Suggestions for non-euclidean geometry (analytical) books?

  • Thread starter Dj Sneaky Whiskers
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  • #1
Dj Sneaky Whiskers
1089 and all that. A Journey into Mathematics.

David Acheson

Oxford University Press (www.oup.com)

ISBN 0-19-851623-1

Introduction: You like maths don't you? Of course you do, because you would be making haste to get out of my sight before I damn your eyes with a strategically and excruciatingly placed cigarette if you didn't. That and you're reading the review section for Maths books, so you obviously like it. In fact, I bet you love it. Maybe you can't get enough of it! Well, if you love it so much, prove it. Yeah, you read right. Prove it. Maths doesn't want you swanning into its life, fumbling with its unmentionables for a couple of hours and then feverishly bragging to your friends in sordidly candid terms about how you sorted out a right tasty looking differential equation the other night. Maths is too classy for that, and don't need no playah who's just looking for a casual ego boost. It wants a hero, not a zero...well, it does most definitely want a zero but...you know what I mean!

But how does one snuggle with maths? Simple, One reads a book like this. A charming, lighthearted overview of the subject of Mathematics, lightly peppered with sparkling anecdote and delivered by the amiable voice of a razor sharp mind.

Audience: Anyone and everyone with a genuine interest in Mathematics, from high school students to graduates and even crusty old dons (should any be reading this forum).

Pros: Charming, witty, and conversational in tone without ever once faltering from presenting major topics in mathematics in an understandable and occassionally impressively detailed light. An absolute gorgeous little tome about no higher than the length of an average sized hand and only 178 pages in length.

Cons: There are no cons to snuggling, ever. Especially when it's with mathematics.

Conclusion: This book is intended as a gentle introduction to the major topics in Mathematics, from elementary geometry to chaos and catastrophe. Even though the introduction is indeed gentle, it sometimes amazes to see how much maths is actually discussed in such a small and informal book. For high school students thinking of studying maths I would make it compulsory reading. If you're an undergraduate, graduate, etc, you're not going to discover anything new, but, lets face it, how many times do you get the chance to enjoy reading about mathematics? Yeah, yeah, I'm sure you "Found 'Principles of Mathematical Analysis' deeply stimulating". But you didn't 'enjoy' it.

In short, it's somewhat akin to attending that general lecture that we all hoped for whilst making our way to university. You know what I'm talking about: the oak panelling; the smell of chalk thick in the air; the jovial, red cheeked, tweed clad, and generally 'hearty' in manner (as well as in circumfrence) professor slipping in and out of anecdote with seemingly practiced ease as he navigates his way through the subject matter.

It also explains the indian rope trick, sort of.

Disclaimer: The above does not represent a description of the author of this book. I have never met the man, but, if I did, I'm sure I would be compelled to comment on his youthful and thoroughly virile appearance. Indeed, I suspect I would find myself even a little attracted to him and would afterwards have to spend hours recouperating on the sofa and contemplating what implications this incident held for my sexuality.
 

Answers and Replies

  • #2
Dj Sneaky Whiskers
A Mathematician's Apology

G.H. Hardy

Cambridge University Press

ISBN 0-521-42706-1


INTRODUCTION: Although this isn't strictly a Mathematical text, it is a book on the subject of Mathematics, or, more specifically, the thoughts of an eminent Mathematician, then at the end of his career, on the subject of his chosen field. G.H. Hardy was the 'purest of the pure' of Mathematicians, concentrating upon Number Theory and Analysis, despite having (in collaboration with Weinberg) managed to establish the mathematical basis for population genetics. Following a career, during which he became a fellow of the Royal Society and served as president of the London Maths Society, he wrote his 'apology'.

It serves as a rare insight into the mind of such a figure (at least in the world of Mathematics); describing his view of the significance of Mathematics, the distinctions between pure and applied Maths, and whether or not Mathematics is a worthwhile preoccupation, amongst other topics.

PROS/CONS: It would be unfair to the author to discuss pros and cons, being, as 'A Mathematician's Apology' is, a largely personal account. At most it can be said that some will enjoy the narrative style (being that of a classically educated Cambridge man of the Edwardian generation in the 1940's), whilst some will undoubtably not.

CONCLUSION: A highly enjoyable account of Mathematics. Both witty and at times poignant, such a book serves as a useful reminder to the more...over-zealous...of undergraduates that Maths is ultimately a human endeavour which, in practice, is at times more akin to a creative act than the performance of soulless calculation. It was not simply out of a desire to flatter that Graham Greene called it 'the best account of what it is like to be a creative artist'.
 
  • #3
Dj Sneaky Whiskers
Philosophy of Mathematics

An Introduction to the World of Proofs and Pictures

Author: James Robert Brown

Publisher: Routledge

ISBN: 0-415-12275-9

---------------------------------------------------------------------

Rambling Preamble:

Now, gentle reader, I know you're interested in the Philosophy of Mathematics. How do I know this? Have I been rifling through your things? Maybe. Have I been stealing your underwear whilst rifling through your things? Quite possibly, and, if you're interested, it's rather a snug fit. But that is not how I know you're interested in what this book has to offer. I know because you're interested in Mathematics, otherwise you wouldn't be looking in this section. If you're interested in Mathematics, you are undoubtably interested in the philosophy of Mathematics.

I stress that there is no possibility here of me being wrong, no matter what you might think. If, against all that is good and decent in the world, you are not interested in this then you are not interested in Mathematics at all! You're merely a calculator, interested in calculation! Interested in little but the execution of routine algorythms and the manipulation of data! We have no need for you, we have computers, with flashing lights and everything, to do that for us! Get out, get out I say, nay, demand! Leave the sanctity of this place, head into the woods and offer yourself to the wolves. I might add that you had better pray that God has mercy upon your soul, for you shall find none here.

Anyway, now that we have gotten the pleasantries out of the way, I shall continue.

Mathematics and Philosophy are inseparable. When we wish to prove something, it is necessary to first ask "What constitutes 'proof'?", this, in turn, entails that we have some concept of 'certainty'. Unlike in the physical sciences, in Mathematics there are no 'approximations' to the 'truth' as there are in Physics (i.e. approximating the magnitude of actual forces acting between planets, using Newton's Law of Gravitation). A Mathematical statement is either correct, or incorrect. How we determine whether or not such a statement is correct, and how we can be certain in either case are questions that have had continued relevence in Mathematics since the Classical era (a contemporary issue being what role computers have in the derivation of proofs). It is this rigour of definition, and the concept of a 'Mathematical Reality' (such a reality is the only place where a circle can be found, such an object does not exist either in the 'mind's eye' or in physical reality, it exists only as a mathematical defintion of an object in geometry) that sets Mathematics apart from other subjects and prevents it from being little but a useful tool for the Sciences.

Introduction:

This book is, I would say, a 'first text' introduction and sourcebook to and for the study of the Philosphy of Mathematics. It describes and comments upon various matters, including: the meaning of a 'definition' in Mathematics; Computation and Proof, different approaches (constructivism etc.); the role of 'picture proofs', the connection with 'Applied Mathematics'; as well as discussing the contributions made by Philosophers such as Wittgenstein, Frege, Kant and Plato to the debate surrounding proof, certainty and Mathematical reality.

Pros: A succinct, yet comprehensive introduction to the major issues in the Philosophy of Mathematics. A 'conversational' and entertaining prose that occassionally discusses practical applications of Mathematics and their implications (such as the sometimes objectionable use of Mathematical concepts in the social sciences). Such is the variety of subject matter and the skill with which it is written, there is rarely a dull moment in my humble, and quite possibly dull opinion. Happily, a great many prominant philosophers are mentioned, often with quotes, allowing many a gleefull hour of name dropping to be had with your contempories.

Cons: Whilst the passion with which the author writes helps to carry the reader through subjects that he/she may otherwise find dry and inaccessible, without ever being overbearing, this really does represent one point of view. Although in many areas this may be of little consequence, in others, such as the discussion of Platonism and the potential of picture-proofs, it may be beneficial to suspend judgement until a wider collection of views have been read (that's not to say that I disagree with Mr Brown's conclusions).

Cover: Yes, this issue now has its very own section. The cover is, quite frankly, amazing. This book has the equivalent of high art, cinematic sex with the 1812 overture as its soundtrack wrapped around its pages. Compared with the banal, middle aged, self loathing marital romp that can be symbolised by most Springer publication covers, and the grubby, pedestrian sub soft-porn fumble of most Mathematical texts' covers (oh look! a roller coaster with superimposed, relevant equation, bravo.), this cover truly is a wonder. Want to know why? Because Routledge have it on the money as far as covers are concerned. A bookcase of Routledge is as beautiful to the eye as the content of the books themselves is beautiful to the mind.

Conclusion: Philosophy of Mathematics, an Introduction to the World of Proofs and Pictures, is an excellent book with which to become acquainted with the subject. Regardless of whether or not it acts as a catalyst to a growing interest in the Philosophy of Mathematics, or represents the reader's first and last encounter with such ideas as are contained within, reading it can only prove a beneficial and, thanks to the style adopted throughout, an enjoyable experience to those who are interested in Mathematics beyond the mundane matter of plugging in numbers and rearranging ever more complex equations. Having said that, it is not, and does not pretend to be the definitive word on the matter, but, thanks to a comprehensive bibliography, could easily be used as a springboard for those who wish to further their interest beyond the book's limitations (a personal recommendation is 'Philosophy of Mathematics, an Anthology', edited by Dale Jacquette and published by Blackwell Philosophy Anthologies).
 
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  • #4
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Hey.

In a few weeks I will start whit a 7.5 point cours in MathPhycis. We will cover something like this topics:
A. Representation theory: A1. Fouriertransformation. A2. SO(3) and SU(2) there Lie-algebraer. A3. Tensorproduckt
B. Quantum Mechanics: B1. The free Laplace operator, momentum representation, domain question. B2. Schrødinger operatoren for the free harmoniske oscillator. B3. Rotational symmetrically potentiel and the hydrogen atom
C.
differential forms on Rn: C1. Outer produckt. closed forms, and *-operationen. C2. Exampels.

The above translated from danish, just fast, sry for any mistakes.

The questions is now if some of you can recommend a book that covers all, or some, of the abov topics.

/Stefan
 
  • #5
Hello.

I'm doing my senior presentation on analytical (not axiomatic) hyperbolic geometry on the unit disk model (Poincaré disk), and I haven't had any courses directly pertaining to this topic. I won't be doing this until next fall, but I need to do as much reading on the subject as possible in the mean time.

Can anyone suggest some good books? The library at my uni (surprisingly) had virtually nothing on analytical non-euclidean geometry. They had plenty books on an axiomatic approach, but that's not what I'm looking for. After my advisor and I scoured the library for anything on this topic, he lent me a couple of his books, which have been helpful to get me started. These books are:

Modern Geometries - The Analytic Approach by Michael Henle.

The Poincaré Half-Plane - A Gateway to Modern Geometry by Saul Stahl.

The latter, as the title denotes, deals with the half plane, and has limited content as far as the disk model goes. The former has been helpful, but I could use something to supplement it. I would very much appreciate it if anyone could offer some suggestions.
 
  • #6
Hi Everyone,

I'm a physics graduate student (going for my masters) currently taking math physics looking for useful books on math physics. I really want to find books/websites that shows the step by step process of how to do for example PDE's, ODE's, cauchy-riemann equations, tips, tricks, involving math in physics. I also would like the books/websites to explain the purpose of the mathematics. I went into graduate school for physics with knowledge of Calc 1 through 3, discrete math, and linear algebra. Sadly I wish I had time for other math classes but now I need to gain this knowledge on my own. So any advice will be helpful. I'm not afraid of hard work to understand the concepts and methods. I'm working very hard to understand everything I can so I can make my future classes more enjoyable.

Key notes: I need books/websites with great in depth examples. Books/websites without heavy math terminology like a book for dummies. The books/websites should also explain the purpose of the math used. <-- I know this is like looking for the fountain of youth but maybe there are great guides out there.

Any advice on what worked for you when you didn't have knowledge on a specific math class will be helpful too.

Thank you
 
  • #7
"Negative Math" book by Martinez - any feedback?

Hi,

I'm about half-way through the book "Negative Math - How Mathematical Rules
Can Be Positively Bent" by Alberto a. Martinez.

The history of passionate rhetoric by even the great mathematicians is fascinating!
This provides a wealth of information about the centuries of arguments relating to
negative and imaginary numbers! (Disclaimer: I am not associated in any way with
this book - other than a passionate reader.)

What is not clear to me is if any of this material is still relevant to "modern" math?
I'm about half through this book and am not clear if these arguments have ever been
settled. It's obvious that most math teachers could not justify many of the arguments
for and against negative number interpretation. If I tried some of them in the
classroom, I would certainy be branded a trouble-maker.

So, do variations in math such as described in this book prove useful? Do useful or
even recreational math problems exist using a solution where +1 * -1 = -1?
Or any other of the many strange (to me) interpretations?

What do other people think of this book's historical usefulness?

Thanks
 
  • #8
191
3
Can anyone recommend a good text to learn the maths behind fractal geometry? I've written a simple program to generate the Mandelbrot set but I would like to extend this, significantly! So to that end, I'm looking for a book to give me a good grounding in the maths required to generate images of various fractal sets. I'm looking on Amazon, but it's hard to determine the best one!

https://www.amazon.com/dp/0716711869/?tag=pfamazon01-20&tag=pfamazon01-20 - Benoit Mandelbrot (I will read this eventually, but I think it might not be the book I'm after right now)

https://www.amazon.com/dp/0470848626/?tag=pfamazon01-20&tag=pfamazon01-20 - Kenneth Falconer (this has great reviews, and I get the feeling it may well serve my needs)

But there's plenty of other ones, so I'll ask if anyone has used and/or can recommend one?
 
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  • #9
918
16
You can read my review at https://www.amazon.com/dp/3540347623/?tag=pfamazon01-20&tag=pfamazon01-20

What follows is a copy:

I have the second (2009) printing of volume I, and the first printing (2008) of volume II of this book. Page numbers may differ if you have different printings.

I fear that I cannot say enough bad things about this book in the short space of this review. According to the author physicists need to extend their methods further than can be justified by rigorous mathematics. In such cases formal methods are used. On the other hand, mathematicians have strayed away from the kinds of intuitive insights that come from solving problems in the physical world. Why isn't there more cooperation? Because they don't speak the same language. These books are intended as a bridge between mathematicians and physicists. The reader would expect at least some translation between the two languages and sure enough, on page 332 of Vol II, such a lexicon exists. In it there are 5 examples. For instance, what is called a connection in mathematics is called potential in physics. That's it, 5 translations. If only physicists and mathematicians were aware of these 5 translations the channels of communication would be opened up and the mathematicians could find rigorous proofs of the methods used by the physicists.

It is not made clear how the other 6000 (I'm extrapolating) pages of the book form a bridge. One idea presented is that when rigorous math can only take you so far, you must use heuristic, or formal methods. However, mathematical rigor is not defined and examples of it are not given. The reader may not know what the author is talking about. One problem in this regard is that the author sometimes uses formal arguments even when rigorous mathematical methods already exist.

The material is presented in a way that is foreign to both mathematicians and physicists. The notation is different. For instance, the double arrow symbol meaning "implies" is used to indicate mappings. The vocabulary is different. For instance, single-valued is used to mean one-to-one, and repulsive force is used to mean restorative force. The definitions are different. For instance the Laplace transform is different from the standard one used by both mathematicians and physicists.

There are assymetries in the text that are inexplicable from any point of view. To denote the dual space of a linear space one finds a superscripted d, asterisk, or prime depending on the page number. Variables that commute are commuted in the middle of a derivation without any reason. What are we to make of the first page of the preface to the first volume? Are we to understand that the gravitational and electromagnetic forces are not between elementary particles but that the strong and weak forces are?

The typos come hot and heavy. Professor Zeidler is not a native speaker of English and you would think that he would have the book proof-read by someone who was. There are too many spelling and grammatical errors for that to be the case. For that matter, he doesn't seem to be a native speaker of the mathematical language either and that is a major problem because of the purpose of the book. Typos in the equations are depressingly common and the figures are wrong too. Figure 5.7b on page 234 of volume I subtracts 1000 words from the discussion it illustrates because one of the arrows points in the wrong direction. Figure 8.1b on page 700 of vol II is truly breathtaking as it depicts a comet being repelled by the gravitational force of the sun.
 
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  • #10
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Hello. I am very well versed in my mathematics(Mary Boas, Mathematical Methods of the Physical Sciences, all problems done, Courant Hilbert, Methods of Mathematical Physics,Real and Complex analysis Walter Rudin)

I want an optics text that will assume all knowledge of mathematics but that does not assume previous knowledge on optics.

I would also appreciate a thermodynamics text that assumes all knowledge on mathematics but nothing on thermodynamics.(classic examples of the wrong book include fermi's book on thermodynamics. it is assumed that you know all elementary results of thermodynamics)
 
  • #11
Im reviewing my discrete math because I want to take some theoretical comp sci courses next semester.
The book Im using is called "A logical approach to discrete math". The book doesnt have a solutions manual but I really like it because I find it highly readable (compared to that terrible book by Grimaldi that my university used). Also I didnt do too well on proofs in my discrete math course, and I think learning how to work with logic will help me get better in that aspect.
Im wondering, has anyone here had any experience with this book and would it be suitable to ask for help on these forums if I get stuck on a question?
 
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  • #12
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Hello everyone

I have been studying math throughout this summer by watching videos on the internet and completing this book called "Basic Mathematics, by Serge Lang". I will start at University next summer to study physics or math. I already own a copy of Feynmann's lectures on physics which I plan on starting to read once i "understand" calculus.
Now for the actual question... Would any of you mind recommending books on math that would ensure I understand all of the basics that is needed to have success studying the first year of undergraduate? And even books on calculus as well. I have several hours a day that I want to spend on math for the next months to come and money on material is not a problem.
I am sorry I cannot give better information on what I can except that I have completed that particular book. If you have any questions please go ahead :-)

Thanks in advance
Andreas
 
  • #13
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3,291

Table of Contents:
Code:
[LIST]
[*] Introduction
[*] Bibliography
[*] Basic Stuff
[LIST]
[*] Trigonometry
[*] Parametric Differentiation
[*] Gaussian Integrals
[*] erf and Gamma
[*] Differentiating
[*] Integrals
[*] Polar Coordinates
[*] Sketching Graphs
[/LIST]
[*] Infinite Series
[LIST]
[*] The Basics
[*] Deriving Taylor Series
[*] Convergence
[*] Series of Series
[*] Power series, two variables
[*] Stirling's Approximation
[*] Useful Tricks
[*] Diffraction
[*] Checking Results
[/LIST]
[*] Complex Algebra
[LIST]
[*] Complex Numbers
[*] Some Functions
[*] Applications of Euler's Formula
[*] Geometry
[*] Series of cosines
[*] Logarithms
[*] Mapping
[/LIST]
[*] Differential Equations
[LIST]
[*] Linear Constant-Coeffcient
[*] Forced Oscillations
[*] Series Solutions
[*] Some General Methods
[*] Trigonometry via ODE's
[*] Green's Functions
[*] Separation of Variables
[*] Circuits
[*] Simultaneous Equations
[*] Simultaneous ODE's
[*] Legendre's Equation
[*] Asymptotic Behavior
[/LIST]
[*] Fourier Series
[LIST]
[*] Examples
[*] Computing Fourier Series
[*] Choice of Basis
[*] Musical Notes
[*] Periodically Forced ODE's
[*] Return to Parseval
[*] Gibbs Phenomenon
[/LIST]
[*] Vector Spaces
[LIST]
[*] The Underlying Idea
[*] Axioms
[*] Examples of Vector Spaces
[*] Linear Independence
[*] Norms
[*] Scalar Product
[*] Bases and Scalar Products
[*] Gram-Schmidt Orthogonalization
[*] Cauchy-Schwartz inequality
[*] Infnite Dimensions
[/LIST]
[*] Operators and Matrices
[LIST]
[*] The Idea of an Operator
[*] Definition of an Operator
[*] Examples of Operators
[*] Matrix Multiplication
[*] Inverses
[*] Rotations, 3-d
[*] Areas, Volumes, Determinants
[*] Matrices as Operators
[*] Eigenvalues and Eigenvectors
[*] Change of Basis
[*] Summation Convention
[*] Can you Diagonalize a Matrix?
[*] Eigenvalues and Google
[*] Special Operators
[/LIST]
[*] Multivariable Calculus
[LIST]
[*] Partial Derivatives
[*] Chain Rule
[*] Differentials
[*] Geometric Interpretation
[*] Gradient
[*] Electrostatics
[*] Plane Polar Coordinates
[*] Cylindrical, Spherical Coordinates
[*] Vectors: Cylindrical, Spherical Bases
[*] Gradient in other Coordinates
[*] Maxima, Minima, Saddles
[*] Lagrange Multipliers
[*] Solid Angle
[*] Rainbow
[/LIST]
[*] Vector Calculus 1
[LIST]
[*] Fluid Flow
[*] Vector Derivatives
[*] Computing the divergence
[*] Integral Representation of Curl
[*] The Gradient
[*] Shorter Cut for div and curl
[*] Identities for Vector Operators
[*] Applications to Gravity
[*] Gravitational Potential
[*] Index Notation
[*] More Complicated Potentials
[/LIST]
[*] Partial Differential Equations
[LIST]
[*] The Heat Equation
[*] Separation of Variables
[*] Oscillating Temperatures
[*] Spatial Temperature Distributions
[*] Specifed Heat Flow
[*] Electrostatics
[*] Cylindrical Coordinates
[/LIST]
[*] Numerical Analysis
[LIST]
[*] Interpolation
[*] Solving equations
[*] Differentiation
[*] Integration
[*] Differential Equations
[*] Fitting of Data
[*] Euclidean Fit
[*] Differentiating noisy data
[*] Partial Differential Equations
[/LIST]
[*] Tensors
[LIST]
[*] Examples
[*] Components
[*] Relations between Tensors
[*] Birefringence
[*] Non-Orthogonal Bases
[*] Manifolds and Fields
[*] Coordinate Bases
[*] Basis Change
[/LIST]
[*] Vector Calculus 2
[LIST]
[*] Integrals
[*] Line Integrals
[*] Gauss's Theorem
[*] Stokes' Theorem
[*] Reynolds Transport Theorem
[*] Fields as Vector Spaces
[/LIST]
[*] Complex Variables
[LIST]
[*] Differentiation
[*] Integration
[*] Power (Laurent) Series
[*] Core Properties
[*] Branch Points
[*] Cauchy's Residue Theorem
[*] Branch Points
[*] Other Integrals
[*] Other Results
[/LIST]
[*] Fourier Analysis
[LIST]
[*] Fourier Transform
[*] Convolution Theorem
[*] Time-Series Analysis
[*] Derivatives
[*] Green's Functions
[*] Sine and Cosine Transforms
[*] Wiener-Khinchine Theorem
[/LIST]
[*] Calculus of Variations
[LIST]
[*] Examples
[*] Functional Derivatives
[*] Brachistochrone
[*] Fermat's Principle
[*] Electric Fields
[*] Discrete Version
[*] Classical Mechanics
[*] Endpoint Variation
[*] Kinks
[*] Second Order
[/LIST]
[*] Densities and Distributions
[LIST]
[*] Density
[*] Functionals
[*] Generalization
[*] Delta-function Notation
[*] Alternate Approach
[*] Differential Equations
[*] Using Fourier Transforms
[*] More Dimensions
[/LIST]
[*] Index
[/LIST]
 
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  • #14
22,089
3,291

Table of Contents:
Code:
[LIST]
[*] Fundamentals of Discrete Mathematics
[LIST]
[*] Fundamental Principles of Counting
[LIST]
[*] The Rules of Sum and Product
[*] Permutations
[*] Combinations: The Binomial Theorem
[*] Combinations with Repetition
[*] The Catalan Numbers (Optional)
[*] Summary and Historical Review
[/LIST]
[*] Fundamentals of Logic
[LIST]
[*] Basic Connectives and Truth Tables
[*] Logical Equivalence: The Laws of Logic
[*] Logical Implication: Rules of Inference
[*] The Use of Quantifiers
[*] Quantifiers, Definitions, and the Proofs of Theorems
[*] Summary and Historical Review
[/LIST]
[*] Set Theory
[LIST]
[*] Sets and Subsets
[*] Set Operations and the Laws of Set Theory
[*] Counting and Venn Diagrams
[*] A First Word on Probability
[*] The Axioms of Probability (Optional)
[*] Conditional Probability: Independence (Optional)
[*] Discrete Random Variables (Optional)
[*] Summary and Historical Review
[/LIST]
[*] Properties of the Integers: Mathematical Induction
[LIST]
[*] The Well-Ordering Principle: Mathematical Induction
[*] Recursive Definitions
[*] The Division Algorithm: Prime Numbers
[*] The Greatest Common Divisor: The Euclidean Algorithm
[*] The Fundamental Theorem of Arithmetic
[*] Summary and Historical Review
[/LIST]
[*] Relations and Functions
[LIST]
[*] Cartesian Products and Relations
[*] Functions: Plain and One-to-One
[*] Onto Functions: Stirling Numbers of the Second Kind
[*] Special Functions
[*] The Pigeonhole Principle
[*] Function Composition and Inverse Functions
[*] Computational Complexity
[*] Analysis of Algorithms
[*] Summary and Historical Review
[/LIST]
[*] Languages: Finite State Machines
[LIST]
[*] Language: The Set Theory of Strings
[*] Finite State Machines: A Frst Encounter
[*] Finite State Machines: A Second Encounter
[*] Summary and Historical Review
[/LIST]
[*] Relations: The Second Time Around
[LIST]
[*] Relations Revisited: Proper ies of Relations
[*] Computer Recognition: Zero- One Matrices and Directed Graphs
[*] Partial Orders: Hasse Diagrams
[*] Equivalence Relations and Partitions
[*] Finite State Machines: The Minimization Process
[*] Summary and Historical Review
[/LIST]
[/LIST]
[*] Further Topics in Enumeration
[LIST]
[*] The Principle of Inclusion and Exclusion
[LIST]
[*] The Principle of Inclusion aid Exclusion
[*] Generalizations of the Principle
[*] Derangements: Nothing Is in Its Right Place
[*] Rook Polynomials
[*] Arrangements with Forbidden Positions
[*] Summary and Historical Review
[/LIST]
[*] Generating Functions
[LIST]
[*] Introductory Examples
[*] Definition and Examples: Calculational Techniques
[*] Partitions of Integers
[*] The Exponential Generating Function
[*] The Summation Operator
[*] Summary and Historical Review
[/LIST]
[*] Recurrence Relations
[LIST]
[*] The First-Order Linear Recurrence Relation
[*] The Second-Order Linear Homogeneous Recurrence Relation with Constant Coefficients
[*] The Nonhomogeneous Recurrence Relation
[*] The Method of Generating Functions
[*] A Special Kind of Nonlinear Recurrence Relation (Optional)
[*] Divide-and-Conquer Algorithm (Optional)
[*] Summary and Historical Review
[/LIST]
[/LIST]
[*] Graph Theory and Applications
[LIST]
[*] An Introduction to Graph Theory
[LIST]
[*] Definitions and Examples
[*] Subgraphs, Complements, and Graph Isomorphism
[*] Vertex Degree: Euler Trails and Circuits
[*] Planar Graphs
[*] Hamilton Paths and Cycles
[*] Graph Coloring and Chromatic Polynomials
[*] Summary and Historical Review
[/LIST]
[*] Trees
[LIST]
[*] Definitions, Properties, and Examples
[*] Rooted Trees
[*] Trees and Sorting
[*] Weighted Trees and Prefix Codes
[*] Biconnected Components and Articulation Points
[*] Summary and Historical Review
[/LIST]
[*] Optimization and Matching
[LIST]
[*] Dijkstra's Shortest-Path Algorithm
[*] Minimal Spanning Trees: The Algorithms of Kruskal and Prim
[*] Transport Networks: The Max-Flow Min-Cut Theorem
[*] Matching Theory
[*] Summary and Historical Review
[/LIST]
[/LIST]
[*] Modern Applied Algebra
[LIST]
[*] Rings and Modular Arithmetic
[LIST]
[*] The Ring Structure: Definition and Examples
[*] Ring Properties and Substructures
[*] The Integers Modulo n
[*] Ring Homomorphisms and Isomorphisms
[*] Summary and Historical Review
[/LIST]
[*] Boolean Algebra and Switching Functions
[LIST]
[*] Switching Functions: Disjunctive and Conjunctive Normal Forms
[*] Gating Networks: Minimal Sums of Products: Karnaugh Maps
[*] Further Applications: Don't-Care Conditions
[*] The Structure of a Boolean Algebra (Optional)
[*] Summary and Historical Review
[/LIST]
[*] Groups, Coding Theory, and Polya's Method of Enumeration
[LIST]
[*] Definition, Examples, and Elementary Properties
[*] Homomorphisms, Isomorphisms, and Cyclic Groups
[*] Cosets and Lagrange's Theorem
[*] The RSA Cryptosystem (Optional)
[*] Elements of Coding Theory
[*] The Hamming Metric
[*] The Parity-Check and Generator Matrices
[*] Group Codes: Decoding with Coset Leaders
[*] Hamming Matrices
[*] Counting and Equivalence: Burnside's Theorem
[*] The Cycle Index
[*] The Pattern Inventory: Polya's Method of Enumeration
[*] Summary and Historical Review
[/LIST]
[*] Finite Fields and Combinatorial Designs
[LIST]
[*] Polynomial Rings
[*] Irreducible Polynomials: Finite Fields
[*] Latin Squares
[*] Finite Geometries and Affine Planes
[*] Block Designs and Projective Planes
[*] Summary and Historical Review
[/LIST]
[/LIST]
[*] Appendix: Exponential and Logarithmic Functions
[*] Appendix: Matrices, Matrix Operations, and Determinants
[*] Appendix: Countable and Uncountable Sets
[*] Contents
[*] Solutions
[*] Index 
[/LIST]
 
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  • #15
Hello all.

Long time back, I was looking for a book on psycology of problem solving. I was trying to understand as to what it takes to be able to solve contest problems like that in the International Mathematical Olympiads. I wish to share here with you a great book that I discovered. It is titled "Mathematical Problem Solving' by Alan Schoenfeld. For me, the main takeaway from this book are the following points.

1. Metacognition is a must have quality for problem solving. He calls it "Good Control". Just reading about it in his book helped me immensely to be able to solve tough IMO level problems. Typically, a novice simply tries one approach and then keeps trying that approach senselessly without even realizing that he has to give up that approach and try another one. He calls it the "Wild Goose Chase". An expert on the other hand looks at one approach, then backs off. He/she then thinks as to what other approach might work. Finally, an attempt is made only after they see a light or get an insight into the problem. These behaviors between an expert and a novice are very nicely explained in this book and in very great detail.

2. Practice alone won't help. Good training is necessary for preparing for IMO. Usually, the problem appears daunting. Kids look at it, spend most of the time being frightened and also fascinated thinking "I solved my math text exercises orally..... he must be a real genius whoever solves such problems". Then after a day's work, he gives up. Looks at the solution. This will happen a few times and then he eventually gives up thinking that he won't be able to make a beginning. How then is practice going to help? All the student does is just stare, get more frightened, and then give up and look at the solution. There is more to it than just doing the above. But what is that quality which is required of a student to become a good problems solver?

The main quality that kids develop with training is to achieve great control. There are books written by G. Polya and according to me, his books titled "Mathematical Discovery" I and II are the best in the series; however, it is not possible to become a good problem solver just by knowing the heuristics. Good control is necessary which comes only with training.

3. Having control alone won't help. Sometimes, a problem might need an insight which alone can help in the solution of the problem. This insight comes only with good preparation in terms of good foundation in basic math. He talks about it referring to the insight as "Resources".

When Polya's "How to Solve it" came out, it became a big hit. I think it is the "Mathematical Discovery" I and II which are the more useful books for contests. But one thing that people found was that it helped the already good problem solvers become more familiar with heuristics but it did not help a whole lot for a novice. I solved exercises from my math textbook in my tenth grade orally. All of them. I could not solve a single IMO problem. Not even a single problem. It is the same case with a LOT of people that I have known. What makes the difference? Does Polya's book help for contests? No. It does not. Training is a MUST. Alan Schoenfeld, a professor at UC Berkeley found that this is a question that needed an answer. Why does the heuristics approach not work for novice students? Mind you. A novice is defined as someone not used to problem solving. Solving exercises does not make a kid an expert. Problems are different from exercises.

Hence Prof. Schoenfeld took up the research to figure it out. He then found answers which are given in this book. I found it amazing and I felt like sharing it with you. It is something which can potentially make a difference for someone who is preparing for a contest or for a math major. I am not sure if a highschool kid will be able to fully appreciate the content of this book if he is not an expert problem solver already; however, he will realize that it is not his lack of ability but it is due to lack of training and guidance because of which he is failing in contests. It is a common sensical thing but it took a while for me to figure it out. It became more obvious to me after I read this book.

I could never solve a single problem from the IMO. I have solved many problems from Putnam and IMO lately. Thanks to this book. I know that some of you might say "well the first problem from these contests are generally easy...." but I have solved many of the tough ones. This book is really very useful. I think that it will help anyone who is a math major and is banging his head against the wall as to why he cannot be better at solving the tougher problems in say Analysis. I highly recommend this book. It is truly one of a kind and I have not found anybody else who has even attempted to shed light into this matter. He may not talk about Analysis or any higher level math but his main focus is on the qualities that training/course on problem solving can impart on the students which is what truly makes the difference between an expert and a novice problem solver.
 
  • #16
153
3
''The third edition of this popular and effective textbook provides in one volume a unified treatment of topics essential for first year university students studying for degrees in mathematics. Students of computer science, physics and statistics will also find this book a helpful guide to all the basic mathematics they require. It clearly and comprehensively covers much of the material that other textbooks tend to assume, assisting students in the transition to university-level mathematics.

Expertly revised and updated, the chapters cover topics such as number systems, set and functions, differential calculus, matrices and integral calculus. Worked examples are provided and chapters conclude with exercises to which answers are given. For students seeking further challenges, problems intersperse the text, for which complete solutions are provided. Modifications in this third edition include a more informal approach to sequence limits and an increase in the number of worked examples, exercises and problems.'' Amazon
https://www.amazon.com/dp/0857092235/?tag=pfamazon01-20&tag=pfamazon01-20
 
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