Expected invariants of simplicial complexes obtained from random point samples

In this paper, we study the expectation values of topological invariants of the Vietoris–Rips complex and ?ech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.

     

Effects of satellite positioning errors and Earth’s multipole moments in the detection of the gravitomagnetic field with an orbiting gravity gradiometer

The paper deals with a dynamical system analysis related to phantom cosmological model. Here gravity is coupled to phantom scalar field having scalar coupling function and a potential. The field equations are reduced to an autonomous dynamical system by a suitable redefinition of the basic variables and assuming some suitable form of the potential function. Finally, critical points are evaluated, their nature have been analyzed and corresponding cosmological scenario has been discussed.     

Synthesis of an amphiphilic spiro-multiblock copolymer via thiol-ene click chemistry

A multiblock [poly(ethylene oxide)-b-spiro-polystyrene] ([(PEO-b-spiro-PS)]) copolymer with a topologically novel architecture was synthesized using thiol-ene step-growth polymerization reaction. Spiro-PS with dimercapto groups as the hard segment was synthesized in three main steps: (a) preparation of tetra-arm PS by atom transfer radical polymerization and the conversion of the chain-end group to azide functionality, (b) alkyne-azide click coupling reaction to synthesize a tricyclic PS, and (c) tactical ring opening of the tricyclic PS through disulfide/thiol redox reaction. The PEO soft segment was obtained as chain-ends modified with norbornene groups. Finally, the hydrothiolation of the highly reactive norbornene chain-ends of polyethylene glycol with the dimercapto groups of spiro-PS produced the multiblock ([(PEO-b-spiro-PS)]) copolymer in quantitative yield. The multiblock copolymer was characterized using size-exclusion chromatography, proton nuclear magnetic resonance spectroscopy, Fourier-transform infrared spectroscopy, X-ray photoelectron spectroscopy, and Raman spectroscopy. © 2019 Wiley Periodicals, Inc. J. Polym. Sci. 2020 , 58, 132–138     

Power-Law Behavior in Geometric Characteristics of Full Binary Trees

Natural river networks exhibit regular scaling laws in their topological organization. Here, we investigate whether these scaling laws are unique characteristics of river networks or can be applicable to general binary tree networks. We generate numerous binary trees, ranging from purely ordered trees to completely random trees. For each generated binary tree, we analyze whether the tree exhibits any scaling property found in river networks, i.e., the power-laws in the size distribution, the length distribution, the distance-load relationship, and the power spectrum of width function. We found that partially random trees generated on the basis of two distinct types of deterministic trees, i.e., deterministic critical and supercritical trees, show contrasting characteristics. Partially random trees generated on the basis of deterministic critical trees exhibit all power-law characteristics investigated in this study with their fitted exponents close to the values observed in natural river networks over a wide range of random-degree. On the other hand, partially random trees generated on the basis of deterministic supercritical trees rarely follow scaling laws of river networks.     

Detection of the gravitomagnetic field using an orbiting superconducting gravity gradiometer: principle and experimental considerations

The angular momentum of the Earth produces gravitomagnetic components of the Riemann curvature tensor, which are of the order of 10⁻¹⁰ of the Newtonian terms arising from the mass of the Earth. Due to the dragging of the local inertial frame by the spinning Earth, there are also secular terms, which grow in time. These fields can be detected in principle by a set of orbiting superconducting gravity gradiometers. The Riemann tensor components for various spacecraft orientations have been computed and the principle of detecting the gravitomagnetic tidal force has been published. In this paper, we review the conclusions of the earlier analyses and discuss the feasibility of the gravity gradiometer experiment.