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Course Information
Roger Chalkley
Professor
822A Old Chemistry Building
513-556-4074
roger.chalkley@uc.edu
http://math.uc.edu/~chalklr/
Professional Summary
Since 1957, my research interests have been concerned mainly with ordinary differential equations having meromorphic coefficients on a region of the complex plane.
Education
Ph.D., University of Cincinnati, 1958.
Research Interests
My most recent publications are:
Basic Global Relative Invariants for Nonlinear Differential Equations, Memoirs of the American Mathematical Society 190 (November 2007), Number 888 (pages 1-365 + xii).
Basic Global Relative Invariants for Homogeneous Linear Differential Equations, Memoirs of the American Mathematical Society 156 (March 2002), Number 744 (pages 1-204 + xi).
Lazarus Fuchs' transformation for solving rational first-order differential equations, Journal of Mathematical Analysis and Applications, 187 (1994) 961 - 985.
A persymmetric determinant, Journal of Mathematical Analysis and Applications, 187 (1994) 107 - 117.
Semi-invariants and relative invariants for homogeneous linear differential equations, Journal of Mathematical Analysis and Applications, 176 (1993) 49 - 75.
A formula giving the known relative invariants for homogeneous linear differential equations, Journal of Differential Equations, 100 (1992) 379 - 404.
The differential equation Q = 0 in which Q is a quadratic form in y", y', and y having meromorphic coefficients, Proceedings of the American Mathematical Society , 116 (1992) 427 - 435.
Relative invariants for homogeneous linear differential equations, Journal of Differential Equations, 80 (1989) 107 - 153.
New contributions to the related work of Paul Appell, Lazarus Fuchs, Georg Hamel, and Paul Painlevé on nonlinear differential equations whose solutions are free of movable branch points, Journal of Differential Equations, 68 (1987) 72 - 117.
My principal research during the years 1995-2001 was published in March of 2002 as the Memoir of the American Mathematical Society (Number 744) titled “Basic Global Relative Invariants for Homogeneous Linear Differential Equations.” That Memoir presents remarkable new results of a rigorous nature for a subject that defied adequate treatment by mathematicians during the years from 1888 to1989. It is completely self-contained and consists of 204 + eleven pages. In particular, it presents simple explicity formulas for all of the basic relative invariants possessed by homogeneous linear differential equations. During the years 2001-2006, those results were extended to provide all of the basic relative invariants for general classes of nonlinear differential equations. The latter results were published in November of 2007 as the Memoir of the American Mathematical Society (Number 888) titled "Basic Global Relative Invariants for Nonlinear Differential Equations." It is also completely self-contained and it consists of 365 + twelve pages.
Peer Reviewed Publications
(1960). An IBM–704 code for a harmonics method applied to two-region spherical reactors. Oak Ridge National Laboratory Report, 2080.
(1960). On the second-order homogeneous quadratic differential equation. Mathematische Annalen, 141, 87-98.
(1963). A certain homogeneous-differential-equation transformation. Arch. Math. (Basel), 14, 186-192.
(1974). Cardan’s formulas and biquadratic equations. Math. Mag., 47, 8-14.
(1975). A lattice of cyclotomic fields. Math. Mag., 48, 42-44.
(1975). Bounds for difference quotients and derivates. Amer. Math. Monthly, 82, 277-279.
(1975). Circulant matrices and algebraic equations. Math. Mag., 48, 73-80.
(1975). Quartic equations and tetrahedral symmetries. Math. Mag., 48, 211-215.
(1975). Algebraic differential equations of the first order and the second degree. J. Differential Equations, 19, 70-79.
(1976). Matrices derived from finite abelian groups. Math. Mag., 49, 121-129.
(1977). A first-order algebraic differential equation. J. Differential Equations, 26, 458-466.
(1978). The perfect nth power which divides a nonzero polynomial. Proc. Amer. Math. Soc., 68, 147-148.
(1979). Analytic solutions of algebraic differential equations. SIAM J. Math. Anal., 10, 778-782.
(1980). Explicit solutions of algebraic differential equations. J. Differential Equations, 35, 275-290.
(1981). Information about group matrices. Linear Algebra Appl., 38, 121-133.
(1987). New contributions to the related work of Paul Appell, Lazarus Fuchs, Georg Hamel, and Paul Painlev´e on nonlinear differential equations whose solutions are free of movable branch points. J. Differential Equations, 68, 72-117.
(1989). Relative invariants for homogeneous linear differential equations. J. Differential Equations, 80, 107-153.
(1992). The differential equation Q = 0 in which Q is a quadratic form in y00, y0, y having meromorphic coefficients. Proc. Amer. Math. Soc., 116, 427-435.
(1992). A formula giving the known relative invariants for homogeneous linear differential equations. J. Differential Equations, 100, 379-404.
(1993). Semi-invariants and relative invariants for homogeneous linear differential equations. J. Math. Anal. Appl., 176, 49-75.
(1994). A persymmetric determinant. J. Math. Anal. Appl., 187, 107-117.
(1994). Lazarus Fuch’s transformation for solving rational first-order differential equations. J. Math. Anal. Appl., 187, 961-985.
(2002). Basic global relative invariants for homogeneous linear differential equations. Mem. Amer. Math. Soc., 156(744), 1, 204.
(2007). Basic Global Relative Invariants for Nonlinear Differential Equations. Mem. Amer. Math. Soc., 190(888), 1, 365.
Books
Roger Chalkley (2007). Basic Global Relative Invariants for Nonlinear Differential Equations. Providence, Rhode Island: American Mathematical Society.
Roger Chalkley (2002). Basic Global Relative Invariants for Homogeneous Linear Differential Equations. Providence, Rhode Island: American Mathematical Society.
