By Edward Burger

2 DVD set with 24 lectures half-hour every one for a complete of 720 minutes...Performers: Taught via: Professor Edward B. Burger, Williams College.Annotation Lectures 1-12 of 24."Course No. 1495"Lecture 1. quantity conception and mathematical examine -- lecture 2. average numbers and their personalities -- lecture three. Triangular numbers and their progressions -- lecture four. Geometric progressions, exponential development -- lecture five. Recurrence sequences -- lecture 6. The Binet formulation and towers of Hanoi -- lecture 7. The classical thought of best numbers -- lecture eight. Euler's product formulation and divisibility -- lecture nine. The leading quantity theorem and Riemann -- lecture 10. department set of rules and modular mathematics -- lecture eleven. Cryptography and Fermat's little theorem -- lecture 12. The RSA encryption scheme.Summary Professor Burger starts with an outline of the high-level recommendations. subsequent, he presents a step by step clarification of the formulation and calculations that lay on the center of every conundrum. via transparent causes, pleasing anecdotes, and enlightening demonstrations, Professor Burger makes this exciting box of analysis available for an individual who appreciates the attention-grabbing nature of numbers. -- writer.

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**Example text**

What can we hear about a lattice? We have said that a lattice property is audible if it is determined by the theta function 8L(Z) = E qN(v), vEL where N (v) is the squared length of v. Now geometrically, the lattice determinant d is the square of the volume of the fundamental parallelotope. It follows that the number of lattice points inside a large ball will be roughly the volume of the ball divided by Vd, and so (since we can "hear" the number of v that have N(v) < R for any R): The determinant is an audible property.

When the titles for the lectures on which this book is based were chosen, this problem was still unsolved. By the time they were given, it had been solved by Gordon, Webb, and Wolpert, who made use of some previous work by Sunada and Buser. It is always exciting to see a classical problem solved, and in this case the solution can be made particularly easy, so although it has little to do with the main topic of these lectures, we give a simple solution to the Kac problem in this lecture. It is perhaps fortunate that the solution took so long to fmd, because the consideration of the problem has led to a lot of interesting mathematics.

The Dirichlet boundary condition is that cl> must vanish on each boundary segment. Using the reflection principle, this is equivalent to the assertion that cl> would change sign if continued as a smooth eigenfunction across any boundary segment, as also indicated in (e). In (0, we show how to obtain from cl> another eigenfunction of eigenvalue A, this time for the right-handed propeller. In the central tri- 48 THE SENSUAL (quadratic) FORM angle, we place the function A(x )+B(x )+C(x). Now we see from (e) that the functions A(x), B (x), C (x) continue smoothly across dashed lines into copies of the functions -d(x), - B (x), -b(x) respectively, so that their sum continues into - [d (x) + B (x) + b(x)] as shown.