Download Mathematical Recurrence Relations: Visual Mathematics Series, by Kiran R. Desai Ph.D.
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Mathematical Recurrence Relations: Visual Mathematics Series, by Kiran R. Desai Ph.D.
Download Mathematical Recurrence Relations: Visual Mathematics Series, by Kiran R. Desai Ph.D.
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This book is about arranging numbers in a two dimensional space. It illustrates that it is possible to create many different regular patterns of numbers on a grid that represent meaningful summations. It uses a color coding scheme to enhance the detection of the underlying pattern for the numbers. Almost all arrangements presented are scalable or extensible, in that the matrix can be extended to larger size without the need to change existing number placements. The emphasis in this book is about the placement and summation of all the numbers for recursive embeddings. In many cases, visual charts are used to provide a higher level view of the topography, and to make the recurrence relations come alive. Number arrangements are represented for many well known multi-dimensional numbers, polygonal numbers, and various polynomials defined by recurrence relations based on equations that are a function of an integer variable n. The solutions for the recurrence relations can also be checked by adding the numbers in the arrangements presented. It is also possible to create a recurrence relation by starting with any polynomial equation using induction principles. Studying the terms in the recurrence relation helps design of the matrix and the number arrangement. This book has shown arrangements for exact powers of two, three, four, and five. Higher powers are indeed conceivable in two or three dimensional space and could be a topic for further study. Number arrangements for equations with different polynomial degree are seen to differ in the rate of change between values at adjacent levels. These have been elaborated at various places in the book. The study of recurrence relations is then steered towards arrangements for multiplication tables and linear equations in two variables. When enumerated on a coordinate graph, linear equations are seen as planar surfaces in space, and also allow solving a system of such equations visually. Although intended for college or advanced high school level students, for the majority audience this book serves as a treatise on the beauty inherent in numbers.
- Sales Rank: #4319812 in Books
- Published on: 2013-04-29
- Original language: English
- Number of items: 1
- Dimensions: 10.00" h x .24" w x 8.00" l, .48 pounds
- Binding: Paperback
- 100 pages
Most helpful customer reviews
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Very Small Book but Rare Up to Date Snapshot of VISUAL RR's
By Let's Compare Options Preptorial
Recurrence relations used to focus most heavily on difference equations, with the most frequent modern application being the ubiquitous filtering aspect of digital signal processing. Historically, their beauty was seen in the best known of all RR's-- The Fibonacci numbers or sequence, used to model critter populations. You've probably seen pyramids, spirals and other forms of these sequences, and this author takes that tiny slice of this huge topic-- the graphical and "beauty" aspects of mathematical feedback series.
That was then. Now = time series in dynamical systems, RRs in algorithms, and compressed translation RRs in linear algebra matrices, vectors and tensors. The DE aspect of RRs is still used in some "rule of thumb" filter calculations by engineers, but much of that has been relegated to software running LA in the background.
This little monograph touches a little on many areas of RRs, but mostly those that can be expressed graphically with relative ease. You might argue that graphs are the essence of RRs from a topology point of view, but that is only one, albeit important, aspect of RRs in use today. Graphs and topology come in when you move from one dimensional sequences to grids in higher dimensions.
Don't expect detail on the recent "meat" of RRs-- eigenvalues, z-transforms, linear algebra, etc. as this is more of a brief survey of some "attractive" results rather than a reference or pedagogical work. Although the graphic logic can easily be applied to more traditional applications like population growth and econ, the author's purpose is not to teach those areas as much as demonstrate additional ways to "view" (literally) the whole idea of recurrence. Since most of the cutting edge research in topics like robotics and vision relate to filters and feedback, and given their usefulness in algorithms, understanding more about RR's is a good time investment. If you're a visual learner, there is definitely value in looking at what you normally would see as grids, matrices and sequences in more complete patterns. From that narrow but current viewpoint, this is excellent.
A note on format: This is more of a "coffee table" style book, 4 color, very thin (only 100 pages), large trim (custom 8 x 10) and print on demand. The author doesn't take a lot of time explaining anything about the equations-- he gives a brief intro, the equation, the solution, and lets you enjoy the 4-color pyramids, diamonds, bar charts, spheres etc. produced by the recurrence, with many examples showing "before and after" the recurrence effect. Although he touches on polynomials, the book is more organized by shape and form-- polygons, simple series, numeric (visual or graph) solutions to some simple linear equations, etc. Just know that this is mostly a "picture book" and you won't be disappointed. In fact, if you already grasp the basics, you'll be delighted!
I disagree with the publisher about a High School audience, UNLESS a HS math teacher, coach, parent or tutor explains the functions. This is more a second year undergrad exposition. If you're in HS or buying for a bright HS student, be aware that most of the functions will be difficult for that level if doing self study. With tutoring, however, the visual presentation is well worth the effort required to get what's going on with the solutions.
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