Some Mathematical Gifts. Ron Rivest is one of the most famous computer scientists in the world. While he has done many important things in many aspects of computer theory, he is best known as the “R” in RSA. This is the renowned public key system based on the hardest of factoring created by Ron and Adi Shamir and Len Adleman. They won the Turing Award for this terrific achievement. Today I thought I would exchange some gifts with you. These are simple mathematical facts that are fun. One of the “gifts” is from Ron, and the others are from various places and people. In this spirit I hope you enjoy a small number of fun items that I view as gifts from mathematics to all of us. Gift I Daniel Kirsch’s gift. One of the coolest sites I have seen in years is this. And got this Very cool. (Let us not forget that TeX itself was a gift. Gift II Ron’s gift. Ron had a brilliant idea that help solve an important open problem called the Aanderaa-Karp-Rosenberg Conjecture.
Is a boolean function. . Or Call a boolean function . Define . Fixed-Odds Betting Arbitrage « Rod Carvalho. Proofs without words. This should really be a comment on Marco Radeschi's answer from Feb 22 involving the area formula for spherical triangles, but since I'm new here I don't have the reputation to leave comments yet. In reply to Igor's comment (on Marco's answer) wondering about an analogous proof for the area formula of hyperbolic triangles: there is one along similar lines, and you're rescued from non-compactness by the fact that asymptotic triangles have finite area. In particular, the proof in the spherical case relies on the fact that the area of a double wedge with angle α is proportional to α; in the hyperbolic case, you need to replace the double wedge with a doubly asymptotic triangle (one vertex in the hyperbolic plane and two vertices on the ideal boundary) and show that if the angle at the finite vertex is α, then the area is proportional to π−α.
(That picture is slightly modified from p. 221 of this book, which has the whole proof in more detail.) Stochastic Processes : An Elementary Introduction. Next: The Black-Scholes Equation Up: thesis Previous: Derivatives and Options Contents Subsections In this chapter, I will present the theory of stochastic processes in an elementary manner sufficient for understanding the theory presented in the following chapters. A reader interested in a more rigorous approach could consult Ross[4]. This chapter mostly follows Roepstorff[5] which has a more physical description of stochastic processes. This is probably the simplest example of a discrete time stochastic process. One way of defining it would be : which simply means that the variable (the subscript stands for the time) has an equal probability of increasing or decreasing by 1 at each time step. This process is both homogeneous (since the transition probability is only dependent on the distance between the initial and final points) and isotropic (the transition probability is independent of the direction of movement).
The random walk can be taken to be a Markov chain with a transition matrix. Ornstein–Uhlenbeck process. The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move back towards a central location, with a greater attraction when the process is further away from the centre. The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process. Representation via a stochastic differential equation[edit] An Ornstein–Uhlenbeck process, xt, satisfies the following stochastic differential equation: where and are parameters and denotes the Wiener process. The above representation can be taken as the primary definition of an Ornstein–Uhlenbeck process.[1][citation needed]. Fokker–Planck equation representation[edit] The probability density function ƒ(x, t) of the Ornstein–Uhlenbeck process satisfies the Fokker–Planck equation is and variance Application in physical sciences[edit]
. , the length Solution[edit] Lyapunov function. In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control theory. A similar concept appears in the theory of general state space Markov Chains, usually under the name Foster-Lyapunov functions. Informally, a Lyapunov function is a function that takes positive values everywhere except at the equilibrium in question, and decreases (or is non-increasing) along every trajectory of the ODE. The principal merit of Lyapunov function-based stability analysis of ODEs is that the actual solution (whether analytical or numerical) of the ODE is not required. Definition of a Lyapunov candidate function[edit] Let be a continuous scalar function. is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e. with , such that: So the new system of on.
Fuzzy sphere. In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is generated by spherical harmonics whose spin l is at most equal to some j. The terms in the product of two spherical harmonics that involve spherical harmonics with spin exceeding j are simply omitted in the product. -dimensional non-commutative algebra. The simplest way to see this sphere is to realize this truncated algebra of functions as a matrix algebra on some finite-dimensional vector space. That form a basis for the j dimensional irreducible representation of the Lie algebra SU(2)
. , where is the totally antisymmetric symbol with , and generate via the matrix product the algebra of j dimensional matrices. Where I is the j-dimensional identity matrix. See also[edit] Fuzzy torus Notes[edit] J. Mark Sapir's Homepage. My older daughter's web page is here. My sister's Web page is here. My mother's Web page is here. Here is how I looked like in 1980-1981 (I was a graduate student then). My Google scholar profile Me in the Russian Internet. Mathoverflow. Scientifically, I am one of about 105000 descendants of Nilos Kabasilas who was a bishop of Thessalonika, Greece, in the 14th century AD. Mark V. Non-commutative algebra: syntax and semantics. The book can be found here. Conferences International Conference on Geometric, Combinatorial and Dynamics aspects of Semigroup and Group Theory, June 11-14, 2013, Bar Ilan, Israel.
Teaching: In the Spring of 2014, I am teaching Math 194, Linear Algebra. Words of encouragement I often hear complains that I don't encourage my students enough. Linear Algebra WebNotes Click here to see the WebBook on Linear Algebra that I wrote in 1996-1997. Editorial boards. Stirling's approximation. The ratio of (ln n!) To (n ln n − n) approaches unity as n increases. The formula as typically used in applications is The next term in the O(ln(n)) is (1/2)ln(2πn); a more precise variant of the formula is therefore Being an asymptotic formula, Stirling's approximation has the property that Sometimes, bounds for rather than asymptotics are required: one has, for all so for all the ratio is always e.g. between 2.5066 and 2.7183.
Derivation[edit] The formula, together with precise estimates of its error, can be derived as follows. The right-hand side of this equation minus is the approximation by the trapezoid rule of the integral and the error in this approximation is given by the Euler–Maclaurin formula: where Bk is a Bernoulli number and Rm,n is the remainder term in the Euler–Maclaurin formula. Denote this limit by y. Where we use Big-O notation, combining the equations above yields the approximation formula in its logarithmic form: .
With an integral: An alternative derivation[edit] one gets then . Puzzles. Here are some mathematical puzzles that I have enjoyed. Most of them are of the kind that you can discuss and solve at a dinner table, usually without pen and paper. So as not to spoil your fun, no solutions are given on this page, but for some problems I have provided some hints. Picking the larger of two cards [Roger Wattenhofer told me this puzzle] Someone picks, at their will, two cards from a deck of cards. Enclosing land by fence pieces [I got this puzzle from Serdar Tasiran.] You are given one 44-meter piece of fence and 48 one-meter pieces of fence. Planar configuration of straight connecting lines [Radu Grigore told me this problem.
Given an even number of points in general positions on the plane (that is, no three points co-linear), can you partition the points into pairs and connect the two points of each pair with a single straight line such that the straight lines do not overlap? Reducing nearby enemies [I got this puzzle from Jason Koenig.] Transporting bananas Chomp Dropping eggs. Absolutely useless. Multifractals by Jan Röman. This work has been supported by the Swedish Natural Science Research Council, which is gratefully acknowledged. I also would like to thank my supervisor Prof. Stig Lundqvist for introducing me to the field of dynamical systems, chaos and fractals. My other supervisors Prof. Mats Jonson and Dr. Predrag Cvitanovic, is gratefully acknowledged for all their advice and discussions that came up under this work. I also want to thank Dr. Gunnar Nicklasson for introducing me to fractal aggregate and practical use of fractals and Dr.
One question one may ask about fractals is: Why do we need non-integer dimensions? A fractal object in three-dimensional space is characterized by a large surface-to-volume ratio. The theory of fractals radically differs from traditional Euclidean geometry, fractal geometry describes objects that are self-similar, or scale invariant. The simplest fractal is the Cantor bar (named after the 19th- century German mathematician Georg Cantor). 2.1: Measuring general sets.