Two-parameter Family Research Articles

Decision-making under interval uncertainty revisited

Purpose In many real-life situations, we do not know the exact values of the expected gain corresponding to different possible actions, we only have lower and upper bounds on these gains – i.e., in effect, intervals of possible gain values. The purpose of this study is to describe all possible ways to make decisions under such interval uncertainty. Design/methodology/approach The authors used both natural invariance and additivity requirements. Findings The authors demonstrated that natural requirements – invariance or additivity – led to a two-parametric family of possible decision-making strategies. Originality/value This is a first description of all reasonable strategies for decision-making under interval uncertainty – strategies that satisfy natural requirements of invariance or additivity.

Nonminimal dyons with regular gravitational, electric and axion fields

We consider the distribution of the axionic dark matter and the profiles of the axionically induced electric field on the background of a regular magnetic monopole. The background static spherically symmetric spacetime is considered to be regular at the center of the object; it has no event horizons, the nonminimal interaction of strong gravitational and monopole type magnetic fields being the source of this regularity. We assume the pseudoscalar (axion) field to be nonminimally coupled to the electromagnetic and gravitational fields, and we search for solutions to the extended equations of nonminimal axion electrostatics, which satisfy two requirements: first, the axionically induced radial electric field vanishes at the center; second, the pseudoscalar field is regular at the center of the formed axionic dyon. We have found the set of parameters, describing the nonminimal axion-photon coupling, for which both requirements are satisfied. Properties of the two-parameter family of regular solutions to the equations of nonminimal axion electrostatics are analyzed numerically. The regular axionically induced electric field is shown to have the structure, which is typical for a simple electric layer.

Family of chaotic maps from game theory

ABSTRACT From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is, for every . This solves a known open question whether there exists a ‘natural’ nontrivial smooth map other than with centres of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.

The geometric classification of nilpotent ℭ𝔇-algebras

We give a geometric classification of complex [Formula: see text]-dimensional nilpotent [Formula: see text]-algebras. The corresponding geometric variety has dimension 18 and decomposes into [Formula: see text] irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras. In particular, there are no rigid [Formula: see text]-dimensional complex nilpotent [Formula: see text]-algebras.

A functional identity for Mahler measures of non-tempered polynomials

ABSTRACT We establish a functional identity for Mahler measures of the two-parametric family . Our result extends an identity proven in a paper of Lalín, Zudilin and Samart. As a by-product, we obtain evaluations of for some algebraic values of a and c in terms of special values of L-functions and logarithms. We also give a sufficient condition for validity of a certain identity between the elliptic integrals of the first and the third kind, which implies several identities for .

The Star Formation Rate–Radius Connection: Data and Implications for Wind Strength and Halo Concentration

Abstract This paper is one in a series that explores the importance of radius as a second parameter in galaxy evolution. The topic investigated here is the relationship between star formation rate (SFR) and galaxy radius ( ) for main-sequence star-forming galaxies. The key observational result is that, over a wide range of stellar mass and redshift in both CANDELS and SDSS, there is little correlation between SFR and at fixed stellar mass. The Kennicutt–Schmidt law, or any similar density-related star formation law, then implies that smaller galaxies must have lower gas fractions than larger galaxies (at fixed ), and this is supported by observations of gas in local star-forming galaxies. We investigate the implications by adopting the equilibrium “bathtub” model: the ISM gas mass is assumed to be constant over time, and the net SFR is the difference between the accretion rate of gas onto the galaxy from the halo and the outflow rate due to winds. To match the observed null correlation between SFR and radius, the bathtub model requires that smaller galaxies at fixed mass have weaker galactic winds. Our hypothesis is that galaxies are a two-parameter family whose properties are set mainly by halo mass and concentration. These determine the radius and gas accretion rate, which in turn predict how wind strength needs to vary with to keep the SFR constant.

Experimentally Accessible Orbits Near a Bykov Cycle

This paper reports numerical experiments done on a two-parameter family of vector fields which unfold an attracting heteroclinic cycle linking two saddle-foci. We investigated both local and global bifurcations due to symmetry breaking in order to detect either hyperbolic or chaotic dynamics. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is still out of reach, using a combination of theoretical tools and computer simulations we have uncovered some complex patterns. We have selected suitable initial conditions to analyze the bifurcation diagrams, and regarding these solutions we have located: (a) an open domain of parameters with regular dynamics; (b) infinitely many parabolic-type curves associated to homoclinic Shilnikov cycles which act as organizing centers; (c) a crisis region related to the destruction or creation of chaotic attractors; (d) a large Lebesgue measure set of parameters where chaotic regimes are dominant, though sinks and chaotic attractors may coexist, and in whose complement we observe shrimps.

Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

Heegner points in Coleman families

We construct two-parameter analytic families of Galois cohomology classes interpolating the etale Abel--Jacobi images of generalised Heegner cycles, with both the modular form and Grossencharacter varying in p-adic families.

Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the equilibria do not intersect. Under some conditions on the parameters and on the eigenvalues of the linearisation of the vector field at the saddle-foci, we prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus to Henon-like strange attractors, as a consequence of the Torus-Breakdown Theory. The mechanism for the creation of horseshoes and strange attractors is also discussed. Theoretical results are applied to show the occurrence of strange attractors in some analytic unfoldings of a Hopf-zero singularity.

Variation of complex structures and variation of Lie algebras II: new Lie algebras arising from singularities

Finite dimensional Lie algebras are semi-direct product of the semi-simple Lie algebras and solvable Lie algebras. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. It is extremely important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this article, a new natural connection between the set of complex analytic isolated hypersurface singularities and the set of finite dimensional solvable (nilpotent) Lie algebras has been constructed. We construct finite dimensional solvable (nilpotent) Lie algebras naturally from isolated hypersurface singularities. These constructions help us to understand the solvable (nilpotent) Lie algebras from the geometric point of view. Moreover, it is known that the classification of nilpotent Lie algebras in higher dimensions ($\gt 7$) remains to be a vast open area. There are one-parameter families of non-isomorphic nilpotent Lie algebras (but no two-parameter families) in dimension seven. Dimension seven is the watershed of the existence of such families. It is well-known that no such family exists in dimension less than seven, while it is hard to construct one-parameter family in dimension greater than seven. In this article, we construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $11$ (resp. $10$) from $\tilde{E}_7$ singularities and show that the weak Torelli-type theorem holds. We shall also construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $12$ (resp. $11$) from $\tilde{E}_8$ singularities and show that the Torelli-type theorem holds. Moreover, we investigate the numerical relation between the dimensions of the new Lie algebras and Yau algebras. Finally, the new Lie algebras arising from fewnomial isolated singularities are also computed.

Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

Abstract We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps T c,ɛ (x) = (−1 + c|x| 1−ɛ ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

Bifurcation Analysis in Planar Quadratic Differential Systems with Boundary

Given a planar quadratic differential system delimited by a straight line, we are interested in studying the bifurcation phenomena that can arise when the position on the boundary of two tangency points are considered as parameters of bifurcation. First, under generic conditions, we find a two-parametric family of quadratic differential systems with at least one tangency point. After that, we find a normal form for this parameterized family. Next, we study two subfamilies, one of them characterized by the existence of two fold points of different nature, and the other one, characterized by the existence of one fold point and one boundary equilibrium point. For the first family, we find sufficient conditions for the existence of stationary bifurcations: saddle-node, transcritical and pitchfork, while for the second family, the existence of the called transcritical Bogdanov–Takens bifurcation is proved. Finally, the results are illustrated with two examples.

Fungal Tip Growth Arising Through a Codimension-1 Global Bifurcation

Tip growth is a growth stage which occurs in fungal cells. During tip growth, the cell exhibits continuous extreme lengthwise growth while its shape remains qualitatively the same. A model for single celled fungal tip growth is given by the Ballistic Aging Thin viscous Sheet (BATS) model, which consists of a five-dimensional system of first-order differential equations. The solutions of the BATS model that correspond to fungal tip growth arise through a codimension-1 global bifurcation in a two-parameter family of solutions. In this paper we derive a toy model from the BATS model. The toy model is given by two-dimensional system of first-order differential equations which depend on a single parameter. The main achievement of this paper is a proof that the toy model exhibits an analogue of the codimension-1 global bifurcation in the BATS model. An important ingredient of the proof is a topological method which enables the identification of the bifurcation points. Finally, we discuss how the proof may be generalized to the BATS model.

Was GW170817 a Canonical Neutron Star Merger? Bayesian Analysis with a Third Family of Compact Stars

We investigate the possibility that GW170817 was not the merger of two conventional neutron stars (NS), but involved at least one if not two hybrid stars with a quark matter core that might even belong to a third family of compact stars. To this end, we develop a Bayesian analysis method for selecting the most probable equation of state (EoS) under a set of constraints from compact star physics, which now also include the tidal deformability from GW170817 and the first result for the mass and radius determination for PSR J0030+0451 by the NICER Collaboration. We apply this method for the first time to a two-parameter family of hybrid EoS based on the DD2 model with nucleonic excluded volume for hadronic matter and the color superconducting generalized nlNJL model for quark matter. The model has a variable onset density for deconfinement and can mimic the effects of pasta phases with the possibility of producing a third family of hybrid stars in the mass-radius diagram. The main findings of this study are that: (1) the presence of multiple configurations for a given mass (twins or even triples) corresponds to a set of disconnected lines in the Λ 1 – Λ 2 diagram of tidal deformabilities for binary mergers, so that merger events from the same mass range may result in a probability landscape with different peak positions; (2) the Bayesian analysis with the above observational constraints favors an early onset of the deconfinement transition, at masses of M onset ≤ 0.8 M ⊙ with an M–R relationship that in the range of observed neutron star masses is almost indistinguishable from that of a soft hadronic Akmal, Pandharipande, and Ravenhall (APR) EoS; (3) a few, yet fictitious measurements of the NICER experiment two times more accurate than the present value and a different mass and radius that would change the posterior likelihood so that hybrid EoS with a phase transition onset in the range M onset = 1.1–1.6 M ⊙ would be favored.

Gluing affine Yangians with bi-fundamentals

The affine Yangian of {mathfrak{gl}}_1 is isomorphic to the universal enveloping algebra of {mathcal{W}}_{1+infty } and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of mathcal{N} = 2 supersymmetric {mathcal{W}}_{infty } algebra was constructed by “gluing” two affine Yangians of {mathfrak{gl}}_1 using operators that transform as (□, overline{square} ) and ( overline{square} , □) w.r.t. the two affine Yangians. In this paper we realize a similar (but non-isomorphic) two-parameter gluing construction where the gluing operators transform as (□, □) and ( overline{square} , overline{square} ) w.r.t. the two affine Yangians. The corresponding representation space consists of pairs of plane partitions connected by a common leg whose cross-section takes the shape of Young diagrams, offering a more transparent geometric picture than the previous construction.

The Rovella attractor is asymptotically sectional-hyperbolic**The first author was partially supported by Proyecto FONDECYT 1181183, Chile. The second author was supported by CONICYT grant 21180163, Chile.

The Rovella attractor is a compact invariant set for a vector field X0 that mimics the construction of the geometric Lorenz attractor, but considering a central contractive singularity instead of a central expansive one. In this paper we will prove that the Rovella attractor is asymptotically sectional-hyperbolic. Furthermore, it is proved that for a generic two-parameter family of vector fields containing X0, asymptotically sectional-hyperbolicity is an almost 2-persistent property.

On the Solvability of a Class of Discrete Matrix Equations with Cubic Nonlinearity

We study and solve one class of discrete matrix equations with cubic nonlinearity. The existence of a two-parameter family of monotone and bounded solutions is proved. Under certain additional conditions, we determine the asymptotic behavior of the constructed solutions. The obtained results are extended to the corresponding inhomogeneous discrete matrix equations and to some more general cases of nonlinearity.

Generalised solutions to the Benjamin problem

We generalise the classical Benjamin solution (Benjamin, J. Fluid Mech. , vol. 31, 1968, pp. 209–248) modelling the flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the surface detaches from the ceiling of the duct as a free surface. It is shown that the Benjamin solution belongs to a one-parameter family of similar solutions, which are divided into two types: solutions that describe potential flows where the free surface of the fluid is deflected from the duct ceiling at a zero angle; and solutions that admit the formation of a vortex flow region in the vicinity of the point of fluid separation from the duct ceiling. It is shown that this one-parameter family of solutions is the limit of a two-parameter family of solutions in which part of the uniform flow energy is converted into energy of the small-scale fluid motion. Based on the local hydrostatic approximation, the applicability of the constructed solutions is discussed.

New optical solitons of Kundu-Eckhaus equation via λ-symmetry

Abstract New closed-form exact solutions for the nonlinear Kundu-Eckahus (KE) equation with generalized coefficients are obtained. A travelling wave transformation reduces the KE equation to a second-order ordinary differential equation that is completely integrated by using the λ-symmetry approach. A one-parameter family of singular solutions of the reduced equation provides a unified expression for a class of solutions for the KE equation which contains, as particular cases, most of the exact solutions derived during the last years by using a great variety of powerful integration methods. The general solution of the reduced equation permits to construct a two-parameter family of exact solutions for the KE equation, providing a rich class of new exact solutions that, to the best or our knowledge, have not been reported before.