Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Discrete Mathematics → Combinatorics ↓


Recurrence Relations


Recurrence relations are a fundamental concept in mathematics, particularly in the fields of combinatorics and discrete mathematics. They are essential for understanding sequences and analyzing algorithms in computer science. A recurrence relation is an equation that recursively defines a sequence or multidimensional array of values. Given one or more initial terms, each subsequent term of the sequence is defined as a function of the preceding terms.

Introduction to recurrence relations

Sequences of numbers can often be described by expressing each element in terms of earlier elements. Such an expression is known as a recurrence relation. Formally, the recurrence relation for a sequence {a_n} is an equation that expresses n term of the sequence a_n in terms of one or more of the previous terms such as a_{n-1}, a_{n-2}, etc.

For example, the well-known Fibonacci sequence is defined by the recurrence relation:

F(n) = F(n-1) + F(n-2)

with initial conditions:

F(0) = 0, F(1) = 1

Here, each term is defined as the sum of the two terms preceding it.

Types of recurrence relations

Linear recurrence relations

A linear recurrence relation is one in which each term is a linear combination of the previous terms. The coefficients are constants. For example, the sequence describing compound interest is a linear recurrence relation.

Homogeneous recurrence relations

A recurrence relation is homogeneous if it can be written in the form:

a_n = c_1 * a_{n-1} + c_2 * a_{n-2} + ... + c_k * a_{nk}

where c_1, c_2, ..., c_k are constants. The Fibonacci sequence is a homogeneous linear recurrence relation.

Non-homogeneous recurrence relations

A non-homogeneous recurrence relation consists of terms that are not linear combinations of preceding terms. An example is:

a_n = a_{n-1} + n^2

which includes the extra term n^2.

Solution of recurrence relations

Solving a recurrence relation involves finding a formula, called the closed form, that defines the nth term of the sequence in terms of n only. Various methods are used to solve recurrence relations, including iteration, characteristic equations, and generating functions.

Iterative method

The iterative method involves expanding the recurrence relation step by step so that a_n can be expressed in terms of initial conditions and then finding a pattern. Consider the recurrence relation:

a_n = 2a_{n-1} + 3

Starting from the initial condition, say a_0 = 1, we calculate:

a_1 = 2*1 + 3 = 5 a_2 = 2*5 + 3 = 13 a_3 = 2*13 + 3 = 29

Observe the pattern and try to describe it using a formula.

Characteristic equation method

For linear homogeneous recurrence relations, we can often use the characteristic equation approach. If the recurrence relation is:

a_n = c_1 * a_{n-1} + c_2 * a_{n-2} + ... + c_k * a_{nk}

We consider a solution of the form a_n = r^n, which gives the following characteristic equation:

r^n = c_1 * r^{n-1} + c_2 * r^{n-2} + ... + c_k * r^{nk}

Simplifying this gives a polynomial in terms of r. Solving this polynomial gives the roots which are used to write the general solution.

Generating function

Generating functions turn a recurrence relation into an algebraic equation. The generating function for the sequence {a_n} is a formal power series:

G(x) = a_0 + a_1 * x + a_2 * x^2 + ... + a_n * x^n + ...

By manipulating the series, we can obtain a formula for a_n.

Examples of recurrence relations

Example 1: Fibonacci numbers

A classical example is the Fibonacci sequence, which is defined as:

F(n) = F(n-1) + F(n-2) F(0) = 0, F(1) = 1

The first few terms are: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Example 2: Factorial

The factorial function, written as n! can be defined recursively as:

n! = n * (n-1)! 0! = 1

This is a simple linear recurrence relation.

Example 3: Towers of Hanoi

The Hanoi Towers problem, which involves moving a stack of disks, is solved recursively with the following relation:

T(n) = 2T(n-1) + 1 T(1) = 1

Here, T(n) is the minimum number of moves required to move n discs.

Visual example

Visualizing the Fibonacci sequence

Consider looking at the Fibonacci sequence and its recursive nature.

11235

The above visual sequence represents the blocks, and demonstrates the recursive nature of its expansion.

Conclusion

Recurrence relations are powerful tools for understanding complex sequences and algorithm performance. By expressing terms in sequences recursively, they provide insight into the behavior of systems, making them indispensable in both theoretical and applied mathematics.


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