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1
The language of mathematics and prompting
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1.1 The art of prompting
1.2 Black box warnings
1.3 Computer algebra (and python)
1.4 Numbers
1.5 Propositional logic
1.6 More on sets
1.7 Ordering numbers
1.8 Proof by induction
1.9 The concept of a function
2
Linear equations
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2.1 One linear equation with one unknown
2.2 Several linear equations with several unknowns
2.3 Gauss elimination
2.4 Polynomials
2.5 Applications of linear equations to polynomials
2.6 Shamir secret sharing
2.7 Fitting data
3
Matrices
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3.1 Matrices
3.2 Linear maps
3.3 Matrix multiplication
3.4 Matrix arithmetic
3.5 The inverse matrix
3.6 The transposed matrix
3.7 Symmetric matrices
4
What is optimization?
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4.1 What is an optimization problem?
4.2 General definition
4.3 Convex optimization
4.4 Linear optimization
4.5 Fourier-Motzkin elimination
4.6 Application in machine learning and data science
5
Euclidean vector spaces
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5.1 Vectors in the plane
5.2 Higher dimensions
5.3 The unreasonable effectiveness of the dot product
5.4 Pythagoras and the least squares method
5.5 The Cauchy-Schwarz inequality
5.6 Special subsets of euclidean spaces
5.7 Continuous functions
5.8 Important and special results for continuous functions
6
Convex functions
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6.1 Strictly convex functions
6.2 Why are convex functions interesting?
6.3 Differentiable functions
6.4 Taylor polynomials
6.5 Differentiable convex functions
7
Several variables
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7.1 Introduction
7.2 Vector functions
7.3 Differentiability
7.4 Newton-Raphson in several variables!
7.5 Local extrema in several variables
7.6 The chain rule
7.7 Logistic regression
7.8 3Blue1Brown
7.9 Lagrange multipliers
7.10 Optimization using the interior and boundary of a subset
8
The Hessian
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8.1 Introduction
8.2 Several variables
8.3 Newton's method for finding critical points
8.4 The Hessian and critical points
8.5 Differential convex functions of several variables
8.6 How to decide the definiteness of a matrix
8.7 A schematic procedure for transforming symmetric matrices
9
Convex optimization
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9.1 Finding the optimal hyperplane separating data
9.2 Logarithmic barrier functions
9.3 A geometric optimality criterion
9.4 KKT
9.5 Computing with KKT
9.6 Optimization exercises
Introduction to Mathematics and Optimization
Niels Lauritzen
