Канал: Zachary Kilpatrick

Apr 23: Wandering Bumps in Neural Fields

Apr 23: Wandering Bumps in Neural Fields

Apr 21: Amplitude Equation for Neural Pattern Formation

Apr 21: Amplitude Equation for Neural Pattern Formation

Apr 19: Neural Field Equations

Apr 19: Neural Field Equations

Apr 14: Multidimensional Homogenization Ctd

Apr 14: Multidimensional Homogenization Ctd

Apr 16: Homogenization and Averaging for Traveling Waves

Apr 16: Homogenization and Averaging for Traveling Waves

Apr 12: Homogenization: Multidimensional Elliptic Case

Apr 12: Homogenization: Multidimensional Elliptic Case

Apr 9: Intro to Homogenenization

Apr 9: Intro to Homogenenization

Apr 7: WKB -- Wave Propagation & Energy Methods

Apr 7: WKB -- Wave Propagation & Energy Methods

Apr 5: WKB Turning Points - Pt2

Apr 5: WKB Turning Points - Pt2

Mar 31: WKB Turning Points - Pt1

Mar 31: WKB Turning Points - Pt1

Mar 29: WKB Approx -- Intro

Mar 29: WKB Approx -- Intro

Mar 24: Nonlinear Diffusion

Mar 24: Nonlinear Diffusion

Mar 22: Multiple Scales and Wave Turbulence

Mar 22: Multiple Scales and Wave Turbulence

Mar 19: Linear + Nonlinear Waves -- Multiple Scales

Mar 19: Linear + Nonlinear Waves -- Multiple Scales

Mar 17: Mathieu Eqn -- Stability Boundaries

Mar 17: Mathieu Eqn -- Stability Boundaries

Mar 15: Rayleigh Oscillator -- Approach to a Limit Cycle

Mar 15: Rayleigh Oscillator -- Approach to a Limit Cycle

Mar 12: Forcing Near Resonance

Mar 12: Forcing Near Resonance

Mar 8: Matched Asymptotics for PDE. Intro to Multiple Scales.

Mar 8: Matched Asymptotics for PDE. Intro to Multiple Scales.

Mar 5: Interior + Corner Layers

Mar 5: Interior + Corner Layers

Mar 3: Initial and Multiple Layers; Interior Layers Intro

Mar 3: Initial and Multiple Layers; Interior Layers Intro

Mar 1: Boundary Layer Examples

Mar 1: Boundary Layer Examples

Feb 26: Matched Asymptotics -- One Boundary Layer

Feb 26: Matched Asymptotics -- One Boundary Layer

Feb 24: Intro to Matched Asymptotics

Feb 24: Intro to Matched Asymptotics

Feb 22 (Pt2): Method of Steepest Descent (Last Example)

Feb 22 (Pt2): Method of Steepest Descent (Last Example)

Feb 22: Method of Steepest Descent Examples

Feb 22: Method of Steepest Descent Examples

Feb 19: Method of Steepest Descent

Feb 19: Method of Steepest Descent

Feb 15: Long Timescale Asymptotics & Method of Steepest Descent

Feb 15: Long Timescale Asymptotics & Method of Steepest Descent

Feb 12: Method of Stationary Phase

Feb 12: Method of Stationary Phase

Feb 10: Laplace's Method -- Additional Examples

Feb 10: Laplace's Method -- Additional Examples

Feb 8: Laplace's Method

Feb 8: Laplace's Method

Feb 5: Watson's Lemma

Feb 5: Watson's Lemma

Feb 3: Integration by Parts

Feb 3: Integration by Parts

Feb 1: Intro to Asymptotics for Integrals

Feb 1: Intro to Asymptotics for Integrals

Jan 29: Asymptotic Series -- Divergence & Remainders

Jan 29: Asymptotic Series -- Divergence & Remainders

Jan 20: Rescaling & Non-Integral Powers

Jan 20: Rescaling & Non-Integral Powers

Jan 27: Asymptotic Order

Jan 27: Asymptotic Order

Jan 25: Eigenvalue Problems

Jan 25: Eigenvalue Problems

Jan 22: Logarithms & Convergence: Iterative Methods

Jan 22: Logarithms & Convergence: Iterative Methods

Jan 15: Algebraic Eqns; Regular/Singular Expansions

Jan 15: Algebraic Eqns; Regular/Singular Expansions

Nov 23: Sine Gordon Practice Problem

Nov 23: Sine Gordon Practice Problem

Nov 20 (Pt6): Bistable Eqn with the Cubic Nonlinearity

Nov 20 (Pt6): Bistable Eqn with the Cubic Nonlinearity

Nov 20 (Pt5): Traveling Waves in the Bistable Eqn

Nov 20 (Pt5): Traveling Waves in the Bistable Eqn

Nov 20 (Pt4): Intro to Bistable Eqn

Nov 20 (Pt4): Intro to Bistable Eqn

Nov 20 (Pt1): Intro to Fishers Eqn

Nov 20 (Pt1): Intro to Fishers Eqn

Nov 20 (Pt3): Traveling Waves in Fisher's Eqn

Nov 20 (Pt3): Traveling Waves in Fisher's Eqn

Nov 20 (Pt2): Stationary Waves in Fishers Eqn

Nov 20 (Pt2): Stationary Waves in Fishers Eqn

Nov 18 (Pt3): KdV Soliton Phase Plane

Nov 18 (Pt3): KdV Soliton Phase Plane

Nov 18 (Pt2): KdV Dispersion Relation

Nov 18 (Pt2): KdV Dispersion Relation

Nov 18 (Pt1): Viscous Burgers Review/KdV Eqn Intro

Nov 18 (Pt1): Viscous Burgers Review/KdV Eqn Intro

Nov 16: Greens Fn Problem

Nov 16: Greens Fn Problem

Nov 11 (Pt3): Nonlinear Waves Intro

Nov 11 (Pt3): Nonlinear Waves Intro

Nov 11 (Pt4): Viscous Burgers

Nov 11 (Pt4): Viscous Burgers

Nov 11 (Pt5): Inviscid Burgers -- Shock Solution

Nov 11 (Pt5): Inviscid Burgers -- Shock Solution

Nov 11 (Pt1): Method of Images: Neumann Boundary

Nov 11 (Pt1): Method of Images: Neumann Boundary

Nov 11 (Pt2): Poisson Eqn -- Neumann Boundary

Nov 11 (Pt2): Poisson Eqn -- Neumann Boundary

Nov 9 (Pt1): Greens Function -- Real Line

Nov 9 (Pt1): Greens Function -- Real Line

Nov 9 (Pt2): Method of Images -- Unit Ball

Nov 9 (Pt2): Method of Images -- Unit Ball

Nov 9 (Pt3): Laplace Eqn -- Unit Ball

Nov 9 (Pt3): Laplace Eqn -- Unit Ball

Nov 6 (Pt2): Poisson Eqn: Greens Function Soln

Nov 6 (Pt2): Poisson Eqn: Greens Function Soln

Nov 6 (Pt1): Greens Function Setup

Nov 6 (Pt1): Greens Function Setup

Nov 6 (Pt3): Method of Images

Nov 6 (Pt3): Method of Images

Nov 4 (Pt1): Properties of Distributions

Nov 4 (Pt1): Properties of Distributions

Nov 4 (Pt2): Shock Waves/Rankie-Hugoniot

Nov 4 (Pt2): Shock Waves/Rankie-Hugoniot

Nov 4 (Pt3): Greens Functions Continued

Nov 4 (Pt3): Greens Functions Continued

Oct 30 (Pt2): Weak Max Principle Proof

Oct 30 (Pt2): Weak Max Principle Proof

Oct 30 (Pt1): Heat Eqn: Weak Max Principle Theorem

Oct 30 (Pt1): Heat Eqn: Weak Max Principle Theorem

Oct 30 (Pt1): Greens Function Properties

Oct 30 (Pt1): Greens Function Properties

Oct 30 (Pt3): Test Functions and Distributions

Oct 30 (Pt3): Test Functions and Distributions

Oct 30 (Pt2): Greens Function Boundaries

Oct 30 (Pt2): Greens Function Boundaries

Oct 28 (Pt2): Poisson Kernel Remarks/Intro to Greens Functions

Oct 28 (Pt2): Poisson Kernel Remarks/Intro to Greens Functions

Oct 28 (Pt3): 1D Poisson Equation

Oct 28 (Pt3): 1D Poisson Equation

Oct 28 (Pt1): Deriving the Poisson Kernel

Oct 28 (Pt1): Deriving the Poisson Kernel

Oct 26 (Pt3): Separation of Variables: Laplace's Eqn

Oct 26 (Pt3): Separation of Variables: Laplace's Eqn

Oct 26 (Pt2): Bounding and Uniqueness: Laplace's Eqn

Oct 26 (Pt2): Bounding and Uniqueness: Laplace's Eqn

Oct 26 (Pt1): Maximum Principle: Laplace

Oct 26 (Pt1): Maximum Principle: Laplace

Oct 23 (Pt2): Mean Value Property (MVP)

Oct 23 (Pt2): Mean Value Property (MVP)

Oct 23 (Pt1): Green's ID 1: Warmup

Oct 23 (Pt1): Green's ID 1: Warmup

Oct 23 (Pt3): MVP ctd and Max Principle

Oct 23 (Pt3): MVP ctd and Max Principle

Oct 21 (Pt3): Poisson Eqn Fund Soln 2

Oct 21 (Pt3): Poisson Eqn Fund Soln 2

Oct 21 (Pt1): Laplace Eqn Fundamental Soln

Oct 21 (Pt1): Laplace Eqn Fundamental Soln

Oct 21 (Pt2): Poisson Eqn Fund Soln

Oct 21 (Pt2): Poisson Eqn Fund Soln

Oct 16 (Pt2): Solve Forced Heat Eqn: Sep of Vars

Oct 16 (Pt2): Solve Forced Heat Eqn: Sep of Vars

Oct 16 (Pt1): Forced Heat Eqn Setup

Oct 16 (Pt1): Forced Heat Eqn Setup

Oct 19 (Pt2): Symmetries of Laplace's Eqn

Oct 19 (Pt2): Symmetries of Laplace's Eqn

Oct 14 (Pt2): Fourier Series L2 Completeness

Oct 14 (Pt2): Fourier Series L2 Completeness

Oct 19 (Pt1): Examples of Laplace's Eqn

Oct 19 (Pt1): Examples of Laplace's Eqn

Oct 16 (Pt2): Uniqueness of Solns to Wave Eqn

Oct 16 (Pt2): Uniqueness of Solns to Wave Eqn

Oct 16 (Pt1): Wave Eqn on an Inhomogeneous String

Oct 16 (Pt1): Wave Eqn on an Inhomogeneous String

Oct 14 (Pt3): Gibb's Phenomenon

Oct 14 (Pt3): Gibb's Phenomenon

Oct 14 (Pt1): Symmetric SL Problems

Oct 14 (Pt1): Symmetric SL Problems

Oct 12 (Pt1): Sturm-Liouville & Self-Adjointness

Oct 12 (Pt1): Sturm-Liouville & Self-Adjointness

Oct 12 (Pt2): Sturm-Liouville Eigenvalues

Oct 12 (Pt2): Sturm-Liouville Eigenvalues

Oct 12 (Pt3): L2 convergence

Oct 12 (Pt3): L2 convergence

Oct 9 (Pt2): Implicit Eigenvalue Eqn

Oct 9 (Pt2): Implicit Eigenvalue Eqn

Oct 9 (Pt1): Heat Eqn -- Robin BCs

Oct 9 (Pt1): Heat Eqn -- Robin BCs

Oct 9 (Pt3): Roots of Eval Eqn

Oct 9 (Pt3): Roots of Eval Eqn

Oct 9 (Pt4): Intro to Sturm-Liouville Theory

Oct 9 (Pt4): Intro to Sturm-Liouville Theory

Oct 9 (Pt3): Robin Boundaries + Inhomogeneous Heat Eqns

Oct 9 (Pt3): Robin Boundaries + Inhomogeneous Heat Eqns

Oct 9 (Pt2): Heat Eqn Convergence

Oct 9 (Pt2): Heat Eqn Convergence

Oct 9 (Pt1): M Test / Convergence

Oct 9 (Pt1): M Test / Convergence

Oct 7 (Pt3): Initial Condition Convergence

Oct 7 (Pt3): Initial Condition Convergence

Oct 7 (Pt2): Check FS Satisfies PDE

Oct 7 (Pt2): Check FS Satisfies PDE

Oct 7 (Pt1): Heat Eqn: Fourier Series Coeffs

Oct 7 (Pt1): Heat Eqn: Fourier Series Coeffs

Oct 2 (Pt3): Wave Quarter Plane Energy

Oct 2 (Pt3): Wave Quarter Plane Energy

Oct 2 (Pt2): Wave Quarter Plane Solution

Oct 2 (Pt2): Wave Quarter Plane Solution

Oct 2 (Pt1): Wave Quarter Plane: Domain of Dependence

Oct 2 (Pt1): Wave Quarter Plane: Domain of Dependence

Oct 2 (Pt4): Separation of Variables: Heat Eqn Intro

Oct 2 (Pt4): Separation of Variables: Heat Eqn Intro

Oct 2 (Pt2): Heat Eqn Uniqueness (w/Growth Condition)

Oct 2 (Pt2): Heat Eqn Uniqueness (w/Growth Condition)

Oct 2 (Pt3): Heat Eqn Uniqueness (w/o Growth Condition)

Oct 2 (Pt3): Heat Eqn Uniqueness (w/o Growth Condition)

Oct 2 (Pt1): Heat Eqn Inhomogeneous Soln

Oct 2 (Pt1): Heat Eqn Inhomogeneous Soln

Sept 30 (Pt2): Maximum Principle -- Proof

Sept 30 (Pt2): Maximum Principle -- Proof

Sept 30 (Pt3): Heat Eqn -- Continuous Dependence & Uniqueness

Sept 30 (Pt3): Heat Eqn -- Continuous Dependence & Uniqueness

Sept 30 (Pt1): Maximum Principle -- Remarks

Sept 30 (Pt1): Maximum Principle -- Remarks

Sept 28 (Pt2): Fourier Transform Examples

Sept 28 (Pt2): Fourier Transform Examples

Sept 28 (Pt3): Maximum Principle Setup

Sept 28 (Pt3): Maximum Principle Setup

Sept 28 (Pt1): Fourier Transform Definition

Sept 28 (Pt1): Fourier Transform Definition

Sept 25 (Pt4): Decaying Diffusion Eqn 2

Sept 25 (Pt4): Decaying Diffusion Eqn 2

Sept 25 (Pt3): Decaying Diffusion Eqn 1

Sept 25 (Pt3): Decaying Diffusion Eqn 1

Sept 25 (Pt2): Duhamel's Principle Question

Sept 25 (Pt2): Duhamel's Principle Question

Sept 25 (Pt1): Heat Quarter Plane Question

Sept 25 (Pt1): Heat Quarter Plane Question

Sept 25 (Pt3): Heat Soln Uniqueness on R and in n-dims

Sept 25 (Pt3): Heat Soln Uniqueness on R and in n-dims

Sept 25 (Pt2): Heat Soln Uniqueness: Bdd Domain

Sept 25 (Pt2): Heat Soln Uniqueness: Bdd Domain

Sept 25 (Pt1): Heat Soln Recap

Sept 25 (Pt1): Heat Soln Recap

Sept 23 (Pt3): Heat Soln Remarks

Sept 23 (Pt3): Heat Soln Remarks

Sept 23 (Pt4): Wave Eqn Soln Properties

Sept 23 (Pt4): Wave Eqn Soln Properties

Sept 23 (Pt1): Heat Soln Properties 1 + 2

Sept 23 (Pt1): Heat Soln Properties 1 + 2

Sept 23 (Pt2): Heat Soln -- Delta Fn Property

Sept 23 (Pt2): Heat Soln -- Delta Fn Property

Sept 21 (Pt2): Heat Eqn: Fundamental Soln

Sept 21 (Pt2): Heat Eqn: Fundamental Soln

Sept 21 (Pt3): Properties of Fundamental Soln

Sept 21 (Pt3): Properties of Fundamental Soln

Sept 21 (Pt1): Self Similar ODE

Sept 21 (Pt1): Self Similar ODE

Sept 18 (Pt3): Wave Soln Jump Discontinuity

Sept 18 (Pt3): Wave Soln Jump Discontinuity

Sept 18 (Pt2): Radially Symmetric Wave Soln

Sept 18 (Pt2): Radially Symmetric Wave Soln

Sept 18 (Pt1): Waves in R3 Problem

Sept 18 (Pt1): Waves in R3 Problem

Sept 18 (Pt1): Waves in n-dimensions: Euler-Poisson-Darboux Equation

Sept 18 (Pt1): Waves in n-dimensions: Euler-Poisson-Darboux Equation

Sept 16 (Pt1): Duhamel's Principle

Sept 16 (Pt1): Duhamel's Principle

Sept 16 (Pt3): Waves in Higher Dimensions

Sept 16 (Pt3): Waves in Higher Dimensions

Sept 18 (Pt2): Wave Eqn Summary

Sept 18 (Pt2): Wave Eqn Summary

Sept 18 (Pt3): Heat Eqn Intro

Sept 18 (Pt3): Heat Eqn Intro

Sept 16 (Pt2): Duhamel's Summary

Sept 16 (Pt2): Duhamel's Summary

Sept 14 (Pt1): Conservation of Energy -- Wave Equation

Sept 14 (Pt1): Conservation of Energy -- Wave Equation

Sept 14 (Pt4): HW Questions

Sept 14 (Pt4): HW Questions

Sept 14 (Pt3): Duhamel's Principle

Sept 14 (Pt3): Duhamel's Principle

Sept 14 (Pt2): Uniqueness -- Wave Equation

Sept 14 (Pt2): Uniqueness -- Wave Equation

Sept 11 (Pt3): Wave Eqn in Quarter Plane

Sept 11 (Pt3): Wave Eqn in Quarter Plane

Sept 11 (Pt2): Support + Domain of Dependence

Sept 11 (Pt2): Support + Domain of Dependence

Sept 9 (Pt2): d'Alembert's Formula

Sept 9 (Pt2): d'Alembert's Formula

Sept 11 (Pt1): Continuous Dependence of Solns to Wave Eqn

Sept 11 (Pt1): Continuous Dependence of Solns to Wave Eqn

Sept 9 (Pt1): Wave Eqn String Derivation Ctd

Sept 9 (Pt1): Wave Eqn String Derivation Ctd

Sept 7: Meth of Chars Ex 3

Sept 7: Meth of Chars Ex 3

Sept 4 (Pt4): Derive Wave Eqn from String

Sept 4 (Pt4): Derive Wave Eqn from String

Sept 4 (Pt3): Wave Eqn Characteristics

Sept 4 (Pt3): Wave Eqn Characteristics

Sept 2 (Pt2): Method of Chars -- Ex. 1

Sept 2 (Pt2): Method of Chars -- Ex. 1

Sept 4 (Pt2): Shocks

Sept 4 (Pt2): Shocks

Sept 2 (Pt1): General Method of Characteristics

Sept 2 (Pt1): General Method of Characteristics

Sept 2 (Pt3): Method of Chars -- Ex.2

Sept 2 (Pt3): Method of Chars -- Ex.2

Sept 4 (Pt1): Method of Chars. -- Ex.2

Sept 4 (Pt1): Method of Chars. -- Ex.2

Aug 31 (Pt3): Method of Characteristics Motivation

Aug 31 (Pt3): Method of Characteristics Motivation

Aug 31 (Pt1): Warm Up + Implicit Fn Thm

Aug 31 (Pt1): Warm Up + Implicit Fn Thm

Aug 31 (Pt4): Q+A

Aug 31 (Pt4): Q+A

Aug 31 (Pt2): Transport Eqn Characteristics

Aug 31 (Pt2): Transport Eqn Characteristics

Aug 28 (Pt4): Transport Eqn Ex

Aug 28 (Pt4): Transport Eqn Ex

Aug 28 (Pt1): Classifying 2nd Order PDE

Aug 28 (Pt1): Classifying 2nd Order PDE

Aug 28 (Pt3): Balance Laws 2

Aug 28 (Pt3): Balance Laws 2

Aug 28 (Pt2): Balance Laws 1

Aug 28 (Pt2): Balance Laws 1

Aug 26 (Pt2): Well-Posedness: Uniqueness

Aug 26 (Pt2): Well-Posedness: Uniqueness

Aug 26 (Pt3): Well-Posedness: Continuous Dependence

Aug 26 (Pt3): Well-Posedness: Continuous Dependence

Aug 26 (Pt1): Linear Transport Waves

Aug 26 (Pt1): Linear Transport Waves

Aug 24 (Pt5): Question about Transport Equation

Aug 24 (Pt5): Question about Transport Equation

Aug 24 (Pt4): Notation + Transport Eqn Solns

Aug 24 (Pt4): Notation + Transport Eqn Solns

Aug 24 (Pt3): Main Questions

Aug 24 (Pt3): Main Questions

Aug 24 (Pt2): Warm Up on Transport Eqn

Aug 24 (Pt2): Warm Up on Transport Eqn

Aug 24 (Pt1): Syllabus

Aug 24 (Pt1): Syllabus

Reservoir computing (or training recurrent neural networks)

Reservoir computing (or training recurrent neural networks)

Homogenizing traveling waves in inhomogeneous media

Homogenizing traveling waves in inhomogeneous media

Motivating the sigmoid in feedforward neural networks

Motivating the sigmoid in feedforward neural networks

Two layer perceptron: Backpropagation (part 2)

Two layer perceptron: Backpropagation (part 2)

Two layer perceptron: Backpropagation (part 1)

Two layer perceptron: Backpropagation (part 1)

Backpropagation intro & simple example

Backpropagation intro & simple example

Single Layer Perceptron: Logical operations

Single Layer Perceptron: Logical operations

Two layer perceptron

Two layer perceptron

"Keeping up with the Joneses' Opinions", ZP Kilpatrick: Press Talk at APS 20

"Keeping up with the Joneses' Opinions", ZP Kilpatrick: Press Talk at APS 20

Bayesian evidence accumulation on social networks: APS 20 (ZPK, Karamched, Stolarczyk, Josić)

Bayesian evidence accumulation on social networks: APS 20 (ZPK, Karamched, Stolarczyk, Josić)