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### The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves

00:48:55Arthur MattuckWe revisit the shortest paths problem, considering the case where the input is a directed minor-free graph with negative arc lengths (but no negative-length cycles).In Lecture 14, we saw almost-linear-time algorithms for the case of planar and bounded-genus graphs. Currently, comparable bounds for minor-free graphs are not known. We shall discuss Goldberg's algorithm, a shortest-path algorithm for general graphs with integer lengths, whose running time depends logarithmically on the magnitude of the largest negative arc length. By exploiting separators (Lecture 6), it runs faster on minor-free graphs than on general graphs, but it still requires superlinear time.

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### Euler's Numerical Method for y'=f(x,y) and its Generalizations

00:50:43Arthur MattuckWe revisit the shortest paths problem, considering the case where the input is a directed minor-free graph with negative arc lengths (but no negative-length cycles).In Lecture 14, we saw almost-linear-time algorithms for the case of planar and bounded-genus graphs. Currently, comparable bounds for minor-free graphs are not known. We shall discuss Goldberg's algorithm, a shortest-path algorithm for general graphs with integer lengths, whose running time depends logarithmically on the magnitude of the largest negative arc length. By exploiting separators (Lecture 6), it runs faster on minor-free graphs than on general graphs, but it still requires superlinear time.

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### Solving First-order Linear ODE's; Steady-state and Transient Solutions

00:50:22Arthur MattuckWe revisit the shortest paths problem, considering the case where the input is a directed minor-free graph with negative arc lengths (but no negative-length cycles).In Lecture 14, we saw almost-linear-time algorithms for the case of planar and bounded-genus graphs. Currently, comparable bounds for minor-free graphs are not known. We shall discuss Goldberg's algorithm, a shortest-path algorithm for general graphs with integer lengths, whose running time depends logarithmically on the magnitude of the largest negative arc length. By exploiting separators (Lecture 6), it runs faster on minor-free graphs than on general graphs, but it still requires superlinear time.

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### First-order Substitution Methods: Bernouilli and Homogeneous ODE's

00:50:11Arthur Mattuck - Play ►
### First-order Autonomous ODE's: Qualitative Methods, Applications

00:45:44Arthur Mattuck - Play ►
### Complex Numbers and Complex Exponentials

00:45:26Arthur Mattuck - Play ►
### First-Order Linear with Constant Coefficients

00:41:10Arthur Mattuck - Play ►
### Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models

00:50:35Arthur Mattuck - Play ►
### Solving Second-Order Linear ODE's with Constant Coefficients

00:49:58Arthur Mattuck - Play ►
### Complex Characteristic Roots; Undamped and Damped Oscillations

00:46:23Arthur Mattuck - Play ►
### Second-Order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians

00:50:31Arthur Mattuck - Play ►
### Inhomogeneous ODE's; Stability Criteria for Constant-Coefficient ODE's

00:46:24Arthur Mattuck - Play ►
### Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials

00:47:54Arthur Mattuck - Play ►
### Interpretation of the Exceptional Case: Resonance

00:44:25Arthur Mattuck - Play ►
### Introduction to Fourier Series; Basic Formulas for Period 2(pi)

00:49:31Arthur Mattuck - Play ►
### More General Periods; Even and Odd Functions; Periodic Extension

00:49:28Arthur Mattuck - Play ►
### Finding Particular Solutions via Fourier Series; Resonant Terms

00:45:44Arthur Mattuck - Play ►
### Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's

00:51:05Arthur Mattuck - Play ►
### Convolution Formula: Proof, Connection with Laplace Transform, Application

00:44:19Arthur Mattuck - Play ►
### Using Laplace Transform to Solve ODE's with Discontinuous Inputs

00:44:08Arthur Mattuck - Play ►
### Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions

00:44:54Arthur Mattuck - Play ►
### First-Order Systems of ODE's; Solution by Elimination, Geometric Interpretation

00:47:02Arthur Mattuck - Play ►
### Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues

00:49:05Arthur Mattuck - Play ►
### Continuation: Repeated Real Eigenvalues, Complex Eigenvalues

00:46:36Arthur Mattuck - Play ►
### Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients

00:50:25Arthur Mattuck - Play ►
### Matrix Methods for Inhomogeneous Systems

00:46:52Arthur Mattuck - Play ►
### Matrix Exponentials; Application to Solving Systems

00:48:51Arthur Mattuck - Play ►
### Decoupling Linear Systems with Constant Coefficients

00:47:05Arthur Mattuck - Play ►
### Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories

00:47:09Arthur Mattuck - Play ►
### Limit Cycles: Existence and Non-existence Criteria

00:45:52Arthur Mattuck - Play ►
### Non-Linear Systems and First-Order ODE's

00:50:09Arthur Mattuck