05887nam 2200673Ia 450 991046188790332120220126212021.00-88385-955-6(CKB)2670000000205163(EBL)3330416(OCoLC)923220108(SSID)ssj0000577632(PQKBManifestationID)11376743(PQKBTitleCode)TC0000577632(PQKBWorkID)10562363(PQKB)11199582(UkCbUP)CR9780883859551(MiAaPQ)EBC3330416(Au-PeEL)EBL3330416(CaPaEBR)ebr10729387(OCoLC)929120334(EXLCZ)99267000000020516320001102d2000 uy 0engur|n|---|||||txtccrThe geometry of numbers[electronic resource] /C.D. Olds, Anneli Lax, Giuliana P. DavidoffWashington, DC Mathematical Association of Americac20001 online resource (193 p.)The Anneli Lax new mathematical library ;v. 41Description based upon print version of record.0-88385-643-3 Includes bibliographical references and index.""Cover ""; ""Title Page""; ""Contents""; ""Preface""; ""Part I Lattice Points and Number Theory""; ""1 Lattice Points and Straight Lines""; ""1.1 The Fundamental Lattice""; ""1.2 Lines in Lattice Systems""; ""1.3 Lines with Rational Slope""; ""1.4 Lines with Irrational Slope""; ""1.5 Broadest Paths without Lattice Points""; ""1.6 Rectangles on Paths without Lattice Points""; ""Problem Set for Chapter 1""; ""References""; ""2 Counting Lattice Points""; ""2.1 The Greatest Integer Function, [x ]""; ""Problem Set for Section 2.1""; ""2.2 Positive Integral Solutions of ax + by = n""""Problem Set for Section 2.2""""2.3 Lattice Points inside a Triangle""; ""Problem Set for Section 2.3""; ""References""; ""3 Lattice Points and the Area of Polygons""; ""3.1 Points and Polygons""; ""3.2 Pick's Theorem""; ""Problem Set for Section 3.2""; ""3.3 A Lattice Point Covering Theorem for Rectangles""; ""Problem Set for Section 3.3""; ""References""; ""4 Lattice Points in Circles""; ""4.1 How Many Lattice Points Are There?""; ""4.2 Sums of Two Squares""; ""4.3 Numbers Representable as a Sum of Two Squares""; ""Problem Set for Section 4.3""""4.4 Representations of Prime Numbers as Sums of TwoSquares""""4.5 A Formula for R(n)""; ""Problem Set for Section 4.5""; ""References""; ""Part II An Introduction to the Geometry of Numbers""; ""5 Minkowski's Fundamental Theorem""; ""5.1 Minkowski's Geometric Approach""; ""Problem Set for Section 5.1""; ""5.2 Minkowski M-Sets""; ""Problem Set for Section 5.2""; ""5.3 Minkowski's Fundamental Theorem""; ""Problem Set for Section 5.3""; ""5.4 (Optional) Minkowski's Theorem in n Dimensions""; ""References""; ""6 Applications of Minkowski's Theorems""; ""6.1 Approximating Real Numbers""""6.2 Minkowski's First Theorem""""Problem Set for Section 6.2""; ""6.3 Minkowski's Second Theorem""; ""Problem for Section 6.3""; ""6.4 Approximating Irrational Numbers""; ""6.5 Minkowski's Third Theorem""; ""6.6 Simultaneous Diophantine Approximations""; ""Reading Assignment for Chapter 6""; ""References""; ""7 Linear Transformations and Integral Lattices""; ""7.1 Linear Transformations""; ""Problem Set for Section 7.1""; ""7.2 The General Lattice""; ""7.3 Properties of the Fundamental Lattice""; ""Problem Set for Section 7.3""; ""7.4 Visible Points""""8 Geometric Interpretations of Quadratic Forms""""8.1 Quadratic Representation""; ""8.2 An Upper Bound for the Minimum Positive Value""; ""8.3 An Improved Upper Bound""; ""8.4 (Optional) Bounds for the Minima of Quadratic Formsin More Than Two Variables""; ""8.5 Approximating by Rational Numbers""; ""8.6 Sums of Four Squares""; ""References""; ""9 A New Principle in the Geometry of Numbers""; ""9.1 Blichfeldt's Theorem""; ""9.2 Proof of Blichfeldt's Theorem""; ""9.3 A Generalization of Blichfeldt's Theorem""; ""9.4 A Return to Minkowski's Theorem""""9.5 Applications of Blichfeldt's Theorem""The Geometry of Numbers presents a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice-points on lines, circles, and inside simple polygons in the plane. Little mathematical expertise is required beyond an acquaintance with those objects and with some basic results in geometry. The reader moves gradually to theorems of Minkowski and others who succeeded him. On the way, he or she will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres. An appendix by Peter Lax gives a lovely geometric proof of the fact that the Gaussian integers form a Euclidean domain, characterizing the Gaussian primes, and proving that unique factorization holds there. In the process, he provides yet another glimpse into the power of a geometric approach to number theoretic problems.Anneli Lax new mathematical library ;v. 41.Geometry of numbersNumber theoryElectronic books.Geometry of numbers.Number theory.512/.75Olds C. D(Carl Douglas),1912-1979.1074126Lax Anneli42255Davidoff Giuliana P622037MiAaPQMiAaPQMiAaPQBOOK9910461887903321The geometry of numbers2572438UNINA02266nam 2200385 450 991068842690332120230629173615.0(CKB)5400000000040783(NjHacI)995400000000040783(EXLCZ)99540000000004078320230629d2020 uy 0engur|||||||||||txtrdacontentcrdamediacrrdacarrierThermosoftening plastics /edited by Gülşen Akın Evingür, Önder Pekcan, Dimitris S. AchiliasLondon :IntechOpen,[2020]©20201 online resource (124 pages) illustrations1-83880-614-8 Includes bibliographical references.Thermosoftening Plastics are polymers that can be manipulated into different shapes when they are hot, and the shape sets when it cools. If we were to reheat the polymer again, we could re-shape it once again. Modern thermosoftening plastics soften at temperatures anywhere between 65 oC and 200 oC. In this state, they can be moulded in a number of ways. They differ from thermoset plastics in that they can be returned to this plastic state by reheating. They are then fully recyclable because thermosoftening plastics do not have covalent bonds between neighbouring polymer molecules. Methods of shaping the softened plastic include: injection moulding, rotational moulding, extrusion, vacuum forming, and compression moulding. The scope of this book covers three areas of thermosoftening plastics, thermoplastic materials, and their characterization. The following tests are covered in the book: thermal analysis (differential scanning calorimetry, heat deflection temperature test), optical properties tests (fluorescence spectroscopy, UV spectroscopy), and mechanical properties tests (thermogravimetry, rheometry, short term tensile test).ThermoplasticsThermoplastics.668.423Evingur Gulsen AkinPekcan ÖnderAchilias Dimitris S.NjHacINjHaclBOOK9910688426903321Thermosoftening Plastics2211032UNINA