Painlevé–Gullstrand form of the Lense–Thirring Spacetime

The standard Lense–Thirring metric is a century-old slow-rotation large-distance approximation to the gravitational field outside a rotating massive body, depending only on the total mass and angular momentum of the source. Although it is not an exact solution to the vacuum Einstein equations, asymptotically the Lense–Thirring metric approaches the Kerr metric at large distances. Herein we shall discuss a specific variant of the standard Lense–Thirring metric, carefully chosen for simplicity, clarity, and various forms of improved mathematical and physical behaviour, (to be more carefully defined in the body of the article). We shall see that this Lense–Thirring variant can be viewed as arising from the linearization of a suitably chosen tetrad representing the Kerr spacetime. In particular, we shall construct an explicit unit-lapse Painlevé–Gullstrand variant of the Lense–Thirring spacetime, one that has flat spatial slices, a very simple and physically intuitive tetrad, and extremely simple curvature tensors. We shall verify that this variant of the Lense–Thirring spacetime is Petrov type I, (so it is not algebraically special), but nevertheless possesses some very straightforward timelike geodesics, (the “rain” geodesics). We shall also discuss on-axis and equatorial geodesics, ISCOs (innermost stable circular orbits) and circular photon orbits. Finally, we wrap up by discussing some astrophysically relevant estimates, and analyze what happens if we extrapolate down to small values of r; verifying that for sufficiently slow rotation we explicitly recover slowly rotating Schwarzschild geometry. This Lense–Thirring variant can be viewed, in its own right, as a “black hole mimic”, of direct interest to the observational astronomy community.

T-Spanner Problem

The t-spanner problem is a popular combinatorial optimization problem and has different applications in communication networks and distributed systems. This chapter considers the problem of constructing a t-spanner subgraph H in a given undirected edge-weighted graph G in the sense that the distance between every pair of vertices in H is at most t times the shortest distance between the two vertices in G. The value of t, called the stretch factor, quantifies the quality of the distance approximation of the corresponding t-spanner subgraph. This chapter studies two variations of the problem, the Minimum t-Spanner Subgraph (MtSS) and the Minimum Maximum Stretch Spanning Tree(MMST). Given a value for the stretch factor t, the MtSS problem asks to find the t-spanner subgraph of the minimum total weight in G. The MMST problem looks for a tree T in G that minimizes the maximum distance between all pairs of vertices in V (i.e., minimizing the stretch factor of the constructed tree). It is easy to conclude from the literatures that the above problems are NP-hard. This chapter presents genetic algorithms that returns a high quality solution for those two problems.

Convolutional Neural Network Based Approach to In Silico Non-Anticipating Prediction of Antigenic Distance for Influenza Virus

Evaluation of the antigenic similarity degree between the strains of the influenza virus is highly important for vaccine production. The conventional method used to measure such a degree is related to performing the immunological assays of hemagglutinin inhibition. Namely, the antigenic distance between two strains is calculated on the basis of HI assays. Usually, such distances are visualized by using some kind of antigenic cartography method. The known drawback of the HI assay is that it is rather time-consuming and expensive. In this paper, we propose a novel approach for antigenic distance approximation based on deep learning in the feature spaces induced by hemagglutinin protein sequences and Convolutional Neural Networks (CNNs). To apply a CNN to compare the protein sequences, we utilize the encoding based on the physical and chemical characteristics of amino acids. By varying (hyper)parameters of the CNN architecture design, we find the most robust network. Further, we provide insight into the relationship between approximated antigenic distance and antigenicity by evaluating the network on the HI assay database for the H1N1 subtype. The results indicate that the best-trained network gives a high-precision approximation for the ground-truth antigenic distances, and can be used as a good exploratory tool in practical tasks.