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Arithmetic
Educator Paul Lockhart's goal is to demystify arithmetic: to bring the subject to life in a fun and accessible way, and to reveal its profound and simple beauty, as seen through the eyes of a modern research mathematician. The craft of arithmetic arises from our natural desire to count, arrange, and compare quantities. Over the centuries, humans have devised a wide variety of strategies for representing and manipulating numerical information: tally marks, rocks and beads, marked-value and place-value systems, as well as mechanical and electronic calculators. Arithmetic traces the history and development of these various number languages and calculating devices and examines their comparative advantages and disadvantages, providing readers with an opportunity to develop not only their computational skills but also their own personal tastes and preferences. The book is neither a training manual nor an authoritative history, but rather an entertaining survey of ideas and methods for the reader to enjoy and appreciate. Written in a lively conversational style, Arithmetic is a fun and engaging introduction to both practical techniques as well as the more abstract mathematical aspects of the subject.-- Provided by publisher.
Book
Motivic Euler Products and Motivic Height Zeta Functions
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes,
in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the
study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant
compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a
topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic
behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above
moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools
for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of
Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with
exponentials constructed using Denef and Loeser’s motivic vanishing cycles.
eBook
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.
eBook
VALUES OF ZETA FUNCTIONS OF ARITHMETIC SURFACES AT
We show that the conjecture of [27] for the special value at$s=1$of the zeta function of an arithmetic surface is equivalent to the Birch–Swinnerton–Dyer conjecture for the Jacobian of the generic fibre.
Journal Article
Fast fact division
\"Anyone who knows how to share equally knows how to divide. It's important to understand that division is more than a part of a math class; it's a math operation used every day in many ways. This valuable book reviews 'fast facts' of division, such as the zero property which states that zero divided by any number is zero. The accessible text and illustrative photographs help make concepts like this more tangible. Learners can test their understanding by completing the questions in the Math Mania! boxes throughout the text, which are accompanied by answer keys for self-assessment opportunities\"--Provided by the publisher.
Book
Global Smooth Solutions for the Inviscid SQG Equation
In this paper, we show the existence of the first non trivial family of classical global solutions of the inviscid surface
quasi-geostrophic equation.
eBook
Automorphisms of Riemann surfaces, subgroups of mapping class groups and related topics
Automorphism groups of Riemann surfaces have been widely studied for almost 150 years. This area has persisted in part because it has close ties to many other topics of interest such as number theory, graph theory, mapping class groups, and geometric and computational group theory. In recent years there has been a major revival in this area due in part to great advances in computer algebra systems and progress in finite group theory.This volume provides a concise but thorough introduction for newcomers to the area while at the same time highlighting new developments for established researchers. The volume starts with two expository articles. The first of these articles gives a historical perspective of the field with an emphasis on highly symmetric surfaces, such as Hurwitz surfaces. The second expository article focuses on the future of the field, outlining some of the more popular topics in recent years and providing 78 open research problems across all topics. The remaining articles showcase new developments in the area and have specifically been chosen to cover a variety of topics to illustrate the range of diversity within the field.
eBook