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Partial differential equations of parabolic type

1965-06-01
John S. Maybee

Matrices, Digraphs, and Determinants

1989-10-01
John S. Maybee , D.D. Olesky , Driessche P. van den +1 more
A detailed account of various determinantal formulas is presented in a graph-theoretic form involving paths and cycles in the digraph of the matrix. For cases in which the digraph has … A detailed account of various determinantal formulas is presented in a graph-theoretic form involving paths and cycles in the digraph of the matrix. For cases in which the digraph has special local properties, for example, a cutpoint or a bridge, particular formulas are given that are more efficient for computing the determinant than simply using the matrix representation. Applications are also given to characteristic deter- minants, general minors, and cofactors.
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Algebraic multiplicity of the eigenvalues of a tournament matrix

1992-05-01
D. de Caen , David A. Gregory , Steve Kirkland +2 more

Combinatorially symmetric matrices

1974-12-01
John S. Maybee

A characterization of graphs of competition number m

1983-09-01
J. Richard Lundgren , John S. Maybee

Tournament matrices and their generalizations, I.

1990-10-01
John S. Maybee , Norman J. Pullman
If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 … If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n − 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n∉ {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.

Sign solvable graphs

1980-04-01
John S. Maybee

Graph Theoretic Methods for the Qualitative Analysis of Rectangular Matrices

1981-09-01
Harvey J. Greenberg , J. Richard Lundgren , John S. Maybee
In the past few years, many large models including several energy models have been represented by rectangular matrices, and graphs appear to be valuable in investigating connectivity and other properties … In the past few years, many large models including several energy models have been represented by rectangular matrices, and graphs appear to be valuable in investigating connectivity and other properties of these models. It is the purpose of this paper to establish some of the basic foundations for the use of graphs and digraphs to investigate properties of rectangular matrices. A variety of graphs and digraphs associated with rectangular matrices are introduced, and several theorems related to connectivity and tearing are proved. There are also a few applications to the area of computer-assisted analysis.

On a smoothing operator for the wave equation.

1960-10-01
John S. Maybee
On a smoothing operator for the wave equation. On a smoothing operator for the wave equation.
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New Generalizations of Jacobi Matrices

1966-09-01
John S. Maybee
Previous article Next article New Generalizations of Jacobi MatricesJohn S. MaybeeJohn S. Maybeehttps://doi.org/10.1137/0114083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Marvin Marcus and , Henryk Minc, A survey of matrix theory … Previous article Next article New Generalizations of Jacobi MatricesJohn S. MaybeeJohn S. Maybeehttps://doi.org/10.1137/0114083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Marvin Marcus and , Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon Inc., Boston, Mass., 1964xvi+180 MR0162808 0126.02404 Google Scholar[2] F. R. Gantmacher and , M. G. Krein, Oscillation Matrices, Oscillation Kernels and Small Vibrations of Mechanical Systems, Moscow-Leningrad, 1950 Google Scholar[3] J. H. Wilkinson, Handbook Series Linear Algebra. Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection, Numer. Math., 4 (1962), 362–367 10.1007/BF01386333 MR0148208 CrossrefGoogle Scholar[4] Alston S. Householder and , Friedrich L. Bauer, On certain methods for expanding the characteristic polynomial, Numer. Math., 1 (1959), 29–37 10.1007/BF01386370 MR0100962 0089.11802 CrossrefGoogle Scholar[5] James Quirk and , Richard Ruppert, Qualitative economics and the stability of equilibrium, Review of Economic Studies, 32 (1965), 311–326 CrossrefISIGoogle Scholar[6] J. S. Maybee , Remarks on the theory of cycles in matrices , to appear Google Scholar[7] F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959Vol. 1, x+374 pp. Vol. 2, ix+276 MR0107649 0085.01001 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Estimating a Falsified ModelAdvances in Pure Mathematics, Vol. 06, No. 08 Cross Ref Structural sign patterns and reduced form restrictionsEconomic Modelling, Vol. 29, No. 2 Cross Ref Falsifying economic modelsEconomic Modelling, Vol. 22, No. 5 Cross Ref From real to complex sign pattern matrices17 April 2009 | Bulletin of the Australian Mathematical Society, Vol. 57, No. 1 Cross Ref Qualitative comparative staticsJournal of Mathematical Economics, Vol. 28, No. 2 Cross Ref Qualitatively Stable Matrices and Convergent Matrices Cross Ref Some possible new directions for combinatorial matrix analysisLinear Algebra and its Applications, Vol. 107 Cross Ref On two classes of matrices with positive diagonal solutions to the Lyapunov equationLinear Algebra and its Applications, Vol. 59 Cross Ref QUALITATIVE STABILITY OF MATRICES AND ECONOMIC THEORY: A SURVEY ARTICLE11I would like to thank the Department of Energy for support of the research underlying this paper, as well as Dick Ruppert for his helpful comments and corrections, and Leslie Fort, who did a superb job of typing this paper. Cross Ref Structural Properties in the Stability Problem of Interconnected SystemsIFAC Proceedings Volumes, Vol. 13, No. 6 Cross Ref Stability of nonlinear systemsAutomatica, Vol. 15, No. 5 Cross Ref Stability of Nonlinear Systems: A Structural ApproachIFAC Proceedings Volumes, Vol. 11, No. 1 Cross Ref Combinatorially symmetric matricesLinear Algebra and its Applications, Vol. 8, No. 6 Cross Ref Mechanical vibration treesJournal of Mathematical Analysis and Applications, Vol. 45, No. 3 Cross Ref A CLASS OF GENERALIZED METZLERIAN MATRICES Cross Ref Remarks on the inertia of a matrixLinear Algebra and its Applications, Vol. 3, No. 1 Cross Ref Qualitative Problems in Matrix Theory17 February 2012 | SIAM Review, Vol. 11, No. 1AbstractPDF (2334 KB) Volume 14, Issue 5| 1966SIAM Journal on Applied Mathematics History Submitted:10 December 1965Published online:13 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114083Article page range:pp. 1032-1037ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

On maximal sign-nonsingular matrices

1996-11-01
Thomas Lundy , John S. Maybee , James Van Buskirk

Some possible new directions for combinatorial matrix analysis

1988-08-01
John S. Maybee

On signed digraphs with all cycles negative

1985-10-01
Frank Harary , J. Richard Lundgren , John S. Maybee
View PDF Chat

Vanishing minor conditions for inverse zero patterns

1993-01-01
Charles R. Johnson , John S. Maybee

Inverting Signed Graphs

1984-06-01
Harvey J. Greenberg , J. Richard Lundgren , John S. Maybee
This paper addresses the question of determining the class of rectangular matrices having a given signed graph as a signed row or column graph. We also determine equivalent conditions on … This paper addresses the question of determining the class of rectangular matrices having a given signed graph as a signed row or column graph. We also determine equivalent conditions on a given pair of signed graphs in order for them to be the signed row and column graphs of some rectangular matrix. In connection with these signed graph inversion problems we discuss the concept of minimality and illustrate how to invert a pair of signed graphs.

Qualitatively Stable Matrices and Convergent Matrices

1989-01-01
John S. Maybee
We review the basic known facts about stable matrices. Prom these and other, related, results we show some of the relationships that exist between qualitatively (sign) stable matrices and Hicksian … We review the basic known facts about stable matrices. Prom these and other, related, results we show some of the relationships that exist between qualitatively (sign) stable matrices and Hicksian stable matrices. We also show that the relationship between stable matrices and convergent matrices can be viewed essentially as a problem on inverses. Finally we derive a variety of results about inverses of sign stable matrices and use these to obtain information about the properties of corresponding convergent matrices.

A Stability Theorem for a Class of Damped Dynamic Systems and some Applications

1966-01-01
J. Genin , John S. Maybee
A Stability Theorem for a Class of Damped Dynamic Systems and some Applications Get access J. GENIN, J. GENIN 1School of Aeronautics, Astronautics, and Engineering Sciences, Purdue UniversityLafayette, Indiana, U.S.A. … A Stability Theorem for a Class of Damped Dynamic Systems and some Applications Get access J. GENIN, J. GENIN 1School of Aeronautics, Astronautics, and Engineering Sciences, Purdue UniversityLafayette, Indiana, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar J. S. MAYBEE J. S. MAYBEE 2Division of Mathematical Sciences, Purdue UniversityLafayette, Indiana, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Applied Mathematics, Volume 2, Issue 4, December 1966, Pages 343–357, https://doi.org/10.1093/imamat/2.4.343 Published: 01 December 1966 Article history Received: 31 January 1966 Revision received: 11 July 1966 Published: 01 December 1966

FORECASTING THE LOAD DURATION CURVE USING BOX-JENKINS TIME SERIES ANALYSIS

1978-01-01
Noel D. Uri , John S. Maybee
This paper endeavors to develop a forecasting model for discrete approximations to the load duration curve using Box-Jenkins time series analysis. A model is structured and estimated for six, twenty. … This paper endeavors to develop a forecasting model for discrete approximations to the load duration curve using Box-Jenkins time series analysis. A model is structured and estimated for six, twenty. and fifty discrete approximations for two electric utility regions in the United States. The forecast results are extremely good. indicating that while a larger number of approximations performs marginally better, there is nothing to mitigate using only a six-step approximation.

The theory of combined‐arms lanchester‐type models of warfare

1985-05-01
John S. Maybee
Abstract The mathematical theory necessary to solve combined arms models of military combat is presented here. We show how to apply the theory of positive operators to such models. Most … Abstract The mathematical theory necessary to solve combined arms models of military combat is presented here. We show how to apply the theory of positive operators to such models. Most of the results are purely qualitative in character showing that many properties of such systems are independent of the actual numerical values of the coefficients. Finally, we discuss in some detail an example of such a system.

Nearly sign-nonsingular matrices

1995-04-01
George M. Lady , Thomas Lundy , John S. Maybee

Normal and Quasi-Normal Modes in Damped Linear Dynamic Systems

1966-06-01
John S. Maybee
A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist … A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist is that certain matrices associated with the system commute. Necessary and sufficient conditions of the same type are also obtained for the classical normal modes, but under more restrictive conditions.

A characterization of graphs with interval two-step graphs

1995-03-01
J. Richard Lundgren , Sarah K. Merz , John S. Maybee +1 more
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Some structural theorems for partial difference operators

1965-02-01
John S. Maybee

Spectra and inverse sign patterns of nearly sign-nonsingular matrices

1996-12-01
Thomas Lundy , John S. Maybee , D.D. Olesky +1 more
View PDF Chat

Nullspaces of matrices and their compounds

1993-05-01
Michael J. Tsatsomeros , John S. Maybee , D.D. Olesky +1 more
We consider compound matrices and exterior products in order to generalize a fundamental relation between the eigenspace of a matrix A corresponding to a simple eigenvalue λ and certain minors … We consider compound matrices and exterior products in order to generalize a fundamental relation between the eigenspace of a matrix A corresponding to a simple eigenvalue λ and certain minors of A − λI of order n − 1. By means of the Laplace Expansion Theorem, we show that if the geometric multiplicity of λ is k > 1, then the exterior product of k linearly eigenvectors corresponding to λ is uniquely (up to scalar multiples) determined by certain minors of A − λI of order n−k. Additional results are included on the nullspace of a compound matrix.
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Remarks on the follower problem and the Euler method in static stability

1971-11-01
J. Genin , John S. Maybee

<i>L</i>-Functions and Their Inverses

1987-01-01
John S. Maybee , Gerry Wiener
Using concepts from qualitative matrix theory, we introduce a class of nonlinear mappings from $\mathbb{R}^n \to \mathbb{R}^n $ called L-functions. These generalize the L-matrices in much the same way that … Using concepts from qualitative matrix theory, we introduce a class of nonlinear mappings from $\mathbb{R}^n \to \mathbb{R}^n $ called L-functions. These generalize the L-matrices in much the same way that M-functions generalize M-matrices. We prove some global inverse function theorems for L-functions on several different types of domains without assuming that such functions are differentiable. Thus we do not make use of the Jacobian matrix. We also obtain interesting qualitative relations which must hold between an L-function and its inverse. Finally we prove a global implicit function theorem for L-functions, again without assuming differentiability.

Nonconservative Linear Systems with Constant Coefficients

1971-01-01
J. Genin , John S. Maybee
Journal Article Nonconservative Linear Systems with Constant Coefficients Get access JOSEPH GENIN, JOSEPH GENIN School of Aeronautics, Astronautics and Engineering Sciences, Purdue UniversityLafayette, Indiana 47907, U.S.A. Search for other works … Journal Article Nonconservative Linear Systems with Constant Coefficients Get access JOSEPH GENIN, JOSEPH GENIN School of Aeronautics, Astronautics and Engineering Sciences, Purdue UniversityLafayette, Indiana 47907, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar JOHN S. MAYBEE JOHN S. MAYBEE Department of Mathematics, University of ColoradoBoulder, Colorado 80302, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Applied Mathematics, Volume 8, Issue 3, December 1971, Pages 358–370, https://doi.org/10.1093/imamat/8.3.358 Published: 01 December 1971 Article history Received: 19 March 1971 Revision received: 05 July 1971 Published: 01 December 1971

Uniformly one-connected matrices and their inverses

1998-01-01
Thomas Lundy , John S. Maybee

Boundedness theorem for a nonlinear Mathieu equation

1970-10-01
J. Genin , John S. Maybee
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Biclique Covers and Partitions of Unipathic Digraphs

2017-07-12
Kim A.S. Hefner , J. Richard Lundgren , John S. Maybee
A digraph D is strongly unipathic if it has exactly one directed path from any vertex to any other. We show that the only bicliques in such a digraph are … A digraph D is strongly unipathic if it has exactly one directed path from any vertex to any other. We show that the only bicliques in such a digraph are claws, and so it follows that the claw cover, biclique cover and biclique partition numbers of D are all equal. Equivalently, the term, Boolean and nonnegative integer ranks of its adjacency matrix A(D) are all equal. We extend the unipathic property to sets of vertex disjoint paths, and prove that the above ranks are all equal to the ordinary real rank of A(D). Also, in the course of the paper, we survey a number of useful characterizations of directed versions of the clique covering and partition numbers that recur in Pullman's work.

Combinatorial Properties of Nearly Principal Minors

1992-01-01
John S. Maybee , Gerard Sierksma

Biclique Covers and Partitions of Unipathic Digraphs

2017-07-12
Kim A.S. Hefner , J. Richard Lundgren , John S. Maybee
A digraph D is strongly unipathic if it has exactly one directed path from any vertex to any other. We show that the only bicliques in such a digraph are … A digraph D is strongly unipathic if it has exactly one directed path from any vertex to any other. We show that the only bicliques in such a digraph are claws, and so it follows that the claw cover, biclique cover and biclique partition numbers of D are all equal. Equivalently, the term, Boolean and nonnegative integer ranks of its adjacency matrix A(D) are all equal. We extend the unipathic property to sets of vertex disjoint paths, and prove that the above ranks are all equal to the ordinary real rank of A(D). Also, in the course of the paper, we survey a number of useful characterizations of directed versions of the clique covering and partition numbers that recur in Pullman's work.

Uniformly one-connected matrices and their inverses

1998-01-01
Thomas Lundy , John S. Maybee

Spectra and inverse sign patterns of nearly sign-nonsingular matrices

1996-12-01
Thomas Lundy , John S. Maybee , D.D. Olesky +1 more
View PDF Chat

On maximal sign-nonsingular matrices

1996-11-01
Thomas Lundy , John S. Maybee , James Van Buskirk

Nearly sign-nonsingular matrices

1995-04-01
George M. Lady , Thomas Lundy , John S. Maybee

A characterization of graphs with interval two-step graphs

1995-03-01
J. Richard Lundgren , Sarah K. Merz , John S. Maybee +1 more
View PDF Chat

Nullspaces of matrices and their compounds

1993-05-01
Michael J. Tsatsomeros , John S. Maybee , D.D. Olesky +1 more
We consider compound matrices and exterior products in order to generalize a fundamental relation between the eigenspace of a matrix A corresponding to a simple eigenvalue λ and certain minors … We consider compound matrices and exterior products in order to generalize a fundamental relation between the eigenspace of a matrix A corresponding to a simple eigenvalue λ and certain minors of A − λI of order n − 1. By means of the Laplace Expansion Theorem, we show that if the geometric multiplicity of λ is k > 1, then the exterior product of k linearly eigenvectors corresponding to λ is uniquely (up to scalar multiples) determined by certain minors of A − λI of order n−k. Additional results are included on the nullspace of a compound matrix.
View PDF Chat

Vanishing minor conditions for inverse zero patterns

1993-01-01
Charles R. Johnson , John S. Maybee

Algebraic multiplicity of the eigenvalues of a tournament matrix

1992-05-01
D. de Caen , David A. Gregory , Steve Kirkland +2 more

Combinatorial Properties of Nearly Principal Minors

1992-01-01
John S. Maybee , Gerard Sierksma

Tournament matrices and their generalizations, I.

1990-10-01
John S. Maybee , Norman J. Pullman
If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 … If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n − 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n∉ {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.

Matrices, Digraphs, and Determinants

1989-10-01
John S. Maybee , D.D. Olesky , Driessche P. van den +1 more
A detailed account of various determinantal formulas is presented in a graph-theoretic form involving paths and cycles in the digraph of the matrix. For cases in which the digraph has … A detailed account of various determinantal formulas is presented in a graph-theoretic form involving paths and cycles in the digraph of the matrix. For cases in which the digraph has special local properties, for example, a cutpoint or a bridge, particular formulas are given that are more efficient for computing the determinant than simply using the matrix representation. Applications are also given to characteristic deter- minants, general minors, and cofactors.
View PDF Chat

Qualitatively Stable Matrices and Convergent Matrices

1989-01-01
John S. Maybee
We review the basic known facts about stable matrices. Prom these and other, related, results we show some of the relationships that exist between qualitatively (sign) stable matrices and Hicksian … We review the basic known facts about stable matrices. Prom these and other, related, results we show some of the relationships that exist between qualitatively (sign) stable matrices and Hicksian stable matrices. We also show that the relationship between stable matrices and convergent matrices can be viewed essentially as a problem on inverses. Finally we derive a variety of results about inverses of sign stable matrices and use these to obtain information about the properties of corresponding convergent matrices.

Some possible new directions for combinatorial matrix analysis

1988-08-01
John S. Maybee

<i>L</i>-Functions and Their Inverses

1987-01-01
John S. Maybee , Gerry Wiener
Using concepts from qualitative matrix theory, we introduce a class of nonlinear mappings from $\mathbb{R}^n \to \mathbb{R}^n $ called L-functions. These generalize the L-matrices in much the same way that … Using concepts from qualitative matrix theory, we introduce a class of nonlinear mappings from $\mathbb{R}^n \to \mathbb{R}^n $ called L-functions. These generalize the L-matrices in much the same way that M-functions generalize M-matrices. We prove some global inverse function theorems for L-functions on several different types of domains without assuming that such functions are differentiable. Thus we do not make use of the Jacobian matrix. We also obtain interesting qualitative relations which must hold between an L-function and its inverse. Finally we prove a global implicit function theorem for L-functions, again without assuming differentiability.

On signed digraphs with all cycles negative

1985-10-01
Frank Harary , J. Richard Lundgren , John S. Maybee
View PDF Chat

The theory of combined‐arms lanchester‐type models of warfare

1985-05-01
John S. Maybee
Abstract The mathematical theory necessary to solve combined arms models of military combat is presented here. We show how to apply the theory of positive operators to such models. Most … Abstract The mathematical theory necessary to solve combined arms models of military combat is presented here. We show how to apply the theory of positive operators to such models. Most of the results are purely qualitative in character showing that many properties of such systems are independent of the actual numerical values of the coefficients. Finally, we discuss in some detail an example of such a system.

Inverting Signed Graphs

1984-06-01
Harvey J. Greenberg , J. Richard Lundgren , John S. Maybee
This paper addresses the question of determining the class of rectangular matrices having a given signed graph as a signed row or column graph. We also determine equivalent conditions on … This paper addresses the question of determining the class of rectangular matrices having a given signed graph as a signed row or column graph. We also determine equivalent conditions on a given pair of signed graphs in order for them to be the signed row and column graphs of some rectangular matrix. In connection with these signed graph inversion problems we discuss the concept of minimality and illustrate how to invert a pair of signed graphs.

A characterization of graphs of competition number m

1983-09-01
J. Richard Lundgren , John S. Maybee

Graph Theoretic Methods for the Qualitative Analysis of Rectangular Matrices

1981-09-01
Harvey J. Greenberg , J. Richard Lundgren , John S. Maybee
In the past few years, many large models including several energy models have been represented by rectangular matrices, and graphs appear to be valuable in investigating connectivity and other properties … In the past few years, many large models including several energy models have been represented by rectangular matrices, and graphs appear to be valuable in investigating connectivity and other properties of these models. It is the purpose of this paper to establish some of the basic foundations for the use of graphs and digraphs to investigate properties of rectangular matrices. A variety of graphs and digraphs associated with rectangular matrices are introduced, and several theorems related to connectivity and tearing are proved. There are also a few applications to the area of computer-assisted analysis.

Sign solvable graphs

1980-04-01
John S. Maybee

FORECASTING THE LOAD DURATION CURVE USING BOX-JENKINS TIME SERIES ANALYSIS

1978-01-01
Noel D. Uri , John S. Maybee
This paper endeavors to develop a forecasting model for discrete approximations to the load duration curve using Box-Jenkins time series analysis. A model is structured and estimated for six, twenty. … This paper endeavors to develop a forecasting model for discrete approximations to the load duration curve using Box-Jenkins time series analysis. A model is structured and estimated for six, twenty. and fifty discrete approximations for two electric utility regions in the United States. The forecast results are extremely good. indicating that while a larger number of approximations performs marginally better, there is nothing to mitigate using only a six-step approximation.

Combinatorially symmetric matrices

1974-12-01
John S. Maybee

Remarks on the follower problem and the Euler method in static stability

1971-11-01
J. Genin , John S. Maybee

Nonconservative Linear Systems with Constant Coefficients

1971-01-01
J. Genin , John S. Maybee
Journal Article Nonconservative Linear Systems with Constant Coefficients Get access JOSEPH GENIN, JOSEPH GENIN School of Aeronautics, Astronautics and Engineering Sciences, Purdue UniversityLafayette, Indiana 47907, U.S.A. Search for other works … Journal Article Nonconservative Linear Systems with Constant Coefficients Get access JOSEPH GENIN, JOSEPH GENIN School of Aeronautics, Astronautics and Engineering Sciences, Purdue UniversityLafayette, Indiana 47907, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar JOHN S. MAYBEE JOHN S. MAYBEE Department of Mathematics, University of ColoradoBoulder, Colorado 80302, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Applied Mathematics, Volume 8, Issue 3, December 1971, Pages 358–370, https://doi.org/10.1093/imamat/8.3.358 Published: 01 December 1971 Article history Received: 19 March 1971 Revision received: 05 July 1971 Published: 01 December 1971

Boundedness theorem for a nonlinear Mathieu equation

1970-10-01
J. Genin , John S. Maybee
View PDF Chat

New Generalizations of Jacobi Matrices

1966-09-01
John S. Maybee
Previous article Next article New Generalizations of Jacobi MatricesJohn S. MaybeeJohn S. Maybeehttps://doi.org/10.1137/0114083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Marvin Marcus and , Henryk Minc, A survey of matrix theory … Previous article Next article New Generalizations of Jacobi MatricesJohn S. MaybeeJohn S. Maybeehttps://doi.org/10.1137/0114083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Marvin Marcus and , Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon Inc., Boston, Mass., 1964xvi+180 MR0162808 0126.02404 Google Scholar[2] F. R. Gantmacher and , M. G. Krein, Oscillation Matrices, Oscillation Kernels and Small Vibrations of Mechanical Systems, Moscow-Leningrad, 1950 Google Scholar[3] J. H. Wilkinson, Handbook Series Linear Algebra. Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection, Numer. Math., 4 (1962), 362–367 10.1007/BF01386333 MR0148208 CrossrefGoogle Scholar[4] Alston S. Householder and , Friedrich L. Bauer, On certain methods for expanding the characteristic polynomial, Numer. Math., 1 (1959), 29–37 10.1007/BF01386370 MR0100962 0089.11802 CrossrefGoogle Scholar[5] James Quirk and , Richard Ruppert, Qualitative economics and the stability of equilibrium, Review of Economic Studies, 32 (1965), 311–326 CrossrefISIGoogle Scholar[6] J. S. Maybee , Remarks on the theory of cycles in matrices , to appear Google Scholar[7] F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959Vol. 1, x+374 pp. Vol. 2, ix+276 MR0107649 0085.01001 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Estimating a Falsified ModelAdvances in Pure Mathematics, Vol. 06, No. 08 Cross Ref Structural sign patterns and reduced form restrictionsEconomic Modelling, Vol. 29, No. 2 Cross Ref Falsifying economic modelsEconomic Modelling, Vol. 22, No. 5 Cross Ref From real to complex sign pattern matrices17 April 2009 | Bulletin of the Australian Mathematical Society, Vol. 57, No. 1 Cross Ref Qualitative comparative staticsJournal of Mathematical Economics, Vol. 28, No. 2 Cross Ref Qualitatively Stable Matrices and Convergent Matrices Cross Ref Some possible new directions for combinatorial matrix analysisLinear Algebra and its Applications, Vol. 107 Cross Ref On two classes of matrices with positive diagonal solutions to the Lyapunov equationLinear Algebra and its Applications, Vol. 59 Cross Ref QUALITATIVE STABILITY OF MATRICES AND ECONOMIC THEORY: A SURVEY ARTICLE11I would like to thank the Department of Energy for support of the research underlying this paper, as well as Dick Ruppert for his helpful comments and corrections, and Leslie Fort, who did a superb job of typing this paper. Cross Ref Structural Properties in the Stability Problem of Interconnected SystemsIFAC Proceedings Volumes, Vol. 13, No. 6 Cross Ref Stability of nonlinear systemsAutomatica, Vol. 15, No. 5 Cross Ref Stability of Nonlinear Systems: A Structural ApproachIFAC Proceedings Volumes, Vol. 11, No. 1 Cross Ref Combinatorially symmetric matricesLinear Algebra and its Applications, Vol. 8, No. 6 Cross Ref Mechanical vibration treesJournal of Mathematical Analysis and Applications, Vol. 45, No. 3 Cross Ref A CLASS OF GENERALIZED METZLERIAN MATRICES Cross Ref Remarks on the inertia of a matrixLinear Algebra and its Applications, Vol. 3, No. 1 Cross Ref Qualitative Problems in Matrix Theory17 February 2012 | SIAM Review, Vol. 11, No. 1AbstractPDF (2334 KB) Volume 14, Issue 5| 1966SIAM Journal on Applied Mathematics History Submitted:10 December 1965Published online:13 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114083Article page range:pp. 1032-1037ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

Normal and Quasi-Normal Modes in Damped Linear Dynamic Systems

1966-06-01
John S. Maybee
A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist … A generalization of the concept of classical normal modes in damped linear systems is presented. It is then shown that a necessary and sufficient condition that such quasi-normal modes exist is that certain matrices associated with the system commute. Necessary and sufficient conditions of the same type are also obtained for the classical normal modes, but under more restrictive conditions.

A Stability Theorem for a Class of Damped Dynamic Systems and some Applications

1966-01-01
J. Genin , John S. Maybee
A Stability Theorem for a Class of Damped Dynamic Systems and some Applications Get access J. GENIN, J. GENIN 1School of Aeronautics, Astronautics, and Engineering Sciences, Purdue UniversityLafayette, Indiana, U.S.A. … A Stability Theorem for a Class of Damped Dynamic Systems and some Applications Get access J. GENIN, J. GENIN 1School of Aeronautics, Astronautics, and Engineering Sciences, Purdue UniversityLafayette, Indiana, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar J. S. MAYBEE J. S. MAYBEE 2Division of Mathematical Sciences, Purdue UniversityLafayette, Indiana, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Applied Mathematics, Volume 2, Issue 4, December 1966, Pages 343–357, https://doi.org/10.1093/imamat/2.4.343 Published: 01 December 1966 Article history Received: 31 January 1966 Revision received: 11 July 1966 Published: 01 December 1966

Partial differential equations of parabolic type

1965-06-01
John S. Maybee

Some structural theorems for partial difference operators

1965-02-01
John S. Maybee

On a smoothing operator for the wave equation.

1960-10-01
John S. Maybee
On a smoothing operator for the wave equation. On a smoothing operator for the wave equation.
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Coauthor Papers Together
J. Richard Lundgren 6
J. Genin 4
Thomas Lundy 4
D.D. Olesky 3
Harvey J. Greenberg 2
Gerry Wiener 2
P. van den Driessche 2
Norman J. Pullman 2
Michael J. Tsatsomeros 1
Driessche P. van den 1
Noel D. Uri 1
Gerard Sierksma 1
Charles R. Johnson 1
Craig W. Rasmussen 1
Frank Harary 1
George M. Lady 1
David A. Gregory 1
Kim A.S. Hefner 1
Steve Kirkland 1
D. de Caen 1
James Van Buskirk 1
Sarah K. Merz 1

Qualitative Problems in Matrix Theory

1969-01-01
John Maybe , J. P. Quirk
Previous article Next article Qualitative Problems in Matrix TheoryJohn Maybe and James QuirkJohn Maybe and James Quirkhttps://doi.org/10.1137/1011004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] John Maybes, Remarks on the theory of … Previous article Next article Qualitative Problems in Matrix TheoryJohn Maybe and James QuirkJohn Maybe and James Quirkhttps://doi.org/10.1137/1011004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] John Maybes, Remarks on the theory of cycles in matrices, Research paper, Purdue University, Lafayette, Indiana, 1966 Google Scholar[2] John S. Maybee, New generalizations of Jacobi matrices, SIAM J. Appl. Math., 14 (1966), 1032–1037 10.1137/0114083 MR0214606 (35:5455) 0168.02701 LinkISIGoogle Scholar[3] John S. Maybee, Matrices of class ${\cal J}\sb{2}$, J. Res. Nat. Bur. Standards Sect. B, 71B (1967), 215–224 MR0227181 (37:2766) 0159.32403 CrossrefGoogle Scholar[4] Carl Goldberg, Random notes on matrices, J. Res. Nat. Bur. Standards, 60 (1958), 321–325 MR0092756 (19,1154a) 0089.00901 CrossrefISIGoogle Scholar[5] S. Parter, On the eigenvalues and eigenvectors of a class of matrices, J. Soc. Indust. Appl. Math., 8 (1960), 376–388 10.1137/0108024 MR0112894 (22:3740) 0115.24804 LinkISIGoogle Scholar[6] Marvin Marcus and , Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon Inc., Boston, Mass., 1964xvi+180 MR0162808 (29:112) 0126.02404 Google Scholar[7] Richard S. Varga, Matrix iterative analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+322 MR0158502 (28:1725) 0133.08602 Google Scholar[8] Kelvin Lancaster, The scope of qualitative economics, Rev. Economic Studies, 29 (1962), 99–132 CrossrefISIGoogle Scholar[9] Kelvin Lancaster, Partitionable systems and qualitative economics, Rev. Economic Studies, 31 (1964), 69–72 CrossrefISIGoogle Scholar[10] Kelvin Lancaster, The theory of qualitative linear systems, Econometrica, 33 (1965), 395–408 MR0180369 (31:4604) CrossrefISIGoogle Scholar[11] Kelvin Lancaster, The solution of qualitative comparative statics problems, Quart. J. Economics, 53 (1966), 278–295 CrossrefISIGoogle Scholar[12] Terrence Gorman, More scope for qualitative economics, Rev. Economic Studies, 31 (1964), 65–68 CrossrefISIGoogle Scholar[13] Lowell Bassett, , John Maybee and , James Quirk, Qualitative economics and the scope of the correspondence principle, Econometrica, 36 (1968), 544–563 MR0237165 (38:5456) 0217.26802 CrossrefISIGoogle Scholar[14] J. R. Hicks, Value and Capital, Oxford University Press, Oxford, 1946 Google Scholar[15] M. Morishima, On the laws of change of the price system in an economy which contains complementary commodities, Osaka Economic Papers, 1 (1952), 101–113 Google Scholar[16] Gerard Debreu and , I. N. Herstein, Nonnegative square matrices, Econometrica, 21 (1953), 597–607 MR0059240 (15,496f) 0051.00901 CrossrefISIGoogle Scholar[17] Helmut Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z., 52 (1950), 642–648 10.1007/BF02230720 MR0035265 (11,710g) 0035.29101 CrossrefGoogle Scholar[18] M. G. Krei˘n and , M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.), 3 (1948), 3–95 MR0027128 (10,256c) Google Scholar[19] F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959Vol. 1, x+374 pp. Vol. 2, ix+276 MR0107649 (21:6372c) Google Scholar[20] J. Quirk and , R. Ruppert, Qualitative economics and the stability of equilibrium, Rev. Economic Studies, 32 (1965), 311–326 CrossrefISIGoogle Scholar[21] J. Quirk, The correspondence principle: A macroeconomic application, Internat. Economic Rev., 9 (1968), 294–306 0186.25502 CrossrefGoogle Scholar[22] Lowell Bassett, , Hamid Habibagahi and , James Quirk, Qualitative economics and Morishima matrices, Econometrica, 35 (1967), 221–233 MR0239843 (39:1200) 0159.49105 CrossrefISIGoogle Scholar[23] L. McKenzie, Arrow Karlin and , Scarf, The matrix with dominant diagonal and economic theoryMathematical Methods in the Social Sciences, Stanford University Press, Stanford, California, 1960 0392.60041 Google Scholar[24] P. Samuelson, The Foundations of Economic Analysis, Harvard University Press, Cambridge, 1955 Google Scholar[25] J. Quirk, Comparative stalks under Walras' law: The case of strong dependence, Rev. Economic Studies, 35 (1968), 11–22 CrossrefISIGoogle Scholar[26] J. Quirk, The competitive equilibrium: A qualitative analysis, Research Papers in Theoretical and Applied Economics, 14, University of Kansas, Lawrence, 1967 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Tipping cyclesLinear Algebra and its Applications, Vol. 646 Cross Ref Minimum number of non-zero-entries in a stable matrix exhibiting Turing instabilityDiscrete and Continuous Dynamical Systems - S, Vol. 15, No. 9 Cross Ref Polynomial stability and potentially stable patternsLinear Algebra and its Applications, Vol. 613 Cross Ref Stabilizing the Metzler matrices with applications to dynamical systems27 November 2019 | Calcolo, Vol. 57, No. 1 Cross Ref Minimum number of non-zero-entries in a 7 × 7 stable matrixLinear Algebra and its Applications, Vol. 572 Cross Ref Linear preservers for the q-permanent, cycle q-permanent expansions, and positive crossings in digraphsLinear Algebra and its Applications, Vol. 561 Cross Ref Aggregates of Monotonic Step Response Systems: A Structural ClassificationIEEE Transactions on Control of Network Systems, Vol. 5, No. 2 Cross Ref Qualitative Economics15 February 2018 Cross Ref Aggregates of positive impulse response systems: A decomposition approach for complex networks Cross Ref Qualitative Stability of Nonlinear Networked SystemsIEEE Transactions on Automatic Control, Vol. 62, No. 8 Cross Ref Sign properties of Metzler matrices with applicationsLinear Algebra and its Applications, Vol. 515 Cross Ref Structural conditions for oscillations and multistationarity in aggregate monotone systems Cross Ref Design and analysis of a synthetic aptamer-based oscillator Cross Ref Some results on the structure and spectra of matrix-productsLinear Algebra and its Applications, Vol. 474 Cross Ref Predictable Dynamics of Opinion Forming for Networks With Antagonistic InteractionsIEEE Transactions on Automatic Control, Vol. 60, No. 2 Cross Ref Reaction Kinetics Cross Ref A Structural Classification of Candidate Oscillatory and Multistationary Biochemical Systems18 September 2014 | Bulletin of Mathematical Biology, Vol. 76, No. 10 Cross Ref Stability analysis of diagonally equipotent matricesAutomatica, Vol. 49, No. 9 Cross Ref Nonpositive Eigenvalues of Hollow, Symmetric, Nonnegative MatricesZachary B. Charles, Miriam Farber, Charles R. Johnson, and Lee Kennedy-Shaffer26 September 2013 | SIAM Journal on Matrix Analysis and Applications, Vol. 34, No. 3AbstractPDF (243 KB)Constructions for potentially stable sign patternsLinear Algebra and its Applications, Vol. 436, No. 12 Cross Ref Structural sign patterns and reduced form restrictionsEconomic Modelling, Vol. 29, No. 2 Cross Ref The inertia sets of symmetric tridiagonal sign-patternsLinear Algebra and its Applications, Vol. 436, No. 6 Cross Ref P 0-Matrix Products of Matrices Cross Ref From Structure to Dynamics in Biological Networks3 April 2011 Cross Ref Allow problems concerning spectral properties of sign pattern matrices: A surveyLinear Algebra and its Applications, Vol. 430, No. 11-12 Cross Ref Qualitative permanence of Lotka–Volterra equations3 June 2008 | Journal of Mathematical Biology, Vol. 57, No. 6 Cross Ref The scope of the LeChatelier PrinciplePhysica A: Statistical Mechanics and its Applications, Vol. 381 Cross Ref Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models31 May 2007 | Journal of Mathematical Biology, Vol. 55, No. 1 Cross Ref Stability of Complex Reaction Networks14 March 2007 Cross Ref Falsifying economic modelsEconomic Modelling, Vol. 22, No. 5 Cross Ref On a conjecture about the μ-permanentLinear and Multilinear Algebra, Vol. 53, No. 3 Cross Ref Constructing symmetric nonnegative matrices via the fast fourier transformComputers & Mathematics with Applications, Vol. 45, No. 10-11 Cross Ref Roles of positive and negative feedback in biological systemsComptes Rendus Biologies, Vol. 325, No. 11 Cross Ref Topics in nonparametric comparative statics and stabilityInternational Advances in Economic Research, Vol. 6, No. 1 Cross Ref From real to complex sign pattern matrices17 April 2009 | Bulletin of the Australian Mathematical Society, Vol. 57, No. 1 Cross Ref Qualitative comparative staticsJournal of Mathematical Economics, Vol. 28, No. 2 Cross Ref Optimal control for competing coalitionsNonlinear Analysis: Theory, Methods & Applications, Vol. 28, No. 8 Cross Ref Tree sign pattern matrices that require zero eigenvalues17 April 2009 | Bulletin of the Australian Mathematical Society, Vol. 55, No. 1 Cross Ref Qualitative Sensitivity Analysis Cross Ref On consistent symbolic representations of general dynamic systemsIEEE Transactions on Systems, Man, and Cybernetics, Vol. 25, No. 8 Cross Ref Robust economic modelsJournal of Economic Dynamics and Control, Vol. 19, No. 3 Cross Ref Qualitative comparative statics and audits of model performanceLinear Algebra and its Applications, Vol. 217 Cross Ref THE SCOPE OF DIMENSIONAL ANALYSIS IN QUALITATIVE REASONING2 April 2007 | Computational Intelligence, Vol. 10, No. 2 Cross Ref On the recognition of S-systemsLinear Algebra and its Applications, Vol. 192 Cross Ref Robust Stability and Diagonal Liapunov FunctionsEugenius Kaszkurewicz and Amit Bhaya17 July 2006 | SIAM Journal on Matrix Analysis and Applications, Vol. 14, No. 2AbstractPDF (1422 KB)Idempotence for sign-pattern matricesLinear Algebra and its Applications, Vol. 180 Cross Ref Spectra with positive elementary symmetric functionsLinear Algebra and its Applications, Vol. 180 Cross Ref Ranks of zero patterns and sign patterns *Linear and Multilinear Algebra, Vol. 34, No. 1 Cross Ref Tournament matrices and their generalizations, I.Linear and Multilinear Algebra, Vol. 28, No. 1-2 Cross Ref A graph-theoretic approach to investigate structural and qualitative properties of systems: A surveyNetworks, Vol. 20, No. 4 Cross Ref Robust stability and diagonal Liapunov functions Cross Ref Extensions of graph inversion to support an artificially intelligent modeling environmentAnnals of Operations Research, Vol. 21, No. 1 Cross Ref Matrices, Digraphs, and DeterminantsJohn S. Maybee, D. D. Olesky, Driessche P. van den, and G. Wiener17 July 2006 | SIAM Journal on Matrix Analysis and Applications, Vol. 10, No. 4AbstractPDF (2263 KB)Qualitative stability of discrete-time systemsLinear Algebra and its Applications, Vol. 117 Cross Ref Sign-Patterns and Stability Cross Ref -N- Party Democracy: The Role of the Minimal State Cross Ref Interpolation of D -stability and sign stability2 April 2008 | Linear and Multilinear Algebra, Vol. 23, No. 4 Cross Ref From qualitative matrices to quantitative restrictionsLinear and Multilinear Algebra, Vol. 22, No. 3 Cross Ref Regular Domains of Tridiagonal Matrices Cross Ref Substitutes and Complements in Constrained Linear ModelsJ. Scott Provan17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 8, No. 4AbstractPDF (2107 KB)Regular and singular orthants of tridiagonal matricesLinear Algebra and its Applications, Vol. 94 Cross Ref Qualitative stability of linear systemsLinear Algebra and its Applications, Vol. 87 Cross Ref L-Functions and Their InversesJohn S. Maybee and Gerry M. Wiener17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 8, No. 1AbstractPDF (1107 KB)Qualitative Economics27 November 2016 Cross Ref Mixed qualitative calculus as a tool in policy modeling: A dynamic simulation model of urban declineJournal of Policy Modeling, Vol. 8, No. 1 Cross Ref Volterra Multipliers IRay Redheffer17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 6, No. 4AbstractPDF (2432 KB)A theorem concerning certain sign symmetric matrices whose inverses are MorishimaLinear Algebra and its Applications, Vol. 68 Cross Ref Some results concerning invertible Morishima and anti-Morishima matrices whose associated digraphs are treesLinear Algebra and its Applications, Vol. 66 Cross Ref Qualitative Spatial Data Analysis: A Compendium of Approaches Cross Ref Inverting graphs of rectangular matricesDiscrete Applied Mathematics, Vol. 8, No. 3 Cross Ref On two classes of matrices with positive diagonal solutions to the Lyapunov equationLinear Algebra and its Applications, Vol. 59 Cross Ref Signsolvability revisitedLinear Algebra and its Applications, Vol. 59 Cross Ref Qualitatively invertible matricesMathematical Social Sciences, Vol. 6, No. 3 Cross Ref An algorithm for identifying Morishima and anti-Morishima matrices and balanced digraphsMathematical Social Sciences, Vol. 6, No. 1 Cross Ref Determinacy in Linear Systems and NetworksJ. Scott Provan17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 4, No. 2AbstractPDF (1895 KB)Rectangular Matrices and Signed GraphsHarvey J. Greenberg, J. Richard Lundgren, and John S. Maybee17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 4, No. 1AbstractPDF (1270 KB)Stochastic eigenvectors for qualitative stochastic matricesDiscrete Mathematics, Vol. 43, No. 2-3 Cross Ref A Class of Matrices Connected with Volterra Prey-Predator EquationsRay Redheffer and ZhiMing Zhou31 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 3, No. 1AbstractPDF (1309 KB)LU decomposition of M-matrices by elimination without pivotingLinear Algebra and its Applications, Vol. 41 Cross Ref Global asymptotic stability for a class of many-variable volterra prey-predator systemsNonlinear Analysis: Theory, Methods & Applications, Vol. 5, No. 12 Cross Ref Graph Theoretic Methods for the Qualitative Analysis of Rectangular MatricesHarvey J. Greenberg, J. Richard Lundgren, and John S. Maybee2 August 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 2, No. 3AbstractPDF (1533 KB)Stability Investigations of Linearized Systems with Random StructureBiometrical Journal, Vol. 23, No. 8 Cross Ref QUALITATIVE STABILITY OF MATRICES AND ECONOMIC THEORY: A SURVEY ARTICLE11I would like to thank the Department of Energy for support of the research underlying this paper, as well as Dick Ruppert for his helpful comments and corrections, and Leslie Fort, who did a superb job of typing this paper. Cross Ref SIGN SOLVABILITY Cross Ref QUALITATIVE MATRICES: STRONG SIGN-SOLVABILITY AND WEAK SATISFIABILITY11The first author's research was supported in part by the Office of Naval Research (N0014-76-C-0423) and the second author's research was supported in part by the National Science Foundation (MCS77-02474). Cross Ref IMPLEMENTATION ASPECTS OF MODEL MANAGEMENT: A FOCUS ON COMPUTER-ASSISTED ANALYSIS Cross Ref MEASURING COMPLEMENTARITY AND QUALITATIVE DETERMINACY IN MATRICIAL FORMS Cross Ref The existence of globally stable equilibria of ecosystems of the generalized Volterra typeJournal of Mathematical Biology, Vol. 10, No. 4 Cross Ref Sign solvable graphsDiscrete Applied Mathematics, Vol. 2, No. 1 Cross Ref Structural Properties in the Stability Problem of Interconnected SystemsIFAC Proceedings Volumes, Vol. 13, No. 6 Cross Ref Stability of nonlinear systemsAutomatica, Vol. 15, No. 5 Cross Ref REFERENCES Cross Ref Mathematical Approaches to Culture Change Cross Ref Global stability of ecosystems of the generalized volterra typeMathematical Biosciences, Vol. 42, No. 1-2 Cross Ref Stability of linear systems with uncertain parameters27 April 2007 | International Journal of Systems Science, Vol. 9, No. 9 Cross Ref Stability of Nonlinear Systems: A Structural ApproachIFAC Proceedings Volumes, Vol. 11, No. 1 Cross Ref Linear algorithms for testing the sign stability of a matrix and for findingZ-maximum matchings in acyclic graphsNumerische Mathematik, Vol. 28, No. 3 Cross Ref Book reviewsLinear and Multilinear Algebra, Vol. 5, No. 1 Cross Ref Book Reviews30 May 2007 | Linear and Multilinear Algebra, Vol. 5, No. 2 Cross Ref Some Aspects of the Theory of $PN$-MatricesJohn S. Maybee12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 31, No. 2AbstractPDF (1221 KB)The Hadamard-Fischer inequality for a class of matrices defined by eigenvalue monotonicity30 May 2007 | Linear and Multilinear Algebra, Vol. 4, No. 3 Cross Ref Combinatorially symmetric matricesLinear Algebra and its Applications, Vol. 8, No. 6 Cross Ref Sufficient conditions for D-stabilityJournal of Economic Theory, Vol. 9, No. 1 Cross Ref Mechanical vibration treesJournal of Mathematical Analysis and Applications, Vol. 45, No. 3 Cross Ref A CLASS OF GENERALIZED METZLERIAN MATRICES Cross Ref On the existence of diagonal solutions to the Lyapunov equation for a third order system Cross Ref Extension of Levins loop analysis to transient and periodic disturbances Cross Ref Volume 11, Issue 1| 1969SIAM Review History Submitted:26 August 1966Accepted:21 August 1968Published online:17 February 2012 InformationCopyright © 1969 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1011004Article page range:pp. 30-51ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Matrices, Digraphs, and Determinants

1989-10-01
John S. Maybee , D.D. Olesky , Driessche P. van den +1 more
A detailed account of various determinantal formulas is presented in a graph-theoretic form involving paths and cycles in the digraph of the matrix. For cases in which the digraph has … A detailed account of various determinantal formulas is presented in a graph-theoretic form involving paths and cycles in the digraph of the matrix. For cases in which the digraph has special local properties, for example, a cutpoint or a bridge, particular formulas are given that are more efficient for computing the determinant than simply using the matrix representation. Applications are also given to characteristic deter- minants, general minors, and cofactors.
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Sign-nonsingular matrices and even cycles in directed graphs

1986-03-01
Carsten Thomassen

New Generalizations of Jacobi Matrices

1966-09-01
John S. Maybee
Previous article Next article New Generalizations of Jacobi MatricesJohn S. MaybeeJohn S. Maybeehttps://doi.org/10.1137/0114083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Marvin Marcus and , Henryk Minc, A survey of matrix theory … Previous article Next article New Generalizations of Jacobi MatricesJohn S. MaybeeJohn S. Maybeehttps://doi.org/10.1137/0114083PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Marvin Marcus and , Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon Inc., Boston, Mass., 1964xvi+180 MR0162808 0126.02404 Google Scholar[2] F. R. Gantmacher and , M. G. Krein, Oscillation Matrices, Oscillation Kernels and Small Vibrations of Mechanical Systems, Moscow-Leningrad, 1950 Google Scholar[3] J. H. Wilkinson, Handbook Series Linear Algebra. Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection, Numer. Math., 4 (1962), 362–367 10.1007/BF01386333 MR0148208 CrossrefGoogle Scholar[4] Alston S. Householder and , Friedrich L. Bauer, On certain methods for expanding the characteristic polynomial, Numer. Math., 1 (1959), 29–37 10.1007/BF01386370 MR0100962 0089.11802 CrossrefGoogle Scholar[5] James Quirk and , Richard Ruppert, Qualitative economics and the stability of equilibrium, Review of Economic Studies, 32 (1965), 311–326 CrossrefISIGoogle Scholar[6] J. S. Maybee , Remarks on the theory of cycles in matrices , to appear Google Scholar[7] F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959Vol. 1, x+374 pp. Vol. 2, ix+276 MR0107649 0085.01001 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Estimating a Falsified ModelAdvances in Pure Mathematics, Vol. 06, No. 08 Cross Ref Structural sign patterns and reduced form restrictionsEconomic Modelling, Vol. 29, No. 2 Cross Ref Falsifying economic modelsEconomic Modelling, Vol. 22, No. 5 Cross Ref From real to complex sign pattern matrices17 April 2009 | Bulletin of the Australian Mathematical Society, Vol. 57, No. 1 Cross Ref Qualitative comparative staticsJournal of Mathematical Economics, Vol. 28, No. 2 Cross Ref Qualitatively Stable Matrices and Convergent Matrices Cross Ref Some possible new directions for combinatorial matrix analysisLinear Algebra and its Applications, Vol. 107 Cross Ref On two classes of matrices with positive diagonal solutions to the Lyapunov equationLinear Algebra and its Applications, Vol. 59 Cross Ref QUALITATIVE STABILITY OF MATRICES AND ECONOMIC THEORY: A SURVEY ARTICLE11I would like to thank the Department of Energy for support of the research underlying this paper, as well as Dick Ruppert for his helpful comments and corrections, and Leslie Fort, who did a superb job of typing this paper. Cross Ref Structural Properties in the Stability Problem of Interconnected SystemsIFAC Proceedings Volumes, Vol. 13, No. 6 Cross Ref Stability of nonlinear systemsAutomatica, Vol. 15, No. 5 Cross Ref Stability of Nonlinear Systems: A Structural ApproachIFAC Proceedings Volumes, Vol. 11, No. 1 Cross Ref Combinatorially symmetric matricesLinear Algebra and its Applications, Vol. 8, No. 6 Cross Ref Mechanical vibration treesJournal of Mathematical Analysis and Applications, Vol. 45, No. 3 Cross Ref A CLASS OF GENERALIZED METZLERIAN MATRICES Cross Ref Remarks on the inertia of a matrixLinear Algebra and its Applications, Vol. 3, No. 1 Cross Ref Qualitative Problems in Matrix Theory17 February 2012 | SIAM Review, Vol. 11, No. 1AbstractPDF (2334 KB) Volume 14, Issue 5| 1966SIAM Journal on Applied Mathematics History Submitted:10 December 1965Published online:13 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114083Article page range:pp. 1032-1037ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

Characterization of even directed graphs

1987-02-01
Paul Seymour , Carsten Thomassen

Some possible new directions for combinatorial matrix analysis

1988-08-01
John S. Maybee

Combinatorially symmetric matrices

1974-12-01
John S. Maybee

On the characteristic roots of tournament matrices

1968-01-01
Alfred Brauer , Ivey C. Gentry
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Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift

1991-06-26
Bernd Sturmfels , Peter Gritzmann

A theorem on inverse of tridiagonal matrices

1979-10-01
Wayne Barrett

Impossibility of Decomposing the Complete Graph on <i>n</i> Points into $n - 1$ Isomorphic Complete Bipartite Graphs

1989-02-01
D. de Caen , D. G. Hoffman
It is proved that $K_{2rs} $ cannot be edge-partitioned into copies of $K_{r,s} $ if r and s are greater than one. This answers a question raised in a recent … It is proved that $K_{2rs} $ cannot be edge-partitioned into copies of $K_{r,s} $ if r and s are greater than one. This answers a question raised in a recent paper of Granville, Moisiadis, and Rees. [Congr. Numer., 61 (1988), pp. 241–248].

Graph Theoretic Methods for the Qualitative Analysis of Rectangular Matrices

1981-09-01
Harvey J. Greenberg , J. Richard Lundgren , John S. Maybee
In the past few years, many large models including several energy models have been represented by rectangular matrices, and graphs appear to be valuable in investigating connectivity and other properties … In the past few years, many large models including several energy models have been represented by rectangular matrices, and graphs appear to be valuable in investigating connectivity and other properties of these models. It is the purpose of this paper to establish some of the basic foundations for the use of graphs and digraphs to investigate properties of rectangular matrices. A variety of graphs and digraphs associated with rectangular matrices are introduced, and several theorems related to connectivity and tearing are proved. There are also a few applications to the area of computer-assisted analysis.

On sign-nonsingular matrices and the conversion of the permanent into the determinant

1991-06-26
Richard A. Brualdi , Bryan L. Shader

Food webs, competition graphs, and the boxicity of ecological phase space

1978-01-01
Fred S. Roberts

Linear hyperbolic partial differential equations with constant coefficients

1951-01-01
Lars Gårding
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Inverses of banded matrices

1981-12-01
Wayne Barrett , Philip Feinsilver

The number of linear, directed, rooted, and connected graphs

1955-01-01
Frank Harary
1. Introduction. Seifert and Threlfall [20, p. 4] have described the principal problem of topology as zu entscheiden, ob zwei vorgelegte Figuren hom6omorph sind und wo moglich alle Klassen nichthomoomorpher … 1. Introduction. Seifert and Threlfall [20, p. 4] have described the principal problem of topology as zu entscheiden, ob zwei vorgelegte Figuren hom6omorph sind und wo moglich alle Klassen nichthomoomorpher Figuren aufzuzahlen. Our object is to obtain the number of nonisomorphic linear graphs with p points and k lines, and also to count various kinds of generalizations of graphs. These include directed graphs (digraphs), rooted graphs, multiply rooted graphs, and two other generalizations which will be called graphs of strength s and graphs of type t. The fundamental theorem used to secure these results is due to Polya [151 and will be reviewed very briefly in the next section. The author is happy to take this opportunity to thank Professor Polya for kindly permitting the presentation of his unpublished formula for the number of linear graphs in this paper. The form of the solution in every case will be the counting polynomial. Thus gp(x) defined by: p(p-l)/2 gp(X) = j gpkX' k-O where gpk iS the number of graphs with p points and k lines, serves to count all graphs of p points. After counting several generalizations of graphs, we shall derive a formula for the number of connected graphs of any given topological type in terms of the total number of (connected as well as disconnected) graphs of this type. The number of connected graphs in terms of the total number of graphs, which first appeared in Riddell [16] and then in Riddell and Uhlenbeck [18], as well as the number of weakly connected digraphs obtained by Polya (unpublished) will follow as corollaries. A simple variation of the method enables one to count the rooted connected graphs of any given type in terms of the unrooted connected ones and the total number of such graphs. To illustrate the method, the number of forests and rooted forests will be found in terms of the known number of trees. The final section states some unsolved combinatorial problems.
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Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models

1970-12-01
George E. P. Box , David A. Pierce
Abstract Many statistical models, and in particular autoregressive—moving average time series models, can be regarded as means of transforming the data to white noise, that is, to an uncorrected sequence … Abstract Many statistical models, and in particular autoregressive—moving average time series models, can be regarded as means of transforming the data to white noise, that is, to an uncorrected sequence of errors. If the parameters are known exactly, this random sequence can be computed directly from the observations; when this calculation is made with estimates substituted for the true parameter values, the resulting sequence is referred to as the "residuals," which can be regarded as estimates of the errors. If the appropriate model has been chosen, there will be zero autocorrelation in the errors. In checking adequacy of fit it is therefore logical to study the sample autocorrelation function of the residuals. For large samples the residuals from a correctly fitted model resemble very closely the true errors of the process; however, care is needed in interpreting the serial correlations of the residuals. It is shown here that the residual autocorrelations are to a close approximation representable as a singular linear transformation of the autocorrelations of the errors so that they possess a singular normal distribution. Failing to allow for this results in a tendency to overlook evidence of lack of fit. Tests of fit and diagnostic checks are devised which take these facts into account.

Riemann’s method and the problem of Cauchy. II. The wave equation in 𝑛 dimensions

1952-06-01
J. B. Díaz , M. H. Martin
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A Semi-Definite Lyapunov Theorem and the Characterization of Tridiagonal D-Stable Matrices

1982-09-01
David Carlson , Biswa Nath Datta , Charles R. Johnson
A new necessary and sufficient condition is given for an $n \times n$ complex matrix A to be stable. It involves a positive semi-definite image under a Lyapunov map and … A new necessary and sufficient condition is given for an $n \times n$ complex matrix A to be stable. It involves a positive semi-definite image under a Lyapunov map and the real and imaginary parts of A. This condition is then used to characterize the real tridiagonal matrices which are D-stable, and those which are totally D-stable.

The square of a chordal graph

1994-04-01
Frank Harary , Terry A. McKee

<i>Lectures on Cauchy's Problem in Linear Partial Differential Equations</i>

1953-08-01
Jacques Hadamard , Philip Μ. Morse

The symmetrization of matrices by diagonal matrices

1962-02-01
Seymour V. Parter , J Youngs

On maximal sign-nonsingular matrices

1996-11-01
Thomas Lundy , John S. Maybee , James Van Buskirk

Sign determinacy of M-matrix minors

1987-06-01
Charles R. Johnson , D.D. Olesky , B. G. Robertson +1 more

Spectra with positive elementary symmetric functions

1993-02-01
Charles R. Johnson , D.D. Olesky , Michael J. Tsatsomeros +1 more
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When the sign pattern of a square matrix determines uniquely the sign pattern of its inverse

1989-07-01
Carsten Thomassen

Matrices with Sign Symmetric Diagonal Shifts or Scalar Shifts

1987-01-01
Daniel Hershkowitz , Volker Mehrmann , Hans Schneider
We generalize the concepts of sign symmetry and weak sign symmetry by defining k-sign symmetric matrices. For a positive integer k, we show that all diagonal shifts of an irreducible … We generalize the concepts of sign symmetry and weak sign symmetry by defining k-sign symmetric matrices. For a positive integer k, we show that all diagonal shifts of an irreducible matrix are k-sign symmetric if and only if the matrix is diagonally similar to a Hermitian matrix. A similar result holds for scalar shifts, but requires an additional condition in the case $k = 1$. Extensions are given to reducible matrices.

Graph-theoretical approach to qualitative solvability of linear systems

1982-12-01
Rachel Manber

On certain methods for expanding the characteristic polynomial

1959-12-01
A. S. Householder , Friedrich L. Bauer

Vanishing minor conditions for inverse zero patterns

1993-01-01
Charles R. Johnson , John S. Maybee

A theorem concerning certain sign symmetric matrices whose inverses are Morishima

1985-07-01
Gerry Wiener

The Jacobian matrix and global univalence of mappings

1965-04-01
David Gale , Hukukane Nikaidō

Finite-Difference Methods for Partial Differential Equations

1962-02-01
Preston C. Hammer , George E. Forsythe , Wolfgang Wasow

Permanents of cyclic matrices

1960-09-01
Marion Tinsley
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A characterization of tridiagonal matrices

1969-04-01
Miroslav Fiedler

On signed digraphs with all cycles negative

1985-10-01
Frank Harary , J. Richard Lundgren , John S. Maybee
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Two-step graphs of trees

1993-08-01
J. Richard Lundgren , Craig W. Rasmussen

Matrices eigenvalues, and directed graphs

1982-01-01
Richard A. Brualdi
We show how several classical results concerning inclusion regions and estimates for the eigenvalues of matrices can be unified and generalized by the use of directed graphs. Applications to nonnegative … We show how several classical results concerning inclusion regions and estimates for the eigenvalues of matrices can be unified and generalized by the use of directed graphs. Applications to nonnegative matricesM-matrices, and the spectra of graphs are given.

Ordering Properties of Linear Successive Iteration Schemes Applied to Multi-Diagonal Type Linear Systems

1957-12-01
Jack Heller

Representative bibliography on load forecasting

1977-03-01
Manoj Sachdev , R. Billinton , Cadence Peterson
Load forecasting has always been an integral part of power system planning and operation. However, it did not receive as much attention in the past as it deserves because the … Load forecasting has always been an integral part of power system planning and operation. However, it did not receive as much attention in the past as it deserves because the fuel supplies, especially hydrocarbons, were cheap and abundant, and utilities could find funds for erecting enough gas/oil generating plants at relatively short lead times. In the last few years, the conditions have considerably changed and the past practices will have to be suitably modified. Load forecasting will assume greater importance and, therefore, it will receive more attention. To assist engineers working in this area, a representative cross section of publications related to load forecasting are listed in this paper.

Treediagonal matrices and their inverses

1982-02-01
Douglas J. Klein

Spectral multiplicity and splitting results for a class of qualitative matrices

1984-09-01
Gerry Wiener

On a class of approximate iterative processes

1967-01-01
James M. Ortega , Werner C. Rheinboldt
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On simple characterizations of k-trees

1974-01-01
Donald J. Rose

Tournament matrices and their generalizations, I.

1990-10-01
John S. Maybee , Norman J. Pullman
If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 … If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n − 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n∉ {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.

L'intégrale de Riemann-Liouville et le problème de Cauchy

1949-01-01
Marcel Riesz
kvant- kvant-
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Conversion of the permanent into the determinant

1971-03-01
Peter M. Gibson
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-square <inline-formula content-type="math/mathml"> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-square <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrix with positive permanent. It is shown that if the permanent of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be converted into a determinant by affixing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus"> <mml:semantics> <mml:mo>±</mml:mo> <mml:annotation encoding="application/x-tex">\pm</mml:annotation> </mml:semantics> </mml:math> </inline-formula> signs to the elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n squared plus 3 n minus 2 right-parenthesis slash 2"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">({n^2} + 3n - 2)/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> positive entries. Corollaries of this result are given.
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Automorphism groups of graphs

1983-09-01
Johannes Siemons

The connectivity function of a graph

1967-12-01
Lowell W. Beineke , Frank Harary
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