Two-parameter Family Research Articles

Fully numerical computation of heteroclinic connection families in the spatial three-body problem

Heteroclinic connections in the spatial circular restricted three-body problem play an important role in astrodynamics. This paper presents a fully numerical methodology for the computation of the two-parameter families of connections that exist between families of invariant tori in the three-body problem. The computation of connections is presented as a two-point boundary value problem in which the initial and final states belong to unstable and stable manifolds of two respective normally hyperbolic invariant manifolds. A flow map torus computation procedure is modified to compute particular trajectories within these spaces, and a robust two-parameter continuation scheme is paired with a shooting technique to compute the entire connection family. Results are provided in the Sun–Earth and Earth–Moon representations of the spatial three-body problem. The Sun–Earth connection family works to verify the methodology with previously developed semi-analytical connection computation techniques. The Earth–Moon example demonstrates the methodology in a setting where fully numerical techniques are required to study the connection family.

A two-parametric family of high rank Mordell curves

We present an elementary construction of an explicit two-parametric family of elliptic curves over [Formula: see text] of the form [Formula: see text] such that the rank of each member of the family is at least seven. The main new ingredient is Ramanujan’s identity for sums of rational cubes.

Thin-Film Equations with Singular Potentials: An Alternative Solution to the Contact-Line Paradox

In the regime of lubrication approximation, we look at spreading phenomena under the action of singular potentials of the form P(h)≈h1-m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P(h)\\approx h^{1-m}$$\\end{document} as h→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\rightarrow 0^+$$\\end{document} with m>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>1$$\\end{document}, modeling repulsion between the liquid–gas interface and the substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, we report that for any m>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>1$$\\end{document} and any value of the speed (both positive and negative) there exists a three-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line). A two-parameter family of advancing “linear-log” fronts also exists, having a logarithmically corrected linear behavior in the liquid bulk. All these fronts have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, fronts have microscopic contact angle equal to π/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi /2$$\\end{document} for all m>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>1$$\\end{document} and finite energy for all m<3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m<3$$\\end{document}. We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique (up to translation) linear-log front for each positive speed. Numerical evidence suggests that, fixed the speed and the frictional coefficient, its shape depends on the spreading coefficient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting.

Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals.

A mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier μ. By means of their specific parameterization they comprise local features of both periodic and aperiodic crystals. In particular, chiral spiral cyclic twins are composed of discrete variants of continuous curves known as circle involutes, each discrete spiral being generated from an integer inclination sequence. The geometry of circle involutes does not only provide for a constant orthogonal separation distance between adjacent spiral branches but also yields an approximate delineation of the intrinsically periodic twin domains as well as a single aperiodic core domain interconnecting them. Apart from its mathematical description and analysis, e.g. concerning its circle packing densities, the toy model is studied in association with the crystallography and crystal chemistry of α-uranium and CrB-type crystal structures.

VC dimensions of group convolutional neural networks

We study the generalization capacity of group convolutional neural networks. We identify precise estimates for the VC dimensions of simple sets of group convolutional neural networks. In particular, we find that for infinite groups and appropriately chosen convolutional kernels, already two-parameter families of convolutional neural networks have an infinite VC dimension, despite being invariant to the action of an infinite group.

Two-parameter family of distributions: Properties, estimation, and applications

In this paper, a new flexible two-parameter family of distributions is introduced. Some mathematical properties, such as asymptotic behavior, moments, stochastic order, entropy, and quantile function, are examined. A special case of this family is introduced called Gemeay–Zeghdoudi distribution. The unknown parameters of the Gemeay–Zeghdoudi distribution are estimated via many estimators. The efficiency of the estimators is compared with a Monte Carlo simulation. The applicability of the new distribution is illustrated by the annual maximum flood and survival times of patients with breast cancer real data analyses.

3D Mirror Symmetry for Instanton Moduli Spaces

We prove that the Hilbert scheme of k points on {mathbb {C}}^2 (hbox {Hilb}^k[{mathbb {C}}^2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the {mathbb {C}}^times _hbar -action. First, we find a two-parameter family X_{k,l} of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of hbox {Hilb}^k[{mathbb {C}}^2] is obtained via direct limit llongrightarrow infty and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted hbar -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-N sheaves on {mathbb {P}}^2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

A Pentagonal Number Theorem for Tribone Tilings

Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we introduce a two-parameter family of roughly hexagonal regions in the hexagonal grid and show that a tiling by tribones exists if and only if the two parameters associated with the region are the paired pentagonal numbers $k(3k \pm 1)/2$.

Exact solutions for submerged von Kármán point vortex streets cotravelling with a wave on a linear shear current

New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy &amp; Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

Densely branching trees as models for Hénon-like and Lozi-like attractors

Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For the Hénon maps this applies to a positive Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters, the Wang-Young parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For the Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967. We also show that no simpler 1-dimensional models exist.

Elliptic finite-band potentials of a non-self-adjoint Dirac operator

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by q≡Adn(x;m), where m∈(0,1) and A can be taken to be positive without loss of generality, for a non-self-adjoint Dirac operator L, which connects two well-known limiting cases of the plane wave (m=0) and of the sech potential (m=1). We show that, if A∈N, then the spectrum consists of R plus 2A Schwarz symmetric segments (bands) on iR. This characterization of the spectrum is obtained by relating the periodic and antiperiodic eigenvalue problems for the Dirac operator to corresponding eigenvalue problems for tridiagonal operators acting on Fourier coefficients in a weighted Hilbert space, and to appropriate connection problems for Heun's equation. Conversely, if A∉N, then the spectrum of L consists of infinitely many bands in C. When A∈N, the corresponding potentials generate finite-genus solutions for all the positive and negative flows associated with the focusing nonlinear Schrödinger hierarchy, including the modified Korteweg-deVries equation and the sine-Gordon equation.

Splitting and coexistence of 2-D strange attractors in a general family of Expanding Baker Maps

We consider a two-parameter family of Expanding Baker Maps on the plane, being a > 1 and an expansion rate and a rotation angle, respectively. We prove that exhibits strange attractors for every a sufficiently close to 1. We also study how such attractors may split into other ones of a larger number of connected pieces as a decreases to 1 and is a rational number. The study of the family is strongly motivated by the rich dynamics observed for the quadratic family .

Point Pair-Based Expression of Cutter Swept Envelopes in Five-Axis Milling

This paper proposes an efficient method of formulating cutter swept envelopes in five-axis milling based on conformal geometric algebra. The surface of the rotary cutter is represented as a canal surface, as the envelope of a one-parameter family of spheres. The swept envelopes generated by the general rigid motion of the cutter can be regarded as the envelopes of a two-parameter family of spheres. With the help of the dual representations of spheres and planes in conformal space, a point pair-based expression of the swept envelopes is formulated analytically, which represents two surfaces uniformly. By using the present method, the swept surfaces of a conical cutter, a drum-shaped cutter, and a toroidal cutter are illustrated. As an application, two types of collision detection methods for 5-axis milling are presented based on the point pair-based envelope expression. The benchmark examples demonstrate that the use of the proposed envelope expression significantly improves computational efficiency.

A generically singular type of saddle–node bifurcation that occurs for relativistic shock waves

This paper is concerned with two-parameter families of planar dynamical systems in which a saddle–node bifurcation interacts with a singular perturbation. We identify bifurcation curves along which the non-saddle point switches from node to focus and establish the existence of heteroclinic connections in either case. As an application, the result is used to show the existence of oscillatory shock profiles in a model of relativistic fluid dynamics.

Stable singularity formation for the inviscid primitive equations

The primitive equations (PEs) model large-scale dynamics of the oceans and the atmosphere. While it is by now well known that the three-dimensional viscous PEs are globally well posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blow-up mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic-in-time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameter family of nonsmooth profiles, and is stable only by smoother perturbations.

Stieltjes Transforms and R-Transforms Associated with Two-Parameter Lambert-Tsallis Functions.

In this paper, we study a two-parameter family of Stieltjes transformations related to holomorphic Lambert-Tsallis functions, which are a two-parameter generalization of the Lambert function. Such Stieltjes transformations appear in the study of eigenvalue distributions of random matrices associated with some growing statistically sparse models. A necessary and sufficient condition on the parameters is given for the corresponding functions being Stieltjes transformations of probabilistic measures. We also give an explicit formula of the corresponding R-transformations.

A Two-Parameter Family of Basic State in Porous Media Leading to Darcy–Bénard Convection

A Two-Parameter Family of Basic State in Porous Media Leading to Darcy–Bénard Convection

Families of Orbits Produced by Three-Dimensional Central and Polynomial Potentials: An Application to the 3D Harmonic Oscillator

We study three-dimensional potentials of the form V=U(xp+yp+zp), where U is an arbitrary function of C2-class, and p∈Z, which produces a preassigned two-parametric family of spatial regular orbits given in the solved form f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const). These potentials have to satisfy two linear PDEs, which are the basic equations of the 3D inverse problem of Newtonian dynamics. The functions f and g can be represented uniquely by the ”slope functions” α(x,y,z) and β(x,y,z). The orbital functions α(x,y,z) and β(x,y,z) have to satisfy three differential conditions according to the theory of the inverse problem. If these conditions are satisfied, then we can find such a potential analytically. We offer pertinent examples of potentials that are mainly used in physical problems. The values obtained for p lead to cases of well-known potentials, such as the Newtonian, cored, logarithmic, polynomial and quadratic ones. New families of orbits produced by the 3D harmonic oscillator are found. Pertinent examples are given and cover all cases. Two-dimensional potentials belong to a special category of potentials and are studied separately. The families of straight lines in 3D space are also examined.

Classical integrability of root- TT¯ flows

The root-$T\overline{T}$ flow was recently introduced as a universal and classically marginal deformation of any two-dimensional translation-invariant field theory. The flow commutes with the (irrelevant) $T\overline{T}$ flow, and it can be integrated explicitly for a large class of actions, leading to nonanalytic Lagrangians reminiscent of the four-dimensional modified-Maxwell theory (ModMax). It is not a priori obvious whether the root-$T\overline{T}$ flow preserves integrability, as is the case for the $T\overline{T}$ flow. In this paper we demonstrate that this is the case for a large class of classical models by explicitly constructing a deformed Lax connection. We discuss the principal chiral model and the nonlinear sigma models on symmetric and semisymmetric spaces, without or with the Wess-Zumino term. We also construct Lax connections for the two-parameter families of theories deformed by both root-$T\overline{T}$ and $T\overline{T}$ for all of these models.

On the Construction of Mathematical Models of the Membrane Theory of Convex Shells

Introduction. The paper considers the issues of constructing mathematical models of the momentless equilibrium stress state of elastic convex shells using methods of the complex analysis. At the same time, shells with a piecewise smooth (ribbed) lateral surface were considered for the first time. The work objective was to find classes of shells for which it is possible to build meaningful mathematical models.Materials and Methods. Using the methods of the theory of the discontinuous Riemann-Hilbert problem for generalized analytic functions, a criterion for the unconditional solvability of the corresponding static problem for the equilibrium equation of a convex shell with a ribbed lateral surface has been obtained. This criterion, combined with the methods of the theory of generalized analytical functions, is a tool for constructing mathematical models of the state of momentless stress equilibrium of elastic convex shells.Results. A method has been developed for constructing mathematical models of the momentless equilibrium stress state of a convex shell under the action of a variable external load and the condition of stress concentration at the corner points of the median surface. The introduction of a vector parameter, as well as the concepts of “order of quasi-correctness” and “quasi-stability”, into the boundary condition provided both quantitative and qualitative comparison of mathematical models. Classes of shells have been found for which the description of mathematical models is given in terms of the geometry of the boundary in the vicinity of the corner points of the median surface. The obtained result, when applied to shallow convex shells, provides a geometric criterion of quasi-stability. It is established that for a shallow shell, which is not quasi-stable, the only adequate mathematical model is a probabilistic one.Discussion and Conclusions. The proposed method for constructing a two-parameter family of problems with a modified boundary condition makes it possible to simulate the momentless equilibrium stress state for fairly wide classes of convex shells with a piecewise-smooth lateral surface under a sleeve connection. At the same time, the developed algorithm for calculating the boundary condition index allowed us to answer the question of the existence of an adequate mathematical model for a shell with a side surface of an arbitrary configuration, and for shells of a special type (specifically, shallow or shells of revolution), to formulate a geometric criterion for the existence of a mathematical model.