Positive Quadrant Research Articles

Study of a drawing-quality sheet steel. II: Forming-limit curves by experiments and micromechanical simulations

In Part I of the current paper, we showed the results of uniaxial-tension tests, through-thickness and plane-strain compression experiments, quantitative texture – orientation distribution function – evaluations and Lankford coefficient measurements. These data were used for calibration and verification of a visco-plastic self-consistent (VPSC) polycrystal-plasticity simulation code for predicting a steel sheet’s ability to be stretched and deep drawn. Lankford coefficients are one, although incomplete, measure of a steel’s drawing quality. In order to obtain a deeper insight and better verification of the simulation code, we measured the forming-limit curve, FLC, for the same steel sheet. To make these measurements we stretched circle-gridded sheets of material with a punch and die. Samples had both a flat-sided and hourglass geometry and ranged from 20 to 80 mm in width. The 80 mm wide sample completely filled the die. With this range of sample sizes, we spanned all of the stress states applicable to a FLC, from uniaxial to biaxial tension. Our FLC curve had the classic “V” shape typical of drawing-quality steel, with a minimum safe forming strain of about 0.35 in plane-strain deformation and a safe forming strain of nearly 0.45 in balanced biaxial stretching. To model the FLC behavior, we used the same VPSC model and calibration employed in Part I. In order to obtain a necking instability in the calculation, a Marciniak defect was implemented into the VPSC model. The severity of the defect was adjusted to match the measured instability strain, 0.35, in plane-strain deformation. Both hardening laws fit in Part I were used to calculate the FLC. In the positive biaxial quadrant of the FLC, the limit strains predicted by the power law closely follow the measured uniform deformations, while the saturation law appears to over predict the limit strains. In uniaxial-tension, it was the opposite. The power-law hardening predictions seemed excessive. However, if we consider the FLC curve to be a band of finite width, both hardening laws and the VPSC formulation capture the essence of the FLC data.

On limit theorems for random fields

A complete separable metric space of functions defined on the positive quadrant of the plane is constructed. The characteristic property of these functions is that at every point x there exist two lines intersecting at this point such that limits limy→x f (y) exist when y approaches x along any path not intersecting these lines. A criterion of compactness of subsets of this space is obtained.

Stability of nonlinear microbial bioconversion system concerning glycerol’s active transport and 1,3-PD’s passive transport

In this study, a new nonlinear mathematical model is developed to describe the conversion of glycerol to 1,3- propanediol by Klebsiella pneumoniae and the changes of concentrations of intracellular substances are taken into consideration. We consider the cases that glycerol passes the membrane by active transport and 1,3-propanediol passes the membrane by passive transport. We firstly study the solution’s existence and uniqueness and prove that the solution is bounded in a positive quadrant. Secondly, we discuss the existence of equilibrium solutions and calculate them, dependent on some given steady accuracy by Newton’s Method. Lastly, we analyze the asymptotical behavior and the global stability of the mathematical model.

Some Inequalities for Whitney–Tutte Polynomials

We introduce and prove a family of inequalities satisfied by the Whitney rank generating function of a matroid in the positive quadrant of ℝ2. These can be interpreted as correlation inequalities at those points where the polynomial is known to count the number of independent sets, bases or spanning sets of the matroid. Our proofs also introduce an idea of rank dominating bijections in matroids, which are then used to obtain some simple extensions of the submodular property of matroid ranks.

Statistical Analysis of Illness–Death Processes and Semicompeting Risks Data

In many instances, a subject can experience both a nonterminal and terminal event where the terminal event (e.g., death) censors the nonterminal event (e.g., relapse) but not vice versa. Typically, the two events are correlated. This situation has been termed semicompeting risks (e.g., Fine, Jiang, and Chappell, 2001, Biometrika 88, 907-939; Wang, 2003, Journal of the Royal Statistical Society, Series B 65, 257-273), and analysis has been based on a joint survival function of two event times over the positive quadrant but with observation restricted to the upper wedge. Implicitly, this approach entertains the idea of latent failure times and leads to discussion of a marginal distribution of the nonterminal event that is not grounded in reality. We argue that, similar to models for competing risks, latent failure times should generally be avoided in modeling such data. We note that semicompeting risks have more classically been described as an illness-death model and this formulation avoids any reference to latent times. We consider an illness-death model with shared frailty, which in its most restrictive form is identical to the semicompeting risks model that has been proposed and analyzed, but that allows for many generalizations and the simple incorporation of covariates. Nonparametric maximum likelihood estimation is used for inference and resulting estimates for the correlation parameter are compared with other proposed approaches. Asymptotic properties, simulations studies, and application to a randomized clinical trial in nasopharyngeal cancer evaluate and illustrate the methods. A simple and fast algorithm is developed for its numerical implementation.

Nekhoroshev theorem for the periodic Toda lattice

The periodic Toda lattice with N sites is globally symplectomorphic to a two parameter family of N−1 coupled harmonic oscillators. The action variables fill out the whole positive quadrant of RN−1. We prove that in the interior of the positive quadrant as well as in a neighborhood of the origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev’s theorem applies on (almost) all parts of phase space (2000 Mathematics Subject Classification: 37J35, 37J40, 70H06).

Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks

A double quasi-birth-and-death (QBD) process is the QBD process whose background process is a homogeneous birth-and-death process, which is a synonym of a skip-free random walk in the two-dimensional positive quadrant with homogeneous reflecting transitions at each boundary face. It is also a special case of a 0-partially homogenous chain introduced by Borovkov and Mogul'skii. Our main interest is in the tail decay behavior of the stationary distribution of the double QBD process in the coordinate directions and that of its marginal distributions. In particular, our problem is to get their rough and exact asymptotics from primitive modeling data. We first solve this problem using the matrix analytic method. We then revisit the problem for the 0-partially homogenous chain, refining existing results. We exemplify the decay rates for Jackson networks and their modifications.

A Kolmogorov-Smirnov Type Test for Positive Quadrant Dependence

We consider a consistent test, that is similar to a Kolmogorov-Smirnov test, of the complete set of restrictions that relate to the copula representation of positive quadrant dependence. For such a test we propose and justify inference relying on a simulation based multiplier method and a bootstrap method. We also explore the finite sample behavior of both methods with Monte Carlo experiments. A first empirical illustration is given for US insurance claim data. A second one examines the presence of positive quadrant dependence in life expectancies at birth of males and females among countries.

Comparison of hip joint range of motion in professional youth and senior team footballers with age-matched controls: An indication of early degenerative change?

Objectives To determine if there is evidence of abnormal hip joint range of motion (ROM) in youth and senior team professional footballers compared with matched controls. Design A case control study design was used. Participants 40 professional footballers (20 youth and 20 senior team) and 40 matched control subjects. Main outcome measures Bilateral measurements of passive hip internal rotation (IR), external rotation (ER), flexion, abduction and extension were made together with Faber's test and the hip quadrant. Results Youth and senior footballers had significantly less IR and Faber's range and significantly higher abduction than their respective controls ( p < 0.001). Senior footballers also had significantly reduced IR ( p < 0.05) and Faber's ( p < 0.001) than the youth team. A higher proportion of senior footballers had positive hip quadrants (45% of all hips) compared to all other groups. No significant difference in hip ROM was found between dominant and non-dominant legs. Conclusions A specific pattern of hip ROM was found in professional footballers which appeared to be different from controls. These changes may demonstrate the early stages of hip degeneration to which it has been shown ex-professional players are prone to. Hip joint ROM exercises may be necessary in these players to restore normal movement and prevent the onset of hip osteoarthritis (OA).

The renormalization transformation for two-type branching models

Cet article etudie des systemes denombrables de diffusions en interaction hierarchiques et lineaires vivant dans le quadrant positif. De tels systemes apparaissent dans la dynamique d’individus de deux types qui migrent tout en interagissant dans des colonies. Le comportement a grande echelle et temps long peut etre etudie en utilisant le programme de renormalisation. Ce programme, qui a permis de resoudre d’autres cas (principalement uni-dimensionnels) est base sur la construction et l’analyse d’une transformation de renormalisation non lineaire, agissant sur la fonction de diffusion des composants du systeme et connectant l’evolution de blocs moyennes sur le temps a differentes echelles. Nous identifions une classe generale de fonctions de diffusion dans le quadrant positif pour lequel la transformation de renormalisation est bien definie et qui, sous une conjecture de comportement aux bords, peut-etre iteree. A l’interieur de certaines sous-classes, nous identifiens les points fixes de la transformation et etudions leurs domaines d’attraction. Ces domaines d’attraction constitutent les classes d’universalite du systeme apres changement d’echelle dans le temps et l’espace.

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cram\'{e}r light-tail assumption on the claim size distribution.

On Conditional Independence, Mixing, and Association

This is a follow-up to a recent article by Prakasa Rao [15] on conditional independence, conditional mixing and conditional association. The purpose of this article is to derive rigorously some results following from conditioning. To this end, a brief review is presented of the concepts of conditional independence of events, classes of events, and random variables, followed by a conditional version of a factorization theorem, as well as a first installment of some basic results. Next, the concepts of conditional covariance and variance are introduced, and a second installment of basic results follows. Furthermore, a certain representation of the covariance is established in detail, followed by a conditional version of it, as well as a generalization. The concept of the conditional characteristic function is also recalled, and a certain inequality is established. Finally, the concept of conditional positive (negative) quadrant dependence, as well as that of conditional positive (negative) association are introduced. The article concludes with the derivation of the conditional versions of some known results, regarding positive (negative) association. This is done anticipating that conditional association (and also conditional mixing) will prove to be of significant applicability.

Qualitative and numerical investigations of the impact of a novel pathogen on a seabird colony

Understanding the dynamics of novel pathogens in dense populations is crucial to public and veterinary health as well as wildlife ecology. Seabirds live in crowded colonies numbering several thousands of individuals. The long-term dynamics of avian influenza H5N1 virus in a seabird colony with no existing herd immunity are investigated using sophisticated mathematical techniques. The key characteristics of seabird population biology and the H5N1 virus are incorporated into a Susceptible-Exposed-Infected-Recovered (SEIR) model. Using the theory of integral manifolds, the SEIR model is reduced to a simpler system of two differential equations depending on the infected and recovered populations only, termed the IR model. The results of numerical experiments indicate that the IR model and the SEIR model are in close agreement. Using Lyapunov's direct method, the equilibria of the SEIR and the IR models are proven to be globally asymptotically stable in the positive quadrant.

Inapproximability of the Tutte polynomial

The Tutte polynomial of a graph G is a two-variable polynomial T ( G ; x , y ) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T ( G ; x , y ) . Jaeger et al. have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #P-hard, except along the hyperbola ( x - 1 ) ( y - 1 ) = 1 and at four special points. We are interested in determining for which points ( x , y ) there is a fully polynomial randomised approximation scheme (FPRAS) for T ( G ; x , y ) . Under the assumption RP ≠ NP , we prove that there is no FPRAS at ( x , y ) if ( x , y ) is in one of the half-planes x < - 1 or y < - 1 (excluding the easy-to-compute cases mentioned above). Two exceptions to this result are the half-line x < - 1 , y = 1 (which is still open) and the portion of the hyperbola ( x - 1 ) ( y - 1 ) = 2 corresponding to y < - 1 which we show to be equivalent in difficulty to approximately counting perfect matchings. We give further intractability results for ( x , y ) in the vicinity of the origin. A corollary of our results is that, under the assumption RP ≠ NP , there is no FPRAS at the point ( x , y ) = ( 0 , 1 - λ ) when λ > 2 is a positive integer. Thus, there is no FPRAS for counting nowhere-zero λ flows for λ > 2 . This is an interesting consequence of our work since the corresponding decision problem is in P for example for λ = 6 . Although our main concern is to distinguish regions of the Tutte plane that admit an FPRAS from those that do not, we also note that the latter regions exhibit different levels of intractability. At certain points ( x , y ) , for example the integer points on the x-axis, or any point in the positive quadrant, there is a randomised approximation scheme for T ( G ; x , y ) that runs in polynomial time using an oracle for an NP predicate. On the other hand, we identify a region of points ( x , y ) at which even approximating T ( G ; x , y ) is as hard as #P.

Strong law of large numbers for pairwise positive quadrant dependent random variables

We give the rate of convergence in the strong law of large numbers for pairwise positive quadrant dependent random variables and contemporaneous functions of these variables. Several examples of applications are given.

The dynamics of Pythagorean Triples

We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite and is equal to the address of the PPT on Barning's (1963) ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugate to a modified Euclidean algorithm. The invariant measure is identified, and the system is shown to be conservative and ergodic. We also show, based on a result of Aaronson and Denker (1999), that the dynamical system can be obtained as a factor map of a cross section of the geodesic flow on a quotient space of the hyperbolic plane by the group Γ(2), a free subgroup of the modular group with two generators.

Nonparametric Tests for Positive Quadrant Dependence

We consider distributional free inference to test for positive quadrant dependence, i.e. for the probability that two variables are simultaneously small (or large) being at least as great as it would be were they dependent. Tests for its generalisation in higher dimensions, namely positive orthant dependences, are also analysed. We propose two types of testing procedures. The first procedure is based on the specification of the dependence concepts in terms of distribution functions, while the second procedure exploits the copula representation. For each specification a distance test and an intersection-union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US and Danish insurance claim data. Practical implications for the design of reinsurance treaties are also discussed.

Locating the equilibrium points of a predator–prey model by means of affine state feedback

Abstract A predator–prey model with prey-dependent functional response is considered. The set of all points in the positive quadrant of the state plane that can be made equilibrium points by means of an affine state-feedback control law is determined, and the values of the control parameters ensuring the desired equilibria are provided. It is shown how the asymptotic stability of the equilibrium points depends on simple geometric conditions. The problem of stabilizing unstable equilibrium points is also briefly discussed.

Corners and Records of the Poisson Process in Quadrant

The scale-invariant spacings lemma due to Arratia, Barbour and Tavare establishes the distributional identity of a self-similar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant

Targeting Inflammatory Demyelinating Lesions to Sites of Wallerian Degeneration

In Theiler’s murine encephalomyelitis virus (TMEV) infection, an animal model for multiple sclerosis (MS), axonal injury precedes inflammatory demyelinating lesions, and the distribution of axonal damage present during the early phase of infection corresponds to regions where subsequent demyelination occurs during the chronic phase. We hypothesized that axonal damage recruits inflammatory cells to sites of Wallerian degeneration, leading to demyelination. Three weeks after TMEV infection, axonal degeneration was induced in the posterior funiculus of mice by injecting the toxic lectin Ricinus communis agglutinin (RCA) I into the sciatic nerve. Neuropathology was examined 1 week after lectin injection. Control mice, infected with TMEV but receiving no RCA I, had inflammatory demyelinating lesions in the anterior/lateral funiculi. Other control mice that received RCA I alone did not develop inflammatory lesions. In contrast, RCA I injection into TMEV-infected mice induced lesions in the posterior funiculus in addition to the anterior/lateral funiculi. We found no differences in lymphoproliferative responses or antibody titers against TMEV among the groups. This suggests that axonal degeneration contributes to the recruitment of inflammatory cells into the central nervous system by altering the local microenvironment. In this scenario, lesions develop from the axon (inside) to the myelin (outside) (Inside-Out model).