text:
Conditional Operator

Welcome! This is a lesson in the Introductory Discrete Mathematics for Computer Science course here at Wikiversity.

Previous lesson: Logical XOR

Your Next-to-Last Operator!
The conditional operator looks like this: $$\to$$

It is a dyadic operator.

Perhaps I should explain conditional operators through a story. Let's suppose that Joe Lion is given a wildebeest if he beats Handsome Dan XVII. Thus, $$p$$ becomes "Joe Lion beats Handsome Dan" and $$q$$ becomes "Joe Lion is given a wildebeest." The order of p and q are important, and so p and q are given special terms when the conditional operator is used. p is the antecedent. q is the consequent.

There are four possible outcomes in this story:

1. Joe Lion beats Handsome Dan, and is given a wildebeest. (p is true and q is true) 2. Joe Lion beats Handsome Dan, but is not given a wildebeest. (p is true and q is false) 3. Joe Lion does not beat Handsome Dan, yet is given a wildebeest. (p is false and q is true) 4. Joe Lion does not beat Handsome Dan, and is not given a wildebeest. (p is false and q is false)
 * $$p$$ $$\to$$ $$q$$ is true.
 * $$p$$ $$\to$$ $$q$$ is false. In other words, the condition (that Joe Lion is given a wildebeest if he beats Handsome Dan) is violated.
 * $$p$$ $$\to$$ $$q$$ is true. The conditional statement (that Joe Lion is given a wildebeest if he beats Handsome Dan) has not been violated, because violation only occurs if the antecedent is true.
 * $$p$$ $$\to$$ $$q$$ is true. The conditional statement (that Joe Lion is given a wildebeest if he beats Handsome Dan) has not been violated, because violation only occurs if the antecedent is true.

Next Lesson
The next lesson is called Biconditional Operator.

title: Conditional Operator


text:
User talk:Brad123

 Hello Brad123, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --mikeu talk 17:04, 23 January 2009 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Brad123


text:
Biconditional Operator

Welcome! This is a lesson in the Introductory Discrete Mathematics for Computer Science course here at Wikiversity.

Previous lesson: Conditional Operator

Your Last Operator!
The biconditional operator looks like this: $$\leftrightarrow$$

It is a dyadic operator. You'll learn about what it does in the next section.

Compound Propositions and Logical Equivalence
Now you will be introduced to the concepts of logical equivalence and compound propositions.
 * Compound propositions involve the assembly of multiple statements, using multiple operators.
 * Logical equivalence means that the truth tables of two statements are the same.

The biconditional operator is sometimes called the "if and only if" operator. $$p \leftrightarrow q$$ = TRUE means that the truth values of p and q are the same. "You will see the notes for this class if and only if someone shows them to you" is an example of a biconditional statement.
 * 1) If someone shows you the notes and you see them, the statement is true.
 * 2) If someone shows you the notes and you do not see them, the biconditional statement is violated. Therefore, a value of "false" is returned.
 * 3) If no one shows you the notes and you see them, the biconditional statement is violated. Therefore, a value of "false" is returned.
 * 4) If no one shows you the notes and you do not see them, a value of true is returned.

 * The biconditional statement $$p \leftrightarrow q$$ is logically equivalent to $$\neg(p \oplus q)$$!

Next Lesson
The next lesson is called Compound Propositions and Useful Rules.

title: Biconditional Operator


text:
User:Bip014

Business and Its Publics - Spring 2009 - Section .014
Instructions: For terms not contained in the Course Glossary:

1. Click on "edit this page" (above)

2. One student will enter a business term, followed by his/her name and Cohort name. For example, EPS (J. Kim, Gramercy).

3. Another student will provide the definition for the term, followed by his/her name and Cohort name. For example: Earnings per share; a company's earnings divided by... (A. Young, Midtown)

4. Professors from Inquiry and Discourse sessions may occasionally enter terms that need definitions.

Course Glossary
Markup (Sanghee Lee, Gramercy)

Definition: the difference between the cost of a good or service and its selling price. A markup is added on to the total cost incurred by the producer of a good or service in order to create a profit (Li Shen, Gramercy).

Kerb Market (NamHyuk Steve Chung, Gramercy)

Definition: In financial markets the term kerb is used to describe trading outside official market hours. The expression comes from trading literally taking place on the kerb outside the stock exchange (Li Shen, Gramercy).

Disintermediation (Li Shen, Gramercy)

Definition: The removal of funds from a financial institution such as a bank for direct purchase of financial instruments; "cutting out the middleman" (Sanghee Lee, Gramercy)

Giffen Good (NamHyuk Steve Chung, Gramercy)

A Giffen Good is a good that exeriences increased demand for when the price rises and decreased demand for when the price falls. Mainly a theoretical concept, but there have arguably been goods with this property. (Dongmin Kim, Gramercy)

Articles of Incorporation (Dongmin Kim, Gramercy)

Definition: The primary rules governing the management of a corporation in the United States. They vary widely from corporation to corporation and jurisdiction to jurisdiction (Ben Granas, Gramercy)

Write-Down (Ben Granas, Gramercy)

Definition: To reduce the value at which an asset is carried on the books to reflect a decline in value. (NamHyuk Steve Chung, Gramercy)

Liquidity (Ben Granas, Gramercy)

Definition: The ability of an investment to be easily converted into cash with little or no loss of capital and minimum delay. (NamHyuk Steve Chung, Gamercy)

Slippage (Li Shen, Gramercy)

Definition: the difference between estimated transaction costs and the amount actually paid. (Sanghee Lee, Gramercy)

Social Return on Investment, or SROI (Sanghee Lee, Gramercy)

Definition: SROI is an approach to understanding and managing the impacts of a project, an organization or a policy. It is based on stakeholders and puts financial values on the important impacts identified by stakeholders which do not have market values (Li Shen, Gramercy).

Adverse Selection (NamHyuk Steve Chung, Gramercy)

Definition: The tendency of those in dangerous jobs or high risk lifestyles to get life insurance. (Dongmin Kim, Gramercy).

Microstructuring (Li Shen, Gramercy)

A method of money laundering in which larger transactions are broken down into multiple smaller transactions to evade detection by authorities. For example, making multiple (under US$10,000) deposits in banks and withdrawing it in overseas countries. (Li Shen, Gramercy)

title: User:Bip014


text:
User:Egm6322.mafia

=General Non-Linear PDEs=

Defining Variables
For these notes, we will assume we have n independent variables:

$$  \left \{ {x}_{i} \right \}=   \left \{ {x}_{1},...,{x}_{n}   \right \} $$.

An example of this would be single dimensional case with a time variable.

' $$ (  x,t )=( {x}_{1}, { x}_{2 } )  $$'

where $$ {x}_{1} = {x}  $$, a spatial variable

and $$ {x}_{2} = {t}  $$, a temporal variable.

A more complex situation would be a three dimensional, time dependent problem:

$$(x,y,z,t)=({x}_{1},...,{x}_{4})$$.

In this case, $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

Definition of Functions
The unknown function $$ u $$ maps numbers from the domain $$ \Omega $$ to the real numbers $$ \mathbb{R} $$ In two dimensions, this means:

$$ \left( x,y \right) \in \Omega $$

$$ u \left( x,y \right) \in \mathbb{R} $$

and

$$ \Omega \rightarrow \mathbb{R}$$

$$ \left( x,y \right) \mapsto u \left( x,y \right) $$

which means that $$\Omega $$ is the domain of u, $$ \mathbb{R}$$ is the range of u, and $$u \left ( \Omega \right )$$ is the image of $$\Omega $$ under the mapping of $$ u $$.

Example Functions
For these notes, we will assume we have unknown functions: $$ {u}_{1},  {u}_{2} ,...  $$.

The Navier Stokes Equation
An example of this is the Navier-Stokes Equations in 3-D:

$$ {u}_{i} ( \left \{  {x}_{j} \right \} ) $$

where $$ {u}_{i} $$ is the velocity field in x, y, z; $$ {u}_{1}, {u}_{2}, {u}_{3} $$, dependent upon variables $$ {x}_{1}, ... {x}_{4}$$, where $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

One Unknown Function
For our present discussion, this can be restricted to one unknown function $$ u $$, containing $$ n $$ independent variables $$ {{x}_{i}} i=1,...,n $$

The mth partial derivative of this function can be expressed as:

$$ \frac{\partial^m u}{\partial x_i\ ,...,\partial x_m} $$ where

$$ {{i}_{1},...,{i}_{m}} $$ is the subset of m indices among n possible indices

$$ {i}_{i}, ..., {i}_{m}=1,...,n $$.

Another Example
Another example of a non-linear PDE is:

$$ F \left ( \left \{ {x}_{i} \right \}, \left \{ \frac{\partial u}{\partial x_i} \right \}, \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}, ... \right ) =0$$

where $$ \left \{ {x}_{i} \right \}$$ contains $$ n $$ arguments,

$$ \left \{ \frac{\partial u}{\partial x_i} \right \}$$ are the components of the $$ \nabla u$$, and

$$ \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$ are the components of the Hessian of $$ u $$.

The Hessian
The Hessian is a symmetric $$ n \times n $$ matrix, thus H = HT

and $${H}_{n \times n}$$ := $$ {\left [ {H}_{ij}\right ]}_{n \times n}$$.

The Hessian is defined as:

$$ {H}_{ij} := \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$.

This has many interesting properties.

The Laplace Equation
The Laplace Equation is a single function of two (or more) variables:

$$ F \left ( \left (x,y \right) ,u,\left ({u}_{x},{u}_{y} \right), \left ({u}_{xx},{u}_{xy},{u}_{yy} \right ), ...   \right )=0 $$

Example of the Laplace Equation include:

$$2{u}_{xx}+3{u}_{yy}=a{x}^{2}+bx$$

$${u}_{xx} +{u}_{yy}=0$$

The Definition of Linearity
If $$ u $$ is an unknown function, the operator $$ L $$ is considered a linear operator with respect to $$ u $$ if:

$$ L \left ( \alpha u + \beta v \right ) = \alpha L \left ( u \right )+ \beta L \left ( v \right ) $$

Example of Checking for Linearity
Let's take the function:

$$2{u}_{xx}+3{u}_{yy}-7{x}^{2}+x=0$$

The operator is :

$$ L= 2 \frac{\partial^2 }{\partial x^2} + 3 \frac{\partial^2 }{\partial y^2} -7 x^2-x$$

where $$-7 x^2-x:= f(x)$$

To check the linearity, we assume $$\alpha$$ and $$\beta$$ are real number constants, and evaluate: $$ L(\alpha u+ \beta v)= 2 \frac{\partial^2 (\alpha u+ \beta v)}{\partial x^2} + 3 \frac{\partial^2 (\alpha u+ \beta v)}{\partial y^2} + f(x)$$

which is equivalent to:

$$ L(\alpha u+ \beta v)=2 {\left (\alpha u + \beta v \right )}_{xx}+3 {\left (\alpha u +\beta v\right )}_{yy}+f(x) $$

If we distribute the integers, as well as the second derivative operators, the equation becomes:

$$ L(\alpha u+ \beta v)=2 \alpha {u}_{xx}+2 \beta {v}_{xx}+3 \alpha {u}_{yy} + 3 \beta {v}_{yy}+f(x) $$.

Grouping the terms by the constants $$\alpha$$ and $$\beta$$ concludes:

$$ L(\alpha u+ \beta v)= \alpha \left [2 {u}_{xx}+3{u}_{yy} \right ] +\beta \left [ 2 {v}_{xx} +3 {v}_{yy} \right ] +f(x) $$.

The operator $$ L $$ is nonlinear because

$$ \alpha L(u)+ \beta L(v)= \alpha \left [2 {u}_{xx} +3 {u}_{yy} +f(x) \right ] +\beta \left [2 {v}_{xx}+3 {v}_{yy} + f(x) \right ] $$

and

$$ f(x) \ne \alpha f(x) +\beta f(x) $$

 * $$J=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n} \end{bmatrix}. $$

Definition of the order of PDE
Order of highest derivative in the PDE.

Example of first order PDE
linear: $$5u_x-7u_y=0$$

non-linear: $$6(u_x)^3+2(u_y)^2+(u)^{1/2}+x^2+sin xy=0$$

we can assume that $$D(u)=6(u_x)^3+2(u_y)^2+(u)^{1/2}$$

Homework
Show D(u) is not linear.

(Solution)

Again, $$D \left( u \right) = 6 \left( u_x \right)^3 + 2 \left( u_y \right)^2 + \left( u \right)^{1/2}$$.

D(u) is linear if the following formula holds:

$$D \left( \alpha\cdot u + \beta\cdot v \right) = \alpha\cdot D \left( u \right) + \beta\cdot D \left( v \right) $$

OR,

$$6 \left( \alpha\cdot u + \beta\cdot v \right)^3 + 2 \left( \alpha\cdot u + \beta\cdot v  \right)^2 +  \left( \alpha\cdot u + \beta\cdot v  \right)^{1/2} = \alpha\cdot \left [ 6 \left( u_x  \right)^3 + 2 \left( u_y  \right)^2 +  \left( u  \right)^{1/2} \right ] + \beta\cdot \left [ 6 \left( v_x  \right)^3 + 2 \left( v_y  \right)^2 +  \left( v  \right)^{1/2} \right ]$$

The expressions on either side of the equality sign in the above equation are NOT identical. Therefore, D(u) is NOT linear.

Example of second order PDE
$$div(gradu)+f(x,y)=0$$

Notice: tensorial notation = coord free notation.

Cartesian Coordinate (x,y)
Let's take the function:

$$u_xx+u_yy+f(x,y)=0$$

We can also write it as div( $$\kappa$$ $$\cdot$$$$grad u$$$$+f(x,y)=0$$

Here, $$\kappa$$ is a continuity tensor (2nd order tensor).

$$\kappa$$ = $$\kappa$$ $$e_i$$ $$\otimes$$ $$e_j$$ 2nd-order tensor

$$v$$ = $$v_i$$ $$e_i$$ vector,1st-order tensor

Operators
$$grad$$ $$u$$ $$=$$ $$\frac{\partial u}{\partial x_i}$$ $$e_i$$

$$div$$ $$v$$ $$=$$ $$\frac{\partial v_i}{\partial x_i}$$ $$e_i$$

$$\kappa$$ $$\cdot$$ $$grad$$ $$u$$ =( $$\kappa_{ij}$$ $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ ($$\frac{\partial u_k}{\partial x_k}$$ $$e_k$$ )

=$$\kappa_{ij}$$ $$\frac{\partial u}{\partial x_k}$$ ( $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ $$e_k$$

Here, $$e_i$$ $$\otimes$$ $$e_j$$ $$\cdot$$ $$e_k$$ = $$e_i$$ ( $$e_j$$ $$\cdot$$ $$e_k$$ )

Kronecker delta
Definition: If $$e_j$$ $$\otimes$$ $$e_k$$ = $$\delta_{jk}$$, $$\delta_{jk}$$ is called Kronecker delta.

$$ \delta_{jk} = \begin{cases} 1 & for \ j=k \\ 0 & for \ j \neq k \end{cases} $$ = $$\kappa_{jk} \frac{\partial u}{\partial x_i} $$ $$e_i$$

$$u(x,y)$$ is a scaler function $$\Rightarrow$$ 0th order tensor $$gradu$$ is a vector field $$\Rightarrow$$ 1st order tensor

$$grad$$  ($$\cdot$$) increases tensor order by 1

$$div$$  ($$\cdot$$) decreases tensor order by 1

We can let $$v$$ : = $$\kappa$$ $$\cdot$$ $$gradu$$

$$div$$ $$v$$ = $$\frac{\partial }{\partial x_i}$$ $$\kappa_{jk}$$ $$\frac{\partial u}{\partial x_j}$$

Homework
Expand the function above.

What's more
$$\kappa_ij$$ = $$\begin{bmatrix} \kappa_{11} & \kappa_{12} \\ \kappa_{21} & \kappa_{22} \end{bmatrix} $$

Notice: in PDE, $$\kappa_{ij}=\kappa_{ji}$$ or $$\kappa^T=$$ $$\kappa$$

$$v$$ = $$v_i$$ $$e_i$$ $$\rightarrow$$ $$\begin{Bmatrix} v_i\end{Bmatrix} _{3 \times 1 or 2 \times 1} $$

And if we change $$e_i$$ coordinate into different system $$\underline{\overline{e_i}}$$

$$\overline{v_i}$$ $$\underline{\overline{e_i}}$$ $$\to$$ $$\overline{\begin{Bmatrix} v_i\end{Bmatrix}}$$

Linearity
if $$\kappa$$ $$= const$$ $$\Rightarrow$$ is a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y,u)$$ $$\Rightarrow$$ is a quasilinear 2nd order PDE

Homework
Show that $$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE

title: User:Egm6322.mafia


text:
Compound Propositions and Useful Rules

Welcome! This is a lesson in the Introductory Discrete Mathematics for Computer Science course here at Wikiversity.

Previous lesson: Biconditional Operator

Compound Propositions

 * A compound proposition is a proposition that involves the assembly of multiple statements. This concept was also discussed a bit in the previous lesson.

Writing Truth Tables For Compound Propositions
To write the truth table for a compound proposition, it's best to calculate the statement's truth value after each individual operator. For example, in the statement $$p \vee \neg q \to q$$, it's best to solve for $$\neg q$$, then for $$p \vee \neg q$$, and finally for the statement as a whole:

Showing Logical Equivalence
Logical equivalence means that the truth tables for two statements are the same. This was also discussed a bit in the previous lesson. In order to prove logical equivalence, simply draw the truth tables for all the statements in question and show that they are the same.

For example, you can show that $$\neg p \vee q$$ is logically equivalent to $$p \to q$$.

The Contrapositive, Inverse and Converse
The contrapositive of conditional statement $$p \to q$$ is $$\neg q \to \neg p$$. A conditional is logically equivalent to its contrapositive. In other words, if q did not occur, then we can assume p also did not occur.

The inverse is $$\neg p \to \neg q$$.

The converse is $$q \to p$$.

The converse and inverse are logically equivalent. They are one another's contrapositives...

Tautology
A tautology is a statement that is always true. Another wording is that a tautology is a statement that is logically equivalent to the constant truth.

$$p \to q \leftrightarrow \neg q \to \neg p$$ is an example of a tautology.

Congratulations
Congratulations! You have finished the first lesson of the course.

Review this lesson

Return to main page

title: Compound Propositions and Useful Rules


text:
User:Bip016

Business and Its Publics - Spring 2009 - Section .016
Instructions: For terms not contained in the Course Glossary:

1. Click on "edit this page" (above)

2. One student will enter a business term, followed by his/her name and Cohort name. For example, EPS (J. Kim, Gramercy).

3. Another student will provide the definition for the term, followed by his/her name and Cohort name. For example: Earnings per share; a company's earnings divided by... (A. Young, Midtown)

4. Professors from Inquiry and Discourse sessions may occasionally enter terms that need definitions.

title: User:Bip016


text:
Talk:Middle School Science

Scientific Method, or perhaps Model
The intrusion of social sciences into the scientific method, or perhaps model, is making a fuss. The "ivory tower" is giving way to the "action research" model, which initially recruited prostitutes as researchers in the research of their sub-culture giving them the ability to construct the "steps" for their escape from their self-imposed dangers.

The difficulty in writing about education for youngsters is in making the writing applicable to both the teachers as well as the youngsters. Perhaps the scientific method information can move to its own page, leaving behind something kids can discuss in class.

title: Talk:Middle School Science


text:
User:Abbie corona

first day trying this out, let's see what happens.

title: User:Abbie corona


text:
Talk:MediaWiki

This needs to be framed as a class, and (in my opinion) should be built with cooperation with the Mediawiki discussion mailing list, perhaps as a faq.--John Bessa 18:42, 23 January 2009 (UTC)
 * I think we should create three mostly self-study classes in which the student follows a step-by-step procedure for learning how to (1) use MediaWiki, (2) set up and administer a MediaWiki installation, and (3) develop MediaWiki. So, basically MediaWiki 101, 201, and 301:

 * 101: MediaWiki editing should require the student to create and use some "hello world"-style templates and, when comfortable enough with internal and external wikilinks and other features of wiki markup, create a wiki page using the full array of features (sidebars, tables, etc.) I'm a little fuzzy on exactly what exercises would be best for training students to do this stuff; if given a step-by-step tutorial, students learn quickly but often forget the material too. Really, there's no substitute for spending a lot of time editing wikis.
 * 201: MediaWiki administration might involve creating your Ubuntu partition (recommended), getting all set up with Apache, MySQL and whatnot, creating your installation in the localhost directory, configuring LocalSettings.php, etc. and as a thesis, the student would create his own public bliki.
 * 301: MediaWiki development might involve developing tag, parser function, hook, special page, skin, magic word, and API extensions, starting out with "hello world" functionality and progressing to more advanced stuff; etc. and as a dissertation, the student should complete a major programming project to fix one or more bugs in Bugzilla. Course credit requires getting Gerrit approval of the submission.
 * Any thoughts on this stuff? I think these classes, especially 201 and 301, could be helpful in recruiting more MediaWiki administrators and developers by making it easier for them. Expanding the administrator community helps with economies of scale, and of course we can always use more devs. Shall we call this MediaWikiversity?

 * Also, I would suggest that users be allowed to test out of the introductory courses. E.g., it would seem to me to make little sense to require one of these people to go through the tedium of MediaWiki 101. Leucosticte (discuss • contribs) 14:17, 3 October 2013 (UTC)

So you want to be a programmer?
http://thenextweb.com/dd/2012/10/21/so-you-want-to-be-a-programmer-huh-heres-25-ways-to-learn-online/ Leucosticte (discuss • contribs) 05:53, 4 December 2013 (UTC)

MediaWiki BootCamp
Yes. Someone is still here waiting for students of {x} to magically appear like future aliens from outer space or time travelers from other dimensions.

checkin

 * instructor -- CQ (discuss • contribs) 04:16, 15 November 2018 (UTC)
 * ... you ...

title: Talk:MediaWiki


text:
User:Cruella369

My name is Denise and I am a firefighter/EMT.

title: User:Cruella369


text:
User talk:Egm6322.s09.three.liu

 Hello Egm6322.s09.three.liu, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --AFriedman 19:32, 23 January 2009 (UTC)

 Hello , we are the Baah and Bahh twins and we'd both like to welcome you to Wikiversity. While you sit back and explore Wikiversity, enjoy some of our fresh goat milk. You can help us learn by creating new learning resources or help us by improving the guided tours and other exploration tools. We look forward to seeing you around. Baaaahbye for now.

Nice work
Very nice mathematics contributions. Have you thought about copying them to pages in School:Mathematics? I think that would be an important and lasting contribution to Wikiversity. --AFriedman 20:52, 6 February 2009 (UTC)

title: User talk:Egm6322.s09.three.liu


text:
User talk:Cruella369

 Hello Cruella369, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --AFriedman 19:32, 23 January 2009 (UTC)

 Woof! My name is Jack(Russell). I am a dog and a Wikiversity mascot. I am pretty new around Wikiversity. Perhaps we can learn together! Tail wag...

title: User talk:Cruella369


text:
User:Egm6322.s09.bit.sahin

$$\Omega$$ is domain of u, $$\mathbb{R}$$ is range of u,

$$u\left(\Omega \right)$$ is image of $$\Omega$$ under mapping $$u$$,

$$v: \Omega \rightarrow \mathbb{R}$$ just anather function like $$u$$,

$$\alpha ,\beta \epsilon \mathbb{R}$$ (2 real numbers)

Example:

$$2u_{xx}+3u_{yy}-7x^{2}-x=0$$

$$F\left(x,y,u,u_{x},u_{y}... \right)=0$$

Therefore,

$$\mathcal{L}\left(\cdot \right)=2{\frac{\partial ^{2}\left(\cdot  \right)}{\partial x^2}}{}+3{\frac{\partial ^{2}\left(\cdot  \right)}{\partial y^2}}{}-7x^{2}-x$$ and

$$\mathcal{L}\left(u \right)=2u_{xx}+3u_{yy}-7x^{2}-x$$

$$\mathcal{L}\left(v \right)=2v_{xx}+3v_{yy}-7x^{2}-x$$

let, $$f\left(x \right)=-7x^{2}-x$$

$$\mathcal{L}\left(\alpha u+\beta v \right)=2\left(\alpha u+\beta v \right)_{xx}+3\left(\alpha u+\beta v \right)_{yy}+f\left(x \right)=\alpha \left[2u_{xx}+3u_{yy} \right]+\beta \left[2v_{xx}+3v_{yy} \right]+f\left(x \right)$$

and

$$\alpha \mathcal{L}\left(u \right)+\beta \mathcal{L}\left(v \right)=\alpha \left[2u_{xx}+3u_{yy}+f\left(x \right) \right]+\beta \left[2u_{xx}+3u_{yy}+f\left(x \right) \right]$$

so,

$$\mathcal{L}\left(\alpha u+\beta v \right)\neq\alpha \mathcal{L}\left(u \right)+\beta \mathcal{L}\left(v \right)$$

It is clear that the partial differential equation is non-linear (Figure 1).

If we define,

$$\mathcal{L}\left(\cdot \right)=2\left(\cdot  \right)_{xx}+3\left(\cdot  \right)_{yy}$$

then cleary

$$\mathcal{L}\left(\alpha u+\beta v \right)=\alpha \mathcal{L}\left(u \right)+\beta \mathcal{L}\left(v \right)$$

this partial differential equation is linear (Figure 1).

Order of PDE
The order of a differential equation is the order of the highest derivative that appears in the equation. Followings are both first order PDEs. First one is a linear first order PDE, whereas second one is non-linear.

$$ 5u_{x}-7u_{y}=0$$

$$6u_{x}^{3}+2u_{y}^{2}+u^{1/2}+x^{2}+siny=0$$

An example for the second order PDE is,

$$div\left(grad\ u \right)+f\left(x,y \right)=0$$

This equation can also be written in cartesian coordinates as:

$$u_{xx}+u_{yy}+f\left(x,y \right)=0$$

Another example for 2nd order example is:

$$div\left(grad\ \kappa \cdot u \right)+f\left(x,y \right)=0$$

Here $$\kappa$$ is known as conductivity tensor (2nd order tensor).

$$\kappa =\kappa _{ij}e_{i}\otimes e_{j}$$

Here $$e_{i}$$ shows the unit vectors. Cartesian coordinate system and corresponding unit vectors can be seen in Figure 2.

$$\kappa \cdot gradu=\left(\kappa _{ij} e_{i}\otimes e_{j}\right)\cdot \left(\frac{\partial u}{\partial x_{k}}e_{k} \right)=\kappa _{ij}\frac{\partial u}{\partial x_{k}}\left(e_{i}\otimes e_{j} \right)e_{k}$$

$$=\kappa _{ij}\frac{\partial u}{\partial x_{k}}e_{i}\left(e_{j}\cdot e_{k} \right)$$

Since $$e_{j}\cdot e_{k}=\delta _{jk} $$

$$\kappa \cdot grad u=\kappa _{ij}\frac{\partial u}{\partial x_{j}}e_{i}=: v_{i}e_{i}$$

$$v:=\kappa \cdot gradu$$

$$divv=\frac{\partial }{\partial x_{i}}\kappa _{ij}\frac{\partial u}{\partial x_{j}}$$

The conductivity tensor is a second order tensor and obeys the usual transformation rule when the coordinate system rotated. Moreover it is usually assumed that $$\kappa$$ is symmetric.

$$\left[\kappa _{ij} \right]=\begin{bmatrix} \kappa _{11} & \kappa _{12}\\ \kappa _{21}& \kappa _{22} \end{bmatrix}$$, $$\kappa _{ij}=\kappa _{ji}$$ or $$\kappa ^{T}=\kappa $$

If $$\kappa$$ is constant, the equation is a linear 2nd order PDE.

If $$\kappa=\kappa(x,y)$$, the equation is a linear 2nd order PDE.

If $$\kappa=\kappa(x,y,u)$$, the equation is a quasi-linear 2nd order PDE.

title: User:Egm6322.s09.bit.sahin


text:
Conservation Entrepreneurship/Resources

Links

 * You can find the 'Social Capital' Index and related resources on social capital at this site.
 * A related conferenceon Social Capital Markets is worth exploring.
 * You can see the wiki from the 2008 Social Capital Markets conference here.
 * The lingo page of the wiki gives good, succinct definitions to many basic social capitalism concepts.

 * Interesting Washington Post article on Wal-Mart's innovations in terms of providing health care benefits. Page 2 talks about some of Wal-Mart's Corporate Social Responsibility and reputation concerns.  Page 3 describes how they assessed employee needs, analagously to how retailers assess customer needs in order to meet needs and provide value.

 * The Education Life supplement from Jan 21, 2009 NY Times has a section on student innovations, including several environmental ones.

promote innovation on campuses, noted that more than 5,000 entrepreneurship programs are offered on two- and four-year campuses -- up from just 250 courses in 1985. Full-scale majors, minors or certificates in entrepreneurship have leaped from 104 in 1975 to more than 500 in 2006. Since 2003, the Kauffman Foundation has given nearly $50 million to 19 colleges and universities to build campus programs."
 * In the dreamers and doers article they state "A report issued last year by the Kauffman Foundation, which finances programs to

 * http://en.wikipedia.org/wiki/Shepard_Fairey (About the guy who designed the Obama "hope" poster, etc..)

 * http://www.youtube.com/watch?v=Ou5wby3X9jU (about coffee certification schemes)

 * Here is a link to the ISO (International Organization for Standardization):http://www.iso.org/iso/home.htm

Lichenology 20:56, 2 February 2009 (UTC)jay
 * Wikipedia also has a page on Triple Bottom Line (http://en.wikipedia.org/wiki/Triple_bottom_line)

 * A search on youtube.com for "conservation entrepreneurship" had some interesting results! The second on the list was this, from our very own university..     http://www.youtube.com/watch?v=tlvsR1mabA8     It is about a project a group of students did for a course and for a competition.  They were successful in adding "infinite" value to a re-used bottle of water.  See how they did it!  with love, Bp 14:09, 4 February 2009 (UTC)

 * SOCAP International is a society of customer care professionals.  This is the site of the 2008 SOCAP international conference.  This page has many power point and other exhibits.

 * Here is a link to the article I read about jaguar tourism in Brazil that guaranteed a 97% chance of seeing a jaguar if you stay for 3 or more days:http://www.telegraph.co.uk/travel/destinations/southamerica/brazil/739298/Jaguars-in-Brazil-The-cream-of-cats.html
 * Here is the Jaguar research center that hosts the jaguar-based ecotourism trips: http://www.jaguarresearchcenter.com/

'Interesting Blogs'

Joel Makower on greening business.

TreeHugger focuses on green consumption.

Literature
Peter Drucker, The Essential Drucker

Chapter 1. What is Management? Motivation and goal setting. •	Origins and development •	Goals of management •	Management vs. entreprensuership •	Management in and for non-profits First reading under management.

Chapter 2: continuation of chapter 1; short. Three goals of management: mission, workers, social impact.

Chapter 3. Abstruse / philosophical treatise on profit (I didn’t understand). BUT, second part is good (pp 20-27); recommend only this part. •	Defines innovation •	Define the customer – who, where, what is utility he wants •	Parts of a business plan (objectives in different areas), but rather dense p. 37 on social responsibility is interesting; combine with Chapter 5.

Chapter 4. Non-profit management. •	Mission – motivation … difficulty yet importance of measuring results •	Board •	Volunteers

Chapter 5. Social responsibility. Gives limited support: •	Good when can be a business opportunity. •	Bad if goes beyond business’ competence / authority

Chapter 6. Eclectic. •	Assumptions vs. Paradigm •	P 77: managing people. Theory X vs. Theory Y (McGregor 1960) •	P 92: management x entrepreneurship •	National x multi-national •	Focus on own technology x importance of technology from other industries (e.g. tradeoffs in having company’s own research lab)

Chapter 7. Information. Linking accounting to results ⇒ Difficult in service / knowledge industries (but necessary).

Chapter 8. Management by objectives.

Chapter 9. Hiring. Excellent.

Chapter 10. Entrepreneurship. Read early in course. Focuses on how big companies can develop an entrepreneurial (innovative) part: answer = separate (intrapreneurship).

Chapter 11. New Venture. Management of a start-up. Apply to case studies such as Jupara’. Important nuts and bolts in this chapter; chapter 12 is more the philosophy behind it.

Chapter 12. Strategy; good overview of strategy. Could be read together with 10 and 11. Not sure how applicable to small or community enterprises (focus is on how to dominate an industry). But an excellent overview of business strategy, with many household name companies as examples.

Chapter 13. Effectiveness is a habit. The habit of focusing on doing “the right thing”. Chapter incomplete; needs to go together with 14.

Chapter 14 (read with 13). Effectiveness is based on looking outward to ask what is needed by others to achieve organization’s goals. (= Proactivity, Covey’s Habit 1).

Chapter 15 – strengths, how you work, values ⇒ where you belong. Could do as exercise in class (or for class). Maybe towards end of term (or do mid-term anonymously, and share at end of term).

Chapter 16. Time Management. Complements Covey. Key is time log exercise. Doesn’t give much on specific techniques.

Chapter 17. Decision-making. Interesting but abstruse; lengthy. Not sure how much students will be able to relate to it, but is thought-provoking.

Chapter 21. Not too relevant for course (more relevant to middle-aged professionals). Human life cycle. Second career.

Chapter 22. Interesting discussion of “western tradition” and “educated person”. Great quote p. 292.

Chapter 24. Entrepreneurial society – to replace welfare state.

Chapter 25. Social sector. Community, citizenship, social change.

Chapter 26. Knowledge vs. machine power.

Afterword – same basic idea as chapter 26.

How to Change the World: Social Entrepreneurs and the Power of New Ideas. By David Bornstein. 2007. This book tells the story of Bill Drayton, the Ashoka Foundtation which he founded, and a wide selection of Ashoka Fellow Social Entrepreneurs. Chapter 11 describes the involvement of McKinsey consulting firm, and gives the perspective of a management consulting firm on Social Entrepreneurship, all in the context of Ashoka Fellow Vera Cordeiro who developed a comprehensive health program for favela children (Rio de Janeiro) who were hospitalized, to avoid the constant rehospatilization. In Peter Senge terms, she helped to treat the whole child (and the family-system in which the child is embedded), as opposed to the traditional approach of treat the parts (symptoms), release the child, and wait for the child to return with a new emergency. Chapter 12 gives Lessons Learned and tries to synthesize common approaches taken by successful Social Entrepreneurs, using various Ashoka Fellows as examples. Chapter 13 was a very inspiring example to me, a US Social Entrepreneur who founded College Summit to mentor talented high school students from underprivileged backgrounds. Chapter 16 was the best general chapter, relating Social Entrepreneurship to successful businesses.

Green, Inc.: an environental insider reveals how a good cause has gone bad. 2008. Christine Macdonald. The Lyons Press, Guilford, Conn. A journalist and former employee of Conservation International has written this supposed "inside account" of how large environmental organizations have sold out to large corporations. She describes various corporate and economic initiatives, including conservation easements, green building, REDD, corporate social responsibility, timber and fisheries certification, charitable contributions and marketing partnerships. Only about 20% of the text is actually about specific NGO activities. She raises some areas of legitimate concern, such as high executive salaries, but there is just not that much specific information here. In my opinion the few interesting cases that are presented have limited credibility because of the author's obvious and unsubstantiated presumption that any kind of compromise is unethical and illegitmate. For example, her assertion that NGOs only got concerned about Global Warming in 2006 when it appeared that there would be conservation payments available is outrageous. I also found her repeatedly calling Brazil's savanna ecosystem "El Cerrado" worth a chuckle. Rbusch 20:38, 12 February 2009 (UTC)

Example of a friends Organic Agriculture Org:
This week I thought I would share a bit about a friend of mine who I would most definitely call a conservation entrepreneur. Since I first met my friend Molly in college she has been nonstop in her efforts to support social and environmental initiatives. While in college she supported a number of causes related to farm workers rights in Immokalee to sustainable agriculture and development on campus. However it wasn’t until after college that she really stepped up to the next level of entrepreneurship. Following college she spent a year in Fiji on a Rotary scholarship working to develop the foundation for organic sugar production. That experience led to the creation of The Fiji Organic Project (http://www.fijiorganic.org/), a project that was eventually picked up and supported by the Fiscal Sponsor NGO Earth Island (http://www.earthisland.org/). Her organization has since made huge steps in helping Fiji transition from unsustainable non-organic sugar production to much cleaner organic methods.

You can read a recent article concerning Fiji’s movement towards organic sugar written by Molly here: http://findarticles.com/p/articles/mi_hb6393/is_2_22/ai_n29354432/pg_1?tag=artBody;col1 ).

Recently Molly has moved on to run her own organization (independent from a fiscal sponsor). With The Fiji Organic Project up on its feet and largely run by Fijians, she has since moved on to create a sustainable agriculture NGO in her home state of Missouri called EarthDance. For more information about this endeavor you can visit her organization’s home page here: http://www.earthdancefarms.org/Home_Page.html

Here’s a short biography of Molly that was put together by our undergraduate college. I think it does a good job at highlighting her entrepreneurial spirit: http://www.eckerd.edu/servicelearning/alumni/mollyrockaman.php

Over the past 6 or 7 years Molly has been extremely helpful in supporting me in my pursuit of creating a similar socially/ environmentally aware organization. I am positive that if anyone has any comments or questions concerning her organizations or how she went about building them she would be more than willing to answer them for us.Lynch 21:36, 30 January 2009 (UTC)

WORKER OWNED COOPERATIVES
I got interested in this concept while reading about Equal Exchange, the coffee and chocolate company, which it itself a worker owned cooperative. A number of the ideas discussed in The Fifth Discipline seem to point towards worker owned cooperative structure, particularly ideas such as harnessing the "intrinsic motivation" of humans and creating a creative work place where all are equally valued and feel empowered. Worker owned cooperatives seem in some ways better positioned to embrace many of the ideas proposed in the Fifth Discipline, as the guiding ideas, theory/methods/tools, and infrastructure are all from the outset designed to promote being learning organizations and the management-worker divide is not so pronounced.

In summary the most important characteristics of worker owned cooperatives are:
 * They are cooperatives democratically owned and controlled by workers-owners.
 * "Pure" worker cooperative- All shares of company owned by work-owners (one worker-one share).
 * Less pure- Majority of workforce owns shares and the work force owns the majority of shares, but some shares may be owned by outside investors
 * Direct worker control- All members meet regularly to collectively make decisions, non-hierarchical structure
 * Indirect worker control- Workers elect Board of Directors, who hire management (who are subject to removal by workers), hierarchical structure.
 * Profits (or losses) earned by the worker's cooperative are shared by worker owners. Instead of "profit", there is a "surplus" that is divided among worker-owners
 * Low ratio between lowest to highest paid workers.

The wikipedia page on Worker Owned Cooperative is very good and worth looking at it. It has an interesting comparison of Corporate vs. State Owned vs. Worker Owned enterprise structure. (http://en.wikipedia.org/wiki/Worker_cooperative)

Also there are a number of interesting profiles of worker owned cooperatives on the website of The United States Federation of Worker Cooperatives (http://www.usworker.coop/aboutworkercoops) On this page they have a PDF of WHAT IS A WORKER OWNED COOPERATIVE?, in which they point out: looking not just at the money they make, but at things like their sustainability as a business, their contribution to the community, and the happiness and longevity of their workers."
 * "Some worker cooperatives have what’s called a “multiple bottom line” - that is, they evaluate their success by

title: Conservation Entrepreneurship/Resources


text:
Conservation Entrepreneurship/Local Initiatives

Local Social Entrepreneurship initiatives in or near Gainesville, Florida
On February 4, 2009, we had the opportunity to visit and talk with the directors of 3 local green businesses:

Volta Cafe (They buy their coffee from  Intelligensia who practice Direct Trade)

Indigo Green Store

Alternatives

Visits to local Ecopreneurs -- what do we want to learn? what questions to ask?
Note: although in the form of direct questions, we may want to get at these topics through observation and our own analysis, as well as direct and indirect questioning of the entrepreneur. What is your business? (value proposition; need fulfilled).

Who is your customer? (segmentation; 80-20 principle)

Who are your competitors?

Understand start-up phase. Problems they ran into. How they overcame. How did they get funding?

Understanding the entrepreneur's personal perspective on business, social enterprise. "What do you consider yourself?" Non-profit or for-profit, and why? Any links to conservation? to non-profits?

Working or partnering with communities? (ways, special challenges, lessons). From community perspective: producing commodities or adding value? (through processing or other ways of adding value?) From community perspective: participation; distribution of benefits.

Markets; niche? Certification? What are your views on the benefits and problems of certification? Fairtrade/organic. What about direct trade? How much more difficult/rewarding is it? What are the logistics involved?

How did you decide on Gainesville as a good place of business? Earned Media- how have you obtained it and how has it impacted your business?

Comments and Evaluation
Please address the following questions with regard to each of the businesses visited:

Responses to this assignment are seen below:

 Value proposition  How do they create value for the consumer? What do they offer to the consumer? What need do they fulfill, or what problem do they solve? Try to encapsulate the value proposition of each business in a single sentence, and back it up in a paragraph. Example: “WalMart provides a wide range of consumer products at the lowest possible price. WalMart stores are no-frills (to say the least) and product quality is variable, but they offer a very wide range of product types and a substantial variety of low to mid-range quality products in each category.”

CAFE VOLTA: Volta offers a diversity of gourmet coffees, teas, and chocolates specially prepared to accentuate their unique flavors in an inviting and sophisticated feeling atmosphere in downtown Gainesville.Keeping the road safer and the Earth greener Author: 98.70.62.12 19:11, 6 February 2009 (UTC) jay Cafe Volta strives to offer the best coffees, teas, and chocolates available to consumers in Gainesville. The cafe stresses that coffee is not just coffee, tea not just tea, and chocolate not just chocolate, but rather that there exist many coffees, teas, and chocolates that have different flavor qualities that are brought forth by their proper preparation. The owner of Volta is a connoisseur and invites his clientele to be connoisseurs as well, or at least to feel like one. Value is created for the customer by the cafe providing access to a diversity of artisan coffees, teas, and chocolates not accessible elsewhere in Gainesville. Further value is added by preparing these products for the customers in what is presented as the ideal way to accentuate their subtleties. Further value is created by providing these products in an inviting atmosphere that encourages socializing and a sense of urban sophistication not widely available in Gainesville. Volta offers customers access to goods (that are backed by stories) not easily attainable elsewhere. It offers them in a non-chain atmosphere in downtown Gainesville, thus making it convenient for those who live, work, or socialize in this area of town.

Cafe Volta provides premium coffee-shop goods and services at competitive rates. The cafe is focused on social responsibility, while maintaining a snobby ambience. They partner with smaller-scale companies that source their products from smaller-scale farmers, and price their goods to compete with larger corporations. The service is exceptional and well educated to impress upon the customer that they deserve the best. 70.164.72.98 03:34, 11 February 2009 (UTC)

INDIGO

Author: Kifaro50 20:35, 8 February 2009 (UTC) Indigo offers eco-friendly, sustainable products for the environmental and health-minded consumer. The primary value to the customer comes from products that allow customers to still have a great looking home, but to do so in a way that is environmentally sustainable. A house can have wood flooring or nice carpets, but these products do not have to be harmful to the environment to make. At the same time there are products offered such as dual-flush toilets and sunlight diffusers that help better utilize natural resources to reduce consumption. Indigo also creates value by offering products that are healthier, avoiding toxic chemicals such as formaldehydes in the glue of carpet. Finally, Indigo provides value to its customers through education, increasing awareness about the range of sustainable products, as well as offering classes on “green” topics.

Author: Emiliano 02:36, 11 February 2009 (UTC) Indigo provides environmentally friendly and healthier products to responsible consumers. Indigo creates value by focusing on the green niche of the housing market. The owner states that today it is possible to compete in prices of products to conventional products with the obvious advantages of environmentally responsible products. Author: Lichenology 02:34, 11 February 2009 (UTC)jay Indigo offers goods and materials to those seeking to design, remodel, and sustain homes using non-toxic and sustainable materials. Indigo offers value to its customers by linking them with product choices that are clean and safe for the consumers health and offer the consumer the opportunity to make ecologically responsible purchasing decisions. Indigo offers a relatively wide selection of products in a new and growing niche, products which are not easily found elsewhere in Gainesville. Another value that Indigo offers is that of supporting a local business which does its best in turn to support other local or regional businesses. It also provides value through the displays that it offers in the store for consumers to see and touch the materials that Indigo offers. Yet more value is offered by Indigo through the classes that they offer to the community. Like the other businesses we visited, Indigo offers value partly through the actual product it supplies and partly through the good (responsible morally and ecologically) feeling that customers get when purchasing the materials that they offer.

ALTERNATIVES

Author: Lichenology 19:12, 10 February 2009 (UTC)jay Alternatives offers consumers an alternative to buying products from questionable sources and guarantees to its customers that its products are fair trade. There seem to be two main values that Alternatives provides to consumers. The first again, is the moral value of knowing that the products one is purchasing come from fair trade sources and thus in theory, purchasing them is giving a just income to the producers. The second value Alternatives gives is access to rare and exotic goods not easily attainable elsewhere. Additionally the products frequently come with a story of their origin and the producers who produce them. All of these together provide perhaps the biggest value that consumers get from shopping at Alternatives, to feel good about shopping, to feel that their purchases are having a neutral or better yet positive effect in the world. Lichenology 19:12, 10 February 2009 (UTC)jay

Author: Lynch 23:20, 10 February 2009 (UTC) Alternatives offers consumers a unique variety of worldly products that tell a story not just about the purchaser but also about the maker; all of which is done with a clean social conscience. As was mentioned above, Alternatives offers a real story with every purchase. These stories create a link between the buyer and the designer/ maker that really adds value to the purchase. This value is something that you will rarely find in products offered at larger businesses- you don't buy a shirt at Target and enjoy telling your friends where you bought it and how it was made. In other words, the products at Alternatives allow you to stand out and be unique in your worldly fashion, all while supporting important social/ environmental causes.

 Customers  Who is their target customer?

Volta Volta offers ambiance and culture (in the sense of culto). When we spoke to the owner of Volta, it seemed that his target audience was more a way of thinking than a "type" of person. He seems to target a wide array of people- from the person who likes the modern cafe ambiance to the person who takes flavor (and particularly coffee) seriously. He adds the value of social consciousness with the products that he provides but does not make that his main focus area. His main focus is reaching people like himself...as he put it. ~Mira According to Anthony, the owner of Volta, there is no one ideal customer. The store sees customers from every social and economic group, blue- and white-collar workers, and people of a variety of ethnicities and backgrounds. As Mira said, customers of Volta should represent more of a similar mindset than any obvious demographic characteristic. Volta’s clientele are likely to be those who know about and want quality drinks. These people combine this desire with a feeling of social responsibility and a desire to enjoy their pleasures in a way that does not lead to the exploitation of people in developing nations. They want good coffee and tea but they also want to see that the farmers who produce it are justly compensated and thus, through their purchasing power, to enact social change. Kifaro50 21:39, 8 February 2009 (UTC)

Alternatives Alternatives has a unique customer base. Alternative's target customer is the young to middle aged cultured individual. Well-traveled, fashionably aware students are likely a major target for Alternatives. I think that the owner recognized this when she first opened the store near campus. A second customer that the owner appears to be targetting more recently is the middle-aged yuppie. These individuals are likely more financially stable and willing to pay more for some of the finer unique products found in Alternatives. Lynch 20:45, 7 February 2009 (UTC)

Middle-age women frequent Alternatives, and are the target market for the store. These women are often married to or are themselves people employed by the University of Florida. As such, they are socially aware and willing to pay a premium for responsible clothing and accessories. Other women, such as stay-at-home moms, are targets.

Indigo Indigo is primarily targeting the environmentally conscious homeowner. These people care about sustainability, while also enjoying the health benefits eco-products provide such as fewer chemicals. They are willing to pay a premium for these products. While Indigo also sells eco-friendly cleaning supplies that could attract a wider consumer base, the majority of their products are items such as flooring materials and water conscious toilets that are not likely to be of interest to non-homeowners. This is countered somewhat by the educational classes they teach about eco-friendly practices such as making and using rain barrels and environmental gardening, bringing other potential consumers, such as local college students, into the store. Kifaro50 23:07, 7 February 2009 (UTC)

Builders offer Indigo the market they desire. By gaining business with homeowners (i.e.- potential builders), they seek to create demand for their products in the construction industry of Gainesville. 70.164.72.98 03:34, 11 February 2009 (UTC)

 Competitors  Who are their competitors in terms of fulfilling the need described above? Think in terms of similar products or related needs; also think about different distribution channels for obtaining a similar product.

Indigo While Indigo claims a fairly small 'green household' niche, they are still in direct competition with other non-green household product businesses. When someone is planning to build or remodel a home they have two general options; go green or don't. In this general sense Indigo is in competition with businesses like Home Depot and Lowes. These massive businesses are likely able to draw customers away from Indigo with lower prices and the one stop shopping experience. Lynch 20:36, 7 February 2009 (UTC)

Author: Emiliano 13:43, 11 February 2009 (UTC) Indigo's competitors competitors today are other building material shops like Home Depot and Lowes. Although Indigo seeks a different niche in the market, their local customer base is still skeptic towards buying green because of the common knowledge that these kind of products are more expensive. The owners of Indigo explain that green products are becoming more and more accessible, and if you make projections including savings and advantages of the green products they have definitely competitive prices.

Volta As a coffee shop, Volta is in direct competition with other coffee shops, such as Starbucks. Volta must compete against the great publicity of Starbucks as well as ease of access with locations on the UF campus and around town. Volta strives to be more than just a coffee shop, also serving as a gathering place for people to come together and interact socially. As such, it is in competition with other public spaces that provide opportunity for meetings. This competition is most likely to focus on providing a meeting place for professors, graduate students, and other professionals such as lawyers. Finally, Volta caries socially responsible goods, such as direct trade coffee, tea, and chocolate. This may lead to competition with fair trade suppliers in Gainesville for customers interested in social and environmental responsibility. Kifaro50 23:24, 7 February 2009 (UTC)

Alternatives It is interesting but there are a few places where Alternatives could have competition in a broad sense. There are several stores in Gainesville that offer novelty items, natural fiber clothing and/or crafts and many of them offer crafts from places where the crafts are inexpensive to buy and ship making them a retail item with a decent percentage return (like the one on Newberry just after 39th St). Alternatives offers a wider variety of items and value-added social awareness but at a higher price, in many cases. Depending on how specific the buyer is in their purchase mentality and willingness to pay for labor, effort and quality (including social consciousness and cause), this could be a competitive disadvantage for Alternatives. ~Mira

 Competencies  What core competencies have these business owners mastered?

Cafe Volta
 * 1) Innovation: Cafe Volta fills a niche; that of the non-corporate, local cafe. But it goes beyond this by providing very high quality products, that are very fresh, and are constantly changing.
 * 2) Educating Consumers: Both subtly and not so subtly, Cafe Volta educates its customers about the differences between different coffees, for example, thus encouraging the customer to become curious about all the subtle flavor differences and interested to experiment with them. The tasting charts on the wall of coffee flavor types are part of this as well, as are the public tastings/cuppings.  The customer is educated to appreciate the high quality of Volta's products and find it objectionable to patronize other sub-par establishments.
 * 3) Interested/Engaged/Educated Staff'': By ensuring that the staff is interested in the products, through tastings and constant education, the staff become more than mere servers. Their knowledge and ability to recommend products and coach the customers are strong fortes of the cafe.  This creates value for the customer by feeling engaged in a learning process.  It probably creates profit for the cafe by encouraging customers to try new (and at times more expensive) products.
 * 4) Background in business: Anthony's eclectic background (including business) gives him a number of important skills and perspectives. His business plan would appear to be well researched, thought out, planned, and excecuted.  He mentioned the lesson, which the cafe seems to embody, of it being important to appear more professional, large and wealthy as an enterprise than it may actually be.
 * 5) Finger on the pulse: Because of Anthony's passion (ensuring that he constantly is learning about developments in the products he carries) and because he deals with the cutting edge of small, constantly innovating artisan suppliers, he is able to transfer this to the customer, a feeling a being in the avant-garde of cafe culture. He and his staff also attend regional, national, and international conferences and competitions, ensuring that their finger stays on the pulse.Lichenology 19:45, 6 February 2009 (UTC)jay

Indigo
 * 1) Educating consumers: They offer transparency and advising in eco-friendly products (from construction to cleaning products) in line with customer needs and requirements.  They do outreach to possible consumers (particularly the Builder's Association) in order to double their mission- to expand and to make Gainesville a safer and more environmentally friendly place to live.
 * 2) Community involvement: They offer workshops and capacity building through SFCC and on their own to educate consumers and interested people on how to live in a more healthy, eco-friendly way.  They help them to be more discerning when it comes to false advertising and "green-washing" leading to more informed consumer decisions.
 * 1) Publicity: Classes, advertising, and local business relations all increase the publicity and likely business for Indigo. The 'green clean' award that they created to offer local business who used green cleaning supplies is an excellent way to push for green living as well as a free way to advertise.
 * 2) Education: Classes taught to the general public on green issues is an excellent way to 'spread the word' and also gain exposure. The more that the community understands about green living the more likely they are to buy green produts from Indigo.
 * 3) Networking: Indigo appear to be involved with networking not only with local green product designers, but also with the greater architectural/ construction world. These connections can only increase business for Indigo. If Indigo is mentioned to home owners by the construction companies then Indigo is more likely to see green homegoods being encorporated into more local homes.
 * 4) Staff: The staff at Indigo appear to be very friendly and educated on all issues relating to green living. For this reasons Indigo is a great place to ask questions on a wide variety of green products that otherwise would be scattered across the internet. As a 'middle-man' company this business competency is vital. Lynch 23:35, 10 February 2009 (UTC)

Alternatives As Laurie tells it, she started out with no business training and little experience working with communities or fair trade goods, from there she has had to develop a number of competencies:
 * 1) Recruiting and working with staff: Laurie has had to find employees that share her passion for empowerment of local people. She has had to train them to run the business in her stead as she often relies on them as she travels the world, working with local communities and sourcing products.
 * 2) Integrating for-profit and non-profit work: In addition to Alternatives, Laurie also runs a non-profit group dedicated to working with and empowering local communities. She has learned to balance and integrate the goals of these two entities, while maintaining aspects of their individuality.
 * 3) Sourcing products: Laurie sources many of the products she sells herself, going to communities and building relationships with the people there. For other products, she has suppliers in countries around the world who work directly with local people and she must coordinate and maintain networks with these people as well as communities across the globe.
 * 4) Passion for others: Talking to Laurie for just a few minutes reveals that what really drives her is a desire to help people in communities across the world and to raise awareness in the United States of these people and their needs. This passion is evident in her business and in the personal sacrifices she has made to start up the company and keep it running. Her passion and the stories and vibrancy she attaches to her products is part of what makes her store unique and attractive. Kifaro50 20:51, 8 February 2009 (UTC)

 Execution

How well do they execute their strategy?

Volta Anthony and Volta appear to be the most business savy of the three businesses we visited. The planning, structure, and execution of Volta seem the most stable and well organized. Even though there already existed a number of coffee businesses in the downtown Gainesville area Anthony was able to carve out an even more specialized niche by offering something that he believed no other company was offering- high quality fresh coffee. I feel that once people try Volta coffee/ tea/ chocolate they will understand that it is of better quality than Starbucks or Maud's and Volta will see an increase in business. As long as Volta sticks to the strategy of seperating itself from the other coffee shops in Gainesville, I see him remaining on top of the quality coffee market in town. Anthony has also made a great business decision in his hiring and team management. Even though he did have one negative experience in the beginning, since then Anthony has hired staff that add a lot of personality and energy to the business. He appears to be dedicated to educating his team (and the public) about coffee/ tea/ and chocolate, and this will in turn translate into better business function. Lynch 21:38, 7 February 2009 (UTC)

Volta seems to be executing their strategy superbly. Whether or not the products offered there are as of high quality and sophistication as one is led to believe the consumer is certainly made to feel that they are and the premium is paid for them. The cost of the coffee beans themselves per cup versus the cost to consumer demonstrates that a significant mark up is being made over the cost of the high quality raw materials. The atmosphere and especially the staff do a very good job of presenting the products as the creme du le creme. The location chosen for the business and its spilling into the street seating make it an appealing place for passersby and its central and unique ambiance a convenient meeting place, ironically lacking in Gainesvillle. Volta has quite effectively grabbed an open niche in the market and appears to be snowballing in effect.Lichenology 03:53, 11 February 2009 (UTC)jay

Indigo Indigo seems to do well in its niche market. This currently seems somewhat limited to certain channels. For example, Indigo seems to be excelling in their goal of community education. Through a partnership with Santa Fe Community College they offer classes every weekend on a variety of environmentally responsible topics. According to the owner, these classes are generally booked well in advance, suggesting that the company’s goal of increasing awareness about issues of sustainability is being met. Financially the company seems to also be sustainable. The biggest challenge now will be finding ways to continue to expand their customer base amidst a suffering economy, perhaps through increased marketing. Kifaro50 21:24, 8 February 2009 (UTC)

Alternatives Alternatives seems to do very well at the goal of providing a market for fair trade products, products which unless they have a market are ultimately not useful to the producers/producer groups/NGOs behind them. It also does well at providing a unique mix of goods for consumers who are looking for something fairtrade and/or of high quality and semi-exotic. Laurie mentioned that part of the mission of Alternatives is education and that she felt it was not doing as much in this arena as she wishes. That said she also mentioned doing sales of products with local churches, as well as a number of open houses during the year. Although not on a grand scale Alternatives is educating consumers and for students and residents in Gainesville Laurie is an excellent contact and source of information about fair trade, specifically in regards to crafts. It seems that the move to the strip mall while it has helped the business survive financially has taken it a bit out of the spot light culturally of Gainesville and provides it less of a chance to make a bigger statement particularly to the University community.Lichenology 03:38, 11 February 2009 (UTC)jay

 Opportunities

Growth potential; opportunities to expand either horizontally or vertically.

Volta

There is always a demand for caffeine! And once someone is hooked on quality coffee such as that offered by volta, then there is no going back to your average cup of joe. I can easily see Volta expanding to other locations in the future. The quality coffe/ tea/ chocolate market is prime for the taking. There already exists a plethora of coffee shops in most towns, but I feel that most shops offer the same thing. If a coffee shop is going to make it they have to offer something that the others do not. If Volta is able to gain the repuation of the best cup of coffee in town then Volta will easily be able to expand. When Anthony said that people in Vail were interested in opening another Volta I immediately thought that it would be a good fit-cold weather and wealthy visitors who would be willing to pay for a great cup of coffee. The business model that Volta used is also one that can be adapted by anyone, so as long as Anthony partners with someone with a good business sense then I see additional Voltas succeeding just as the one in Gainesville has thus far. Lynch 21:59, 7 February 2009 (UTC)

Volta may have the opportunity to expand and yet at the same time may best keep its appeal and image by not expanding. Expanding vertically into coffee roasting or chocolate making would perhaps be of interest but given that they carry some of the best quality of these products, it may be hard for them to match the quality of those artisans for whom they are an outlet. In addition the additional space and equipment might make this an unattractive idea. However, all that said, to have the smell of fresh roasting coffee or cocoa would certainly add to the ambiance. Whether this would translate into sales or be worth the cost however is unknown. In terms of horizontal expansion, it would seem that in the years ahead (assuming that the bottom does not drop out of the gourmet market) Volta could quite easily expand horizontally as Anthony mentioned has already been solicited. I would argue however that beyond very modest expansion (to other places that Anthony likes to spend time) horizontal expansion could risk turning the anti-chain boutique into a chain. Thus expansion would need to go slow and be very specific about placement of other stores. The best choice for Anthony may be in fact to enjoy the relatively tranquility and easy control of having one (or two) stores in Gainesville and hiring himself out as a consultant to help other entrepreneurs start Volta-esque endeavors.Lichenology 03:38, 11 February 2009 (UTC)jay

Indigo

I feel like Indigo has great potential to expand into their goal of becoming a "Green Home Depot"- although that is very contingent on many factors like market pressures (cost, accessibility, green incentives) and that many of these factors are leaning in a direction that favors this kind of expansion at the moment (with tax-breaks for environmentally friendly home repairs, updates and alternative energy). Assuming that is true and they are able to expand into a large material and home goods and home project planning store, there is even greater future potential to open up like stores in other locations...

Indigo appears to be at the very beginning of its growth with a good deal of room for expansion if the economic situation does not hamper it too much. The business seems to have filled an important and growing niche market in Gainesville and the region. Though it may be a tougher market, if the housing market recovers, perhaps trying to make in roads into the Ocala area where retirees seem to build and hopefully could be convinced of going green. Other opportunities for expansion will be to make more links with contractors and to figure out how to fit green building materials into stimulus spending and potential future tax incentives for houses built with eco material. Into the near future it seems wisest to keep a small inventory and continue to order on demand to keep overhead down. Perhaps work with suppliers and shippers could lessen to lag time between order and delivery.Lichenology 03:38, 11 February 2009 (UTC)jay

Alternatives In terms of growth of the company, Alternatives’ opportunities are currently limited. The store is filled to capacity so there is not too much room to offer more products. Although Laurie says she does not foresee this happening in the near future, if the location of the store were to move there would be potential for expansion. Even if the location does not change, there is a chance that additional marketing or perhaps a partnership with Santa Fe or UF to increase education about local communities and their needs as well as about the products offered at Alternatives, could increase sales and thus provide opportunity to grow. Laurie has said education is something she would like to see happen more. Kifaro50 21:24, 8 February 2009 (UTC)

 Vulnerabilities  Risks.

Alternatives It appears to me that the focus almost exclusively on "non-essential" products offered by Alternatives could be a risk. Especially with the economic downturn it would appear that disposable income is shrinking and Alternatives could suffer. Then again it is this very non-essential niche that Alternatives fulfills. To deal with "more essential" products would risk filling the store with even more products and competing with other stores like Wards, Mother Earth, Indigo, etc. Perhaps another future risk will be increased transportation costs for goods from far-a-way places in small quantities.Lichenology 21:22, 6 February 2009 (UTC)jay

Volta The above could also be said about Volta as they sell what could be considered luxury products that could be more cheaply replicated at home- they are products that are also consumed on the premises (often) so that one is paying for atmosphere- another luxury and "non-essential" item. Good point, Jay!

Indigo 1. As was mentioned above for Alternatives, economic downturns may result in a decrease in business for Indigo. While green products are in many times economically beneficial in the longterm, they are commonly more expensive up front. During hard times people might opt to buy the non-green product solely based on price.

2. From a purely business standpoint, I feel that Indigo is taking a risk by choosing Gainesville as a business location. A large portion of the young population of Gainesville, who I would associate with more of a green attitude, are renters not home owners. I also feel that the median income per household would be lower in Gainesville than might be expected near a larger city. Relocating somewhere that is not such a college-based town might serve Indigo well. Lynch 20:22, 7 February 2009 (UTC)

I think that the risk that Indigo takes is not spending more on marketing at this point. They are a niche market but have come in at a really good time. Gainesville, as a location (due to the large quantity of builders and the higher property values by the University) is not a bad one at all. And with the presence of the university (green-consciousness) and the fact that they claim to offer comparable prices to Home Depot (for example) it seems that their applicability is pretty good. Their greatest risk, it seems to me, is that they have no inventory- which on the one hand is a benefit since they don't have to put money in up-front but...they have to depend on businesses that don't allow people to order directly through the company or find greater incentives for people to come through their business instead of going directly to the manufacturer. They could play up their value as "knowledgeable" a great deal more, it seems to me (although clearly giving classes through Sta Fe is helpful and good localized advertising).

Other local green businesses
community, but for those that don't...You can be a seed for change that will create more jobs in our town, more outlets for local farmers, and more community building. Purchase a membership for the Citizens co-op now, and help them reach the goal of 500 members before the summer. If you're already a member, go out and recruit just one more. The link to their website for more information, or to sign up is: http://www.citizensco-op.com https://www.growersalliance.com/ Lichenology 02:53, 28 January 2009 (UTC)jay
 * 1) Citizens co-op.  Many of you may already know about this amazing movement in our
 * 1) Sweetwater Coffee
 * 2) Gainesville's Farmers Market located downtown on Wednesdays, from "4 to 7 pm"
 * 3) Book Lovers Cafe (and Books, Inc.)
 * 4) here is a link to another regional social enterprise we may want to visit or invite to visit with us.


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User:Egm6322.s09.mafia

=General Non-Linear PDEs=

Partial differential equations (PDEs) are a class of mathematical expressions that include one or more independent variables and their respective derivatives. These equations describe a host of physical occurrences in the real world. Examples of PDEs in everyday life can be as simple as tracking the change in temperature for a can of soda that is placed on a kitchen counter after being removed from the refrigerator (see Heat Conduction Equation, ). More complex examples of PDEs can be found in Schrodinger's Equation which describes the quantum state of a system, , or, the Monge Ampere equation that occurs most often in studies of differential geometry. A brief introduction to the basic tenets in the study of partial differential equations is presented here. Information given on this Wiki page are from lecture notes of EGM 6322 taught by Dr. Vu-Quoc.

Defining Variables
For these notes, we will assume we have n independent variables:

$$  \left \{ {x}_{i} \right \}=   \left \{ {x}_{1},...,{x}_{n}   \right \} $$.

An example of this would be single dimensional case with a time variable.

' $$ (  x,t )=( {x}_{1}, { x}_{2 } )  $$'

where $$ {x}_{1} = {x}  $$, a spatial variable

and $$ {x}_{2} = {t}  $$, a temporal variable.

A more complex situation would be a three dimensional, time dependent problem:

$$(x,y,z,t)=({x}_{1},...,{x}_{4})$$.

In this case, $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

Definition of Functions
The unknown function $$ u $$ maps numbers from the domain $$ \Omega $$ to the real numbers $$ \mathbb{R} $$ In two dimensions, this means:

$$ \left( x,y \right) \in \Omega $$

$$ u \left( x,y \right) \in \mathbb{R} $$

and

$$ \Omega \rightarrow \mathbb{R}$$

$$ \left( x,y \right) \mapsto u \left( x,y \right) $$

which means that $$\Omega $$ is the domain of u, $$ \mathbb{R}$$ is the range of u, and $$u \left ( \Omega \right )$$ is the image of $$\Omega $$ under the mapping of $$ u $$.

Example Functions
For these notes, we will assume we have unknown functions: $$ {u}_{1},  {u}_{2} ,...  $$.

The Navier Stokes Equation
An example of this is the Navier-Stokes Equations in 3-D:

$$ {u}_{i} ( \left \{  {x}_{j} \right \} ) $$

where $$ {u}_{i} $$ is the velocity field in x, y, z; $$ {u}_{1}, {u}_{2}, {u}_{3} $$, dependent upon variables $$ {x}_{1}, ... {x}_{4}$$, where $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

One Unknown Function
For our present discussion, this can be restricted to one unknown function $$ u $$, containing $$ n $$ independent variables $$ {{x}_{i}} i=1,...,n $$

The mth partial derivative of this function can be expressed as:

$$ \frac{\partial^m u}{\partial x_i\ ,...,\partial x_m} $$ where

$$ {{i}_{1},...,{i}_{m}} $$ is the subset of m indices among n possible indices

$$ {i}_{i}, ..., {i}_{m}=1,...,n $$.

Another Example
Another example of a non-linear PDE is:

$$ F \left ( \left \{ {x}_{i} \right \}, \left \{ \frac{\partial u}{\partial x_i} \right \}, \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}, ... \right ) =0$$

where $$ \left \{ {x}_{i} \right \}$$ contains $$ n $$ arguments,

$$ \left \{ \frac{\partial u}{\partial x_i} \right \}$$ are the components of the $$ \nabla u$$, and

$$ \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$ are the components of the Hessian of $$ u $$.

The Hessian
The Hessian is a symmetric $$ n \times n $$ matrix, thus H = HT

and $${H}_{n \times n}$$ := $$ {\left [ {H}_{ij}\right ]}_{n \times n}$$.

The Hessian is defined as:

$$ {H}_{ij} := \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$.

This has many interesting properties.

The Laplace Equation
The Laplace Equation is a single function of two (or more) variables:

$$ F \left ( \left (x,y \right) ,u,\left ({u}_{x},{u}_{y} \right), \left ({u}_{xx},{u}_{xy},{u}_{yy} \right ), ...   \right )=0 $$

Example of the Laplace Equation include:

$$2{u}_{xx}+3{u}_{yy}=a{x}^{2}+bx$$

$${u}_{xx} +{u}_{yy}=0$$

The Definition of Linearity
If $$ u $$ is an unknown function, the operator $$ L $$ is considered a linear operator with respect to $$ u $$ if:

$$ L \left ( \alpha u + \beta v \right ) = \alpha L \left ( u \right )+ \beta L \left ( v \right ) $$

Example of Checking for Linearity
Let's take the function:

$$2{u}_{xx}+3{u}_{yy}-7{x}^{2}+x=0$$

The operator is :

$$ L= 2 \frac{\partial^2 }{\partial x^2} + 3 \frac{\partial^2 }{\partial y^2} -7 x^2-x$$

where $$-7 x^2-x:= f(x)$$

To check the linearity, we assume $$\alpha$$ and $$\beta$$ are real number constants, and evaluate: $$ L(\alpha u+ \beta v)= 2 \frac{\partial^2 (\alpha u+ \beta v)}{\partial x^2} + 3 \frac{\partial^2 (\alpha u+ \beta v)}{\partial y^2} + f(x)$$

which is equivalent to:

$$ L(\alpha u+ \beta v)=2 {\left (\alpha u + \beta v \right )}_{xx}+3 {\left (\alpha u +\beta v\right )}_{yy}+f(x) $$

If we distribute the integers, as well as the second derivative operators, the equation becomes:

$$ L(\alpha u+ \beta v)=2 \alpha {u}_{xx}+2 \beta {v}_{xx}+3 \alpha {u}_{yy} + 3 \beta {v}_{yy}+f(x) $$.

Grouping the terms by the constants $$\alpha$$ and $$\beta$$ concludes:

$$ L(\alpha u+ \beta v)= \alpha \left [2 {u}_{xx}+3{u}_{yy} \right ] +\beta \left [ 2 {v}_{xx} +3 {v}_{yy} \right ] +f(x) $$.

The operator $$ L $$ is nonlinear because

$$ \alpha L(u)+ \beta L(v)= \alpha \left [2 {u}_{xx} +3 {u}_{yy} +f(x) \right ] +\beta \left [2 {v}_{xx}+3 {v}_{yy} + f(x) \right ] $$

and

$$ f(x) \ne \alpha f(x) +\beta f(x) $$

 * $$J=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n} \end{bmatrix}. $$

Definition of the order of PDE
Order of highest derivative in the PDE.

Example of first order PDE
linear: $$5u_x-7u_y=0$$

non-linear: $$6(u_x)^3+2(u_y)^2+(u)^{1/2}+x^2+sin xy=0$$

we can assume that $$D(u)=6(u_x)^3+2(u_y)^2+(u)^{1/2}$$

More Examples of 1st Order PDEs

Examples of 1st order linear and nonlinear PDEs are given below. For more information, see the Wiki page for Partial Differential Equations.

(LINEAR) $$ \frac{du}{dx}\ + \frac{du}{dy}\ + \frac{du}{dz}\ = 0 $$ (LINEAR) $$ \dot x \ + 42\dot y \ = 0 $$ (LINEAR) $$ t \frac{dy}{dt}\ + 4 \frac{dy}{dx}\ = 0 $$ (NON-LINEAR) $$ \left(\frac{du}{dx}\ \right)^2 + \left(\frac{du}{dy}\ \right)^3  = 0 $$ (NON-LINEAR) $$ u + \sqrt[3]{\left(\frac{du}{dy}\ \right)}  = 0 $$

As an addition proof, it should be noted that the linear PDEs given above are also separable (the non-linear PDEs are not separable expressions). More information about the linearity of PDEs is given in the following references:,.

Homework
Show D(u) is not linear.

(Solution)

Again, $$D \left( u \right) = 6 \left( u_x \right)^3 + 2 \left( u_y \right)^2 + \left( u \right)^{1/2}$$.

D(u) is linear if the following formula holds:

$$D \left( \alpha\cdot u + \beta\cdot v \right) = \alpha\cdot D \left( u \right) + \beta\cdot D \left( v \right) $$

OR,

$$6 \left( \alpha\cdot u + \beta\cdot v \right)^3 + 2 \left( \alpha\cdot u + \beta\cdot v  \right)^2 +  \left( \alpha\cdot u + \beta\cdot v  \right)^{1/2} = \alpha\cdot \left [ 6 \left( u_x  \right)^3 + 2 \left( u_y  \right)^2 +  \left( u  \right)^{1/2} \right ] + \beta\cdot \left [ 6 \left( v_x  \right)^3 + 2 \left( v_y  \right)^2 +  \left( v  \right)^{1/2} \right ]$$

The expressions on either side of the equality sign in the above equation are NOT identical. Therefore, D(u) is NOT linear.

Example of second order PDE
$$div(gradu)+f(x,y)=0$$

Notice: tensorial notation = coord free notation.

Cartesian Coordinate (x,y)
Let's take the function:

$$u_xx+u_yy+f(x,y)=0$$

We can also write it as div( $$\kappa$$ $$\cdot$$$$grad u$$$$+f(x,y)=0$$

Here, $$\kappa$$ is a continuity tensor (2nd order tensor).

$$\kappa$$ = $$\kappa$$ $$e_i$$ $$\otimes$$ $$e_j$$ 2nd-order tensor

$$v$$ = $$v_i$$ $$e_i$$ vector,1st-order tensor

Operators
$$grad$$ $$u$$ $$=$$ $$\frac{\partial u}{\partial x_i}$$ $$e_i$$

$$div$$ $$v$$ $$=$$ $$\frac{\partial v_i}{\partial x_i}$$ $$e_i$$

$$\kappa$$ $$\cdot$$ $$grad$$ $$u$$ =( $$\kappa_{ij}$$ $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ ($$\frac{\partial u_k}{\partial x_k}$$ $$e_k$$ )

=$$\kappa_{ij}$$ $$\frac{\partial u}{\partial x_k}$$ ( $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ $$e_k$$

Here, $$e_i$$ $$\otimes$$ $$e_j$$ $$\cdot$$ $$e_k$$ = $$e_i$$ ( $$e_j$$ $$\cdot$$ $$e_k$$ )

Kronecker delta
Definition: If $$e_j$$ $$\otimes$$ $$e_k$$ = $$\delta_{jk}$$, $$\delta_{jk}$$ is called Kronecker delta.

$$ \delta_{jk} = \begin{cases} 1 & for \ j=k \\ 0 & for \ j \neq k \end{cases} $$ = $$\kappa_{jk} \frac{\partial u}{\partial x_i} $$ $$e_i$$

$$u(x,y)$$ is a scaler function $$\Rightarrow$$ 0th order tensor $$gradu$$ is a vector field $$\Rightarrow$$ 1st order tensor

$$grad$$  ($$\cdot$$) increases tensor order by 1

$$div$$  ($$\cdot$$) decreases tensor order by 1

We can let $$v$$ : = $$\kappa$$ $$\cdot$$ $$gradu$$

$$div$$ $$v$$ = $$\frac{\partial }{\partial x_i}$$ $$\kappa_{jk}$$ $$\frac{\partial u}{\partial x_j}$$

Leopold Kronecker (1823 - 1891)

Kronecker Delta was named after Leopold Kronecker, a 19th century mathematician from Prussia. Born in 1823, Kronecker studied at the University of Berlin and later went on to teach at what is now known as Humboldt University in 1883. He specialized in the areas of number theory, finitism, and continuity before dying in Berlin in 1891. source:

Homework
Expand the function above.

What's more
$$\kappa_ij$$ = $$\begin{bmatrix} \kappa_{11} & \kappa_{12} \\ \kappa_{21} & \kappa_{22} \end{bmatrix} $$

Notice: in PDE, $$\kappa_{ij}=\kappa_{ji}$$ or $$\kappa^T=$$ $$\kappa$$

$$v$$ = $$v_i$$ $$e_i$$ $$\rightarrow$$ $$\begin{Bmatrix} v_i\end{Bmatrix} _{3 \times 1 or 2 \times 1} $$

And if we change $$e_i$$ coordinate into different system $$\underline{\overline{e_i}}$$

$$\overline{v_i}$$ $$\underline{\overline{e_i}}$$ $$\to$$ $$\overline{\begin{Bmatrix} v_i\end{Bmatrix}}$$

Linearity
if $$\kappa$$ $$= const$$ $$\Rightarrow$$ is a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y,u)$$ $$\Rightarrow$$ is a quasilinear 2nd order PDE

Homework
Show that $$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE

title: User:Egm6322.s09.mafia


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User:Egm6322.s09.bit.gk

Further Examples of Linear PDE
{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Examples of Linear and Non Linear PDE

 * A simple example of a linear PDE is the 2-D Laplace Equation

$$L\left( \phi\right)=\frac{\partial^2\left( \phi\right) }{\partial^2 x}+\frac{\partial^2\left( \phi\right) }{\partial^2 y}$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial^2\left( \right) }{\partial^2 x}+\frac{\partial^2\left( \right) }{\partial^2 y}$$

$$\Rightarrow L\left(\alpha \phi \right)=\frac{\partial^2\left(\alpha \phi \right) }{\partial^2 x}+\frac{\partial^2\left(\alpha \phi \right) }{\partial^2 y}$$

Where, $$\phi=\phi\left(x,y\right)$$

$$\Rightarrow L\left(\alpha \phi \right)=\alpha L\left(\phi\right)$$

$$\Rightarrow Linear$$

 * Another example for linear PDE

$$\frac{\partial\left(\ u\right)}{\partial t}+\frac{\partial\left(\ u\right)}{\partial x}=0$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial\left( \right) }{\partial t}+\frac{\partial\left( \right) }{\partial x}$$

$$L\left(\alpha u+\beta v \right)=\frac{\partial\left(\alpha u+\beta v \right) }{\partial t}+\frac{\partial\left(\alpha u+\beta v \right) }{\partial x}$$

$$\Rightarrow L\left( \alpha u+\beta v \right)=\alpha\frac{\partial\left(\ u\right) }{\partial t}+\beta\frac{\partial\left(\ v\right) }{\partial t}+\alpha\frac{\partial\left(\ u\right) }{\partial x}+\beta\frac{\partial\left(\ v\right) }{\partial t}$$

$$\Rightarrow L\left(\alpha u+\beta v \right)=\alpha\left\{\frac{\partial\left(\ u \right) }{\partial t}+\frac{\partial\left(\ u \right) }{\partial x} \right\}+\beta\left\{\frac{\partial\left(\ v\right) }{\partial t}+\frac{\partial\left(\ v\right) }{\partial x} \right\}$$

$$\Rightarrow L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Further Examples of non linear PDE

 * The inviscid Burgers equation

$$\frac{\partial\left(\ u\right)}{\partial t}+u\frac{\partial\left(\ u\right)}{\partial x}=0$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial\left( \right) }{\partial t}+\left(\right)\frac{\partial\left( \right) }{\partial x}$$

$$L\left(\alpha u+\beta v \right)=\frac{\partial\left(\alpha u+\beta v \right) }{\partial t}+\left(\alpha u+\beta v \right)\frac{\partial\left(\alpha u+\beta v \right) }{\partial x}$$

$$\Rightarrow L\left( \alpha u+\beta v \right)=\alpha\frac{\partial\left(\ u\right) }{\partial t}+\beta\frac{\partial\left(\ v\right) }{\partial t}+\alpha^2 u\frac{\partial\left(\ u\right) }{\partial x}+\beta^2 v\frac{\partial\left(\ v\right) }{\partial x}+\alpha\beta\left\{u\frac{\partial\left(\ v\right) }{\partial x}+ v\frac{\partial\left(\ u\right) }{\partial x}\right\}$$

$$\Rightarrow L\left(\alpha u+\beta v\right)\neq\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

 * Another example for non linear PDE

$$L\left( u\right)=\frac{\partial^2\left( u\right) }{\partial^2 t}+\frac{\partial^2\left( u\right) }{\partial^2 x}-u^2$$

Let,

$$L\left( \right):$$ be an operator,such that $$L\left(\ u\right)$$ is linear with respect to $$u$$ if,

$$ L\left(\alpha u+\beta v\right)=\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$

Therefore, in the present problem, we have,

$$L\left( \right)=\frac{\partial^2\left( \right) }{\partial^2 t}+\frac{\partial^2\left( \right) }{\partial^2 x}-\left(\right)^2$$

From the definition of operator $$L\left(\right)$$,

$$L\left(\alpha u+\beta v \right)=\frac{\partial^2\left(\alpha u+\beta v \right) }{\partial^2 t}+\frac{\partial^2\left(\alpha u+\beta v \right) }{\partial^2 x}-\left(\alpha u+\beta v \right)^2$$

Clearly,

$$\Rightarrow L\left(\alpha u+\beta v\right)\neq\alpha L\left(\ u\right)+\beta L\left(\ v\right)$$
 * }

Problem 5
Egm6322.s09.bit.gk 22:37, 26 January 2009 (UTC)

title: User:Egm6322.s09.bit.gk


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Artificial Consciousness/Philosophy of Mind

 Philosophy of Mind

In order to understand consciousness and why it is so difficult for philosophy to deal with we need a little background in philosophy, you can get a good introduction from Historical Introduction to Philosophy

However this is a general introduction, if you don't want to take the time for a general history of philosophy you might prefer one of the following links
 * The Mind-Body Problem
 * Theories of Mind
 * Personal Identity

Caution should be taken following the links in the above list because they are organized for a separate course and you may lose your way.

 * David Chalmers guide to Philosophy of Mind Articles
 * Consciousness Studies A book with a historical account on Consciousness Philosophies

The List is not exhaustive, if only because I am not a philosopher, and so only worry about the philosophy when it directly impacts my work. Others may want to flesh this section out, if only because philosophy related to thought has been a major topic of discussion by philosophers since time immemorial.

Most Proponents of Artificial Consciousness are Monists, in that they don't believe there is anything in the Mind that cannot be explained by the Brain. However some are exotic forms of dualists, that just believe that the Mind Stuff that isn't explained by the brain, can be explained by the software or equivalent that runs on top of the physical brain. There are many schools of philosophy that argue that Artificial Consciousness will be impossible to produce. Whether or not they are right, seems to depend very heavily on what definition you use for consciousness One Hypothesis of Computational Sufficiency, Ray Jackendoff(1987) states:  Every phenomenological distinction is caused by/supported by/projected from a corresponding computational distinction

To support this, consider the following link: David Chalmers paper on Computational Sufficiency

I am strongly of the belief that soft computing, and satisfycing systems can account for some of the functions of the brain, and that these are forms of computation, even if they are not "Truth Preserving Functions" in and of themselves.

One very prolific discussion on PhilPapers is the Explanatory Gap Thread where philosophers from Universities all over the world are discussing the idea that there is an explanatory gap between the physical systems we know the brain uses, and the phenomenal effects such as feelings that the individual experiences. There seem to be three or four opinions,
 * 1) the explanatory gap exists and never will be bridged
 * 2) the explanatory gap exists and will eventually be bridged
 * 3) the explanatory gap doesn't exist, what exists is a philosophical error
 * 4) the explanatory gap doesn't exist, because that assumes that it could be bridged at some point in the future and it won't be because we don't know what it feels like to be a bat and we will never know.

This type of argument is often found when we try to deal with the phenomenal aspects of the mind, without understanding how they map to the Physical Mechanisms and functions we know the mind performs.

--Graeme E. Smith 04:38, 15 May 2009 (UTC)

See Also:

 * Wikiversity Article on Cognitive Science


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User:Gopher65

About me
Hia:). My home wiki is English Wikinews, so if you want to reach me quickly leave a message on my Wikinews talk page instead of here. Thanks.

title: User:Gopher65


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User talk:Europha

 Hello Europha, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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User talk:Kenzie4466

 Hello Kenzie4466, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --mikeu talk 01:36, 24 January 2009 (UTC)

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User talk:Nickteran

 Hello Nickteran, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --mikeu talk 01:37, 24 January 2009 (UTC)

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