Home Page of Roger Chalkley

Also, see    http://asweb.artsci.uc.edu/collegedepts/math/facStaff/profile_details.aspx?ePID=MjU1NTY%3D

       Edmund Laguerre in 1879, Francesco Brioschi in and following 1879, Georges-Henri Halphen during the early 1880's, Andrew Russell Forsyth during the late 1880's, and many other mathematicians from 1879 until present times 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.  For more detail click here.  Our research in this subject developed from an earlier interest in a type of nonlinear differential equation that Paul Appell had studied during the late 1880's.  More detail about this can be found by clicking here.  Explicit formulas that yield all of the basic relative invariants for general classes of nonlinear differential equations have been more recently presented in R. Chalkley, Basic Global Relative Invariants for Nonlinear Differential Equations, Memoirs Amer. Math. Soc. 190 (2007), Number 888, pages 1-365.  For more detail, click here.  Researchers may download  MATHEMATICA  notebooks for the 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. 

          MATHEMATICA, Version 6.0, has an excellent implementation for its  Coefficient  command.  For some observations about earlier versions of this command, click here. 

          For the class Calculus and Analytic Geometry IV (15-MATH-264-001) that meets during the Autumn  Quarter of 2008 on Mondays, Tuesdays, Wednesdays, Thursdays, and Fridays at 8:00-8:50 a.m. in Room 308 of Zimmer Hall, the syllabus may be accessed by clicking here.

        For the class Applied Calculus I (15-MATH-226-003) that meets on Tuesdays and Thursdays at 11:00-12:15 p.m. in Room 3240 of the Campus Recreation Center, the syllabus may be accessed by clicking here.

        Students in Calculus IV or Applied Calculus I will be able to access their individual blackboard  course records through the link http://www.uc.edu/blackboard/ .  Special announcements (if they are necessary) may also be placed there.   

        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).

          Efficient Polynomial Evaluation is suggested 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.

          The Calculus-2-Laboratory has been taught by various teachers in different ways.   One can learn precise details about how Sections 003, 004, 006 of this one-credit hour course were taught during the Winter Quarter of 2003 by clicking here. 

         Students of any Calculus-2-Lab based on the textbook by Selwyn Hollis titled "CalcLabs With Mathematica" can download all of the Input statements for Chapters 1, 2, 3, 4, 5, 6, and 8 by first clicking here.  In this way, one can easily evaluate any Input statement while concentrating on the textbook's explanation about its purpose.   

         Biographies of Mathematicians (with observations about temperament, political pressures, 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.  

         Students of Applied Linear Algebra (from previous quarters) may wish to continue using MATHEMATICA commands (rather than MATLAB commands) .   For that purpose a MATHEMATICA notebook can be accessed by first clicking here.  To merely read the files, one can use MathReader; and the latter can  be downloaded without obligation from http://www.wolfram.com/products/mathreader/

          For obsolete information from previous quarters, click here.

          Since 1957, my research interests have been concerned mainly with ordinary differential equations having meromorphic coefficients on a region of the complex plane.

My most recent publications are:

         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.

         This 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 + five 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 now appear in the 365-page Memoir (Number 888, November 2007) listed above and titled "Basic Global Relative Invariants for Nonlinear Differential Equations."  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 current  development.  

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 an explanatory context of our interest, click here.