Two-parameter Family Research Articles

Relation of the Böttcher Equation with the Parametrized Poisson Integral

The Bottcher functional equation and one of its real generalizations are considered. It is shown that, in some situations, after finding a solution of the generalized equation, other solutions can also be obtained. For example, a three-parameter family of real functional equations for a function of two arguments is described, for which solutions are found. The generalization described has wide application. Many quantities after an appropriately introduced parameterization satisfy the generalized Bottcher equation as functions of parameters. As an illustration, two-parametric families generated by the determinant of a linear combination of second-order matrices are presented. It is shown that the parameterized Poisson integral as a function of its parameters satisfies the generalized Bottcher equation. This made it possible to calculate the Poisson integral and the Euler integral in a new way. In addition, the calculation of the Poisson integral by the method of integral sums is described.

Fixed points and dynamics of two-parameter family of hyperbolic cosine like functions

In this paper, the two-parameter family Fb≡{fλ,μ(z)=λbz+μb−zforz∈C:λ≥μ>0} of transcendental entire functions is considered. By investigating on the existence and nature of the fixed points of fλ,μ, the dynamics of functions fλ,μ∈Fb is studied. It is shown that the Julia set of fλ,μ is nowhere dense and not locally connected whenever parameters λ and μ satisfy 1+4λμ(ln⁡b)2≤ln⁡(1+1+4λμ(ln⁡b)22λln⁡b) and the Julia set is the whole of extended complex plane for 1+4λμ(ln⁡b)2>ln⁡(1+1+4λμ(ln⁡b)22λln⁡b).

Gravitational properties of the Proca field

We study various properties of a Proca field coupled to gravity through minimal and quadrupole interactions, described by a two-parameter family of Lagrangians. Stückelberg decomposition of the effective theory spells out its model-dependent ultraviolet cutoff, parametrically larger than the Proca mass. We present pp-wave solutions that the model admits, consider linear fluctuations on such backgrounds, and thereby constrain the parameter space of the theory by requiring null-energy condition and the absence of negative time delays in high-energy scattering. We briefly discuss the positivity constraints—derived from unitarity and analyticity of scattering amplitudes—that become ineffective in this regard.

Renormalization group flow of the Aharonov–Bohm scattering amplitude

The Aharonov–Bohm elastic scattering with incident particles described by plane waves is revisited by using the phase-shifts method. The formal equivalence between the cylindrical Schrödinger equation and the one-dimensional Calogero problem allows us to show that up to two scattering phase-shifts modes in the cylindrical waves expansion must be renormalized. The renormalization procedure introduces new length scales giving rise to spontaneous breaking of the conformal symmetry. The knowledge of the exact beta function permits us to describe the renormalization group flows within the two-parametric family of renormalized Aharonov–Bohm scattering amplitudes.

Squashed holography with scalar condensates

We evaluate the partition function of the free and interacting O(N) vector model on a two-parameter family of squashed three spheres in the presence of a scalar deformation. We also find everywhere regular solutions of Einstein gravity coupled to a scalar field in AdS and in dS with the same double squashed boundary geometry. Remarkably, the thermodynamic properties of the AdS solutions qualitatively agree with the behavior predicted by the free O(N) model with a real mass deformation. The dS bulk solutions specify the semiclassical ‘no-boundary’ measure over anisotropic deformations of inflationary, asymptotic de Sitter space. Through dS/CFT the partition function of the interacting O(N) model yields a holographic toy model of the no-boundary measure. We find this yields a qualitatively similar probability distribution which is normalizable and globally peaked at the round three sphere, with a low amplitude for strong anisotropies.

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

The quadrature rule of Hammer and Stroud (1956) for cubic polynomials has been shown to be exact for a larger space of functions, namely the C1 cubic Clough–Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle Kosinka and Bartoň (2018). We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of C1 quadratic Powell–Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C1 quadratic Powell–Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in Kosinka and Bartoň (2018). The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three.

M-polynomial revisited: Bethe cacti and an extension of Gutman’s approach

The M-polynomial of a graph G is defined as $$\sum _{i\le j} m_{i,j}(G)x^iy^j$$ , where $$m_{i,j}(G)$$ , $$i,j\ge 1$$ , is the number of edges uv of G such that $$\{d_v(G), d_u(G)\} = \{i,j\}$$ . Knowing the M-polynomial, formulas for bond incident degree indices (an important subclass of degree-based topological indices) can be obtained by means of specific operators defined on differentiable functions in two variables. This is illustrated on three infinite families of Bethe cacti. Gutman’s approach for the computation of the coefficients of the M-polynomial is also recalled and an extension of it is given. This extension is used to determine the M-polynomial of a two-parameter infinite family of lattice graphs.

Spherically symmetric configurations of General Relativity in presence of scalar fields: separation of circular orbits

We study test-body orbits in the gravitational field of a static spherically symmetric object in presence of a minimally coupled nonlinear scalar field. We generated a two-parametric family of scalar field potentials, which allow finding solutions of Einstein's equations in an analytic form. The results are presented by means of hypergeometric functions; they describe either a naked singularity (NS) or a black hole (BH). Our numerical investigation shows that in both cases the stable circular orbits can form separated (non-connected) regions around the configuration. We found existence conditions for such separated regions and present examples for some family parameters in case of NS and BH. The results may be of interest for testing models of the dynamical dark energy.

About the Optimal Replacement of the Lebesque Constant Fourier Operator by a Logarithmic Function

The Lebesgue constant, corresponding to the classical Fourier operator is approaching a two-parameter family of logarithmic functions. The optimal values of these parameters are found in which the best uniform approximation of the Lebesgue constant a well-defined function of this family is achieved. We considered the case when corresponding to these functions the remaining members are strictly decreasing.

Phase transitions of a two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki model

We study spin-2 deformed-AKLT models on the square lattice, specifically a two-parameter family of $O(2)$-symmetric ground-state wavefunctions as defined by Niggemann, Kl\"umper, and Zittartz, who found previously that the phase diagram consists of a N\'eel-ordered phase and a disordered phase which contains the AKLT point. Using tensor-network methods, we not only confirm the N\'eel phase but also find an XY phase with quasi-long-range order and a region adjacent to it, within the AKLT phase, with very large correlation length, and investigate the consequences of a perfectly factorizable point at the corner of that phase.

Orbital Stability of Solitary Waves for Generalized Derivative Nonlinear Schrödinger Equations in the Endpoint Case

We consider the following generalized derivative nonlinear Schr\"odinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2\sigma}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $\sigma\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{\omega,c}(t,x)=\Phi_{\omega,c}(x)e^{i\omega t+\frac{ic}2x-\frac i{2\sigma+2}\int_0^x\Phi_{\omega,c}(y)^{2\sigma}dy},$$ with $(\omega,c)$ satisfying $\omega>c^2/4$, or $\omega=c^2/4$ and $c>0$. The stability theory in the frequency region $\omega>c^2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $\omega=c^2/4$ and $c>0$.

On Radial Schr\xf6dinger Operators with a Coulomb Potential

This paper presents a thorough analysis of one-dimensional Schrödinger operators whose potential is a linear combination of the Coulomb term 1 / r and the centrifugal term 1/r^2. We allow both coupling constants to be complex. Using natural boundary conditions at 0, a two-parameter holomorphic family of closed operators on L^2({mathbb {R}}_+) is introduced. We call them the Whittaker operators, since in the mathematical literature their eigenvalue equation is called the Whittaker equation. Spectral and scattering theory for Whittaker operators is studied. Whittaker operators appear in quantum mechanics as the radial part of the Schrödinger operator with a Coulomb potential.

The complete classification of empty lattice 4-simplices

In previous work we classified all empty 4-simplices of width at least three. We here classify those of width two. There are 2 two-parameter families that project to the second dilation of a unimodular triangle, 29+23 one-parameter families of them that project to hollow 3-polytopes, and 2282 individual ones that do not.

Invariant entanglement and generation of quantum correlations under global dephasing

We investigate the dynamics of quantum entanglement and more general quantum correlations quantified via negativity and local quantum uncertainty, respectively, for two-qubit systems undergoing Markovian collective dephasing. Focusing on a two-parameter family of initial two-qubit density matrices, we study the relation of the emergence of the curious phenomenon of time-invariant entanglement and the dynamical behavior of local quantum uncertainty. Developing an illustrative geometric approach, we demonstrate the existence of distinct regions of quantum entanglement for the considered initial states and identify the region that allows for completely frozen entanglement throughout the dynamics, accompanied by generation of local quantum uncertainty. Furthermore, we present a systematic analysis of different dynamical behaviors of local quantum uncertainty, such as its sudden change or smooth amplification, in relation with the dynamics of entanglement.

Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps

<p style='text-indent:20px;'>We characterize the attractors for a two-parameter class of two-dimensional piecewise affine maps. These attractors are strange attractors, probably having finitely many pieces, and coincide with the support of an ergodic absolutely invariant probability measure. Moreover, we demonstrate that every compact invariant set with non-empty interior contains one of these attractors. We also prove the existence, for each natural number <inline-formula><tex-math id="M1">\begin{document}$ n, $\end{document}</tex-math></inline-formula> of an open set of parameters in which the respective transformation exhibits at least <inline-formula><tex-math id="M2">\begin{document}$ 2^n $\end{document}</tex-math></inline-formula> non connected two-dimensional strange attractors each one of them formed by <inline-formula><tex-math id="M3">\begin{document}$ 4^n $\end{document}</tex-math></inline-formula> pieces.

Field redefinitions and Plebanski formalism for GR

We point out that there exists a family of transformations acting on BF-type Lagrangians of gravity, with Lagrangians related by such a transformation corresponding to classically equivalent theories. A transformation of this type corresponds to a particular field redefinition. We discuss both the chiral and non-chiral cases. In the chiral case there is a one-parameter, and in the non-chiral case a two-parameter family of such transformations. In the chiral setup, we use these transformations to give an alternative derivation of the chiral BF plus potential formulation of general relativity that was proposed recently. In the non-chiral case, we show that there is a new BF plus potential type formulation of GR. We also make some remarks on the non-chiral pure connection formulation.

Gravitational waves from quasinormal modes of a class of Lorentzian wormholes

Quasinormal modes of a two-parameter family of Lorentzian wormhole spacetimes, which arise as solutions in a specific scalar-tensor theory associated with braneworld gravity, are obtained using standard numerical methods. Being solutions in a scalar-tensor theory, these wormholes can exist with matter satisfying the Weak Energy Condition. If one posits that the end-state of stellar-mass binary black hole mergers, of the type observed in GW150914, can be these wormholes, then we show how their properties can be measured from their distinct signatures in the gravitational waves emitted by them as they settle down in the post-merger phase from an initially perturbed state. We propose that their scalar quasinormal modes correspond to the so-called breathing modes, which normally arise in gravitational wave solutions in scalar-tensor theories. We show how the frequency and damping time of these modes depend on the wormhole parameters, including its mass. We derive the mode solutions and use them to determine how one can measure those parameters when these wormholes are the endstate of binary black hole mergers. Specifically, we find that if a breathing mode is observed in LIGO-like detectors with design sensitivity, and has a maximum amplitude equal to that of the tensor mode that was observed of GW150914, then for a range of values of the wormhole parameters, we will be able to discern it from a black hole. If in future observations we are able to confirm the existence of such wormholes, we would, at one go, have some indirect evidence of a modified theory of gravity as well as extra spatial dimensions.

Wavefronts, actions and caustics determined by the probability density of an Airy beam

The main contribution of the present work is to use the probability density of an Airy beam to identify its maxima with the family of caustics associated with the wavefronts determined by the level curves of a one-parameter family of solutions to the Hamilton–Jacobi equation with a given potential. To this end, we give a classical mechanics characterization of a solution of the one-dimensional Schrödinger equation in free space determined by a complete integral of the Hamilton–Jacobi and Laplace equations in free space. That is, with this type of solution, we associate a two-parameter family of wavefronts in the spacetime, which are the level curves of a one-parameter family of solutions to the Hamilton–Jacobi equation with a determined potential, and a one-parameter family of caustics. The general results are applied to an Airy beam to show that the maxima of its probability density provide a discrete set of: caustics, wavefronts and potentials. The results presented here are a natural generalization of those obtained by Berry and Balazs in 1979 for an Airy beam. Finally, we remark that, in a natural manner, each maxima of the probability density of an Airy beam determines a Hamiltonian system.

T-duality and high-derivative gravity theories: the BTZ black hole/string paradigm

We show that the temperature and entropy of a BTZ black hole are invariant under T-duality to next to leading order in M⋆− 2, M⋆ being the scale suppressing higher-curvature/derivative terms in the Lagrangian. We work in the framework of a twoparameter family of theories exhibiting T-duality, which includes (but goes beyond) String Theory. Interestingly enough, the AdS/CFT correspondence enforces quantization conditions on these parameters. In the particular case of bosonic/heterotic string theory, our results extend those of a classical paper by Horowitz and Welch. For generic (albeit quantized) values of the parameters, it suggests that T-duality might be an interesting tool to constrain consistent low-energy effective actions while entailing physical equivalences outside String Theory. Moreover, it generates a new family of regular asymptotically flat black string solutions in three-dimensions.

Virtual walks in spin space: A study in a family of two-parameter models.

We investigate the dynamics of classical spins mapped as walkers in a virtual "spin" space using a generalized two-parameter family of spin models characterized by parameters y and z [de Oliveira etal., J. Phys. A 26, 2317 (1993)JPHAC50305-447010.1088/0305-4470/26/10/006]. The behavior of S(x,t), the probability that the walker is at position x at time t, is studied in detail. In general S(x,t)∼t^{-α}f(x/t^{α}) with α≃1 or 0.5 at large times depending on the parameters. In particular, S(x,t) for the point y=1,z=0.5 corresponding to the Voter model shows a crossover in time; associated with this crossover, two timescales can be defined which vary with the system size L as L^{2}logL. We also show that as the Voter model point is approached from the disordered regions along different directions, the width of the Gaussian distribution S(x,t) diverges in a power law manner with different exponents. For the majority Voter case, the results indicate that the the virtual walk can detect the phase transition perhaps more efficiently compared to other nonequilibrium methods.