Origins: Brain and Self Organization

By Karl Pribram | Go to book overview

Appendix A: Constructing Green's Functions

There are several difficulties in constructing a Greens function for a given differential equation. First of all, the inverse does not necessarily exist. This shortcoming can be redeemed in many cases by some transformation. Even if the inverse exists, its construction is usually more difficult than finding solutions by using a properly chosen conventional trick. However, if the inverse exists, then the problem is equivalent to the task of finding or constructing the kernel, the Green's function of the integral equation (See e.g. G(x, t) in eq. [2.14]). There is more than one way to construct the Green's function associated with a differential equation. Each method may have corresponding physical meaning. Here we presented the one which is the easiest and the most revealing in terms of using the adjointness we have already introduced in our paper. Here we just further refine the basic concepts.

Definition: L* differential operator is formally adjoint to L if L and L* are associated with the following equation

f*(x) Lg(x) dx = [...] + ∫ f(x) L*g(x) dx (A.1)
Definition: L is formally self-adjoint if L = L*.

Definition: L* differential operator is adjoint to L if the associated differential equations of L has homogeneous boundary conditions, that if the eq. (A.1) takes the simple form
f*(x) Lg(x) dx = ∫ f(x) L*g(x) dx. (A.2)
or using the inner-product notation
(f, g) = ∫ f(x)*g(x)dx (A.3)
eq. (A.2) takes the form
(Lf,g) = (f, L*g). (A.4)

The key step to achieve the adjointness is to recognize the importance of elimination of the boundary terms in eq. (A.1). The very same idea leads us to the Green's function ( Greenberg, 1971; pp. 22-26). If we find a G function for a given g, for which
L*(G) = δ(x′-x) (A.5)
G(a, x) = G(x′, b) = 0 (A.6)

-91-

Notes for this page

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this page

Cited page

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited page

Bookmark this page
Origins: Brain and Self Organization
Table of contents

Table of contents

Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this book

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen
/ 718

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Author Advanced search

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.