Home Page of Roger Chalkley

Also, see  https://researchdirectory.uc.edu/p/25556

 

Principal Research Accomplishments

     The subject of invariants for ordinary differential equations has been completely redeveloped during the years 2002--2018 in my monographs that are displayed by a click here or here.   For a brief explanation, click here.  The serious handicaps faced by earlier researchers and how they were overcome are clearly described in Chapters 1, 15, and 18 of my 2018 monograph titled The Research about Invariants of Ordinary Differential Equations.  That monograph is the single best reference on the subject for any first-rate research library because it provides a thorough historical perspective as well as complete details about the redevelopment and all recent advances.   To read the Abstract and Preface, click here.  To read Chapter 1, click here.  To read Chapters 15, click here.  To read Chapter 16, click here.  To read Chapter 18, click here.  In particular, by interacting with a system of computer algebra while reading Chapter 16, anyone knowledgeable about the differential calculus can easily acquire an excellent introduction to results about invariants. 

 

 Viewpoint

     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.  The motivation for that approach was made possible by my use of the various Mathematics Libraries at the University of Cincinnati from 1957 onward in the manner described via a click here

      In particular, an important key to progress was the recognition in 1989 that no one had previously developed adequate formulas for the coefficients of m-th order homogeneous linear differential equations resulting from changes of the independent variable for the context where m is merely a symbol for any positive integer.  Then, after we developed adequate formulas by using a natural notation for the coefficients of the equations, we observed that all of the previous researchers had been handicapped with a counter-productive notation for their writing of homogeneous linear differential equations.  Namely, as explained by a click here or here, all of them had inserted binomial coefficients counter-productively.    For our earlier viewpoint from 2014, click here

 

Supplementary Information

     The Mathematica commands written and presented in The Research about Invariants of Ordinary Differential Equations were checked and found to evaluate as desired by Versions 7, 8, 9, 10, and 11 of Mathematica. Namely, the commands were copied directly from the manuscript and pasted into Mathematica notebooks for evaluation by the various versions.  Those same notebooks may be accessed with the Google Chrome Browser after a click here.  (We have noticed that the Microsoft Internet Explorer Browser and the Mozilla Firefox Browser do not serve to download Mathematica notebooks.  However, at this time, the Google Chrome Browser serves well for that purpose.)

     The principal earlier contributions about invariants for ordinary differential equations were made by James Cockle in numerous short papers during years including 1862--1875, Edmund Laguerre in 1879,  Georges-Henri Halphen in 1880-1884, Andrew Forsyth in 1888, and Paul Appell  in 1889.  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 study of invarriants and Henri Poincar received honorable mentioon.     

     While E. Laguerre, G.-H. Halphen, A. R. Forsyth, and P. Appell were the key contributors to the early development of relative invariants (as the most interesting kind of invariant), numerous mathematicians were interested in the subject.  In regard to Francesco Briochi, click here.

     My interest in the 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.  

     Although each of Edmond Laguerre, Georges-Henri Halphen, Andrew Forsyth, and Paul Appell made interesting discoveries about basic relative invariants for ordinary differential equations during the years 1879--1889, the main problem of obtaining explicit formulas for all of the basic relative invariants was nowhere near a solution during the years from 1879 through 1988.  What happened to enable one person to develop explicit formulas for all of the basic relative?  For the answer, click here and in more detail here.

 

Use of Mathematics Library

     My research has been highly dependent upon resources provided by the various University of Cincinnati Mathematics Libraries from 1957 onward.  The three intensive searches of the particular areas in the mathematical literature that the libraries enabled me to undertake are described on the page obtained by clicking here.  

 

Use of Computer Algebra

     My research has also benefitted from the availability of computer algebra.  It enabled me to do computational experimentation in minutes that could not have been done in years under favorable circumstances by the early researchers about invariants.  Professors Dieter Schmidt and Kenneth Meyer were exceptionally helpful in introducing me to the computer-algebra-system Macsyma and permitting me to employ it on their advanced Symbolics Computer during the years 1988-1996.  Then, when Version 3.0 of Mathematica appeared in 1996, I purchased it (with its large instruction manuel) as well as considerable additional random access memory for my personal computer located at home.  That was an excellent  investment. 

     During various years from 2003 through 2011, I taught short courses about the use of Mathematica as a supplement for students of Calculus II and Calculus III.  During those years, the University-of-Cincinnati Department-of-Mathematical-Sciences occupied the eighth floor level of the Old Chemistry Building.  Then, our offices, the Mathematics Library, and the Computer Laboratory were all on that eighth-floor level and the classrooms were usually near by.  All of that was changed.  Details for my courses in 2009 about a short introduction to Mathematica can be accessed with a click here.   This hyperlink may still be useful for some individuals.   For additional information, click here.     

 

Inappropriate Generalizations  

     When unmotivated abstractions and generalizations are introduced in mathematics, it may be for the purpose of giving someone something to do of a non-demanding nature.  History can then be ignored.  It can also be a way to avoid explicitly admitting ignorance when consulted for help about concrete problems.  Or, perhaps as an occupational deformation, there may be a perceived need to appear brilliant.    

     Zentralblatt MATH, the online successor to the Zentralblatt fuer Mathematik, has had editors that permit their reporters to write obvious nonsense.   For details, click here.

 

Commendable Sites

     Some well-motivated abstractions could be safely accessed in 2018 by clicking here.  

     For various interesting viewpoints, start here.  

 

Commentary written during the year 2007

     Edmund Laguerre in 1879 and many other mathematicians from 1879 onward were fascinated by the subject of relative invariants for homogeneous linear differential equations.   The principal challenge was to discover explicit formulas for all of the basic relative invariants.  Prior to 2002, this had only been done for equations of order m when m = 3, 4, 5, 6, and 7.  We were fortunate to have discovered simple explicit formulas for all of the basic relative invariants in R. Chalkley, Basic Global Relative Invariants for Homogeneous Linear Differential Equations, Memoirs Amer. Math. Soc. 156 (2002), Number 744, pages 1-204.  Explicit formulas that yield all of the basic relative invariants for general classes of nonlinear differential equations were presented in R. Chalkley, Basic Global Relative Invariants for Nonlinear Differential Equations, Memoirs Amer. Math. Soc. 190 (2007), Number 888, pages 1-365.  

     Researchers may download  Mathematica  notebooks for computations on pages 28-30 of my Memoir titled  Basic Global Relative Invariants for Homogeneous Linear Differential Equations  by clicking here. Mathematica notebooks for the computations in my Memoir titled Basic Global Relative Invariants for Nonlinear Differential Equations may be downloaded by clicking here.  Versions 3.0 evaluates these notebooks effectively.  Versions 4.0 and 5.0 evaluate most of the notebooks without modification.  Avoid Version 6.  When notebooks are prepared directly with Version 7.0 to have the same input statements as the ones available here for Version 3.0, the corresponding evaluations occur efficiently for all of them.   (We have more recently found that Versions 8, 9 10, and 11 are also suitable.)  

 

For a Concise History about various Departments of Mathematics at the University of Cincinnati, click here.  

     For a photograph of Professor Louis Brand, click here. Prior to his retirement in 1956, Professor Brand lived in a Swiss-chalet-style home located at the top of the Straight-Street hill on the North-West corner of Straight Street and University Court.  For photographs of it taken via Google-Maps sometime just before its destruction in 2011, click here.       

 

Direct Proof that a Product of Circulant Matrices Is a Circulant Matrix

     On page 191 of the monograph by Philip J. Davis titled Circulant Matrices, (Wiley, 1979), it is stated that generalizations can be found in the paper of mine titled Matrices derived from finite abelian groups (Mathematics Magazine, Volume 49, pages 121-129, 1976).   Rather than use an unmotivated argument applicable only to circulant matrices, that algebraic viewpoint enables one to give a clear direct proof that the product of two n x n circulant matrices over a ring R is an n x n circulant matrix over R and that the multiplication for circulant matrices is commutative when R is a commutative ring.  For such a proof, click here.  As demonstrated there, that argument about closure also applies to any two n x n group matrices of the same type for any finite group and shows that their matrix multiplication is commutative when the multiplications in both R and the group are commutative.    

 

Efficient Polynomial Evaluation

     We suggest Efficient Polynomial Evaluation as a name to help popularize the useful old topic of "Synthetic Division" that is missing from most current elementary textbooks on algebra.  For a  reminder that the possibly  disliked term  "division"  can be completely avoided, click here.  A related observation written about that same time in 2004 can be found here.

 

Differential Equations Solvable by Mathematicians and Not Solvable by Machines 

     For my personal experience about this, visit the yet-to-be-completed web page obtained by clicking here.  

 

Failure to Broaden Perspective at the Monthly

     Volume 1 of the Monthly of the Mathematical Association of America appeared in 1894 and many basic algebraic properties of circulant matrices were discovered before 1894.  The publication of Circulant Matrices by Philip J. Davis in 1979 is undoubtedly responsible for the resurgence of interest in that subject.  For example, the Monthly published a twenty-page article in November of 2001 that may be viewed by clicking here.  Its subject about using circulant matrices to facilitate the derivation of formulas for solving third-degree and fourth-degree polynomial equations bears a remarkably close resemblance to ideas expressed in numerous earlier publications to which it makes no reference.  For instance compare its subject with those of reference0 of 1974, reference1 in 1975, reference2 in 1975, reference3 of 1883,  reference4 of 1883, and many others thereafter.  Also, Mr. Lodge in reference4 points out that Mr. Glaisher in 1879 and Mr. Scott in 1880 had also shared ideas similar to those of Mr. Legoux.  In fact, as shown by reference5 of 1883 and reference6 of 1880, other matrices than circulant ones serve just as well.  Moreover, a nontrivial extension of those ideas had already been made by Arthur Cayley in his paper reference7 of 1882 when, in order to study special situations for the roots of fifth degree polynomial equations, he states in his first sentence a result explainable in terms of the then-known factorization of a 5 x 5 circulant matrix .  

     After reading the November 2001 article in the Monthly about circulant matrices, it became obvious to me that circulant matrices had become widely regarded as curiously-interesting isolated objects unrelated to important historical developments of mainstream mathematics.   Naturally, so I thought, the Monthly would wish to correct faulty misimpressions it helped create.  With that belief, I wrote a paper for that purpose and submitted it to the Monthly on April 26, 2002.  For my letter to the editor about that paper here.  

     For my manuscript of April 2002 titled "Diagonalizations of circulant matrices and analogous reductions for group matrices", click here.  

     For the receipt of my letter and manuscript, click here.  

     For the first referee's response, click here.  

     For the second referee's response, click here.  

     For the editor's final response, namely his rejection of my paper eight months later, click here.  

     The editor did thank me for thinking of the Monthly.  

     At that time, my  thoughts about the Monthly were those summarized here.  

  

Photograph      

     For a photograph of Roger Chalkley taken on September 24, 2005, click here (JPEG 1,024,687 bytes), or here (JPEG 2,107959 bytes), or here (TIF 18,288,748 bytes)

 

Miscellaneous Observations    

     Biographies of Mathematicians (with observations about temperament, political considerations, etc.), Historically Important Curves (illustrated - and given with their corresponding rectangular and polar equations),  Definitions of Mathematical Terms, Maps illustrating where well-known mathematicians were born, and other Interesting features can be accessed from the web site created by John J. O'Connor and Edmund F. Robertson.  

Course Information from the year 2000 onward

     For information about courses taught during previous Semesters or Quarters,  click here.

 

Brief Summary of Mathematical Research Interests

     Since 1957, my concerns have focused mainly on ordinary differential equations having meromorphic coefficients on a region of the complex plane.  My more recent publications are:

bullet The Research about Invariants of Ordinary Differential Equations, Available from Amazon.com and Other Retail Outlets, 2018, 190 + xi pages.  ISBN: 978-1985381193
bullet Relative Invariants from 1879 Onward: Their Evolution for Differential Equations, Llumina Press,  (January of 2014),  145 + xviii pages.  ISBN: 978-`1-62550-120-2
bullet Basic global relative invariants for nonlinear differential equations, Memoirs of the American Mathematical Society, 190  (November of 2007), Number  888, 365 + xii pages.        QA 371.C435   2007        ISBN: 978-0-8218-3991-1
bullet Basic global relative invariants for homogeneous linear differential equations, Memoirs of the American Mathematical Society 156 (March of 2002), Number 744, 204 +xi pages.     QA 3.A57  no. 744    ISBN: 0-8218-2781-2
bullet Lazarus Fuchs' transformation for solving rational first-order differential equations, Journal of Mathematical Analysis and Applications, 187 (1994) 961 - 985.
bullet A persymmetric determinant, Journal of Mathematical Analysis and Applications, 187 (1994) 107 - 117.
bullet Semi-invariants and relative invariants for homogeneous linear differential equations, Journal of Mathematical Analysis and Applications, 176 (1993) 49 - 75.
bullet A formula giving the known relative invariants for homogeneous linear differential equations, Journal of Differential Equations, 100 (1992) 379 - 404.
bullet The differential equation Q = 0 in which Q is a quadratic form in y", y', y having meromorphic coefficients, Proceedings of the American Mathematical Society , 116 (1992) 427 - 435.
bullet Relative invariants for homogeneous linear differential equations, Journal of Differential Equations, 80 (1989) 107 - 153.
bullet New contributions to the related work of Paul Appell, Lazarus Fuchs, Georg Hamel, and Paul Painlev 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 1994-2001 was published in March of 2002 as the Memoir of the American Mathematical Society titled Basic Global Relative Invariants for Homogeneous Linear Differential Equations.  It is identified as  Number 744 (the fifth of 5 numbers) in Volume 156 and it is bound separately as a book that has the International Standard Book Number  ISBN-0-8218-2781-2 and the Library of Congress identification  QA 3.A57 no. 744.  For information about it from the American Mathematical Society, click here.  Various pages from this Memoir can be viewed at the Google website visited by clicking here

         That 2002 Memoir completely redevelops the subject of relative invariants for homogeneous linear differential equations (from 1879 onward).  In particular, for the first time, it presents explicit formulas for all of the basic relative invariants of homogeneous linear differential equations.  Its results are rigorous for a subject that defied significant advances by mathematicians during the years from 1888 to 1989.   In particular, the Memoir is completely self-contained and consists of 204 + xi pages.  

         During the years following 2001, I was able to extend my techniques for homogeneous linear differential equations to ones that also specify all of the basic global relative invariants for general classes of nonlinear differential equations.  Those results are presented in the 365-page Memoir of the American Mathematical Society (Number 888, November 2007) titled Basic Global Relative Invariants for Nonlinear Differential Equations For information about it from the American Mathematical Society, click here Various pages may be viewed by clicking here.  Because the subject for nonlinear equations has received little previous attention due to its difficulty, there are undoubtedly many mathematicians who could be pleasantly surprised by its development.  

         Immediately after the publication of my 2007 Memoir, the principal unsolved problem was that of finding an explicit construction for combining relative invariants to obtain others in such a manner that the basic relative invariants can be used to specify all relative invariants of any given weight.  Suitable explicit constructions for that purpose are presented in my 2014 monograph titled Relative Invariants from 1879 Onward: Their Evolution for Differential Equations.   Thus, my first three monographs solve the principal problems.  My fourth monograph titled The Research about Invariants of Ordinary Differential Equations provides a careful perspectiv for all of the advances.   

 

Here's looking at you kid!

Jumping spiders are not camera-shy!   Their vision is very sharp and they appear to look at the photographer as if they are keenly interested in his intentions.   However, it may be that they like to admire their reflection in the camera lens.   For details, click here.

Department of Mathematical Sciences
University of Cincinnati
Cincinnati, OH 45221-0025

E-Mail: Roger.Chalkley@uc.edu