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E-raamat: Architecture of Mathematics

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(Al-Farabi Kazakh National University, Department of Differential Equations and Control Theory, Almaty, Kazakhstan)
  • Formaat: 394 pages
  • Ilmumisaeg: 11-Aug-2020
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9780429893537
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Architecture of MathematicsSuurem pilt
  • Formaat: 394 pages
  • Ilmumisaeg: 11-Aug-2020
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9780429893537
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Architecture of Mathematics describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject.

Features











All notions and properties are introduced logically and sequentially, to help the reader gradually build understanding.





Focusses on illustrative examples that explain the meaning of mathematical objects and their properties.





Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research.





Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture.

Arvustused

"The author, a talented and broadly erudite mathematician presents an interesting view on the "building" of Mathematics. This book does not require a deep mathematical knowledge. He makes many fundamental mathematical concepts and their interrelations very easy to understand. Students, teachers, professional mathematicians, applied scientists using mathematical models in their research and everyone interested in mathematics can enjoy this book and can benefit from it."

Valerij Romanovskij, Univerza v Mariboru

Preface xi
Introduction xiii
Floor 1 Language
1(10)
Room 1.1 Alphabet
2(2)
Room 1.2 Syntax
4(1)
Room 1.3 Semantics
5(6)
Floor 2 SETS
11(36)
Room 2.1 Sets
12(7)
Room 2.2 Subsets
19(6)
Room 2.3 Set product
25(2)
Room 2.4 Correspondences
27(3)
Room 2.5 Relations
30(4)
Room 2.6 Operators
34(7)
Room 2.7 Equinumerosity
41(6)
Floor 3 Numbers
47(36)
Section I Cardinalities
49(7)
Room 3.1 Zero number
49(1)
Room 3.2 Natural numbers
50(6)
Section II Solutions
56(15)
Room 3.3 Integer numbers
56(8)
Room 3.4 Rational numbers
64(5)
Room 3.5 Algebraic numbers
69(2)
Section III Cuts
71(6)
Room 3.6 Real numbers
71(6)
Section IV Tuples
77(6)
Room 3.7 Complex numbers
77(2)
Room 3.8 Quaternions
79(4)
Floor 4 Objects
83(178)
Block A Ordered Objects
85(28)
Room 4A.1 Preordered sets
87(12)
Room 4A.2 Partially ordered sets
99(8)
Room 4A.3 Special ordered sets
107(6)
Block B Algebraic Objects
113(68)
Section I Operations
114(1)
Room 4B.1 Operations
114(3)
Section II Sets With Interior Composition Laws
117(1)
Subsection 1 Groupoids
118(1)
Room 4B.2 Groupoids
118(11)
Room 4B.3 Monoids
129(4)
Room 4B.4 Groups
133(15)
Subsection 2 Rings
148(1)
Room 4B.5 Rings
148(6)
Room 4B.6 Bodies and fields
154(4)
Subsection 3 Lattices
158(1)
Room 4B.7 Lattices
158(3)
Room 4B.8 Boolean algebras
161(3)
Section III Sets With Exterior Composition Laws
164(1)
Subsection 1 Groups with operators
165(1)
Room 4B.9 Modules
165(1)
Room 4B.10 Vector spaces
166(13)
Subsection 2 Rings with operators
179(1)
Room 4B.11 Algebras
179(2)
Section IV Universal Algebras
181(6)
Room 4B.12 Universal algebras
181(6)
Block C Topological Objects
187(45)
Section I Topological Spaces
189(1)
Room 4C.1 General topological spaces
190(9)
Room 4C.2 Determination of topological spaces
199(7)
Room 4C.3 Special topological spaces
206(14)
Section II Metric Spaces
220(1)
Room 4C.4 General metric spaces
220(6)
Room 4C.5 Special metric spaces
226(6)
Block D Measurable Objects
232(29)
Section I Measurable Spaces
233(1)
Room 4D.1 Measurable spaces
233(5)
Section II Measure Spaces
238(1)
Room 4D.2 Measures
238(15)
Room 4D.3 Integrals
253(8)
Floor 5 Composites
261(28)
Section I Mixed Structures
262(4)
Room 5.1 Consistence of structures
262(4)
Section II Topological Algebraic Objects
266(18)
Room 5.2 Topological groupoids
268(2)
Room 5.3 Topological groups
270(2)
Room 5.4 Topological vector spaces
272(3)
Room 5.5 Normed vector spaces
275(2)
Room 5.6 Banach spaces
277(3)
Room 5.7 Hilbert spaces
280(4)
Section III Applications
284(5)
Room 5.8 Derivatives
284(5)
Floor 6 Synthesis
289(72)
Section I Structures
290(17)
Room 6.1 Scale of sets
290(4)
Room 6.2 Structures
294(13)
Section II Categories
307(54)
Subsection 1 Categories
309(1)
Room 6.3 General categories
309(6)
Room 6.4 Non-concrete categories
315(4)
Room 6.5 Functors
319(12)
Subsection 2 Concepts
331(2)
Room 6.6 Special morphisms
333(4)
Room 6.7 Subobjects and quotient objects
337(5)
Room 6.8 Product of objects
342(3)
Room 6.9 Initial and terminal objects
345(3)
Subsection 3 Beginning
348(1)
Room 6.10 Ordered objects
348(6)
Room 6.11 Algebraic objects
354(5)
Room 6.12 Short addition
359(2)
Index 361
Simon Serovajsky is a professor of differential equations and control theory at al-Farabi Kazakh National University in Kazakhstan. He is the author of many books published in the area of optimisation and optimal control theory, modelling, philosophy and history of mathematics as well as a long list of high-quality publications in learned journals.
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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