrinos momentum asymmetry along the field direction is obtained. It is shown that the neutrino-nucleon scattering gives a contribution to the asymmetry comparable with direct URCA processes. Hence, the neutrino-nucleon scattering should be taken into account in estim...

Cellular lesions form in Bowman's space in both crescentic glomerulonephritis and collapsing glomerulopathy. The pathomechanism and origin of the proliferating cells in these lesions are unknown. In this study, we examined proliferating cells by lineage tracing of either podocytes or parietal epithelial cells (PECs) in the nephrotoxic nephritis model of inflammatory crescentic glomerulonephriti...

This paper focuses on bounding the total communication complexity of collapsing protocols for multiparty pointer jumping problem (MPJ k ). Brody and Chakrabati in [BC08] proved that in such setting one of the players must communicate at least n − 0.5 logn bits. Liang in [Liang13] has shown protocol matching this lower bound on maximum complexity. His protocol, however, was behaving worse than t...

Classical knot theory is about the classification, up to isotopy, of embedded closed curves in three-dimensional space. Two closed curves embedded in three-dimensional space are said to be isotopic if there is a continuous family of embeddings starting with one curve and ending with the other curve. This definition of isotopy captures our intuitive notion of moving a length of elastic rope abou...

We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and ...

A model of a self-interacting directed animal, which also interacts with a solid wall, is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition. The directed animal is confined to a 45° wedge, and it interacts with one of the walls of this wedge. The existence of a thermodynamic limit in this model shown, and the presence of an adsorpti...