algebra   4734

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[1604.01760] Solving Diophantine Equations
In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation {\eta}({\pi}(x)) = {\pi}({\eta}(x)), where {\eta} is the Smarandache function and {\pi} is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and then we check all possible situations, and of course we retain among them only those solutions that verify our equation. In other words, we say that the equation does not have solutions in the search domain, or the equation has n solutions in this domain. This mode of solving is called partial resolution. Partially solving a Diophantine equation may be a good start for a complete solving of the problem. The authors have identified 62 Diophantine equations that impose such approach and they partially solved them. For an efficient resolution it was necessarily that they have constructed many useful tools for partially solving the Diophantine equations into a reasonable time. The computer programs as tools were written in Mathcad, because this is a good mathematical software where many mathematical functions are implemented. Transposing the programs into another computer language is facile, and such algorithms can be turned to account on other calculation systems with various processors.
algebra  number-theory  rather-interesting  books  nudge-targets  consider:looking-to-see  algorithms  numerical-methods 
3 days ago by Vaguery
[1202.4358] Natural Product Xn on matrices
This book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research level.
matrices  algebra  open-questions  rather-interesting  book  nudge-targets  consider:looking-to-see 
3 days ago by Vaguery
immersive linear algebra
The world's first linear algebra book with fully interactive figures.
math  algebra  linear  learning 
5 days ago by _rational
Neanderthal - Fast Native Matrix and Linear Algebra in Clojure
Fast native-speed matrix and linear algebra in Clojure

On the GPU: more than 1000x faster for large matrices than the fastest optimized Java libraries!

Works on AMD, Nvidia, and Intel hardware!

On the CPU: 10x - 60x faster than optimized Java libraries.
algebra  clojure  gpu  math  matrix 
5 days ago by jtth

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