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E-raamat: Mathematical Mechanics: From Particle to Muscle [World Scientific e-raamat]

(Retired Teacher From Rockport High School, Ma, Usa)
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Mathematical Mechanics: From Particle to MuscleSuurem pilt
Teised raamatud teemal:
World Scientific e-raamatudRohkem infot World Scientific e-raamatute kohta
Raamatu kodulehekülg: http://www.worldscientific.com/worldscibooks/10.1142/7520#t=toc
This unprecedented book offers all the details of the mathematical mechanics underlying modern modeling of skeletal muscle contraction. The aim is to provide an integrated vision of mathematics, physics, chemistry and biology for this one understanding. The method is to take advantage of latest mathematical technologies Eilenberg-Mac Lane category theory, Robinson infinitesimal calculus and Kolmogorov probability theory to explicate Particle Mechanics, The Theory of Substances (categorical thermodynamics), and computer simulation using a diagram-based parallel programming language (stochastic timing machinery). Proofs rely almost entirely on algebraic calculations without set theory. Metaphors and analogies, and distinctions between representational pictures, mental model drawings, and mathematical diagrams are offered.AP level high school calculus students, high school science teachers, undergraduates and graduate college students, and researchers in mathematics, physics, chemistry, and biology may use this integrated publication to broaden their perspective on science, and to experience the precision that mathematical mechanics brings to understanding the molecular mechanism vital for nearly all animal behavior.
Acknowledgments xv
Introduction 1(20)
1 Introduction
3(18)
1.1 Why Would I Have Valued This Book in High School?
4(1)
1.2 Who Else Would Value This Book?
5(1)
1.3 Physics & Biology
6(1)
1.4 Motivation
7(3)
1.5 The Principle of Least Thought
10(1)
1.6 Measurement
11(1)
1.7 Conceptual Blending
11(2)
1.8 Mental Model of Muscle Contraction
13(2)
1.9 Organization
15(3)
1.10 What is Missing?
18(1)
1.11 "What is Original?
19(2)
Mathematics 21(124)
2 Ground & Foundation of Mathematics
23(52)
2.1 Introduction
23(3)
2.2 Ground: Discourse & Surface
26(10)
2.2.1 Symbol & Expression
27(1)
2.2.2 Substitution & Rearrangement
28(2)
2.2.3 Diagrams Rule by Diagram Rules
30(1)
2.2.4 Dot & Arrow
30(6)
2.3 Foundation: Category & Functor
36(5)
2.3.1 Category
38(2)
2.3.2 Functor
40(1)
2.3.3 Isomorphism
40(1)
2.4 Examples of Categories & Functors
41(28)
2.4.1 Finite Set
41(2)
2.4.2 Set
43(7)
2.4.3 Exponentiation of Sets
50(1)
2.4.4 Pointed Set
51(2)
2.4.5 Directed Graph
53(1)
2.4.6 Dynamic System
54(2)
2.4.7 Initialized Dynamic System
56(3)
2.4.8 Magma
59(1)
2.4.9 Semigroup
60(1)
2.4.10 Monoid
61(2)
2.4.11 Group
63(1)
2.4.12 Commutative Group
63(1)
2.4.13 Ring
64(1)
2.4.14 Field
65(1)
2.4.15 Vector Space over a Field
66(1)
2.4.16 Ordered Field
67(1)
2.4.17 Topology
68(1)
2.5 Constructions
69(6)
2.5.1 Magma Constructed from a Set
70(1)
2.5.2 Category Constructed from a Directed Graph
71(3)
2.5.3 Category Constructed from a Topological Space
74(1)
3 Calculus as an Algebra of Infinitesimals
75(36)
3.1 Real & Hyperreal
76(3)
3.2 Variable
79(3)
3.2.1 Computer Program Variable
79(1)
3.2.2 Mathematical Variable
79(1)
3.2.3 Physical Variable
80(2)
3.3 Right, Left & Two-Sided Limit
82(1)
3.4 Continuity
83(1)
3.5 Differentiable, Derivative & Differential
83(5)
3.5.1 Partial Derivative
86(2)
3.6 Curve Sketching Reminder
88(1)
3.7 Integrability
89(3)
3.8 Algebraic Rules for Calculus
92(7)
3.8.1 Fundamental Rule
92(1)
3.8.2 Constant Rule
92(1)
3.8.3 Addition Rule
92(1)
3.8.4 Product Rule
92(1)
3.8.5 Scalar Product Rule
93(1)
3.8.6 Chain Rule
93(1)
3.8.7 Exponential Rule
94(1)
3.8.8 Change-of-Variable Rule
94(1)
3.8.9 Increment Rule
94(1)
3.8.10 Quotient Rule
94(1)
3.8.11 Intermediate Value Rule
94(1)
3.8.12 Mean Value Rule
95(1)
3.8.13 Monotonicity Rule
95(1)
3.8.14 Inversion Rule
95(2)
3.8.15 Cyclic Rule
97(2)
3.8.16 Homogeneity Rule
99(1)
3.9 Three Gaussian Integrals
99(2)
3.10 Three Differential Equations
101(2)
3.11 Legendre Transform
103(3)
3.12 Lagrange Multiplier
106(5)
4 Algebra of Vectors
111(34)
4.1 Introduction
111(1)
4.2 When is an Array a Matrix?
112(1)
4.3 List Algebra
113(11)
4.3.1 Abstract Row List
114(1)
4.3.2 Set of Row Lists
114(1)
4.3.3 Inclusion of Row Lists
115(1)
4.3.4 Projection of Row Lists
115(1)
4.3.5 Row List Algebra
115(2)
4.3.6 Monoid Constructed from a Set
117(2)
4.3.7 Column List Algebra & Natural Transformation
119(3)
4.3.8 Lists of Lists
122(2)
4.4 Table Algebra
124(3)
4.4.1 The Empty and Unit Tables
124(1)
4.4.2 The Set of All Tables
124(1)
4.4.3 Juxtaposition of Tables is a Table
125(1)
4.4.4 Outer Product of Two Lists is a Table
126(1)
4.5 Vector Algebra
127(20)
4.5.1 Category of Vector Spaces & Vector Operators
128(1)
4.5.2 Vector Space Isomorphism
129(4)
4.5.3 Inner Product
133(1)
4.5.4 Vector Operator Algebra
134(1)
4.5.5 Dual Vector Space
135(2)
4.5.6 Double Dual Vector Space
137(1)
4.5.7 The Unique Extension of a Vector Operator
137(2)
4.5.8 The Vector Space of Matrices
139(1)
4.5.9 The Matrix of a Vector Operator
139(1)
4.5.10 Operator Composition & Matrix Multiplication
140(1)
4.5.11 More on Vector Operators
141(4)
Particle Mechanics 145(28)
5 Particle Universe
147(26)
5.1 Conservation of Energy & Newton's Second Law
149(1)
5.2 Lagrange's Equations & Newton's Second Law
150(2)
5.3 The Invariance of Lagrange's Equations
152(3)
5.4 Hamilton's Principle
155(5)
5.5 Hamilton's Equations
160(2)
5.6 A Theorem of George Stokes
162(1)
5.7 A Theorem on a Series of Impulsive Forces
163(1)
5.8 Langevin's Trick
164(1)
5.9 An Argument due to Albert Einstein
165(2)
5.10 An Argument due to Paul Langevin
167(6)
Timing Machinery 173(30)
6 Introduction to Timing Machinery
175(12)
6.1 Blending Time & State Machine
177(1)
6.2 The Basic Oscillator
178(1)
6.3 Timing Machine Variable
179(1)
6.4 The Robust Low-Pass Filter
180(1)
6.5 Frequency Multiplier & Differential Equation
180(1)
6.6 Probabilistic Timing Machine
181(1)
6.7 Chemical Reaction System Simulation
182(1)
6.8 Computer Simulation
183(4)
7 Stochastic Timing Machinery
187(16)
7.1 Introduction
187(5)
7.1.1 Syntax for Drawing Models
189(1)
7.1.2 Semantics for Interpreting Models
190(2)
7.2 Examples
192(1)
7.2.1 The Frequency Doubler of Brian Stromquist
192(1)
7.3 Zero-Order Chemical Reaction
193(12)
7.3.1 Newton's Second Law
194(1)
7.3.2 Gillespie Exact Stochastic Simulation
195(1)
7.3.3 Brownian Particle in a Force Field
196(7)
Theory of Substances 203(88)
8 Algebraic Thermodynamics
205(36)
8.1 Introduction
205(2)
8.2 Chemical Element, Compound & Mixture
207(2)
8.3 Universe
209(15)
8.4 Reservoir & Capacity
224(1)
8.5 Equilibrium & Equipotentiality
225(4)
8.6 Entropy & Energy
229(5)
8.7 Fundamental Equation
234(4)
8.8 Conduction & Resistance
238(3)
9 Clausius, Gibbs & Duhem
241(6)
9.1 Clausius Inequality
241(3)
9.2 Gibbs-Duhem Equation
244(3)
10 Experiments & Measurements
247(28)
10.1 Experiments
247(24)
10.1.1 Boyle, Charles & Gay-Lussac Experiment
247(4)
10.1.2 Rutherford-Joule Friction Experiment
251(1)
10.1.3 Joule-Thomson Free Expansion of an Ideal Gas
252(2)
10.1.4 Iron-Lead Experiment
254(4)
10.1.5 Isothermal Expansion of an Ideal Gas
258(2)
10.1.6 Reaction at Constant Temperature & Volume
260(1)
10.1.7 Reaction at Constant Pressure & Temperature
261(4)
10.1.8 Theophile de Donder & Chemical Affinity
265(3)
10.1.9 Gibbs Free Energy
268(3)
10.2 Measurements
271(4)
10.2.1 Balance Measurements
273(2)
11 Chemical Reaction
275(16)
11.1 Chemical Reaction Extent, Completion & Realization
279(2)
11.2 Chemical Equilibrium
281(4)
11.3 Chemical Formations & Transformations
285(1)
11.4 Monoidal Category & Monoidal Functor
286(3)
11.5 Hess' Monoidal Functor
289(2)
Muscle Contraction Research 291(36)
12 Muscle Contraction
293(34)
12.1 Muscle Contraction: Chronology
293(32)
12.1.1 19th Century
293(1)
12.1.2 1930-1939
293(1)
12.1.3 1940-1949
294(2)
12.1.4 1950-1959
296(3)
12.1.5 1960-1969
299(2)
12.1.6 1970-1979
301(3)
12.1.7 1980-1989
304(1)
12.1.8 1990-1999
305(6)
12.1.9 2000-2010
311(14)
12.2 Conclusion
325(2)
Appendices 327(26)
Appendix A Exponential & Logarithm Functions
329(2)
Appendix B Recursive Definition of Stochastic Timing Machinery
331(4)
B.1 Ordinary Differential Equation: Initial Value Problem
331(1)
B.2 Stochastic Differential Equation: A Langevin Equation without Inertia
332(1)
B.3 Gillespie Exact Stochastic Simulation: Chemical Master Equation
333(1)
B.4 Stochastic Timing Machine: Abstract Theory
334(1)
Appendix C MATLAB Code
335(12)
C.1 Stochastic Timing Machine Interpreter
335(3)
C.2 MATLAB for Stochastic Timing Machinery Simulations
338(1)
C.3 Brownian Particle in Force Field
339(5)
C.4 Figures. Simulating Brownian Particle in Force Field
344(3)
Appendix D Fundamental Theorem of Elastic Bodies
347(6)
Bibliography 353(10)
Index 363
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