Scattering in the Poincaré Disk and in the Poincaré Upper Half-Plane

We investigate the scattering of a plane wave in the hyperbolic plane. We formulate the problem in terms of the Lippmann-Schwinger equation and solve it exactly for barriers modeled as Dirac delta functions running along: (i) N−horizontal lines in the Poincaré upper half-plane; (ii) N− concentric circles centered at the origin; and, (iii) a hypercircle in the Poincaré disk.

A novel way of constraining the α-attractor chaotic inflation through Planck data

Defining a scale of k-modes of the quantum fluctuations during inflation through the dynamical horizon crossing condition k = aH we go from the physical t variable to k variable and solve the equations of cosmological first-order perturbations self consistently, with the chaotic α-attractor type potentials. This enables us to study the behaviour of ns , r, nt and N in the k-space. Comparison of our results in the low-k regime with the Planck data puts constraints on the values of the α parameter through microscopic calculations. Recent studies had already put model-dependent constraints on the values of α through the hyperbolic geometry of a Poincaré disk: consistent with both the maximal supergravity model 𝒩 = 8 and the minimal supergravity model 𝒩 = 1, the constraints on the values of α are 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3. The minimal 𝒩 = 1 supersymmetric cosmological models with B-mode targets, derived from these supergravity models, predicted the values of r between 10-2 and 10-3. Both in the E-model and the T-model potentials, we have obtained, in our calculations, the values of r in this range for all the constrained values of α stated above, within 68% CL. Moreover, we have calculated r for some other possible values of α both in low-α limit, using the formula r = 12α/N 2, and in the high-α limit, using the formula r = 4n/N, for n = 2 and 4. With all such values of α, our calculated results match with the Planck-2018 data with 68% or near 95% CL.

Bifurcations of the Riccati Quadratic Polynomial Differential Systems

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

Global Phase Portraits for the Kukles Systems of Degree 3 with ℤ2-Reversible Symmetries

We provide the normal forms, the bifurcation diagrams and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree [Formula: see text] with [Formula: see text]-symmetries.

Global Phase Portraits for a Planar ℤ2-Equivariant Kukles Systems of Degree 3

We provide normal forms and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree [Formula: see text] with [Formula: see text]-equivariant symmetry. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain

We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.

Structure and Dynamic of Global Population Migration Network

People are the most important factors of economy and the primary carriers of social culture. Cross-border migration brings economic and cultural impacts to the origin and destination and is also a key to reflect the international relations of related countries. In fact, the migration relationships of countries are complex and multilateral, but most traditional migration models are bilateral. Network theories could provide a better description of global migration to show the structure and statistical characteristics more clearly. Based on the estimated migration data and disparity filter algorithm, the networks describing the global multilateral migration relationships have been extracted among 200 countries over fifty years. The results show that the global migration networks during 1960–2015 exhibit a clustering and disassortative feature, implying globalized and multipolarized changes of migration during these years. The networks were embed into a Poincaré disk, yielding a typical and hierarchical “core-periphery” structure, which is associated with angular density distribution, and has been used to describe the “multicentering” trend since 1990s. Analysis on correlation and evolution of communities indicates the stability of most communities, yet some structural changes still exist since 1990s, which reflect that the important historical events are contributable to regional and even global migration patterns.

Global Phase Portraits of ℤ2-Symmetric Planar Polynomial Hamiltonian Systems of Degree Three with a Nilpotent Saddle at the Origin

We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin and [Formula: see text]-symmetric with [Formula: see text].