Three-parameter Family Research Articles

A Riemannian approach to strain measures in nonlinear elasticity

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n) of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n). Objectivity requires the Riemannian metric to be left-GL(n)-invariant, isotropy requires the Riemannian metric to be right-O(n)-invariant. The latter two conditions are only satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n). Surprisingly, the final result is basically independent of the chosen parameters.In deriving the result, geodesics on GL(n) have to be parameterized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved.

Improving the sensitivity of a search for coalescing binary black holes with nonprecessing spins in gravitational wave data

We demonstrate for the first time a search pipeline with improved sensitivity to gravitational waves from coalescing binary black holes with spins aligned to the orbital angular momentum by the inclusion of spin effects in the search templates. We study the pipeline recovery of simulated gravitational wave signals from aligned-spin binary black holes added to real detector noise, comparing the pipeline performance with aligned-spin filter templates to the same pipeline with nonspinning filter templates. Our results exploit a three-parameter phenomenological waveform family that models the full inspiral-merger-ringdown coalescence and treats the effect of aligned spins with a single effective spin parameter $\ensuremath{\chi}$. We construct template banks from these waveforms by a stochastic placement method and use these banks as filters in the recently developed gstlal search pipeline. We measure the observable volume of the analysis pipeline for binary black hole signals with ${M}_{\text{total}}$ and $\ensuremath{\chi}\ensuremath{\in}[0,0.85]$. We find an increase in observable volume of up to 45% for systems with $0.2\ensuremath{\le}\ensuremath{\chi}\ensuremath{\le}0.85$ with almost no loss of sensitivity to signals with $0\ensuremath{\le}\ensuremath{\chi}\ensuremath{\le}0.2$. We also show that the use of spinning templates in the search pipeline provides for more accurate recovery of the binary mass parameters as well as an estimate of the effective spin parameter. We demonstrate this analysis on 25.9 days of data obtained from the Hanford and Livingston detectors in LIGO's fifth observation run.

Rotating elegant bessel-gaussian beams

We considered a new three-parameter family of rotating asymmetric Bessel-Gaussian beams (aBG-beams) with integer and fractional orbital angular momentum (OAM). Complex amplitude of aBG-beams is proportional to a Gaussian function and to n-th order Bessel function with complex argument. These beams have finite energy. The asymmetry ratio of the aBG-beam depends on the real parameter c ≥ 0. For c = 0, the aBG-beam coincides with the conventional radially symmetric Bessel-Gaussian beam, with increasing c intensity distribution of the aBG-beam takes the form of Crescent. At c >> 1, the beam is widening along the vertical axis while shifting along the horizontal axis. In initial plane, the intensity distribution of asymmetric Bessel-Gaussian beams has a countable number of isolated zeros located on the horizontal axis. Locations of these zeros correspond to optical vortices with unit topological charges and opposite signs on different sides of the origin. During the beam propagation, centers of these vortices rotate around the optical axis along with the entire beam at a non-uniform rate (for large c >> 1): they will rotate by 45 degrees at a distance of the Rayleigh range, and then by another 45 degrees during the rest distance. For different values of the parameter c zeros in the transverse intensity distribution of the beam change their location and change the OAM of the beam. Isolated intensity zero on the optical axis corresponds to an optical vortex with topological charge of n. Laser beam with the shape of rotating Crescent has been generated by using a spatial light modulator.

Self-similar asymptotics for linear and nonlinear mathematical models of tumor angiogenesis: a review

We show that the long time asymptotic solutions of initial value problems for linear and nonlinear mathematical models of tumor angiogenesisare self-similar spreading solutions. The symmetries of the governing equationsyield three-parameter families of these solutions given in terms of their mass,center of mass, and variance. Unlike the mass and center of mass, the variance,or ”time-shift,” of a solution is not a conserved quantity for the non linear problem

Ceva's triangle inequalities

We characterize triples of cevians which form a triangle independent of the triangle where they are constructed. This problem is equivalent to solving a three-parameter family of inequalities which we call Ceva’s triangle inequalities. Our main result provides the parametrization of the solution set.

Whirling skirts and rotating cones

Steady, dihedrally symmetric patterns with sharp peaks may be observed on a spinning skirt, lagging behind the material flow of the fabric. These qualitative features are captured with a minimal model of traveling waves on an inextensible, flexible, generalized-conical sheet rotating about a fixed axis. Conservation laws are used to reduce the dynamics to a quadrature describing a particle in a three-parameter family of potentials. One parameter is associated with the stress in the sheet, another is the Noether current associated with rotational invariance and the third is a Rossby number which indicates the relative strength of Coriolis forces. Solutions are quantized by enforcing a topology appropriate to a skirt and a particular choice of dihedral symmetry. A perturbative analysis of nearly axisymmetric cones shows that Coriolis effects are essential in establishing skirt-like solutions. Fully nonlinear solutions with three-fold symmetry are presented which bear a suggestive resemblance to the observed patterns.

On the integrability of zero-range chipping models with factorized steady states

The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis, we conjecture an integral formula for the Green function of the evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.

Linear canonical Wigner distribution and its application

Linear canonical transform (LCT) form a three-parameter family of intergral transforms with wide application in optics. In this paper, we investigate the linear canonical Wigner distribution (LCWD) which is based on the LCT and the classical Wigner distribution (WD). Firstly, the definition of LCWD is discussed. Moreover, the transformation law for the LCWD through a first-order optical system is derived. This new phase-space distribution provides analysis of signals in both space and LCT domains simultaneously. Then, the main properties of LCWD are investigated in detail. Finally, the application of the LCWD is presented. The LCWD is found to be the appropriate phase-space distribution function for light-beam characterization in first-order optical system. Moreover, the moment matrix formalism for beam characterization is studied.

A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function

A new three-parameter exponential-type family of distributions which can be used in modeling survival data, reliability problems and fatigue life studies is introduced. Its failure rate function can be constant, decreasing, increasing, upside-down bathtub or bathtub-shaped depending on its parameters. It includes as special sub-models the exponential distribution, the generalized exponential distribution [Gupta, R.D., Kundu, D., 1999. Generalized exponential distributions. Australian and New Zealand Journal of Statistics 41, 173–188] and the extended exponential distribution [Nadarajah, S., Haghighi, F., 2011. An extension of the exponential distribution. Statistics 45, 543–558]. A comprehensive account of the mathematical properties of the new family of distributions is provided. Maximum likelihood estimation of the unknown parameters of the new model for complete sample as well as for censored sample is discussed. Estimation of the stress–strength parameter is also considered. Two empirical applications of the new model to real data are presented for illustrative purposes.

A note on the integrability of a remarkable static Euler–Bernoulli beam equation

It has recently been shown that the fourth-order static Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y −5/3, and the equation has the nonsolvable algebra \({sl(2, \mathbb{R})}\) . We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of \({sl(2, \mathbb{R})}\) remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y −5/3.

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

Although phase transition waves in atomic chains with double-well potential play a fundamental role in materials science, very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewise-quadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the corresponding three-parameter family of waves persists as long as the perturbation is small and localized with respect to the strain variable. As a standard Lyapunov--Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterize the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive on a small ball in a suitable function space. Moreover, we derive a uniqueness result for phase transition waves with certain properties and discuss the kinetic relations.

Direction of light propagation to order G2 in static, spherically symmetric spacetimes: a new derivation

A procedure avoiding any integration of the null geodesic equations is used to derive the direction of light propagation in a three-parameter family of static, spherically symmetric spacetimes within the post-post-Minkowskian approximation. Quasi-Cartesian isotropic coordinates adapted to the symmetries of spacetime are systematically used. It is found that the expression of the angle formed by two light rays as measured by a static observer staying at a given point is remarkably simple in these coordinates. The attention is mainly focused on the null geodesic paths that we call the quasi-Minkowskian light rays. The vector-like functions characterizing the direction of propagation of such light rays at their points of emission and reception are firstly obtained in the generic case where these points are both located at finite distances from the centre of symmetry. The direction of propagation of the quasi-Minkowskian light rays emitted at infinity is then straightforwardly deduced. An intrinsic definition of the gravitational deflection angle relative to a static observer located at a finite distance is proposed for these rays. The expression inferred from this definition extends the formula currently used in VLBI astrometry up to the second order in the gravitational constant G.

New quasi-exactly solvable double-well potentials

A new three-parameter family of quasi-exactly solvable double-well potentials is introduced. We show that the solutions of this family of double-well potentials are expressed in terms of the Heun confluent functions. Under a certain parameter condition, some of the bound-state wavefunctions and associated energies can be found exactly in explicit form. In particular, we develop an analytical method to derive the conditions for the energy eigenvalues of the bound states. It is also shown that our analytical results can be applied to construct exact solutions of the nonlinear Schrödinger equation with double-well potentials and spatially localized nonlinearities.

An example of a piecewise linear ergodic polymorphism

We consider dynamical systems introduced by Vershik and called polymorphisms. In particular, such systems encompass the class of multivalued mappings of a closed interval onto itself which have an invariant measure. Polymorphisms arise in different areas of mathematics and mechanics, for example, in the problem of the destruction of the adiabatic invariant. We are concerned with the ergodic properties of polymorphisms. The first section deals with the main notions. In Secs. 2 and 3, we consider an example of a three-parameter family of ergodic polymorphisms formed by piecewise linear mappings.

Generalized Poincaré algebras, Hopf algebras and κ-Minkowski spacetime

We propose a generalized description for the κ-Poincaré–Hopf algebra as a symmetry quantum group of underlying κ-Minkowski spacetime. We investigate all the possible implementations of (deformed) Lorentz algebras which are compatible with the given choice of κ-Minkowski algebra realization. For the given realization of κ-Minkowski spacetime there is a unique κ-Poincaré–Hopf algebra with undeformed Lorentz algebra. We have constructed a three-parameter family of deformed Lorentz generators with κ-Poincaré algebras which are related to κ-Poincaré–Hopf algebra with undeformed Lorentz algebra. Known bases of κ-Poincaré–Hopf algebra are obtained as special cases. Also deformation of igl(4) Hopf algebra compatible with the κ-Minkowski spacetime is presented. Some physical applications are briefly discussed.

Renormalization group flow of the Holst action

The renormalization group (RG) properties of quantum gravity are explored, using the vielbein and the spin connection as the fundamental field variables. The scale dependent effective action is required to be invariant both under spacetime diffeomorphisms and local frame rotations. The nonperturbative RG equation is solved explicitly on the truncated theory space defined by a three-parameter family of Holst-type actions which involve a running Immirzi parameter. We find evidence for the existence of an asymptotically safe fundamental theory, probably inequivalent to metric quantum gravity constructed in the same way.

Dipole evolution in rotating two-dimensional flow with bottom friction

The evolution of a dipolar vortex in a quasi-two-dimensional rotating flow of barotropic fluid over a flat surface with a no-slip condition in the Ekman bottom layer is considered. The vorticity equation in this case becomes nonlinear. An effect of bottom friction is displayed mainly in cyclone-anticyclone asymmetry, which results in the expansion (diminution) of cyclonic (anticyclonic) local structures and in the stronger decay of positive vorticity. When a lateral viscosity is omitted, the vorticity equation has a solution in the form of vortex patches and hence a contour dynamics method may be used for numerical simulation. An approach of point vortices with decaying strengths is also discussed. In an approximation of two patches of opposite uniform vorticity, a three-parameter family of stationary (in an ideal fluid) orbital dipoles [V. G. Makarov and Z. Kizner, “Stability and evolution of uniform-vorticity dipoles,” J. Fluid Mech. 672, 307 (2011)] https://doi.org/10.1017/S0022112010006026 consisting of patches with unequal areas and absolute values of vorticity is considered. A three-dimensional domain of instability for this family is numerically constructed. It is shown that the evolution of such dipoles in a flow with bottom friction is described with good accuracy by a properly matched trajectory in parameter space of ideal-fluid steady states. Explicit time-dependent formulae for this phase trajectory are obtained. All characteristics (including the patch's shapes) of the evolutionary dipole, and the same characteristics for the corresponding ideal-fluid stationary dipole, almost completely coincide, at least while the phase trajectory remains in the stability region. The evolution of stationary translating dipoles (with zero net circulation) that have continuously distributed vorticity inside a quasi-elliptical finite region is also examined. When the circular Lamb dipole is used as the initial condition, good qualitative agreement is observed between the asymmetric dipoles obtained during evolution and the chain of the exact solutions from the known one-parameter family of non-symmetrical Chaplygin dipoles.

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of non$-$smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called fold-cusp singularity by means of the variation of 3 parameters.

OSCILLATORY WAVES IN DISCRETE SCALAR CONSERVATION LAWS

We study Hamiltonian difference schemes for scalar conservation laws with monotone flux function and establish the existence of a three-parameter family of periodic traveling waves (wavetrains). The proof is based on an integral equation for the dual wave profile and employs constrained maximization as well as the invariance properties of a gradient flow. We also discuss the approximation of wavetrains and present some numerical results.

Magnetohydrodynamics convective flow in a vertical slot through a porous medium

An analysis is presented with magnetohydrodynamics natural convective flow of a viscous Newtonian fluid saturated porous medium in a vertical slot. The flow in the porous media has been modeled using the Brinkman model. The fully-developed two-dimensional flow from capped to open ends is considered for which a continuum of solutions is obtained. The influence of pertinent parameters on the flow is delineated and appropriate conclusions are drawn. The asymptotic behaviour and the volume flux are analyzed and incorporated graphically for the three-parameter family of solution.