mm-2209 http://mathforum.org/students/high/ Gerontius (another old guy with maths as a hobby) I am selling Mathematica 5. Please see: I have tried to choose groups where this would be relevant, but if it is OT, I apologize. could be contacted! Magic Squares are a fascinating area, I have been looking at the literature on the subject and think it would be a great topic to explore with my students in secondary school. However, it seems that there is little work to be found on it mathematically. Most seem to dwell on the methods of construction or go much farther into the area of polytopes etc. 1)Square matrices seems to be a good way to look at it. If we beginby dispensing with the requirement that the entries are successive integers or belong to some particular set of integers, we could investigate their properties. 2)Calling a square n X n matrix A = [a_ij] of real or complex numbers additively good if its row sums and column sums are equal and additively very good if its row sums, column sums and diagonal sums are equal. This leads to the entries in a good matrix having 2n constraints to satisfy, while those in a very good matrix have 2n+2 constraints. If we denote by v(A), the common row and column sum of an additively good matrix A and by u(A) the common row,column and diagonal sum of an additively very good matrix A. 3)What expressions would I put down for v(A),u(A), involving all entries of A? 4)Is it possible to show that very good matrices of any order exist? This would allow me to say good matrices of any order exist.. 5)What would be a good way to show that additively good matrices are closed under addition, scalar multiplication and matrix multiplication? 6)Is it possible to show that additively very good matrices are not necessarily closed under matrix multiplication? I'd be delighted if anyone could show me some insight in the above questions that are bothering me. Brendan If you don't have an antivirus program, you should get one and check your machine. AVG is a free antivirus, it is rated highly and works very well. John Starrett Since this is the first time this has been posted to sci.chem... Get AdAware and SpyBot Search and Destroy and run them also. You possibly could have ended up with malware that one of them will recognize/remove. They both are shareware/freeware. David A. Smith should have unexpected but check your well. Ad-Aware has trust issues. http://www.dslreports.com/forum/remark,12665642~mode=flat Quick summary: Ad-aware's spyware definitions have been getting smaller and people are wondering what is being removed. Someone noticed the WhenUsave adware not being removed anymore [It used to be] and basically raised a storm. ... things every week, but that, as you say, may only the tip of the iceberg. David A. Smith ESBPCS-Stats is a subset of ESBPCS (ESB Professional Computation Suite) containing Components and Routines for Statistical Analysis and Matrix/Vector Manipulation in Borland Delphi and C++ Builder. http://www.esbpcs.com This subset is ideal for people who just want the Stats and/or Matrix/Vector parts of ESBPCS, though you can upgrade to the full version at any time. Also includes Components and routines covering Probability Distributions, Linear Regression, Hypothesis Analysis, Equation Solving and more. The subset includes a good collection of Edits, SpinEdits, ComboBoxes, Memos, CheckBoxes, RadioGroups, CheckGroups as well as a huge collection of routines. Also Includes Data Aware Components, Help and full source. http://www.esbpcs.com/feature-matrix.htm And we have just released a new Trial version of ESBPCS-Stats for Borland Delphi 5, 6, 7 and 2005/Win32, so that way you don't have to download the full trial version: http://www.esbpcs.com/downloads.htm ESBPCS-Stats can be purchased from various sources at: http://www.esbpcs.com/purchase.htm Glenn Crouch mailto:glenn@esbconsult.com ESB Consultancy http://www.esbconsult.com Home of ESBPCS & ESB Calculators Kalgoorlie-Boulder, Western Australia I am not sure about the definition of a Legendre transformation. Is it simply x dy = d(xy) - y dx as used in integratio-by-parts, or does it apply to two or more variable pairs, s dt + x dy = d(ts) + d(xy) - t ds - y dx.? What happens when I transform one variable pair and not the other, particularly if I want to express integral values as total functions i.e. Integral (s dt) + integral (x dy) = U. The partial transformation XY - U does not make sense. The total transformation XY + TS -U = G = integral (t ds) + integral (y dx) does make sense.