devil s staircase in number theory

... functions 1.1 Introduction Additive number theory is concerned with expressing an integer n as a sum of integers from some given set S For example, S might consist of primes, squares, cubes, or ... Analysis ANDERSON/FULLER Rings and Categories of Modules GOLUBITSKY/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure ... constant must be — 140G, and this proves the theorem LÌ 1.9 The Eisenstein series and the invariants is called the Eisenstein series of order n The invariants g, and g3 are the numbers defined

Ngày tải lên: 11/05/2018, 16:46

... fails for at least half of the possible bases b Strong pseudoprimes We now discuss one more type of primality test, which is in one respect even better than the Solovay-Strassen test based on ... integers h and k We now find y - equivalently, s - for which this time estimate is minimal Since r, the number of bits in n, is fixed, this means minimizing Trang 28( r I s Y Is eks with respect ... n = pq in the RSA cryptosystem Remark In many primality tests, if a composite n happens to pass some initial test and then fails a subsequent test, one not only learns that n is composite, but

Ngày tải lên: 30/01/2020, 12:09

... optimistic about the possibilities for knapsack cryptosystems Since the problem of breaking the system is in a very difficult class of problems ( NP-complete problems) , they reasoned, the system should ... analytic number theory, increasing use has been made of algebraic number theory ( primality testing with Gauss and Jacobi sums, cryptosystems based on quadratic fields, the number field sieve) ... possible, so you want to find some subset of the k items that exactly fills the knapsack In other words, you want to find some subset I C { 1, , k } such that E iEI Vi = V, if such a subset exists

Ngày tải lên: 30/01/2020, 12:43

... analytic number theory, increasing use has been made of algebraic number theory (primality testing with Gauss and Jacobi sums, cryptosystems based on quadratic fields, the number field sieve) and ... theoretical research papers on certain types of number theory In part it is the dramatic increase in computer power and tion that has influenced some of the questions being studied by number theorists, ... trying some of the less familiar exercises Depending on the students' background, it should be possible to cover most of the first five chapters in a semester Alternately, if the book is used in

Ngày tải lên: 15/09/2020, 13:08

... skim through the first two chapters, perhaps trying some of the less familiar exercises Depending on the students' background, it should be possible to cover most of the first five chapters in ... other cases as well, especially when we're using the binary system (b = 2), if the choice of base is clear from the context Since it is sometimes useful to work in other bases than 10, one should ... automorphism, 30, 35 ciphertext, 53 classical cryptosystem, 85 coin toss, 87-88, 93, 192-193 complex numbers, 16 Gaussian integers, 16, 41, 155 composite number, 10 composition of cryptosystems, 63,

Ngày tải lên: 15/09/2020, 13:14

... some given set S For example, S might consist of primes, squares, cubes, or other special numbers We ask whether or not a given number can be expressed as a sum of elements of S and, if so, in ... permission ofthe publisher (Springer Science+Business Media, LLC), ехсер! for brief excerpts in connection with reviews or scholarly analysis Use in connection with anу form of information stora- ... Theory Series in Number Theory 9 HUMPHREYS Introduction to Lie Algebras 2nd ed and Representation Theory 42 SERRE Linear Representations of Finite 10 COHEN A Course in Simple Homotopy Groups Theory

Ngày tải lên: 15/09/2020, 13:23

... from some given set S For example, S might consist of primes, squares, cubes, or other special numbers We ask whether or not a given number can be expressed as a sum of elements of Sand, if so, in ... Dedekind's functional equation from Iseki's formula 3.7 Properties of Dedekind sums 3.8 The reciprocity law for Dedekind sums 3.9 Congruence properties of Dedekind sums 3.10 The Eisenstein series ... Weierstrass IJ-function, Eisenstein series of order n, n ~ 3, Eisenstein series of order 2, invariants, values of IJ at the half-periods, discriminant 9~ - 279~ , upper half-plane Im(T) > 0, Klein's

Ngày tải lên: 15/09/2020, 13:24

... letters  since still has much of plaintext structure 18 Polyalphabetic Ciphers  polyalphabetic substitution ciphers  improve security using multiple cipher alphabets  make cryptanalysis harder ... tables of single, double & triple letter frequencies for various languages 10 Use in Cryptanalysis • key concept - monoalphabetic substitution ciphers not change relative letter frequencies • discovered ... hear one person speaking out of hubbub of many, again because of redundancy in aural Language Redundancy and Cryptanalysis • This redundancy is also the reason we can compress text files, the computer

Ngày tải lên: 20/09/2020, 13:57

... end find GCD value and also x & y • if GCD(a,b)=1 these values are inverses Trang 48Finding Inverses• An important problem is to find multiplicative inverses in such finite fields • Can show ... that such inverses always exist, & can extend the Euclidean algorithm to find them as shown • See text for discussion as to why this works Trang 50Inverse of 550 in GF(1759)• Example showing ... pursued the science of numbers with keen interest and astonishing intelligence. — Living with Birds, Len Howard Trang 6• Finite fields have become increasingly important in cryptography • A number

Ngày tải lên: 20/09/2020, 13:59

... some given set S For example, S might consist of primes, squares, cubes, or other special numbers We ask whether or not a given number can be expressed as a sum of elements of S and, if so, in ... permission ofthe publisher (Springer Science+Business Media, LLC), ехсер! for brief excerpts in connection with reviews or scholarly analysis Use in connection with anу form of information stora- ... Theory Series in Number Theory 9 HUMPHREYS Introduction to Lie Algebras 2nd ed and Representation Theory 42 SERRE Linear Representations of Finite 10 COHEN A Course in Simple Homotopy Groups Theory

Ngày tải lên: 26/05/2022, 14:31

... is in S Thus S must be the set ofall positive integers Thus T must be the set of all positive integers since S is asubset of T Trang 131.3 DIVISIBILITY AND THE DIVISION ALGORITHM 13Exercises1 ... building blocks of the theory of numbers This chapter containssomewhat very simple and obvious observations starting with properties of inte-gers and yet the proofs behind those observations are ... positiveintegers As a result, there are some integers that are not contained in S and thusthose integers must have a least element α by the well ordering principle Noticethat α 6= 1 since 1 ∈ S

Ngày tải lên: 07/05/2018, 13:48

... orbites ferm´es des flots g´od´siques e e e e et hyperboliques La troisi`me partie s intitule “Syst`mes dynamiques: Echanges d’intervalles, e e Surfaces plates et Petits diviseurs” Les le¸ons d”Anton ... Born-Infeld interactions of massless strings to their couplings to D branes One seminar by Katrin Wendland concludes this part: “Superconformal field theories associated to very attractive quartics” ... complexes d´finies sur des doe maines sym´triques et invariantes par des groupes arithm´tiques, elles ape e paraissent fr´quemment en th´orie des cordes Boris Pioline a d´velopp´ son e e e e s minaire...

Ngày tải lên: 12/02/2014, 16:20

... viewpoint of classical mechanics these models are the best known examples of classical chaos, their spectral statistics are close to the Poisson statistics typical for integrable models The reason ... Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, ... matrices into non-overlapping classes of conjugated elements the summation should be performed over matrices S such that no two of them can be represented as S2 = sS1 and s belongs to Sg This is...

Ngày tải lên: 12/02/2014, 17:20

... UNDERLYING STRUCTURE The Residue Classes as an Invention The Residue Classes as an Invention The Residue Classes as a Tool The Residue Classes as a Tool The Residue Classes as a Group The Residue Classes ... Conjectures and their Interrelations Splitting the Primes into Equinumerous Classes Primes into Equinumerous Classes Splitting Euler 's Criterion Formulated Euler 's Criterion Formulated Euler 's Criterion ... most immediate thing found in Gauss 's book was not one of these new things; it was a new way of looking a t the old things By this new way we mean the residue classes Gauss begins on page as...

Ngày tải lên: 15/03/2014, 16:03

... cryptosystems for sending messages tend to be slower to implement than the clasical systems that are in current use The number of plaintext message units per second that can be transmitted is less ... of messages would then be sent by the faster, older methods Probabilistic Encryption Most of the number theory based cryptosystems for message transmission are deterministic, in the sense that ... notions Cryptography is the study of methods of sending messages in disguised form so that only the intended recipients can remove the disguise and read the message The message we want to send is...

Ngày tải lên: 31/03/2014, 16:20

... Summary of Results 14.2 A Recursion Formula 14.3 Sums of Two Squares 14.4 Sums of Four Squares 14.5 Sums of Six Squares 14.6 Sums of Eight Squares 14.7 Sums of Ten Squares 14.8 Notes ... proofs of some deep results in number theory We give the Erd s- Selberg proof of the prime number theorem, o Linnik s solution of Waring s problem, Liouville s still mysterious method to obtain ... previously appeared in a text Indeed, many results in this book are new Number theory is an ancient subject, but we still cannot answer the simplest and most natural questions about the integers Important,...

Ngày tải lên: 31/03/2014, 16:21

... prolate spheroidal coordinates is obtained by setting ξ = cosh α and η = cos β so that in terms of Cartesian coordinates x= d d d sinh α sin β cos ϕ, y = sinh α sin β sin ϕ, z = cosh α cos β 2 ... oblate spheroidal coordinates is obtained by setting ξ = sinh α and η = cos β so that in terms of Cartesian coordinates x= d d d cosh α sin β cos ϕ, y = cosh α sin β sin ϕ, z = sinh α cos β, 2 ... purposes The smoothness of the surface, including the presence of singularities such as corners or conical tips, is important in considering the existence and uniqueness of solutions This topic...

Ngày tải lên: 17/03/2014, 14:41

... h`DIdt d `cR S Ô S c xd H g S a S S D d Gpty c S C S a h dpB d H e n0ă H y FD F B t y DrH d w sf P X B `cR S Ô S c yvw S V S A Gwôm X GD nX vGl` d I`BR d aB X y S D d Gc X ` ... I  h { o  h y x Ô ƠhƠhCâ agppsyp !s W f{}gSeT S dqxằxs~}0xx~ắ||qeiĂ sqW}p UẽÔu0~Uabagâ GI S `XX}VQTg {S ppsyq}~|TG!xx{} Tsắq}!{ {nax pQ| ~ ebx i S R bg S  e x Ă c | Ê x Y ~| ... mX S I E S I `cR X c D r `dQ`cR E S I d`Bj AI E GGqDR e S m X ` S I V  V E S I X `cGc uI S B d DBH X RB Hy R w I F R dg X ` S I X h j ÊbH  f `HWaD "r `IB X bH B X B d `cR S C S I...

Ngày tải lên: 31/03/2014, 16:22

... business' In particular, congruence theory has been used in constructing perpetual calendars, scheduling round-robin tournaments, splicing telephone cables, devisin g systematic methods for storing ... Nelson & Slowinski, 197 Slowinski 198 Colquitt & Welsh 198 Slowinski, 198 Slowinski 198 Slowinski & Gage, 199 Slowinski & Gage 199 Slowinski & Gage, 199 Armengaud & Woltman et al (GIMPS) ... practising law in Toulouse, and producing hig h quality work in number theory and other areas of mathematic s as a hobby He published almost nothing revealing most of his results in his extensive...

Ngày tải lên: 31/03/2014, 16:22