Dominant Root Research Articles

Testing unstable econometric models for stability: An empirical study

Econometric models may be tested for stability using the asymptotic distribution of the dominant characteristic root of the system as derived by Theil and Boot (1962) or by Oberhofer and Kmenta (1973). This paper points out that this distribution is known for stable models only which implies that the null hypothesis must always be stability and the power of the test is in question. A Monte Carlo study is performed to investigate the power of the test and the distribution of the test statistic for unstable and stable models in the small sample case.

Propagation of Quercus Seedlings in Bottomless Containers with Osmocote

Shumard oak seedlings (Quercus shumardii) were grown in bottomless, square containers placed on a wire bench. The dominant tap root grew downward and was 'air pruned', producing a fibrous root system. Three heights of containers were cut from quart milk carton stock giving three volumes of media with the same surface area and three different drainage columns. Five rates of Osmocote 18·6·12, based on the surface area of the containers, were applied af­ter the seedlings had emerged. Growth increased significantly with the three highest levels of Osmocote. Seedlings grown in the smallest container were not significantly different from those in the larger containers. Five replications were planted in 2 gallon (7572 cm3) containers and grown for one year. Even though all seedlings were treated alike during this period, those which were propagated with higher fertility rates had an increased survival rate and faster growth suggesting that the nutritional state of liners, not necessarily age or size, strongly influences survival and future growth.

An assessment of hemispheric specialization in monkeys.

Split-brain monkeys learned several sets of visual discriminations with each hemisphere. Some stimuli, such as photographs of monkey's faces, were intended to favor mechanisms similar to those of man's nondominant hemisphere, while other tasks, requiring sequential comparison of visual stimuli, should favor mechanisms similar to ones in the dominant hemisphere of man. The tests uniformly demonstrated hemispheric equivalence for solving all types of problems, regardless of handedness, sex, or side of surgical retraction. A review of the literature also offers little support for the concept of hemispheric specialization in infra-human mammals although a few leads still need to be explored before abandoning the hope of finding the roots of human cerebral dominance in monkeys.

A Dimensioning Criterion for the Shafting of the Variable Frequency Fed Induction Motor

For the variable frequency fed induction motor the possible operating points are limited not only by the feed and motor characteristics but also by the presence of an instability region. The pattern of this instability region and the particular dependence of the extent of the frequency interval bounding it, on the torsional elasticity of the shafting, allow to reduce the extent of the instability region by appropriately dimensioning the shafting. A numerical example shows how the dominant roots locus of the characteristic equation of the linearized system, enables the numerical value of the torsional elasticity to be used in the dimensioning of the shafting, to be determined.

Acknowledgement

Econometric models may be tested for stability using the asymptotic distribution of the dominant characteristic root of the system as derived by Theil and Boot (1962) or by Oberhofer and Kmenta (1973). This paper points out that this distribution is known for stable models only which implies that the null hypothesis must always be stability and the power of the test is in question. A Monte Carlo study is performed to investigate the power of the test and the distribution of the test statistic for unstable and stable models in the small sample case.

Modified Chebyshev filtering functions with low Q-factor

A new numerical algorithm is presented which determines the coefficients of a low-pass nonequal-ripple modified Chebyshev function with multiplicity of the dominant root pair greater than one; as a result its degree is higher than the corresponding Chebyshev polynomial but a much lower dominant root <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> -factor <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q_{c}</tex> is obtained. Intermediate modified Chebyshev functions with higher transition region attenuation and therefore increased <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q_{c}</tex> are also discussed.

Properties of the Leslie population matrix.

A simple similarity transformation of the Leslie matrix renders certain properties obvious. In particular the characteristic polynomial, characteristic vectors and principal vectors can be explicitly written out. Bounds for the dominant root are given.

Derivatives of the dominant root

Treating the greatest eigenvalue of an irreducible matrix as differentiable function of the matrix elements, one gets that the partial derivatives are closely related to the cofactors of the characteristics equation. The eigenvector property of the partial derivatives gives a general differentiation law for the eigenvalue and allows one to treat some special cases of variation of the matrix elements.

A Set of Rational Functions Related to the Euler–Fobenius Polynomials

The asymptotic behavior of the fundamental cardinal spline (Schoenberg’s terminology) of odd degree n is governed by that particular root of the Euler–Frobenius polynomial $\Pi _n $ which lies inside of and nearest to the unit circle. Rigorous bounds for this dominant root and an asymptotic formula for the remaining roots are given. These results are based on the partial fraction expansion of $q_n ( - e^{\pi w} )$, where $q_n (z) = z\Pi _n (z)(1 - z)^{ - n - 1} $ Various other properties of the functions $q_n$ are discussed, as is their use in the representation of the fundamental cardinal splines of odd degree.

Design of active filters using double layer rc distributed lines

AbstractApplication of double layer RC distributed lines in the design of low‐pass, band‐pass and high‐pass filters is considered in this communication. Realization of these filters employing two double layer lines and a difference amplifier using dominant root approach is also presented.

Nonstationary Blocking in Telephone Traffic

Blocking is considered for an N-trunk group of exponential servers with Poisson-offered load whose rate parameter varies with time. The infinite trunk case is solved by means of a rapidly convergent series of Poisson-Charlier polynomials. This solution is used to obtain practical approximations of blocking probability, transition probabilities, and recovery function for general time-variable offered load in the finite trunk-group case. An integral equation is derived satisfied by the blocking probability in the general case. In the situation of constant offered load, two additional methods are derived for providing easily computable approximations; one based on the integral equation, the other based on an approximate inversion of the Laplace transform. To aid in the latter approximation, bounds on the roots of Poisson-Charlier polynomials are obtained; in particular, an approximation is obtained for the dominant root. The inversion of the integral equation is studied with the purpose of providing the basis for future investigations of errors of approximation. Curves are provided for a number of examples permitting comparison of exact and approximate solutions.

The Excitability of Flexible Rotors in Short Sleeve Bearings

Assuming the short bearing approximation and constant lubricant properties, the root loci of the pertinent characteristic function were obtained for the linearized model of a simple symmetric flexible rotor bearing system. Using these loci, design maps consisting of lines of constant damping and vibration frequency pertaining to the dominant roots are presented as a function of the equilibrium eccentricity ratio and a frequency parameter for relevant degrees of flexibility. These maps display undesirable operating regions where external disturbances such as shock or unbalance loading are likely to excite undesirable vibrations, as well as regions of instability. The maps may conveniently be used to determine the effect of changing journal speed, lubricant viscosity and/or bearing clearance. Increased flexibility is seen to reduce the stability threshold in a predictable manner and to reduce damping at the pin-pin critical speed. The approach is applicable to more complex rotor bearing systems. It is felt that the use of such maps will enhance the understanding of rotor bearing system behavior, particularly at operating regions close to the stability threshold.

Watermelon Root Development Affected by Direct Seeding and Transplanting1

Abstract A dominant tap root was apparent in nearly all direct-seeded watermelon plants, Citrullus lanatus (Thunb.) Mansf., but was lacking in those that had been transplanted. Transplants were characterized by a shallow, extensive root system and superior yield in contrast to direct-seeded plants.

Relationship between conventional-control-theory figures of merit and quadratic performance index in optimal control theory for a single-input/single-output system

The paper presents a correlation between the figures of merit used in conventional-control-design theory and the elements of the Q matrix in the quadratic performance index that is utilised for an optimal-control-design theory of single-input/single-output systems. This correlation results in a design procedure that utilises a set of curves (Mp, tp, ts and ωd against qnn) for determining the elements of the Q matrix. In addition to the design procedure, the paper presents an analysis of the steady-state error for ramp inputs; the condition that the dominant roots are the same for any nth-order system, whereas the nondominant root for a specified plant is given by s −K √qnn (under the reproducibility condition defined in the paper) and the necessary conditions for an accurate solution of the Riccati equation.

Estimation of Standard Errors of the Characteristic Roots of a Dynamic Econometric Model

where y(t) represents the vector of endogenous variables, x(t) the vector of exogenous variables, u(t) the vector of stochastic disturbances, and t the tth period of observation. The matrices A, (T = 0, 1, . . . , m) of the structural coefficients are square matrices of order G. It is assumed that the conditions justifying the theorems in [3, Ch. 10] are satisfied, and that there are no nonlinear restrictions on the elements of A.. The stability of the system is determined by reference to the dominant root of the polynomial equation (2) det E Atmt) =0. t=O

The asymptotic form of the extinction probabilities for supercritical multitype branching processes

We consider a positively regular branching process for which there is a finite number T of types. It is assumed that ϱ, the dominant characteristic root of the matrix of first moments, is larger than 1. Then if q( n) i is the probability that the line descended from an ancestor of type i has become extinct by generation n, q i = lim n →∞ q i ( n) , and q i > 0, i = 1,…, T, it can be shown that lim n→∞ (q i − q n i) ϱ ̌ n = d i > 0 , where 0 < ϱ <1. A consequence is that all of the moments of the conditional time to extinction of a line, given ultimate extinction, are finite, regardless of the type of the ancestral individual.

Design curves for linear systems using Hurwitz determinants

By using Hurwitz determinants, boundary curves (α and ζ diagrams) are constructed according to specified stability constraints. From these diagrams, the dominant roots to be used for design purposes can be located.

Synthesis of systems with dominant complex roots†

Multiparameter systems with dominant complex roots constrained by degrees of freedom to construct reduced characteristic equation

A Matrix Approach to the Management of Renewable Resources, with Special Reference to Selection Forests-Two Extensions

In an earlier paper in this Journal (Usher 1966) a mathematical model was developed for renewable resources that could be categorized by size classes. The model was developed to answer the question: 'Given the effect of the site factors on the growth of the resource, what then is the structure of the resource that will achieve the maximum sustained yield?' Various parameters of the organisms-the processes of enlargement, the processes of recruitment to the smallest size classes, and the management decision of how to deal with large organisms-were included in a matrix Q, whose dominant latent root determined the exploitation. The latent vector associated with this root gave the structure of the resource that would give a sustained yield. However, in the Appendix to the paper, one problem of the model was stated as: '. . . any latent root greater than unity of the matrix Q determines a structure that is biologically meaningful. It cannot yet be proved that there is only one latent root greater than unity which satisfies the matrix Q.' There was thus no unique soluton to the model, and the forest manager might have been faced with the problem of which of several structures to use.

A Matrix Model for Forest Management

A model for the management of renewable resources is developed, a variant of a model previously given by Usher [1966] which posed one question-is the manager faced with one or several solutions? The mathematical development here shows that there is only one solution of the model that is biologically meaningful. This is associated with the only latent root of the matrix which is greater than unity. This dominant latent root has a latent vector such that all the elements can be chosen as positive. The result is based on a study of the derivative of the model, and from this a rapid method of solving the model is described. Data for a Scots pine forest are given.