Several chapters from number theory

Elementary number theory in nine chapters

Some Facts from Number Theory

m ƒ qn‚ r , and 0 r < jnj. Not surprisingly, q is the quotient and r is the remainder when m is divided by n.1 We say that n divides m, or n is a divisor of m, when the remainder is zero. The notation n jm is used to signify that n divides m. An integer p is a prime number if 1 < jpj and the only divisors of p are 1 and p. We write a b „mod n… if b a is divisible by n.2 This notion, called cong...

Three Chapters of Measure Theory in Isabelle/HOL

Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to realvalued measures. We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as measure values. After defining the Lebesgue...

Trollope’s Chapters

The unit of the chapter – so fundamental to nineteenth-century prose narrative, yet so thoroughly ignored in accounts of the form of novels – is fundamental to Anthony Trollope’s work. The chapter stems from the textual practices of antiquity and of medieval biblical transmission, ultimately finding a home in the novel form. Chapters are key to Trollope’s understanding of readerly experience, i...

Bernoulli Number Identities from Quantum Field Theory

We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli ...