On Sylvester's Proof of the Reality of the Roots of Lagrange's Determinantal Equation

1897 ◽  
Vol 19 (4) ◽  
pp. 312
Author(s):  
Thomas Muir
Keyword(s):  
Determinantal Equation

Buckling of Oblique Plates with Clamped Edges Under Uniform Shear

1954 ◽  
Vol 5 (1) ◽  
pp. 39-51 ◽  
Author(s):  
W. H. Wittrick
Keyword(s):  
Pure Shear ◽  
Ritz Method ◽  
Critical Value ◽  
Critical Values ◽  
Reverse Direction ◽  
Buckling Mode ◽  
Determinantal Equation ◽  
Uniform System ◽  
Uniform Shear ◽  
Satisfactory Method

SummaryIn a previous paper the author derived, by the Rayleigh-Ritz method, a determinantal equation for obtaining the critical magnitude of any uniform system of edge stress applied to a clamped oblique plate. In the present paper this equation is used to derive values for the buckling stresses of clamped oblique plates subjected to pure shear along and perpendicular to two edges of the plate.As would be expected, it is found that reversal of the direction of the shear produces a change in the critical value. The lower value occurs when the shear is tending to increase the obliquity of the plate and critical values corresponding to this direction are calculated for 45° oblique plates with side ratios varying between 3/5 and 5/3. The critical values obtained for the reverse direction were obviously very inaccurate, due to the inadequacy of the series which was used to represent the buckling mode. As yet no satisfactory method has been found for overcoming this difficulty.

3249. On the Roots of a Certain Determinantal Equation

1970 ◽  
Vol 54 (387) ◽  
pp. 57 ◽  
Author(s):  
Murray S. Klamkin
Keyword(s):  
Determinantal Equation

Multivariate Linear Rational Expectations Models

1997 ◽  
Vol 13 (6) ◽  
pp. 877-888 ◽  
Author(s):  
Michael Binder ◽  
M. Hashem Pesaran
Keyword(s):  
Rational Expectations ◽  
Stable Solution ◽  
Future Expectations ◽  
Determinantal Equation ◽  
Fast Determination ◽  
Recursive Solution ◽  
Dependent Variables ◽  
High Degree ◽  
Theoretical Results

This paper considers the solution of multivariate linear rational expectations models. It is described how all possible classes of solutions (namely, the unique stable solution, multiple stable solutions, and the case where no stable solution exists) of such models can be characterized using the quadratic determinantal equation (QDE) method of Binder and Pesaran (1995, in M.H. Pesaran & M. Wickens [eds.], Handbook of Applied Econometrics: Macroeconomics, pp. 139–187. Oxford: Basil Blackwell). To this end, some further theoretical results regarding the QDE method expanding on previous work are presented. In addition, numerical techniques are discussed allowing reasonably fast determination of the dimension of the solution set of the model under consideration using the QDE method. The paper also proposes a new, fully recursive solution method for models involving lagged dependent variables and current and future expectations. This new method is entirely straightforward to implement, fast, and applicable also to high-dimensional problems possibly involving coefficient matrices with a high degree of singularity.

Asymptotic Cumulants for Distributions of k-State Markov Chains

1962 ◽  
Vol 58 (2) ◽  
pp. 427-430 ◽  
Author(s):  
P. V. Krishna Iyer ◽  
N. S. Shakuntala
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Probability Distributions ◽  
Exact Probability ◽  
Determinantal Equation ◽  
Asymptotic Values ◽  
Asymptotic Cumulants ◽  
Latent Roots

The expectation, variance and covariance for different states of a k-state Markov chain have been given by Patankar (6), Whittle (7), Good (3) and Bhat (l). Patankar's results involve the k latent roots of the determinantal equation. As it is not easy to determine the latent roots when k > 2, the actual asymptotic values of variances and covariances cannot be readily evaluated. Whittle gives exact probability distributions for the transitions, but the moments have been obtained after some gross approximations. Good (3) and Bhat(l) have given the first two moments and product moment for the frequency of different states. By using certain methods developed by Iyer and Kapur(4), the first four cumulants and product cumulants for the transition numbers of a two-state Markov chain were calculated and presented in an earlier publication(5).

The stability of elastico-viscous flow between rotating cylinders. Part 2

1964 ◽  
Vol 19 (4) ◽  
pp. 557-560 ◽  
Author(s):  
R. H. Thomas ◽  
K. Walters
Keyword(s):  
Viscous Flow ◽  
Stability Problem ◽  
Narrow Channel ◽  
Taylor Number ◽  
Wave Number ◽  
Rotating Cylinders ◽  
Orthogonal Functions ◽  
Determinantal Equation ◽  
Coaxial Cylinders ◽  
The Stability

Further consideration is given to the stability of the flow of an idealized elasticoviscous liquid contained in the narrow channel between two rotating coaxial cylinders. The work of Part 1 (Thomas & Walters 1964) is extended to include highly elastic liquids. To facilitate this, use is made of the orthogonal functions used by Reid (1958) in his discussion of the associated Dean-type stability problem. It is shown that the critical Taylor number Tc decreases steadily as the amount of elasticity in the liquid increases, until a transition is reached after which the roots of the determinantal equation which determines the Taylor number T as a function of the wave-number ε become complex. It is concluded that the principle of exchange of stabilities may not hold for highly elastic liquids.

On the Roots of a Determinantal Equation

1899 ◽  
Vol 21 (4) ◽  
pp. 367
Author(s):  
W. H. Metzler
Keyword(s):  
Determinantal Equation
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