Lectures related to Multiplicity (mathematics)

Solutions of Homogeneous Linear Differential Equations

Covers the solutions of homogeneous linear differential equations and how to find linearly independent solutions.

Characteristic Polynomials and Similar Matrices

Explores characteristic polynomials, similarity of matrices, and eigenvalues in linear transformations.

Algebraic Multiplicity, Geometric Multiplicity

Explores algebraic and geometric multiplicities of eigenvalues in linear algebra.

Isogeny Graphs: Eigenvalues and Cryptography

Explores isogeny graphs of supersingular elliptic curves, showing optimal mixing times for random walks and applications to cryptography.

Linear Differential Equations

Explores solving linear differential equations and the principle of superposition with illustrative examples.

Projective Plane Curves

Introduces projective plane curves, degrees, components, multiplicities, intersection numbers, tangents, and multiple points, culminating in the statement of Bézout's theorem and its consequences.

Principle of Superposition in Differential Equations

Explores the principle of superposition in differential equations and the fundamental theorem of algebra.

Polynomial Equations: Determining Coefficients

Covers the process of determining coefficients in polynomial equations.

Characteristic Polynomials: Solutions and Roots

Covers characteristic polynomials, their solutions, and roots for different types of polynomials.

Affine Plane Curves

Covers the study of affine plane curves and their properties, including the concept of multiplicity and its applications.

Multistep methods

Covers multistep methods for solving differential equations, focusing on stability conditions and examples.

Tangent Lines and Equality

Explores the intersection multiplicity of curves and the absence of common tangent lines.

Local Rings and Residues

Covers the proof of theorem 4.2 on multiplicities and the special structure of local rings at a simple point of a plane.

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Linear Algebra: Cofactors and Eigenvectors

Explains cofactors, eigenvectors, eigenvalues, and studying linear applications.

Plane Curves: Singular Points and Multiplicities

Explores plane curves, focusing on singular points, multiplicities, and tangent lines.

Holomorphic Functions: Taylor Series Expansion

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Bézout's Theorem and Cayley-Bacharach

Explores Bézout's theorem and Cayley-Bacharach, discussing intersection multiplicities and linear combinations in projective geometry.

Symmetric Matrices and Eigenvalues

Explores symmetric matrices, eigenvalues, and characteristic polynomials in matrix analysis.

Polynomial Factorization: Field Approach

Covers the factorization of polynomials over a field, including division with remainder and common divisors.