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### Sets, Functions & Limits- Preface

00:32:06Herbert GrossWe 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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### Analytic Geometry

00:37:42Herbert GrossWe 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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### Inverse Functions

00:40:39Herbert GrossWe 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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### Sets, Functions & Limits- Derivatives and Limits

00:45:00Herbert Gross - Play ►
### A More Rigorous Approach to Limits

00:46:13Herbert Gross - Play ►
### Sets, Functions & Limits- Mathematical Inductions

00:29:24Herbert Gross - Play ►
### Derivatives of Some Simple Functions

00:28:17Herbert Gross - Play ►
### Approximations and Infinitesimals

00:34:36Herbert Gross - Play ►
### Composite Functions and the Chain Rule

00:39:16Herbert Gross - Play ►
### Differentiation of Inverse Functions

00:28:55Herbert Gross - Play ►
### Implicit Differentiation

00:39:58Herbert Gross - Play ►
### Differentiation- Continuity

00:22:49Herbert Gross - Play ►
### Differentiation- Curve Plotting

00:31:49Herbert Gross - Play ►
### Differentiation- Maxima and Minima

00:34:53Herbert Gross - Play ►
### Differentiation- Rolle's Theorem and its Consequences

00:30:28Herbert Gross - Play ►
### Differentiation- Inverse Differentiation

00:42:59Herbert Gross - Play ►
### Differentiation- The "Definite" Indefinite Integral

00:29:16Herbert Gross - Play ►
### The Circular Functions

00:36:01Herbert Gross - Play ►
### Inverse Circular Functions

00:26:09Herbert Gross - Play ►
### The Definite Integral

00:36:37Herbert Gross - Play ►
### Marriage of Differential and Integral Calculus

00:30:31Herbert Gross - Play ►
### Three-Dimensional Area

00:42:06Herbert Gross - Play ►
### One-Dimensional Area

00:36:45Herbert Gross - Play ►
### Logarithms without Exponents

00:34:46Herbert Gross - Play ►
### Inverse Logarithms

00:21:37Herbert Gross - Play ►
### What a Difference a Sign Makes

00:27:43Herbert Gross - Play ►
### Inverse Hyperbolic Functions

00:29:55Herbert Gross - Play ►
### More Integration Techniques- Some Basic Recipes

00:30:29Herbert Gross - Play ►
### More Integration Techniques- Partial Functions

00:32:29Herbert Gross - Play ►
### More Integration Techniques- Integration by Parts

00:27:01Herbert Gross - Play ►
### More Integration Techniques- Improper Integrals

00:29:39Herbert Gross - Play ►
### Infinite Series- Positive Series

00:34:50Herbert Gross - Play ►
### Infinite Series- Absolute Convergence

00:21:09Herbert Gross - Play ►
### Infinite Series- Polynomial Approximations

00:32:42Herbert Gross - Play ►
### Infinite Series- Uniform Convergence

00:28:57Herbert Gross - Play ►
### Infinite Series- Uniform Convergence of Power Series

00:27:04Herbert Gross

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