Number theory modular arithmetic stanford university. Anyone who has made a study of di erential equations will know that even supposedly elementary examples can be hard to solve. Sauloy proves the confluence of the socolled birkhoff matrix to the complex monodromy. Birkhoff, the generalized riemann problem for linear differential equations and the allied problems for linear difference and q difference equations, proc.
Local analytic classification of qdifference equations jeanpierre ramis, jacques sauloy, changgui zhang abstract. In particular, we generalize the rellichwittichtype theorem and malmquisttype theorem about differential equations to the case of qdifference differential equations system. This paper surveys recent applications of basic hypergeometric functions to partitions, number theory, finite vector spaces, combinatorial identities and physics. Di erence equations relate to di erential equations as discrete mathematics relates to continuous mathematics. Nicholas katz related some cases to deformation theory in 1972, in a paper where the conjecture was published. Formal and local analytic classification of qdifference. Linear qdifference equations request pdf researchgate.
Difference between algebra and arithmetic algebra vs arithmetic. It is easy to verify that is a linear transformation, and that this operation is inverse to the map 1. There is a galois theory for qdifference equations as well as for differen tial equations. This means that we have characterized the vector space. Real life example to explain the difference between algebra. Introduction to the arithmetic theory of quadratic forms 3 p q p p qp. In this paper we propose a global theory of q difference equations over a finite extension of the field of rational functions k q. What is the difference between arithmetic and algebra. Request pdf linear qdifference equations we prove that a linear. Arithmetic and geometricprogressions mctyapgp20091 this unit introduces sequences and series, and gives some simple examples of each. Modular arithmetic victor adamchik fall of 2005 plan 1. Mixed questions on arithmetic and geometric progression.
A sequence, or series, is a group of numbers that can be written in a particular order, or it can just be. Pdf the purpose of this paper is to develop the theory of ordinary, linear q difference. Arithmetic and algebra are two branches of mathematics. Difference equations that represent arithmetic sequences. On the other hand, algebra uses numbers and variables for solving problems. Grothendiecks conjecture on pcurvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction. Id like to give you an example of why we wanted clarification about what you meant by algebra and arithmetic, or the intended level of audience. C, nonzero, not a root of unity r is calgebra of functions with an action of qz. Number theory, in turn, is foundational for cryptography, the science of keeping ones communications and data secure from eavesdropping third parties. In an ap, the sum of the first 3 terms is 36 and that of the last 3 is 27. The general theory of linear qdifference equations jstor.
Derivation sum of arithmetic series arithmetic sequence is a sequence in which every term after the first is obtained by adding a constant, called the common difference d. This research monograph develops an arithmetic analogue of the theory of ordinary differential equations. The qanalogue of grothendieckkatzs conjecture on pcurvatures. It also applies a version of superrigidity particular to arithmetic groups. Division is notably absent from the above discussion. By using the nevanlinna theory of value distribution, we investigate the existence of solutions of some types of nonlinear qdifference differential equations. Before we begin, we must first define a couple of basic terms. It is equivalent to katzs conjectural description of the generic galois group. If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference. When i think of arithmetic, i think of my recent study of a course in arithmetic by serre. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference. Topic coverage includes numerical analysis, numerical methods, differential equations, combinatorics and discrete modeling.
If there are 10 terms, what are the 1 st term and the common difference respectively. Carmichael,the general theory of linear qdifference equations. A q analogue for difference equations has been proposed. We define a q difference analogue of the theory of gfunctions, answering to a question of y. Carmichael, the general theory of linear q difference equations, amer. Difference equations that represent geometric sequences. Arithmetic, being the most basic of all branches of mathematics, deals with the basic computation of numbers by using operations like addition, multiplication, division and subtraction. Analytic theory of linear q difference equations springerlink. The regular integers are visualized as lying on a number line, where integers to the left are smaller than integers on the right.
Arithmetic theory of q difference equations article pdf available in inventiones mathematicae 1503. Arithmetic differential equations american mathematical society. Arithmetic theory of q difference equations springerlink. Galois theory of fuchsian qdifference equations sciencedirect. Grothendiecks conjecture on pcurvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and. Pdf on ordinary, linear q difference equations, with applications. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Pdf floquet theory for qdifference equations researchgate. From arithmetic to algebra slightly edited version of a presentation at the university of oregon, eugene, or february 20, 2009 h. The existence of solutions of qdifferencedifferential equations.
A sequence, or series, is a group of numbers that can be written in. Arithmetic progression questions and answer hitbullseye. Apart from the stuff given above, if you want to know more about mixed questions on arithmetic and geometric progression. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Local analytic classification of qdifference equations. It is worth to be noted that the qdifference equations theory considered in this book is a. Divisibility and modular arithmetic are foundational to number theory, which is the theory of the integers. Difference equations, second edition, presents a practical introduction to this important field of solutions for engineering and the physical sciences. A sequence of real numbers, indexed by either z or n 0, is written in. It also explores particular types of sequence known as arithmetic progressions aps and geometric progressions gps, and the corresponding series.