## Professional Summary

My fourth monograph titled *The Research about Invariants of Ordinary Differential Equations *was published in March of 2018. Its back cover contains the following explanation.

"Readers acquainted with the differential calculus and the concept of a polynomial can effortlessly acquire an excellent perspective about invariants for differential equations by using a system of computer algebra to interact with the examples in Chapters 16 and 17.

During the years 1879-1889, there were differential equations for which mathematicians found remarkable combinations of the coefficients that possessed an invariant character under unrestricted transformations. In fact, specific examples of basic relative invariants as the most interesting kind were discovered by E. Laguerre in 1879, G.-H. Halphen in 1880-1884, A. R. Forsyth in 1888, and P. Appell in 1889. However, there was little progress after 1889 about the principal problems until the subject was completely redeveloped during 1989-2013. A thorough explanation for this remarkable situation is given in Chapters 15 and 18.

All of the basic relative invariants are now explicitly known for numerous types of differential equations and the main problems have now been solved. This monograph provides details about these developments and gives numerous illustrations to show how the relative invariantss of a given weight are expressible in terms of the basic relative invariants.

Roger Chalkley was awarded the degree of Ch.E. in 1954 at the University of Cincinnati where he also earned an A.M. (mathematics) in 1956 and a Ph.D. (mathematics) in 1958. His Ph.D. thesis advisor, Professor Arno Jaeger, had a deep interest in the subject of differential algebra as developed by J. F. Ritt and E. R. Kolchin. That algebraic viewpoint is evident throughout the current monograph as well as his two Memoirs of the American Mathematical Society that were published as Numbers 744 in 2002 and Number 888 in 2007."

**My research is unusual because, instead of fragmenting mathematics by rushing lots of short papers into publication, I have avoided superficiality by concentrating on difficult problems whose solutions enabled me to completely redevelop and unify the subject of invariants for ordinary differential equations in a series of four monographs published in 2002, 2007, 2014, and 2018 consisting respectively of 204 + xi pages, 365 + xii pages, 145 + xvIII pages, and 190 + xi pages. **

** ** Previous research on that subject was done primarilly during the years 1879-1889 by Edmond Laguerre, Georges-Henri Halphen, Andrew Forsyth, and Paul Appell. They had presented isolated examples of considerable interest. In particular, G.-H. Halphen was awarded the Grand Prize of the French Academy of Sciences in 1880 for his research about invariants.

My interest in that subject began in 1957 when I read some of the 1879-1889 publications. Then, it became compelling when I noticed a lack of general results and became aware of the numerous negative efforts after 1889 to advance the subject. Thus, I was truly fortunate to have found a fascinating area for research and to have had sufficient time to develop it.

For more detail, visit my web-page at

http://homepages.uc.edu/~chalklr/

## Education

Ph.D.: University of Cincinnati Cincinnati, Ohio 45221, 1958 (Mathematics)

Ch.E.: University of Cincinnati Cincinnati, Ohio 45221, 1954 (Chemical Engineering)

A.M.: University of Cincinnati Cincinnati, Ohio 45221, 1956 (Mathematics)

## Research and Practice Interests

My most recent publications (for all of which I am the sole author) are:

The Research about Invariants of Ordinary Differential Equations, Available from Amazon.com and other retail outlets, 2018, ISBN: 978-1985381193

Relative Invariants from 1879 Onward: Their Evolution for Differential Equations, Llumina Press, Plantation, Florida, 2014, ISBN: 978-1-62550-120-2

Basic Global Relative Invariants for Nonlinear Differential Equations, Memoirs of the American Mathematical Society, Number 888, Providence, Rhode Island, 2007, ISBN: 978-0-8218-3991-1

Basic Global Relative Invariants for Homogeneous Linear Differential Equations, Memoirs of the American Mathematical Society, Number 744, Providence Rhode Island, 2002, ISBN: 0-8218-2781-2

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.

For more details about current research interests, click on

http://homepages.uc.edu/~chalklr/

Complete details about the subject of relative invariants for differential equations and its long history are presented in the monographs listed here

http://www.amazon.com/s/ref=ntt_athr_dp_sr_1?ie=UTF8&field-author=Roger+Chalkley&search-alias=books&text=Roger+Chalkley&sort=relevancerank

or here

http://www.alibris.co.uk/search/books/author/Roger-Chalkley

## Positions and Work Experience

1946 -1949 caddy,, Fort Mitchell Country Club,, Fort Mitchell, Kentucky

1950 -1950 Member of Yard Labor Gang,, first co-op employment,, Gardner Board and Carton Company,, Lockland, Ohio

1950 -1952 Laboratory Technician,, second co-op employment,, Container Corporation of America,, 5500 Eastern Avenue, Cincinnati, Ohio

1952 -1953 Assistant to Plant Engineer,, third co-op employment,, Interchemical Corporation,, Dana and Montgomery Avenues, Cincinnati, Ohio

1957 -1958 Instructor of Mathematics,, University of Cincinnati,, Cincinnati, Ohio

1958 -1959 Mathematician, Reactor Experimental Engineering Division,, Research about the two-region spherical reactor,, Oak Ridge National Laboratory, Y-12,, Oak Ridge, Tennessee

1960 -1962 Assistant Professor of Mathematics,, Knox College,, Galesburg, Illinois

1962 -1963 Assistant Professor of Mathematics,, University of Cincinnati,, Cincinnati, Ohio

1963 -1980 Associate Professor of Mathematics,, University of Cincinnati,, Cincinnati, Ohio

1980 -2016 Professsor of Mathematics,, University of Cincinnati,, Cincinnati, Ohio

## Abbreviated Publications

### 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 y'', y', 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.

### Monograph

(2014). Roger Chalkley, *Relative Invariants from 1879 Onward: Their Evolution for Differential Equations*, Llumina Press, Plantation, Florida,

(2007). Roger Chalkley, *Basic Global Relative Invariants for Nonlinear Differential Equations, * American Mathematical Society, Providence, Rhode Island,

(2002). Roger Chalkley, *Basic Global Relative Invariants for Homogeneous Linear Differential Equations, *American Mathematical Society, Providence, Rhode Island,

(2018). Roger Chalkley, *The Research about Invariants of Ordinary Differential Equations*, Available from Amazon.com and other retail outlets,

## Post Graduate Training and Education

1959-1960 Postgraduate Study,, University of Zurich,, , Zurich, Switzerland

1959-1960 Postgraduate Study, , ETH Zurich,, , Zurich, Switzerland

## Keywords

Relative Invariants for Ordinary Differential Equations

## Contact Information

Phone: 513-556-4082