Cayley-Hamilton Research Articles

MULTIGRID METHOD AND LOW-REYNOLDS-NUMBER k - epsilon MODEL FOR TURBULENT RECIRCULATING FLOWS

The numerical procedure based on a finite volume method is developed for solving turbulent recirculating flows. The present method including low-Reynolds-number two-equation turbulence models is based on a non-orthogonal, and fully collocated grid which is applicable to incompressible flows. The turbulence model of Park and Sung [2] is modified to consider of nonequilibrium effects in complex shear flows. Based on the Cayley-Hamilton theorem, modifications are made on the damping function and the model constant C* epsilon1of the dissipation rate equation. Special treatments are introduced for a full multigrid and full approximation storage (FMG/FAS), and a convergence acceleration in calculations with low-Reynolds-number k - epsilon models is obtained. The multigrid performance shows encouraging features in turbulent recirculating flows.

The explicit representation to the principal rotation angle and the principal rotation axis

By using Cayley-Hamilton theorem, two kinds of explicit representation for the rotation tensor are proposed. One contains the lower powers of deformation gradient, by which the formula of the principal rotation angle and the explicit representation of principal axis are obtained; the other, a high efficient method to obtain the rotation tensor, does not contain the complicated coefficients and uses few variables. Some properties about the principal rotation angle and the principal rotation axis are obtained.

The algebra of two symmetric matrices: Proving completeness and deriving syzygies for a set of invariants of the Riemann tensor

A large number of Riemann tensor invariants can be written as traces of products of two 3×3 matrices, representing the Weyl tensor and the Weyl-like square of the Ricci tensor. It is pointed out that finding a complete set, ℐ, for all of these invariants is a simple consequence of the more general problem of finding a complete set of symmetric matrices, ℳ, in terms of which all symmetric matrix polynomials in these two matrices can be expressed. Such a set is constructed and a formal proof of its completeness is given. Several matrix identities and a scalar syzygy, obtained recently by Sneddon, are rederived and their interrelationships clarified. They are shown to be, ultimately, consequences of the Cayley–Hamilton theorem. A “minimal set” of invariants, that must be contained in the complete set of invariants of the general problem, is identified and it is concluded that no set proposed so far is complete.

On representations of the cayley-hamilton equation in the supercase

On representations of the cayley-hamilton equation in the supercase

Newton’s formula and the Cayley-Hamilton theorem for the skew Capelli elements

Newton’s formula and the Cayley-Hamilton theorem for the skew Capelli elements

Cayley-Hamilton theorem for 2 × 2 matrices over the Grassmann algebra

It is shown that the characteristic polynomial of matrices over a Lie nilpotent ring introduced recently by Szigeti is invariant with respect to the conjugation action of the general linear group. Explicit generators of the corresponding algebra of invariants in the case of 2 × 2 matrices over an algebra over a field of characteristic zero satisfying the identity [[ x, y], z] = 0 are described. In this case the coefficients of the characteristic polynomial are expressed by traces of powers of the matrix, yielding a compact form of the Cayley-Hamilton equation of 2 × 2 matrices over the Grassmann algebra.

A Quick Cayley-Hamilton

(1998). A Quick Cayley-Hamilton. The American Mathematical Monthly: Vol. 105, No. 9, pp. 842-844.

The quantum Cayley-Hamilton theorem

Let X = ( x ij ) n× n be the generic matrix of the quantum group K[ GL q ( n)]. First we prove that X satisfies two quantum characteristic equations, both become the classical characteristic equation when q = 1. Second we prove a quantum version of Muir's formula for X.

METHOD OF INITIAL FUNCTIONS FOR THE ANALYSIS OF LAMINATED CIRCULAR CYLINDRICAL SHELLS UNDER AXISYMMETRIC LOADING

The method of initial functions has been used for the static analysis of an infinite and simply supported, orthotropic, and laminated, circular cylindrical shell of revolution subjected to axisymmettic load. In this method the three-dimensional state equations for an individual ply of a laminated shell are established without making any a priori assumptions regarding the distribution of stresses and displacements across the thickness of the shell By using the continuity conditions of displacements and stresses on each interface between adjacent layers, the state equation far the laminate is obtained. Using the Cayley-Hamilton theorem, the transfer matrix that maps the initial state vector into the field is evaluated explicitly, leading to an exact solution of the problem (MIF—exact). Alternatively, depending on the number of terms retained in the series expansion of the transfer matrix, different-order theories of MIF are derived. The results of different-order MIF theories, classical theories, and shear deformation shell theories are compared with the results of MIF—exact to assess their accuracy and limitations.

Transmission Line Modeling: A Circuit Theory Approach

As shown in this paper, the classical transmission line equations for the distributed parameters model can be obtained from standard two-port network matrices avoiding the explicit use of partial differential equations. This is performed through a lemma derived directly from the Cayley--Hamilton theorem. The main advantage of this approach is that the modeling arises naturally in the frequency domain, allowing the consideration of frequency-dependent parameters (as, for instance, the resistance and inductance variations caused by the skin effect), normally not taken into account in time domain models.

On the Computation of An

Previous article Next article On the Computation of AnSaber N. Elaydi and William A. Harris, Jr.Saber N. Elaydi and William A. Harris, Jr.https://doi.org/10.1137/S0036144597319235PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractMethods, which are based on the Cayley--Hamilton theorem, for the computation of An for nonsingular A are presented.[1] Saber Elaydi, An introduction to difference equations, Undergraduate Texts in Mathematics, Springer‐Verlag, 1999xviii+427 1711587 CrossrefGoogle Scholar[2] Edward Fulmer, Computation of the matrix exponential, Amer. Math. Monthly, 82 (1975), 156–159 50:13687 CrossrefISIGoogle Scholar[3] Morris Hirsch and , Stephen Smale, Differential equations, dynamical systems, and linear algebra, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York‐London, 1974xi+358, Pure and Applied Mathematics, Vol. 60 58:6484 Google Scholar[4] Po‐Fang Hsieh, , Mitsuhiko Kohno and , Yasutaka Sibuya, Construction of a fundamental matrix solution at a singular point of the first kind by means of the SN decomposition of matrices, Linear Algebra Appl., 239 (1996), 29–76 98c:65037 CrossrefISIGoogle Scholar[5] Google Scholar[6] R. B. Kirchner, An explicit formula for eAt, Amer. Math. Monthly, 74 (1967), pp. 1200–1204. aml AMMYAE Am. Math. Monthly CrossrefISIGoogle Scholar[7] J. LaSalle, Stability theory for difference equations, Math. Assoc. of America, Washington, D.C., 1977, 0–0, 1–31. Stud. in Math., Vol. 14 58:1789 Google Scholar[8] I. Leonard, The matrix exponential, SIAM Rev., 38 (1996), 507–512 10.1137/S0036144595286488 97d:34009 LinkISIGoogle Scholar[9] E. Putzer, Avoiding the Jordan canonical form in the discussion of linear systems with constant coefficients, Amer. Math. Monthly, 73 (1966), 2–7 33:2845 CrossrefISIGoogle ScholarKeywordsCayley--Hamiltondifference equationCasorati matrix Previous article Next article FiguresRelatedReferencesCited ByDetails Hyers–Ulam stability and discrete dichotomy for difference periodic systemsBulletin des Sciences Mathématiques, Vol. 140, No. 8 | 1 Nov 2016 Cross Ref Laplace and Z transforms of linear dynamical systems and conic sectionsZeitschrift für angewandte Mathematik und Physik, Vol. 67, No. 3 | 2 May 2016 Cross Ref Hyers–Ulam stability and discrete dichotomyJournal of Mathematical Analysis and Applications, Vol. 423, No. 2 | 1 Mar 2015 Cross Ref Dynamics of a Rational System of Difference Equations in the PlaneAdvances in Difference Equations, Vol. 2011, No. 1 | 1 Jan 2011 Cross Ref Ordinary Dierential and Dierence EquationsFundamentals of Linear Systems for Physical Scientists and Engineers | 29 August 2013 Cross Ref Linear Operators and MatricesFundamentals of Linear Systems for Physical Scientists and Engineers | 29 August 2013 Cross Ref Ordinary Differential and Difference EquationsFundamentals of Linear Systems for Physical Scientists and Engineers | 1 Oct 2009 Cross Ref Integers Powers of Certain Asymmetric MatricesComputational Science – ICCS 2009 | 1 Jan 2009 Cross Ref Calculating the matrix exponential of a constant matrix on time scalesApplied Mathematics Letters, Vol. 21, No. 6 | 1 Jun 2008 Cross Ref Reducibility and Stability Results for Linear System of Difference EquationsAdvances in Difference Equations, Vol. 2008 | 1 Jan 2008 Cross Ref On linear matrix differential equationsAdvances in Applied Mathematics, Vol. 39, No. 3 | 1 Sep 2007 Cross Ref Functions of matricesLinear Algebra and its Applications, Vol. 406 | 1 Sep 2005 Cross Ref Avoiding Eigenvalues in Computing Matrix PowersThe American Mathematical Monthly, Vol. 112, No. 5 | 1 February 2018 Cross Ref Remarks on the Calculation of the Power of a MatrixJournal of Difference Equations and Applications, Vol. 10, No. 2 | 25 January 2007 Cross Ref Floquet Theory for Time Scales and Putzer Representations of Matrix LogarithmsJournal of Difference Equations and Applications, Vol. 9, No. 1 | 1 Jan 2003 Cross Ref Remarks on the Notion of Order of Difference EquationsJournal of Difference Equations and Applications, Vol. 8, No. 10 | 1 Jan 2002 Cross Ref Matrix Exponentials---Another ApproachSIAM Review, Vol. 43, No. 4 | 4 August 2006AbstractPDF (497 KB)Optimal Existence Results for nth Order Periodic Boundary Value Difference EquationsJournal of Mathematical Analysis and Applications, Vol. 247, No. 1 | 1 Jul 2000 Cross Ref Existence and approximation of solutions for first-order discontinuous difference equations with nonlinear global conditions in the presence of lower and upper solutionsComputers & Mathematics with Applications, Vol. 39, No. 1-2 | 1 Jan 2000 Cross Ref Volume 40, Issue 4| 1998SIAM Review765-1024 History Published online:02 August 2006 InformationCopyright © 1998 Society for Industrial and Applied MathematicsKeywordsCayley--Hamiltondifference equationCasorati matrixMSC codes15-0115A2139-0139A10PDF Download Article & Publication DataArticle DOI:10.1137/S0036144597319235Article page range:pp. 965-971ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

An extendedk-ε model for numerical simulation of wind flow around buildings

It is assumed in this paper that for a high Reynolds number nearly homogeneous wind flow, the Reynolds stresses are uniquely related to the mean velocity gradients and the two independent turbulent scaling parameters k and e. By applying dimensional analysis and owing to the Cayley-Hamilton theorem for tensors, a new turbulence enclosure model so-called the extended k-e model has been developed. The coefficients of the model expression were determined by the wind tunnel experimental data of homogeneous shear turbulent flow. The model was compared with the standard k-e model in its composition and the prediction of the Reynold's normal stresses. Using the new model the numerical simulation of wind flow around a square cross-section tall building was performed. The results show that the extended k-e model improves the prediction of wind velocities around the building and wind pressures on the building envelope.

The exponential map of GL(N)

A finite expansion of the exponential map for a matrix is presented. The method uses the Cayley - Hamilton theorem for writing the higher matrix powers in terms of those for the first N - 1. The resulting sums over the corresponding coefficients are rational functions of the eigenvalues of the matrix.

Explicit algebraic scalar‐flux models for turbulent reacting flows

AbstractExplicit algebraic scalar‐flux models that are valid for three‐dimensional turbulent flows are derived from a hierarchy of second‐order moment closures. The mathematical procedure is based on the Cayley–Hamilton theorem and is an extension of the scheme developed by Taulbee. Several closures for the pressure–scalar gradient correlations are considered and explicit algebraic relations are provided for the velocity–scalar correlations in both nonreacting and reacting flows. In the latter, the role of the Damköhler number is exhibited in isothermal turbulent flows with nonpremixed reactants. The relationship between these closures and traditional models based on the linear gradientdiffusion approximation is theoretically established. The results of model predictions are assessed by comparison with available laboratory data in turbulent jet flows.

A note on the characteristic equation in the max-plus algebra

We discuss the characteristic equation of a matrix in the max-plus algebra. In their Linear Algebra Appl. paper [101:87–108 (1998)] Olsder and Roos have used a transformation between the max-plus algebra and linear algebra to show that the Cayley-Hamilton theorem also holds in the max-plus algebra. We show that the derivation of Olsder and Roos is not entirely correct, and we give the correct formulas for the coefficients of this alternative version of the max-algebraic characteristic equation. We also give a counterexample for a conjecture of Olsder in which he states necessary and sufficient for the coefficients of the max-algebraic characteristic equation.

A note on the characteristic equation in the max-plus algebra

We discuss the characteristic equation of a matrix in the max-plus algebra. In their Linear Algebra Appl. paper [101:87–108 (1998)] Olsder and Roos have used a transformation between the max-plus algebra and linear algebra to show that the Cayley-Hamilton theorem also holds in the max-plus algebra. We show that the derivation of Olsder and Roos is not entirely correct, and we give the correct formulas for the coefficients of this alternative version of the max-algebraic characteristic equation. We also give a counterexample for a conjecture of Olsder in which he states necessary and sufficient for the coefficients of the max-algebraic characteristic equation.

On the Direct Determination of the Rotation Tensor from the Deformation Gradient

An explicit representation of the rotation tensor is given in terms of the deformation gradient. The derivation does not require the a priori determination of the stretch tensor. Instead, it relies on systematic use of the Cayley-Hamilton theorem and basic properties of proper orthogonal tensors.

Hecke Symmetries and Characteristic Relations on Reflection Equation Algebras

We analyze the relation between the properties of Hecke symmetry (i.e., Hecke type R-matrix) and the algebraic structure of the corresponding reflection equation (RE) algebra. Analogues of the Newton relations and Cayley–Hamilton theorem for the matrix of generators of the RE algebra associated with a finite rank even Hecke symmetry are derived.

Discretization of continuous systems with internal and external point delays through a quasiparametrical ARMA model

In this note, a discrete input/output model, which involves an ARMA part plus a nonrecursive additive term associated with the initial conditions of the free response, is formulated for continuous linear time-invariant systems involving internal and external point delays. The model is obtained from the application of the Cayley-Hamilton theorem to the continuous state-transition matrix. In some particular situations of asymptotic stability of the free system, the additive term associated with the response to initial conditions tends to a constant as time increases to infinity, and can be compensated through feedback so that the closed-loop model becomes a classical ARMA model

New determinants and the Cayley-Hamilton Theorem for matrices over Lie nilpotent rings

New determinants and the Cayley-Hamilton Theorem for matrices over Lie nilpotent rings