RESEARCH INTERESTS

RESEARCH PUBLICATIONS

• 1. On an explicit method for the solution of a Stefan problem, J. Soc. Indus. Appl. Math. 7 (1959), 184-204 (doctoral thesis).
• 2. On periodicities of certain sequences of residues, Amer. Math. Monthly 67 (1960), 652-656.
• 3. Stability of the snowplow differential equation representing the expansion of a plasma bubble, Chap. 10 of Theoretical study of hydromagnetic stability and turbulence, Annual Report #AFSWC-TDR-62-12 for the period January 1-December 31, 1961, Radio Corp. of America.  Missile and Surface Radar Div., Moorestown, N.J., with Martin Kruskal.
• 4. A general class of discrete time-invariant filters, J. Soc. Indust. Appl. Math. 9 (1961), 405-421.
• 5. On the asymptotic behavior of solutions of second order linear differential equations, Proc. Amer. Math. Soc. 14 (1963), 12-14.
• 6. An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Indust. Appl. Math. 12 (1964), 515-522.
• 7. An algorithm for the inversion of finite Hankel matrices, J. Soc. Indust. Appl. Math. 13 (1965), 1102-1107.
• 8. On the extrapolation of a special class of stationary time series, Ann. Math. Stat. 36 (1965), 1426-1432.
• 9. On the inversion of a Hilbert type matrix, SIAM Review 8 (1966), 57-61 (with P. A. Scheinok).
• 10. On the stability of midpoint smoothing with Legendre polynomials, Proc. Amer. Math. Soc. 18 (1967), 191-199.
• 11. Bounds on the generating functions of certain smoothing operations, Proc. Amer. Math. Soc. 18 (1967), 200-206.
• 12. Research in methods of generating Liapunov functions, Final report # NASA-CR-61437, 1967 with R. Drake and E. Norman.
• 13. A data combination method for postflight trajectory analysis, J. Spacecraft and Rockets 41 (1967), 424-427.
• 14. Weighting coefficients for the prediction of stationary time series from the finite past, SIAM J. Appl. Math. 15 (1967), 1502-1510.
• 15. Stability of certain differential equations with homogeneous right hand sides, J. Diff. Eqns. 5 (1969), 470-475 (with E. Norman).
• 16. Continuous smoothing with polynomial weighting, SIAM J. Appl. Math. 19 (1970), 382-389.
• 17. Discrete minimum variance smoothing of a polynomial plus correlated noise, J. Math. Anal. Appl. 35 (1971), 630-645.
• 18. Stability of a class of discrete minimum variance smoothing formulas, SIAM J. Numer. Anal. 9 (1972), 307-315.
• 19. Asymptotic behavior of solutions of Lu=g(t,u,dots,u^(k-1)), J. Diff. Eqns. 11 (1972), 38-48.
• 20. Asymptotic behavior of solutions of perturbed disconjugate equations, J. Diff. Eqns. 11 (1972), 661-671.
• 21. Strong discrete minimum variance estimation of a polynomial plus random noise, J. Math. Anal. Appl. 41 (1973), 20-33.
• 22. Nonnegative and alternating expansions of one set of orthogonal polynomials in terms of another, SIAM J. Math. Anal. 4 (1973), 111-115.
• 23. Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc. 189 (1974), 319-327.
• 24. Inversion of Toeplitz band matrices, Math. Comp. 28 (1974), 1089-1098.
• 25. A sufficient condition for eventual disconjugacy, Proc. Amer. Math. Soc. 52 (1975), 139-146.
• 26. Proof of a conjecture of Askey on orthogonal expansions with positive coefficients, Bull. Amer. Math. Soc. 82 (1975), 954-956.
• 27. Oscillation properties of perturbed disconjugate equations, Proc. Amer. Math. Soc. 52 (1976), 147-155.
• 28. Asymptotic integration of linear differential equations subject to integral smallness conditions involving ordinary convergence, SIAM J. Math. Anal. 7 (1976), 213-221.
• 29. Orthogonal polynomial expansions with nonnegative coefficients, SIAM J. Math. Anal. 7 (1976), 824-833.
• 30. Band matrices with Toeplitz inverses, Linear Algebra Appl. 27 (1979), 199-210 (with T. N. E. Greville).  (Also, Transactions of the Twenty-Fourth Conference of Army Mathematicians, (1978) 365-375, and Selected Papers of T. N. E. Greville, Charles Babbage Research Centre, Winnepeg, Canada, 1984.)
• 31. Interpolating functions associated with second order differential equations, Proc. Amer. Math. Soc. 78 (1980), 253-258.
• 32. Limits of certain sequences associated with cylinder functions, Houston J. Math. 7 (1981), 289-294.
• 33. An oscillation condition for differential equations of arbitrary order, Proc. Amer. Math. Soc. 82 (1981), 548-552.
• 34. Asymptotic theory of perturbed general disconjugate equations, Hiroshima Math J. 12 (1982), 43-58.
• 35. Systems of differential equations subject to mild integral smallness conditions, Proc. Amer. Math. Soc. 87 (1983), 263-270.
• 36. Asymptotic integration of y^(n) +p(t)y^gamma =f(t) under mild integral smallness conditions, Funkcial. Eckvac. 26 (1983), 197-209.
• 37. Eventual disconjugacy of a linear differential equation, Proc. Amer. Math. Soc. 89 (1983), 461-466.
• 38. Asymptotic behavior of solutions of an n-th order differential equation, Rocky Mountain J. Math. 14 (1984), 441-450.
• 39. Asymptotic integration of linear differential equations subject to mild integral conditions, SIAM J. Math. Anal. 15 (1984), 932-942.
• 40. Asymptotic theory of perturbed general disconjugate equations II, Hiroshima Math. J. 14 (1984), 169-187.
• 41. Asymptotics of differential systems with deviating arguments, Proc. Amer. Math. Soc. 92 (1984), 219-224.
• 42. Functional perturbations of second order differential equations, SIAM J. Math. Anal. 16 (1985), 741-756.
• 43. On the eigenvalue problem for Toeplitz band matrices, Linear Algebra Appl. 64 (1985), 199-214.
• 44. On the eigenvalue problem for Toeplitz matrices generated by rational functions, Lin. Multilin. Algebra 19 (1985), 169-185.
• 45. Explicit inversion formulas for Toeplitz band matrices, SIAM J. Alg. Disc. Meth. 6 (1985), 546-554.
• 46. Eventual disconjugacy of a linear differential equation II, Proc. Amer. Math. Soc. 94 (1985), 629-635.
• 47. Global existence theorems for solutions of nonlinear differential equations with prescribed asymptotic behavior, J. London Math. Soc. 31 (1985), 478-486 (with T. Kusano).
• 48. On the eigenvalue problem for a class of band matrices including those with Toeplitz inverses, SIAM J. Alg. Disc. Meth. 7 (1986), 167-179.
• 49. Solution of systems with Toeplitz matrices generated by rational functions, Linear Algebra Appl. 74 (1986), 191-211.
• 50. On t_\infty quasi-similarity of linear systems, Ann. di Mat. Pura ed Appl. 142 (1985), 293-302.
• 51. Existence of global solutions with prescribed asymptotic behavior for nonlinear differential equations, Ann. di Mat. Pura ed Appl. 142 (1985), 381-392 (with T. Kusano).
• 52. Extensions of a theorem of Wintner on systems with asymptotically constant solutions, Trans. Amer. Math. Soc. 293 (1986), 477-483.
• 53. Existence and nonoscillation theorems for an Emden-Fowler equation with deviating argument, Trans. Amer. Math. Soc. 294 (1986), 217-232.
• 54. Linear perturbations of a nonoscillatory second order equation, Proc. Amer. Math. Soc. 97 (1986), 423-428.
• 55. Linear perturbations of general disconjugate equations, Proceedings of Equadiff 6, The Czechoslovak Conference on Differential Equations and Their Applications, Springer-Verlag.
• 56. Global existence of solutions of mixed sublinear-superlinear differential equations, Hiroshima Math. J. 16 (1986), 597-606 (with T. Kusano).
• 57. Linear perturbations of a constant coefficient differential equation subject to mild integral smallness conditions, Czechoslovak Math. J. 36 (1986), 623-633.
• 58. A note on the solution of nearly triangular Toeplitz systems, Linear Algebra Appl. 93 (1987), 57-65.
• 59. First integrals for equations with nonlinearities of the Emden-Fowler type, Int. J. Non-linear Mech. 22 (1987), 217-226 (with L. Y. Bahar).
• 60. Global existence of nonoscillatory solutions of perturbed general disconjugate equations, Hiroshima Math. J. 17 (1987), 415-431 (with T. Kusano)
• 61. Characteristic polynomials of symmetric rationally generated Toeplitz matrices, Lin. Multilin. Alg. 21 (1987), 289-296.
• 62. Toeplitz systems associated with the product of a formal Laurent series and a Laurent polynomial, SIAM J. Matrix Th. Appl. 9 (1988), 181-193.
• 63. Numerical solution of the eigenvalue problem for symmetric rationally generated Toeplitz matrices, in Linear Algebra in Signals, Systems and Control rm ed. by B. N. Datta, et. al. and as an invited paper in a special issue of SIAM J. Matrix Th. Appl. 9 (1988), 291-303.
• 64. Asymptotic integration of a perturbed constant coefficient differential equation under mild integral smallness conditions, SIAM J. Math. Anal. 19 (1988), 444-459.
• 65. Efficient application of the Schauder-Tychonoff theorem to functional perturbations of x^(n) =0, Funkcial. Ekvac. 31 (1988), 161-178.
• 66. Systems of functional differential equations with asymptotically constant solutions, Proc. Amer. Math. Soc. 104 (1988), 1091-1097 (with T. Kusano).
• 67. Numerical solution of the eigenvalue problem for Hermitian Toeplitz matrices, SIAM J. Matrix Anal. Appl. 10 (1989), 135-156.
• 68. General functional differential systems with asymptotically constant solutions, Differential and Integral Equations 2 (1989), 405-410.
• 69. Mixed sublinear, superlinear, and singular systems of functional differential equations, Differential Equations and Applications (Proceedings of the International Conference on Theory and Applications of Differential Equations, Ohio University, 1988), 448-453.
• 70. A note on a Toeplitz inversion formula, Linear Algebra Appl. 129 (1990), 55-62.
• 71. Efficient application of the Schauder-Tychonoff theorem to systems of functional differential equations, J. Math. Anal. Appl. 150 (1990), 246-263.
• 72. Spectral evolution of a one-parameter extension of a real symmetric Toeplitz matrix, SIAM J. Matrix Anal. Appl. 11 (1990), 601-611.
• 73. Explicit weighting coefficients for predicting ARMA time series from the finite past, J. Comp. Appl. Math. 34 (1991), 251-262.
• 74. Numerical solution of the eigenvalue problem for efficiently structured Hermitian matrices, Linear Algebra Appl. 154-156 (1991), 415-432.
• 75. Mild integrability conditions for global solutions of an elliptic equation, J. Comp. Appl. Math. 39 (1992), 39-48.
• 76. Asymptotic behavior of solutions of a linear second order difference equation, J. Comp. Appl. Math. 41 (1992), 95-103.
• 77. Global solutions of nonlinear differential equations, in World Scientific Series in Applicable Analysis, vol. 1 (1992), 543-557, Recent Trends in Differential Equations, dedicated to Professor V. I. Lakshmikantham on his 65th birthday.
• 78. A note on computing eigenvalues of banded Hermitian Toeplitz matrices, SIAM J. Sci. Stat. Comput. 14 (1993), 248-252.
• 79. Existence of solutions of functional-differential systems with prescribed asymptotic behavior, Proc. Fourth International Conference on Differential Equations, 148-158 (Rousse, Bulgaria, 13-19 August, 1989); edited by P. Popivanov and S. Tersian.
• 80. Asymptotic behavior of solutions of functionally perturbed nonoscillatory second order differential equations, Dynamical Systems, Proceedings of the Special Program at Nankai Institute of Mathematics held at Nankai University, Tianjin, People's Republic of China, 1991, 221-236. Published by World Scientific, Singapore, New Jersey, London , Hong Kong; edited by L. Shan-Tao, Y. Yan-Qian, D. Tong-Ren.
• 81. Asymptotic behavior of solutions of Poincar'e difference equations, Proc. Amer. Math. Soc. 119 (1993), 431-438.
• 82. Asymptotic behavior of solutions of Emden-Fowler difference equations with oscillating coefficients, J. Math. Anal. Appl. 179 (1993), 135-153.
• 83. Interlacement of the even and odd spectra of real symmetric Toeplitz matrices, Linear Algebra Appl., 195 (1993), 59-68.
• 84. Some spectral properties of Hermitian Toeplitz matrices, SIAM J. Matrix Anal. Appl. 15 (1994), 938-942.
• 85. Asymptotic behavior of solutions of Poincar'e recurrence systems, Computers Math. Applic., 28 (1994), 317-324.
• 86. Invertibly convergent infinite products of matrices, with applications to difference equations, Computers Math. Applic. 30 (1995), 39-46.
• 87. Asymptotic behavior of solutions of asymptotically constant coefficient systems of linear differential equations, Computers Math. Applic. 30 (1995), 111-117.
• 88. Numerical solution of the inverse eigenvalue problem for real symmetric Toeplitz matrices, SIAM J. Sci. Comp. 18 (1997), 1722-1736.
• 89. Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems, Computers Math. Applic. 36 (1998) 261-267.
• 90. Trench's canonical form for a disconjugate nth-order linear difference equation, Pan American J. Math. 8 (1998), 55-71 (with B.Harris and R.Krueger).
• 91. Invertibly convergent infinite products of matrices, J. Comp. Appl. Math. 101 (1999), 255-263.
• 92. Asymptotic distribution of the spectra of a class of generalized Kac-Murdock-Szeg"o matrices, Linear Algebra Appl. 294 (1999), 181-192.
• 93. Conditional convergence of infinite products, American Mathematical Monthly 106 (1999), 646-651.
• 94. Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices, Linear Algebra Appl. 302-303 (1999), 155-162.
• 95. Linear asymptotic equilibrium for nilpotent systems of difference equations, J. Difference Equations Appl. 5 (1999), 549-556.
• 96. Systems of difference equations with asymptotically constant solutions, Lakshmikantham's Legacy, Nonlinear Analysis, Theory, Methods, and Applications, 40 (2000), 611-615.
• 97. Linear perturbations of a nonoscillatory second order difference equation, J. Math. Anal. Appl. 255 (2001), 627-635.
• 98. Properties of some generalizations of Kac-Murdock-Szego matrices, Structured Matrices in Mathematics, Computer Science II Control, Signal and Image Processing (AMS Contemporary Mathematics Series), 281 (2001), 233-245.
• 99. Spectral distribution of generalized Kac-Murdock-Szeg"o matrices, Linear Algebra Appl. 347 (2002), 251-273.
• 100. Asymptotically constant solutions of functional difference systems, J. Difference Eq. Appl., 8 (2002), 811-821.
• 101. Linear perturbations of a nonoscillatory second order difference equation II, J. Math. Anal. Appl., 273 (2002), 548-557.
• 102. Linear perturbations of a nonoscillatory second order differential equation II, Proc. Amer. Math. Soc., 131 (2003), 1415-1422.
• 103. Absolute equal distribution of the spectra of Hermitian matrices, Linear Algebra Appl., 366 (2003), 417-431.
• 104. Absolute equal distribution of families of finite sets, Linear Algebra Appl. 367 (2003), 131-146.
• 105. Spectral distribution of Hermitian Toeplitz matrices formally generated by rational functions, Fast Algorithms for Structured Matrices: Theory and Applications, (Ed. Vadim Olshevsky), Contemporary Mathematics, Vol. 323, American Mathematical Society, 2003, 323-327.
• 106. A note on asymptotic zero distribution of orthogonal polynomials, Linear Algebra Appl. 375 (2003) 275-281.
• 107. Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear Algebra Appl. 377 (2004) 207-218.
• 108. Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry, Linear Algebra Appl. 380 (2004), 199-211.
• 109. An elementary note on asymptotic properties of Toeplitz and multilevel Toeplitz matrices, Linear Algebra Appl. 382 (2004) 231-235.
• 110. Hermitian, Hermitian R-symmetric, and Hermitian R-skew symmetric Procrustes problems, Linear Algebra Appl. 387 (2004) 83-98.
• 111. Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices, Linear Algebra Appl. 389 (2004) 23-31.
• 112. Simplification and strengthening of Weyl's definition of asymptotic equal distribution of two families of finite sets, Cubo A Mathematical Journal Vol. 06 N 3 (2004), 47-54.
• 113. Asymptotic relationships between singular values of structured matrices similarly generated by different formal expansions of a rational function, Numer. Linear Algebra Appl. 12 (2005), 111-116.
• 114. Characterization and properties of (R,S)-symmetric, (R,S)-skew symmetric, and (R,S)-conjugate matrices, SIAM J. Matrix Anal Appl. 26 (2005) 748-757.
• 115. Multilevel matrices with involutory symmetries and skew symmetries, Linear Algebra Appl. 403 (2005) 53-74.
• 116. Functional perturbations of nonoscillatory second order difference equations, Dynamic Systems and Applications 15 (2006) 155-171.
• 117. Absolute equal distribution of the eigenvalues of discrete Sturm-Liouville problems, J. Math. Anal. Appl. 321 (1) (2006) 299-307.
• 118. Asymptotic spectral properties of totally symmetric multilevel Toeplitz matrices, Linear Algebra Appl. 416 696-709 (2006).
• 119. Characterization and properties of matrices with k-involutory symmetries, Linear Algebra Appl. 429, Issues 8-9 (2008) 2278-2290.
• 120. Properties of unilevel block circulants, Linear Algebra Appl. 430 (2009) 2012-2025.
• 121. Properties of multilevel block circulants, Linear Algebra Appl. 431 (2009) 1833-1847.
• 122. On nonautonomous linear systems of differential and difference equations with R-symmetric coefficient matrices, Linear Algebra Appl. 431 (2009) 2109-2117.

SUBMITTED FOR PUBLICATION

• 1. Characterization and properties of matrices with k-involutory symmetries II.
• 2. The method of Frobenius without the mess and the mystery.

UNPUBLISHED NOTES AND REPORTS

• 1. ON THE RELATIONSHIP OF THE GOHBERG-SEMENCUL AND HEINIG FORMULAS FOR THE INVERSE OF A TOEPLITZ MATRX AND THE TRENCH ALGORITHMS FOR INVERSION OF TOEPLITZ AND HANKEL MATRICES.
• 2. Notes on the real symmetric Toeplitz eigenvalue problem.
• 2. A note on Toeplitz inversion formulas.
• 3. Numerical solution of the eigenvalue problem for Hermitian Toeplitz-like matrices
• 4. A Matlab package for the real symmetric Toeplitz inverse eigenvalue problem.
• 5. Efficient application of the Schauder-Tychonoff theorem to functional differential systems.
• 6. Inverses of lower triangular Toeplitz matrices.
• 7. Spectral decomposition of Kac-Murdock-Szego matrices.
• 8. A Riemann integral proof of a generalized Riemann lemma.
• 9. A polynomial identity motivated by a calculus problem.
• 10. On matrices with rotative symmetries.
• 11. A Matlab procedure for spectral decomposition of real symmetric Toeplitz matrices.
• 12. Linear differential systems with coefficient matrices that commute with a spectrally separated matrix function, Trinity University Mathematics Department Colloquium (with corrections), October 28, 2009.
• 13. Banded symmetric Toeplitz matrices: where linear algebra borrows from difference equations, Trinity University Majors' Seminar (with corrections), October 29, 2009.

RELATED LINKS

• Google Scholar: Toeplitz Matrix
• Google Scholar: The Eigenvalue Problem for Hermitian Toeplitz Matrices
• Google Scholar: The Inverse Eigenvalue Problem for Symmetric Toeplitz Matrices
• Google Scholar: Disconjugate Equations
• Uri Elias, in Encyclopaedia of Mathematics: Disconjugacy
• Wikipedia: Levinson Recursion
• T.R. Hopkins: FORTRAN Subroutines for the Solution of Symmetric and Non-Symmetric Toeplitz Sets of Linear Equations Using the Levinson-Trench Algorithm
• Greg Ammar: Toeplitz Inversion Formulas
• Thomas Kailath: Selected Bibliography on Displacement Structure
• Dirk Laurie: Octave Functions to Work with Toeplitz and Hankel Matrices
• A. Ian McLeod, Hao Yu, and Zinovi L. Krougly: Algorithms for Linear Time Series Analysis
• Wikipedia: Centrosymmetric Matrix
• L. H. Erbe, R. M. Mathsen, and A. Peterson: Factoring Linear Differential Operators on Measure Chains
• Martin Bohner and Stevo Stevic: Trench's Perturbation Theorem for Dynamic Equations
• Earliest Known Uses of Some of the Words of Mathematics ("Time Domain" and "Frequency Domain")