Book on inequalities to help prepare for the Putnam exam

[ehrenfest]

I want to get a book on inequalities to help prepare for the Putnam exam. Its common for me to spend about half an hour getting frustrated on a practice problem and then find that the only way to do the problem is with an inequality that I have never heard of. Does anyone have any recomendations? Here are some I found on amazon:

https://www.amazon.com/dp/0883856034/?tag=pfamazon01-20
https://www.amazon.com/dp/052154677X/?tag=pfamazon01-20
https://www.amazon.com/dp/0521358809/?tag=pfamazon01-20

 

[mathematicsma]

I am an undergraduate student, and I just started a course called "Mathematics of Compound Interest." Most of the students in the class are taking actuarial exams (I myself have not decided what I'm doing), so the course is geared toward that kind of study (it's not proof oriented, etc.). The suggested text is Theory of Interest by Kellison, but the professor told me that given my background (Calc I, currently taking Calc II), it's too advanced for me, and we won't be going that deep in class anyway. So does anyone have a suggestion of a text that follows similar material on a simpler level that I would understand? Thanks a lot. I really like having a textbook in addition to lecture notes.

 

[jhaber]

I'm reading Zee's QFT as self-study and have had trouble with the applications of group theory in Section 2. I'd love a book recommendation to fill in my gaps.

I took algebra in the math department, but there groups were structures to distinguish from rings and such by definition, with no regard to matrix representation. I've read the group theory chapter in Atkins's "Molecular Quantum Mechanics," but it didn't get me far enough in understanding the offhand references here to, say, the appropriate cyclic permutations of a 4 x 4 matrix.

Obviously I don't mean a comprehensive, difficult, separate book on quantum theory from a group-theoretical point of view such as Weyl's. Thank you.

 

[fa2209]

I am going to start my third year of a theoretical physics degree but have always had an interest in pure mathematics so I am currently teaching myself real analysis from the book "Real Analysis" by Howie. I've done some basic introductory set theory including cardinality, countability and Russell's paradox but what can I read to help me go from this level of understanding to the incompleteness theorems. Would I have to read more about set theory or go straight into formal logic?

Any textbook recommendations would be much appreciated.

 

[bennyska]

so it turns out i only need 2 classes next semester before i wrap up my undergrad degree in statistics, but i have a scholarship that requires me to take 4 classes. i was thinking about doing an independent study through the college to help make up one of the ones I'm missing.

2 things i'd be interested in learning: bayesian statistics, and linear algebra specifically for statistics.

i don't really know anything about bayesian stuff, other than the very basic problems we did in intro stats, and i already have a bit of experience with linear, although not from a stats perspective (i.e. stat applications). i just really like linear, and i'd like to develop it more. i have taken both an intro and proof based linear, but i'd like more application.

does anyone have any suggestions? they'll have to be books that my teachers can approve so i can get credit for them. thanks

 

[Greg Bernhardt]

• Author: Michael Sullivan III
• Title: Intermediate Algebra
• Amazon Link: https://www.amazon.com/dp/0321567528/?tag=pfamazon01-20
• Prerequisities:

Table of Contents:

[LIST]
[*] Preface
[*] Real Numbers and Algebraic Expressions
[LIST] 
[*] Success in Mathematics
[*] Sets and Classifications of Numbers
[*] Operations on Signed Numbers; Properties of Real Numbers
[*] Order of Operations
[*] Algebraic Expressions
[/LIST]
[*] Linear Equations and Inequalities
[LIST]
[*] Linear Equations and Inequalities in One Variable
[LIST]
[*] Linear Equations in One Variable
[*] An Introduction to Problem Solving
[*] Using Formulas to Solve Problems
[*] Linear Inequalities in One Variable
[/LIST]
[*] Linear Equations and Inequalities in Two Variables
[LIST]
[*] Rectangular Coordinates and Graphs of Equations
[*] Linear Equations in Two Variables
[*] Parallel and Perpendicular Lines
[*] Linear Inequalities in Two Variables
[/LIST]
[/LIST]
[*] Relations, Functions, and More Inequalities
[LIST]
[*] Relations
[*] An Introduction to Functions
[*] Functions and Their Graphs
[*] Linear Functions and Models
[*] Compound Inequalities
[*] Absolute Value Equations and Inequalities
[*] Variation
[/LIST]
[*] Systems of Linear Equations and Inequalities
[LIST]
[*] Systems of Linear Equations in Two Variables
[*] Problem Solving: System of Two Linear Equations Containing Two Unknowns
[*] Systems of Linear Equations in Three Variables
[*] Using Matrices to Solve Systems
[*] Determinants and Cramer's Rule
[*] System of Linear Inequalities
[/LIST]
[*] Polynomials and Polynomial Functions
[LIST]
[*] Adding and Subtracting Polynomials
[*] Multiplying Polynomials
[*] Dividing Polynomials; Synthetic Division
[*] Greatest Common Factor; Factoring by Grouping
[*] Factoring Trinomials
[*] Factoring Special Products
[*] Factoring: A General Strategy
[*] Polynomial Equations
[/LIST]
[*] Rational Expressions and Rational Functions
[LIST]
[*] Multiplying and Dividing Rational Expressions
[*] Adding and Subtracting Rational Expressions
[*] Complex Rational Expressions
[*] Rational Equations
[*] Rational Inequalities
[*] Models Involving Rational Expressions
[/LIST]
[*] Radicals and Rationals Exponents
[LIST]
[*] [itex]n[/itex]th Roots and Rational Exponents
[*] Simplify Expressions Using the Laws of Exponents
[*] Simplifying Radical Expressions Using Properties of Radicals
[*] Adding, Subtracting, and Multiplying Radical Expressions
[*] Rationalizing Radical Expressions
[*] Functions Involving Radicals
[*] Radical Equations and Their Applications
[*] The Complex Number System
[/LIST]
[*] Quadratic Equations and Functions
[LIST]
[*] Solving Quadratic Equations by Completing the Square
[*] Solving Quadratic Equations by the Quadratic Formula
[*] Solving Equations Quadratic in Form
[*] Graphing Quadratic Functions Using Transformations
[*] Graphing Quadratic Functions Using Properties
[*] Quadratic Inequalities
[/LIST]
[*] Exponential and Logarithmic Functions
[LIST]
[*] Composite Functions and Inverse Functions
[*] Exponential Functions
[*] Logarithmic Functions
[*] Properties of Logarithms
[*] Exponential and Logarithmic Equations
[/LIST]
[*] Conics
[LIST]
[*] Distance and Midpoint Formulas
[*] Circles
[*] Parabolas
[*] Ellipses
[*] Hyperbolas
[*] Systems of Nonlinear Equations
[/LIST]
[*] Sequences, Series, and the Binomial Theorem
[LIST]
[*] Sequences
[*] Arithmetic Sequences
[*] Geometric Sequences and Series
[*] The Binomial Theorem
[/LIST]
[*] Answers to Selected Exercises
[*] Applications Index
[*] Subject Index
[*] Photo Credits
[/LIST]

 

[Greg Bernhardt]

• Author: Gareth A. Jones (Author), Josephine M. Jones (Author)
• Title: Elementary Number Theory
• Amazon Link: https://www.amazon.com/dp/3540761977/?tag=pfamazon01-20
• Prerequisities:

Table of Contents:

[LIST]
[*] Notes to the Reader
[*] Divisibility
[LIST]
[*] Divisors
[*] Bezout's identity
[*] Least common multiples
[*] Linear Diophantine equations
[*] Supplementary exercises
[/LIST]
[*] Prime Numbers
[LIST]
[*] Prime numbers and prime-power factorisations
[*] Distribution of primes
[*] Fermat and Mersenne primes
[*] Primality-testing and factorisation
[*] Supplementary exercises
[/LIST]
[*] Congruences
[LIST]
[*] Modular arithmetic
[*] Linear congruences
[*] Simultaneous linear congruences
[*] Simultaneous non-linear congruences
[*] An extension of the Chinese Remainder Theorem
[*] Supplementary exercises
[/LIST]
[*] Congruences with a Prime-power Modulus
[LIST]
[*] The arithmetic of [itex]\mathbb{Z}_p[/itex]
[*] Pseudoprimes and Carmichael numbers
[*] Solving congruences mod [itex](p^e)[/itex]
[*] Supplementary exercises
[/LIST]
[*] Euler's Function
[LIST] 
[*] Units
[*] Euler's function
[*] Applications of Euler's function
[*] Supplementary exercises
[/LIST]
[*] The Group of Units
[LIST]
[*] The group [itex]U_n[/itex]
[*] Primitive roots
[*] The group [itex]U_{p^e}[/itex], where [itex]p[/itex] is an odd prime
[*] The group [itex]U_{2^e}[/itex]
[*] The existence of primitive roots
[*] Applications of primitive roots
[*] The algebraic structure of [itex]U_n[/itex]
[*] The universal exponent
[*] Supplementary exercises
[/LIST]
[*] Quadratic Residues
[LIST]
[*] Quadratic congruences
[*] The group of quadratic residues
[*] The Legendre symbol
[*] Quadratic reciprocity
[*] Quadratic residues for prime-power moduli
[*] Quadratic residues for arbitrary moduli
[*] Supplementary exercises
[/LIST]
[*] Arithmetic Functions
[LIST]
[*] Definition and examples
[*] Perfect numbers
[*] The Mobius Inversion Formula
[*] An application of the Mobius Inversion Formula
[*] Properties of the Mobius function
[*] The Dirichlet product
[*] Supplementary exercises
[/LIST]
[*] The Riemann Zeta Function
[LIST]
[*] Historical background
[*] Convergence
[*] Applications to prime numbers
[*] Random integers
[*] Evaluating [itex]\zeta(2)[/itex]
[*] Evaluating [itex]\zeta(2k)[/itex]
[*] Dirichlet series
[*] Euler products
[*] Complex variables
[*] Supplementary exercises
[/LIST]
[*] Sums of Squares
[LIST]
[*] Sums of two squares
[*] The Gaussian integers
[*] Sums of three squares
[*] Sums of four squares
[*] Digression on quaternions
[*] Minkowski's Theorem
[*] Supplementary exercises
[/LIST]
[*] Fermat's Last Theorem
[LIST]
[*] The problem
[*] Pythagoras's Theorem
[*] Pythagorean triples
[*] Isosceles triangles and irrationality
[*] The classification of Pythagorean triples
[*] Fermat
[*] The case [itex]n = 4[/itex]
[*] Odd prime exponents
[*] Lame and Kummer
[*] Modern developments
[*] Further reading
[/LIST]
[*] Appendix: Induction and Well-ordering
[*] Appendix: Groups, Rings and Fields
[*] Appendix: Convergence
[*] Appendix: Table of Primes [itex]p < 1000[/itex]
[*] Solutions to Exercises
[*] Bibliography
[*] Index of symbols
[*] Index of names
[*] Index 
[/LIST]