Positive Quadrant Research Articles

Bivariate power Lomax distribution with medical applications.

In this paper, a bivariate power Lomax distribution based on Farlie-Gumbel-Morgenstern (FGM) copulas and univariate power Lomax distribution is proposed, which is referred to as BFGMPLx. It is a significant lifetime distribution for modeling bivariate lifetime data. The statistical properties of the proposed distribution, such as conditional distributions, conditional expectations, marginal distributions, moment-generating functions, product moments, positive quadrant dependence property, and Pearson's correlation, have been studied. The reliability measures, such as the survival function, hazard rate function, mean residual life function, and vitality function, have also been discussed. The parameters of the model can be estimated through maximum likelihood and Bayesian estimation. Additionally, asymptotic confidence intervals and credible intervals of Bayesian's highest posterior density are computed for the parameter model. Monte Carlo simulation analysis is used to estimate both the maximum likelihood and Bayesian estimators.

Wavelet Estimation of a Density From Observations of Almost Periodically Correlated Process Under Positive Quadrant Dependence

In this paper, we construct a new wavelet estimator of density for the component of a finite mixture under positive quadrant dependence. Our sample is extracted from almost periodically correlated processes. To evaluate our estimator we will determine a convergence speed from an upper bound for the mean integrated squared error (MISE). Our result is compared to the independent case which provides an optimal convergence rate.

Probability of total domination for transient reflecting processes in a quadrant

AbstractWe consider two-dimensional Lévy processes reflected to stay in the positive quadrant. Our focus is on the non-standard regime when the mean of the free process is negative but the reflection vectors point away from the origin, so that the reflected process escapes to infinity along one of the axes. Under rather general conditions, it is shown that such behaviour is certain and each component can dominate the other with positive probability for any given starting position. Additionally, we establish the corresponding invariance principle providing justification for the use of the reflected Brownian motion as an approximate model. Focusing on the probability that the first component dominates, we derive a kernel equation for the respective Laplace transform in the starting position. This is done for the compound Poisson model with negative exponential jumps and, by means of approximation, for the Brownian model. Both equations are solved via boundary value problem analysis, which also yields the domination probability when starting at the origin. Finally, certain asymptotic analysis and numerical results are presented.

A goodness-of fit improvement based on τ-preserving transformation for semiparametric family of copulas

In this paper, we study on improving the goodness-of fit for the data by using τ-preserving transform for the semiparametric family of bivariate copulas. We estimate the generator function using the Bézier polynomial function and define τ- preserving transformation on the generator function under positive quadrant dependence assumption. We investigate the performance of our methodology on real data example contained life expectancy study. The findings indicate that the model fitting is improved by using τ- preserving transformation.

Vortex Merger in Shallow Water Model

An inviscid, nonlinear shallow water model is applied to study the formation of filaments and merging of potential vorticity (PV, π). We confirm that misalignment of vorticity with the streamlines is crucial to merger. Flow creates positive PV tendency (πt) on the lee side of π ridge, which increases the angle between streamlines and PV contours and the efficiency of vorticity transport along the parallel but oppositive direction in the positive quadrants of πt. So, the vortices move closer while rotating around each other. Filamentation starts as weak vorticity shatters from the outer edge of vortex core. The filament grows and rotates around the cores, but shows little effects to positive PV advection or vortex merging in the inner core, evidenced by the steady core area integrated PV trends. Consecutive PV transfer from lower to higher interval levels are observed during merger process while elongated filamentation is dissolving into the lower level PV regime. The vortices never merge when negative πt prevails between two vortices. Distance between the vortex cores show direct correspondence to positive πt between them. Hence, the advance of positive πt provides a simple mechanism of merging. The Rossby radius of deformation (LD) is confirmed to be another strong indicator for vortex pair merger where LD comparable or smaller than the initial vortex core separation length scale makes merger more likely due to geostrophic adjustment. The Rossby number (Ro) affected the overall flow interaction speed speed when LD is fixed.

Global phase portraits of a predator–prey system

We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator–prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems.

An Alexandrov-Bakelman-Pucci estimate for an anisotropic Laplacian with positive drift in unbounded domains

We show an Alexandrov-Bakelman-Pucci (ABP) estimate for a PDE with anisotropic Laplacian in two dimensions in unbounded domains, where the drift vector varies in a segment of the positive quadrant. The unbounded domains are assumed to be bounded below in the x-direction, as well as in the y-direction. The constant in the upper estimate of the ABP-estimate, which depends in the usual theorems for bounded domains on the diameter of this domain, depends in our case on the small parameter ϵ, appearing in the anisotropic Laplacian. The result is motivated by certain problems of singular perturbation in stochastic control theory, and our methods are probabilistic.

A new non-parametric test for testing positive quadrant dependence

We develop a simple non-parametric test for testing independence of two random variables against the alternative that they are positive quadrant dependent. The asymptotic distribution of the proposed test statistic is distribution free under the null hypothesis of independence. A Monte Carlo simulation study is carried out to evaluate the finite sample performance of the proposed test. We illustrate the proposed method with a real data set.

On the limit cycle of a Belousov–Zhabotinsky differential systems

In Leonov and Kuznetsov (2013), the authors shown numerically the existence of a limit cycle surrounding the unstable node that system (1) has in the positive quadrant for specific values of the parameters. System (1) is one of the Belousov–Zhabotinsky dynamical models. The objective of this paper is to prove that system (1), when in the positive quadrant Q has an unstable node or focus, has at least one limit cycle, and when , , and ϵ > 0 sufficiently small this limit cycle is unique.

Some results on increments of l ∞-valued random fields

The objective of this paper is to establish superior limit results for the increments of a strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) random fields.

Effect of Torsional Loading in an Axisymmetric Micro-Isotropic, Micro-Elastic Solid

In this paper, an attempt is made to obtain the solution for the problem of torsional loading in an axisymmetric Micro-isotropic, Micro-elastic half-space under the action of an arbitrary load on its boundary. The components of displacement, microrotation, stress, couple stress and stress moment are obtained. These components are also obtained for a particular type of twist and represented graphically in the positive quadrant.

Schrödinger operators periodic in octants

We consider Schrödinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrödinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.

Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori

<p style='text-indent:20px;'>This paper concerns the mathematical analysis of quasi-periodic travelling wave solutions for beam equations with damping on 3-dimensional rectangular tori. Provided that the generators of the rectangular torus satisfy certain relationships, by excluding some values of two model parameters, we establish the existence of small amplitude quasi-periodic travelling wave solutions with three frequencies. Moreover, it can be shown that such solutions are either continuations of rotating wave solutions, or continuations of quasi-periodic travelling wave solutions with two frequencies, and that the set of two model parameters is dense in the positive quadrant.</p>

Nonparametric Estimation and Testing for Positive Quadrant Dependent Bivariate Copula

In many practical scenarios (e.g., finance, system reliability, etc.), it is often of interest to estimate a bivariate distribution and test for some desired association properties like positive quadrant dependent (PQD) or negative quadrant dependent (NQD). Often estimation and testing for PQD/NQD property are performed using copula models as it then eliminates the need for estimating marginal distributions. Many parametric copula families have been used that allow for controlling the PQD/NQD property by a finite dimensional parameter (often just real-valued) and the problem reduces to the straightforward estimation and testing for fixed dimensional parameter using standard statistical methodologies (e.g., maximum likelihood). This article extends such a line of work by dropping any parametric assumptions and provides a fully data-dependent automated approach to estimate a copula and test for PQD property. The estimator is shown to be large-sample consistent under a set of mild regularity conditions. Numerical illustrations based on simulated data are also provided to compare the performance of the proposed testing procedure with some available methods and applications to real case studies are also provided.

Polynomial bivariate copulas of degree five: characterization and some particular inequalities

AbstractBivariate polynomial copulas of degree 5 (containing the family of Eyraud-Farlie-Gumbel-Morgenstern copulas) are in a one-to-one correspondence to certain real parameter triplets (a,b,c), i.e., to some set of polynomials in two variables of degree 1:p(x,y) =ax+by+c. The set of the parameters yielding a copula is characterized and visualized in detail. Polynomial copulas of degree 5 satisfying particular (in)equalities (symmetry, Schur concavity, positive and negative quadrant dependence, ultramodularity) are discussed and characterized. Then it is shown that for polynomial copulas of degree 5 the values of several dependence parameters (including Spearman’s rho, Kendall’s tau, Blomqvist’s beta, and Gini’s gamma) lie in exactly the same intervals as for the Eyraud-Farlie-Gumbel-Morgenstern copulas. Finally we prove that these dependence parameters attain all possible values in ]−1, 1[ if polynomial copulas of arbitrary degree are considered.

Milnor and Topological Attractors in a Family of Two-Dimensional Lotka–Volterra Maps

In this work, we consider a family of Lotka–Volterra maps [Formula: see text] for [Formula: see text] and [Formula: see text] which unfold a map originally proposed by Sharkosky for [Formula: see text] and [Formula: see text]. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region [Formula: see text]. Some properties and bifurcations are described. The [Formula: see text]-axis is invariant, on which the map reduces to the logistic. For any [Formula: see text] an interval of values for [Formula: see text] exists for which all the cycles on the [Formula: see text]-axis are transversely attracting. This invariant set is the source of several kinds of bifurcations. Riddling bifurcations lead to attractors in Milnor sense, not topological but with a stable set of positive measure, which may be the unique attracting set, or coexisting with other topological attractors. The riddling and blowout bifurcations are described related to chaotic intervals on the invariant set, and these global bifurcations have different dynamic results. Chaotic intervals which are not topological attractors may have all the cycles transversely attracting and as Milnor attractors. We show that Milnor attractors may also be related to attracting cycles on the [Formula: see text]-axis at the bifurcation associated with the transverse and parallel eigenvalues. We show particular examples related to topological attractors with very narrow basins of attraction, when the majority of the trajectories are divergent.

Two Nested Limit Cycles in Two-Species Reactions

We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica computer algebra system, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each have two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable, which appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable.

Moderate Deviation and Exit Time Estimates for Stationary Last Passage Percolation

We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $\rho\in (0,1)$ and points $v_N=((1-\rho)^2 N,\rho^2 N)$ going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case $\rho=\frac{1}{2}$ in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on moderate deviation estimates for point-to-point and point-to-line exponential Last Passage Percolation which are obtained via random matrix estimates.

Weakly mixing smooth planar vector field without asymptotic directions

We construct a planar smooth weakly mixing stationary random vector field with nonnegative components such that, with probability 1, the flow generated by this vector field does not have an asymptotic direction. Moreover, for all individual trajectories, the set of partial limiting directions coincides with those spanning the positive quadrant. A modified example shows that a particle in space-time weakly mixing positive velocity field does not necessarily have an asymptotic average velocity.

Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment

In this article, we propose and analyse a predator-prey model where apart from direct predation the prey population is affected by the fear induced from predators. The reproduction of the prey population is reduced as a cost of fear. The predator is assumed to consume the prey according to ratio-dependent functional response and is also involved in intra-specific competition due to limited resources of food. Through model analysis, it has been observed that fear factor regulates the dynamics of the system in a completely different way than in the case where functional response is only prey dependent. Also, intra-specific competition among predators reduces the effect of fear and it forms a different pattern in the system dynamics than that of the effect of fear. Furthermore, the deterministic model has been extended to a stochastic model by perturbing the natural death rates of both prey and predators. It has been observed that the stochastic system possesses a unique positive solution that is globally stable with respect to anywhere in the interior of the positive quadrant. The stochastic extinction and persistence scenario for both the species have been analysed and a detailed comparison between the deterministic and stochastic models have been done through exhaustive numerical simulation. Finally, numerical simulation has been performed to figure out the impact of fear on the population dynamics.