PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD ↓


Combinatorics


Introduction

Combinatorics is a branch of mathematics concerned with the study of counting, arranging, and combining objects. It is a fundamental part of discrete mathematics and has applications in computer science, physics, biology, and other fields. In its simplest form, combinatorics investigates the ways in which objects can be selected and arranged, often under certain restrictions.

Basic concepts

Rule of addition and rule of product

These are the two fundamental principles of combinatorics.

Law of Addition: If there are a ways to do a task and b ways to do another task, and both the tasks cannot be done at the same time, then there are a + b ways to choose one of them.

Law of Product: If there are a ways to do a task and b ways to do another task which are independent of the first, then there are a * b ways to do both the tasks.

Example: Counting options

Suppose you have 3 types of ice cream (vanilla, chocolate, strawberry) and 2 types of cones (wafer, cup). How many different ice cream cones can you have?

Types of Ice Cream: Vanilla, Chocolate, Strawberry (3 options)
 Types of Cones: Wafer, Cup (2 options)

Use of the product rule:

3 * 2 = 6

So, 6 different combinations of ice-cream cones are possible.

Permutation and combination

Permutation

Permutation refers to the different ways in which a group of objects can be sorted or arranged. The order of arrangement is important here.

Permutation of a set

For a set of n objects, the number of permutations is found using the factorial function:

n! = n * (n - 1) * (n - 2) * ... * 1

If you have a set of three letters, say A, B, and C, then the permutations would be as follows:

  • ABC
  • ACB
  • BAC
  • BCA
  • CAB
  • CBA

Thus, the set {A, B, C} has 3! = 6 permutations.

Permutations with restrictions

If you can select only r objects out of the n available, and the order matters, then the formula for a permutation is:

P(n, r) = n! / (n - r)!

Combination

Items are selected in combination, where the order has no importance.

Combination of a set

If you choose r objects out of n and the order is not important, the formula to count the combinations is:

C(n, r) = n! / [r! * (n - r)!]

For example, choosing 2 letters from a set of three {A, B, C} might give us:

  • AB
  • AC
  • BC

Note that AB and BA are the same combination because the order does not matter. So, C(3, 2) = 3 combinations.

Advanced concepts

Binomial theorem

The binomial theorem describes the algebraic expansion of the powers of a binomial expression. It is stated as follows:

(x + y)^n = Σ [C(n, k) * x^(n-k) * y^k], from k = 0 to n

where C(n, k) is the binomial coefficient.

Pigeonhole principle

The pigeonhole principle is a simple but still very powerful principle used in combinatorics. It states that if n items are placed in m containers, where n > m, then at least one container must contain more than one item.

Example

If you have 10 pairs of socks but only 9 drawers, you must store more than one pair of socks in at least one drawer.

Visualization of combinatorics

Let's use some visual diagrams using simple shapes and lines to represent sets, permutations, and combinations.

Visual example: Simple sets and permutations



  
  
  A
  
  B
  
  C

This diagram shows an example set {A, B, C} and the different curved lines or paths you can take that represent different permutations.

Visual example: Combining using a matrix



    
    
    
    Item 1
    Item 2

    
    
    
    
    
    Option A
    Option B
    Option C
    Option D

The grid shows the different possible combinations: choosing between different 'options' for each 'object' in a simple matrix visualization.

Applications of combinatorics

The application of combinatorics occurs in various practical scenarios and areas:

  1. Cryptography: For designing and analyzing security algorithms.
  2. Graph theory: in network design and route optimization.
  3. Coding theory: for error detection and correction in digital communications.
  4. Statistical Physics: In Modeling Particle Distributions.

Conclusion

Understanding combinatorics provides essential insight into how complex arrangements and calculations operate. The basic principles provide the foundational skills to investigate more advanced topics in mathematics and tackle real-world problems. Its applications, though varied, are basically based on the simple principles of counting, selection, and arrangement.


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