Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to obtain for certain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory.
Kazuo Kondo (1911-2001) was Chair of the Department of Mathematical Engineering at the University of Tokyo, Japan. Over a period of 50 years, he and a few colleagues wrote and published a voluminous series of papers and monographs on the applications of analytical geometry within a diverse range of subjects in the natural sciences. Inspired by Otto Fischers attempt at a quaternionic unified theory in the late 1950s he adopted the mathematics of the revered Akitsugu Kawaguchi to produce his own speculative unified theory. The theory appears to successfully apply Kawaguchis mathematics to the full range of natural phenomena, from the structure of fundamental particles to the geometry of living beings. The theories are testable and falsifiable. Kondo and his theories are now almost completely unknown and this paper serves as the barest introduction to his work
This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the extreme, namely we present two proofs of the Extreme Value Theorem: the programmer proof that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values of the given function in a metric space setting, and an abstract space proof (the level-set proof) for semicontinuous functions defined on compact topological spaces. Next, in the intermediate part, we consider the Intermediate Value Theorem, generalize it to a wide class of discontinuous functions, and re-examine the meaning of the intermediate value property. The trek reaches the final frontier when we discuss the Uniform Continuity Theorem, generalize it, re-examine the meaning of uniform continuity, and find the optimal delta of the given epsilon. Have fun!
This is an English translation of the Latin original De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ signum negativum (1775). E596 in the Enestrom index. Let $chi$ be the nontrivial character modulo 4. Euler wants to know what $sum_p chi(p)/p$ is, either an exact expression or an approximation. He looks for analogies to the harmonic series and the series of reciprocals of the primes. Another reason he is interested in this is that if this series has a finite value (which is does, the best approximation Euler gets is 0.3349816 in section 27) then there are infinitely many primes congruent to 1 mod 4 and infinitely many primes congruent to 3 mod 4. In section 15 Euler gives the Euler product for the L(chi,1). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $sum_p frac{chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshevs estimate for the second Chebyshev function (summing the von Mangoldt function).
We discuss the history of the monodromy theorem, starting from Weierstrass, and the concept of monodromy group. From this viewpoint we compare then the Weierstrass , the Legendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in P SL(2,Z) and their monodromy. Then we focus on the birth of the concept of the Jacobian variety, and the geometrization of the theory of Abelian functions and integrals. We end illustrating the methods of complex analysis in the simplest issue, the difference equation $f(z) = g(z+1) - g(z)$ on $mathbb C$.
We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
We give another proof for [ sum_{n=1}^{infty}frac{1}{n^2}=frac{pi^2}{6} ] that basically follows from the theory of difference equations.
This is a typeset version of Alan Turings declassified Second World War paper textit{Paper on Statistics of Repetitions}. See the companion paper, textit{The Applications of Probability to Cryptography}, also available from arXiv at arXiv:1505.04714, for Editors Notes.
This is a typeset version of Alan Turings Second World War research paper textit{The Applications of Probability to Cryptography}. A companion paper textit{Paper on Statistics of Repetitions} is also available in typeset form from arXiv at arXiv:1505.04715. The original papers give a text along with figures and tables. They provide a fascinating insight into the preparation of the manuscripts, as well as the style of writing at a time when typographical errors were corrected by hand, and mathematical expression handwritten into spaces left in the text. Working with the papers in their original format provides some challenges, so they have been typeset for easier reading and access.
Poly-infix operators and operator families are introduced as an alternative for working modulo associativity and the corresponding bracket deletion convention. Poly-infix operators represent the basic intuition of repetitively connecting an ordered sequence of entities with the same connecting primitive.
