OpenLLM-France/wikimedia huggingface dataset:
# Numerical Analysis/Power iteration exercises
## Exercises

Consider the matrix, $A=\left[{\begin{array}{c c c}6&2&-1\\2&5&1\\-1&1&4\end{array}}\right]$, and the vector, ${\textbf {x}}^{(0)}=\left[{\begin{array}{c}1\\1\\1\\\end{array}}\right].$
By hand, do two iterations of the power method starting with ${\textbf {x}}^{(0)}.$
Solution, first iteration:
$A{\textbf {x}}^{(0)}=\left[{\begin{array}{c c c}6&2&-1\\2&5&1\\-1&1&4\end{array}}\right]\left[{\begin{array}{c}1\\1\\1\end{array}}\right]=\left[{\begin{array}{c}7\\8\\4\end{array}}\right],$ so the estimated eigenvalue is $\mu _{1}=8$
and the estimated eigenvector is ${\textbf {x}}^{(1)}={\frac {1}{8}}A{\textbf {x}}^{(0)}=\left[{\begin{array}{c}{\frac {7}{8}}\\1\\{\frac {1}{2}}\end{array}}\right]$.
Solution, second iteration:
$A{\textbf {x}}^{(1)}=\left[{\begin{array}{c c c}6&2&-1\\2&5&1\\-1&1&4\end{array}}\right]\left[{\begin{array}{c}{\frac {7}{8}}\\1\\{\frac {1}{2}}\end{array}}\right]=\left[{\begin{array}{c}{\frac {27}{4}}\\{\frac {29}{4}}\\{\frac {17}{8}}\end{array}}\right],$ so the estimated eigenvalue is $\mu _{2}={\frac {29}{4}}$ and the estimated eigenvector is ${\textbf {x}}^{(2)}={\frac {4}{29}}A{\textbf {x}}^{(1)}=\left[{\begin{array}{c}{\frac {27}{29}}\\1\\{\frac {17}{58}}\end{array}}\right]$.
Use a computational software package to do 50 iterations. What was the result?
Solution, 50 iterations:
The estimated eigenvalue after 50 iterations is $\mu =7.5712$ with estimated eigenvector ${\textbf {x}}^{(50)}=\left[{\begin{array}{c}1.0000\\0.7507\\-0.0698\end{array}}\right]$
# Numerical Analysis/Order of RK methods

* Derivation of a second order RK method
* Derivation of a third order RK method
* Quiz on RK order
* Implicit RK2 on an Autonomous ODE
# Numerical Analysis/Order of RK methods/Implicit RK2 on an Autonomous ODE

We consider an autonomous initial value ODE
    $y'(t)=f(y(t))\quad {\text{with}}\quad y(t_{0})=y_{0}\,.$         (ODE)
Applying the Tradezoidal rule gives the implicit Runge-Kutta method
    $y_{n+1}=y_{n}+{\frac {h}{2}}\left(f(y_{n})+f(y_{n+1})\right)\,.$         (method)
We will show that (method) is second order.
Expanding the true solution $y(t_{n}+h)$ about $t_{n}$ using Taylor series, we have
    $y(t_{n}+h)=y(t_{n})+hy'(t_{n})+{\frac {h^{2}}{2}}y''(t_{n})+{\frac {h^{3}}{3!}}y'''(t_{n})+O(h^{4})\,.$
Since $y(t)$ satisfies (ODE), we can substitute $y'(t)=f(y(t))$ and obtain
    $y(t_{n}+h)=y(t_{n})+hf(y(t_{n}))+{\frac {h^{2}}{2}}f'(y(t_{n}))f(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.$         (true)
In (method) we can assume $y_{n}=y(t_{n})$ since that is the previous data. Subtracting (method) from (true) gives us the local truncation error
    $y(t_{n}+h)-y_{n+1}={\frac {h}{2}}\left[f(y(t_{n}))-f(y_{n+1})\right]+{\frac {h^{2}}{2}}f'(y(t_{n}))f(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.$         (error1)
In order to cancel more terms we need to expand $f(y_{n+1})$. However, $y_{n+1}\not =y(t_{n}+h)$ so we cannot do a regular Taylor expansion. Instead we can plug (method) back into $f$ and then do a Taylor expansion to obtain
    ${\begin{aligned}f(y_{n+1})&=f\left(y(t_{n})+{\frac {h}{2}}\left(f(y(t_{n}))+f(y_{n+1})\right)\right)\\&=f(y(t_{n}))+{\frac {h}{2}}\left(f(y(t_{n}))+f(y_{n+1})\right)f'(y(t_{n}))+{\frac {h^{2}}{8}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}f''(y(t_{n}))+O(h^{3})\,.\end{aligned}}$         (implicit)
Substituting (implicit) into (error1) yields
    ${\frac {h^{2}}{4}}f'(y(t_{n}))\left[f(y(t_{n}))-f(y_{n+1})\right]+{\frac {-h^{3}}{16}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}f''(y(t_{n}))+{\frac {h^{3}}{3!}}\left[f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right]+O(h^{4})\,.$         (error2)
This substitution was productive since the $h$ terms canceled. We can do this trick again, but this time only need (implicit) up to $O(h^{2})$ since everything will be multiplied by at least $h^{2}$ and this can go into the $O(h^{4})$. Substituting (implicit) in for the first occurance of $f(y_{n+1})$ in (error2) yields
    ${\frac {h^{3}}{2}}\left\{f''(y(t_{n}))\left[{\frac {1}{3}}(f(y(t_{n})))^{2}+{\frac {-1}{8}}\left(f(y(t_{n}))+f(y_{n+1})\right)^{2}\right]+(f'(y(t_{n})))^{2}\left[{\frac {1}{3}}f(y(t_{n}))+{\frac {-1}{4}}\left(f(y(t_{n}))+f(y_{n+1})\right)\right]\right\}+O(h^{4})\,.$         (error3)
This substitution was productive since the $h^{2}$ terms canceled. We can do this again, now truncating (implicit) at $O(h)$. Substituting (implicit) into (error3) yields
    ${\frac {-h^{3}}{12}}\left\{f''(y(t_{n}))(f(y(t_{n})))^{2}+(f'(y(t_{n})))^{2}f(y(t_{n}))\right\}+O(h^{4})\,.$
Since the $h^{3}$ term does not cancel, we have shown that the local truncation error is $O(h^{3})$ and thus the method is order 2.
# Numerical Analysis/Other Resources

* Boundary Value Problems/Numerical Methods
* Introduction to Numerical Methods and Matlab Programming, a free online text for a college junior level course Applied Numerical Methods.
* NumaWWW, offers the possibility of testing a lot of methods from numerical computing and optimization interactively, with only minimal programming knowledge needed.
* MA213: Numerical Analysis is a free online course.
* Holistic Numerical Methods collects undergraduate resources and learning materials.
# Numerical Analysis/Polynomial interpolation concept quiz

Choose the best answer for each question:
# Ordered field/a \geq b positive/Ratio/Exercise

Let $K$ be an ordered field and let $x,y$ be positive elements. Show that $x\geq y$ is equivalent with ${\frac {x}{y}}\geq 1$.
# Ordered field/x^2+(x+1)^2 \geq (x+2)^2 for x \geq 3/Exercise

Let $K$ denote an ordered field. Show that for $x\geq 3$ the relation
    $x^{2}+(x+1)^{2}\geq (x+2)^{2}\,$
holds.
# Ordered field/x \geq 1/Inverse/Exercise

Let $K$ be an ordered field and $x\geq 1$. Show that for the inverse element $x^{-1}\leq 1$ holds.
# Ordered field/x negative/Inverse/Exercise

Let $K$ be an ordered field and $x<0$. Show that also the inverse element $x^{-1}$ is negative.
# Ordinary Differential Equations
## Contents

* Basic Concepts
** What is a differential equation?
** Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs)
** Order of an ODE
** Linear and Non-linear ODEs
* ODEs as Models
* Verifying Solutions to ODEs
* Initial Value Problems (IVP)
* First Order ODEs
** Direction Fields
** Separable Equations
** Method of Integrating Factors
** Exact Equations
** Euler's Method
** The Existence and Uniqueness Theorem
* Second Order ODEs
** Solving Constant Coefficient Second Order Homegenous ODEs $ay''+by'+cy=0$
# Ordered field/b bigger 1/Factor decomposition/Exercise

Let $K$ be an ordered field and $b\in K$, $b>1$. Show that there exists elements $c,d>1$ such that $b=cd$.
# Ordinary differential equation/1/Exercise form

Find the solutions to the ordinary differential equation
    $y'=\,$
# Ordered field/x positive/Inverse/Exercise

Let $K$ be an ordered field and $x>0$. Show that also the inverse element $x^{-1}$ is positive.
# Motivation and emotion/Book/2010/Overeating motivation

* Multimedia presentation (5 min)
# Payment Documentation Requirements/Summary

| Unit 4.7-Payment Documentation Requirements | Introduction | Required Documents | Summary | Resources | Activities | Assessment Go to Unit 4.1 | Unit 4.2 | Unit 4.3 | Unit 4.4 | Unit 4.5 | Unit 4.6 | Unit 4.7 | Unit 4.8 | Unit 4.9 Return to Trade Finance Module 4: Payment Methods |

## Summary

An international manager should now be able to describe the major documents utilized in international trade as well as to identify how these documents are used and by whom. For a complete list of documents, issuer, user and uses for each, review the attached International Documentation Listing provided. It is important to note that the major documents and the minimum documents required for every export and import transaction to and from the US are
* commercial invoice
* packing list
* bill of lading
# Payment Methods for International Transactions/Assessment

| Unit 3.4-Payment Methods for International Transactions | Introduction | Methods of Payment | Comparing Methods | Summary | Resources | Activities | Assessment Go to Unit 3.1 | Unit 3.2 | Unit 3.3 | Unit 3.4 | Unit 3.5 Return to Trade Finance Module 3: Commercial Risk |
# Payment Methods for International Transactions/Activities

| Unit 3.4-Payment Methods for International Transactions | Introduction | Methods of Payment | Comparing Methods | Summary | Resources | Activities | Assessment Go to Unit 3.1 | Unit 3.2 | Unit 3.3 | Unit 3.4 | Unit 3.5 Return to Trade Finance Module 3: Commercial Risk |

## Activity

Divide the class in half (exporter and importer). Present each side with the following scenario:
Importers
* You are a new company (in business two years) and you are interested in placing a fairly large order with your foreign supplier.
* Your goal is to purchase the products and resell them to your customers.
* You need the product to be shipped within two weeks, and it will take six weeks for you to receive the product once it ships.
* Once you receive the product, you will repack it and ship it to your customer within one week. You have agreed to receive payment from your customer 60 days after you ship.
* In order to fund this scenario, you have requested open account payment terms from your vendor at Net 120 Days on Invoice and will pay in their currency.
Exporters
* You are a well-established company that has been selling your products overseas for 15 years, although you have never sold to a customer in this new market.
* You currently hold inventory of the product this customer wants to purchase, so you do not need to manufacture additional products.
* You never extend open payment terms to new customers; however, this customer is also talking to your biggest competitor and may place the order there if you do not comply.
Task Negotiate payment terms and discuss each group’s needs and concerns.
Is there a compromise available that would please both companies?
Alternatives To create new scenarios, you can include one or all of the following caveats:
    1. the currency in the importer’s country has devalued by 25%.
    2. there are rumblings of a dock strike in the importer’s country.
    3. the exporter wants to conduct a credit check in order to extend payment terms. The importer has agreed to supply the necessary information and will wait four weeks for the shipment to depart.
    4. the exporter has to manufacture the necessary products and will require funds to purchase raw materials.
    5. the exporter has never sold products overseas.
# Payment Documentation Requirements/Resources

| Unit 4.7-Payment Documentation Requirements | Introduction | Required Documents | Summary | Resources | Activities | Assessment Go to Unit 4.1 | Unit 4.2 | Unit 4.3 | Unit 4.4 | Unit 4.5 | Unit 4.6 | Unit 4.7 | Unit 4.8 | Unit 4.9 Return to Trade Finance Module 4: Payment Methods |

## Resources

International Business: Environments and Operations, J. Daniel, L Radebaugh, 7th ed. Addison-Wesley Publishers.
Exporting: From Start to Finance, 3rd Edition, L. Wells and K Dulat, McGraw Hill.
The Global Entrepreneur, James Foley, 2nd Edition, Jamric Press, 2004.
The Basic Guide to Exporting, US Department of Commerce.
World Trade Press Publications – www.worldtradepress.com
* Dictionary of International Trade
* International Finance
US Department of Commerce – www.export.gov

### Attachments

International Documents Listing
# Payment Methods for International Transactions/Summary

| Unit 3.4-Payment Methods for International Transactions | Introduction | Methods of Payment | Comparing Methods | Summary | Resources | Activities | Assessment Go to Unit 3.1 | Unit 3.2 | Unit 3.3 | Unit 3.4 | Unit 3.5 Return to Trade Finance Module 3: Commercial Risk |

## Summary

This discussion has focused on how to identify a variety of payment methods available to a risk manager. Businesses who sell in the 21st century global market appreciate that secured transactions are not only way transact business when selling internationally if they want to be both competitive and grow their business. A significant responsibility of a credit manager is to understand the use of the various payment terms in a competitive business environment. However, an international manager is not the only person who needs to learn to describe and apply the various payment methods that will apply to buyers. Senior financial management of the seller as well as the “business” people – marketing, operations – are well-advised to seek information as to payment decisions with buyers so that they are able to evaluate payment terms. Payment terms are an important component of the relationship with buyers. A pro-active international manager involves other business units and individuals, such as those in sale management, not only in the process of “why” certain credit terms are determined for a buyer, but also “how” they are determined.
Even though in most organizations the credit risk group has the responsibility of establishing and determining payment terms of customers, good business judgment in an organization should be exercised by bringing in internal customers to the decision process since they have stakes in the relationship with existing or potential buyers.
# Payment Methods for International Transactions/Comparing Methods of Payment and Risk

| Unit 3.4-Payment Methods for International Transactions | Introduction | Methods of Payment | Comparing Methods | Summary | Resources | Activities | Assessment Go to Unit 3.1 | Unit 3.2 | Unit 3.3 | Unit 3.4 | Unit 3.5 Return to Trade Finance Module 3: Commercial Risk |

## Comparing Methods of Payment and Risk

The following chart offers a comparison from lowest to highest risk to an exporter/seller and importer/buyer (permission to use image, granted by Wm Hindon, VP, JP Morgan, 2004). The following chart lists the methods of payment discussed in the prior section in risk order to both the exporter (seller) and importer (buyer).
On the left side of the chart, you see the risk evaluation of the methods of payment from the perspective of an exporter or seller. For example, selling products or services on a “cash in advance” basis has a low uncertainty” or “low risk” to the seller because “cash” or funds are made available to the seller prior to the sale. A letter of credit, next on the “level of certainty/uncertainty” scale, is also a sound method of payment, although there could be discrepancies in the documents that could possibly affect payment. The same applies to the documentary collections. When “open account,” the last item on the list, is considered as to an exporter’s certainty/uncertainty of payment, the international manager should understand that by agreeing to “open account” (selling on net 30-day payment terms) there is a risk element of when and if payment will be made.
On the right side of the chart, you can see the level of uncertainty in terms of payment made by the importer or buyer. For example, if the importer (buyer) pays “cash in advance” for a product or service, the “uncertainty” is that money is paid before the product/service is available. What if there is a quality problem or delay in receiving the product or service which has already been paid for by the buyer? The buyer has no recourse. At the other end of the scale, the buyer has a low level of “uncertainty” if he/she has been “given” open account (30 days to pay) for product or service that has been delivered. The buyer has the product/service but has not yet paid for the invoice.
Note: the image on which this page text depends was deleted, because no license information had been provided. See . This was a technical deletion. If any user wants to work on this page to restore the information in that file, for free use or fair use, I will restore the file for examination, but I'm not going to do that without a request. This notice may be moved to the Talk page here, once there is an involved user willing to work on this resource. Please communicate with me on my user talk page. --Abd 18:23, 6 November 2011 (UTC)
# Payment Methods for International Transactions/Resources

| Unit 3.4-Payment Methods for International Transactions | Introduction | Methods of Payment | Comparing Methods | Summary | Resources | Activities | Assessment Go to Unit 3.1 | Unit 3.2 | Unit 3.3 | Unit 3.4 | Unit 3.5 Return to Trade Finance Module 3: Commercial Risk |

## Resources

US Dept of Commerce – www.export.gov MSU Globaledge Modules – http://globaledge.msu.edu - Export Financing – Receiving Payment
# Payment Methods for International Transactions

| Unit 3.4-Payment Methods for International Transactions | Introduction | Methods of Payment | Comparing Methods | Summary | Resources | Activities | Assessment Go to Unit 3.1 | Unit 3.2 | Unit 3.3 | Unit 3.4 | Unit 3.5 Return to Trade Finance Module 3: Commercial Risk |

## Unit 3.4- Payment Methods for International Transactions

The uncertainty of international sales reinforces the earlier discussion regarding sources of credit information. A significant reason for obtaining and effectively utilizing sources of information is to make the best decision for a seller as far as what payment terms are most appropriate, payment terms that minimize risk while allowing a seller to be competitive in the market place. Each of these payment terms can be measured by levels of risk, and an international credit manager needs to be in a position to understand and describe these methods of payments. The concept of extending credit in the international environment requires than an international credit manager learn methods for mitigating payment risks for the organization. One way to do so is with different methods of payment.
There are a variety of payment methods available for international transactions. Some companies, who are new to selling in overseas markets, believe that the letter of credit or some type of secured transaction is the “best” (or only) way to transact business when selling offshore. However, a pro-active international credit manager recognizes that in an attempt to obtain new business, many different payment terms are important in a competitive business environment. The ways these terms apply as they relate to a seller’s risk of receiving payment from a buyer are examined in this lesson.

### Unit Objective

The goal of this material is to introduce you to the methods of payment available for international transactions and the risks to buyers and sellers for each method of payment. By the end of this unit you will be able to
* identify the methods of payment available for international transactions.
* identify the risks to a buyer associated with each method of payment.
* identify the risks to a seller associated with each method of payment.

### Unit Outline

* Introduction
* Methods of Payment
* Comparing Methods of Payment and Risk
* Summary
* Resources
* Activities
* Assessment
Correlation: Materials from this unit correlate with NASBITE CGCP's Knowledge Statement 04/03/04: Knowledge of methods of payment (e.g., letters of credit, documentary collection, cash in advance, sight draft)
# Rhetoric

Aristotle defined rhetoric as the art of persuasion through all available means, and, generally speaking, the definition stuck. Rhetoric is firmly rooted in the idea that messages are delivered to specific audiences, and that successful communication requires that the speaker or writer appeal to the expectations, values, and conventions of those audiences. Rhetorical preferences have changed with the centuries, although the Modern Enlightenment's emphasis on objectivity generated a preference in speaking, writing, or even now in other communication media, to present your ideas in the clearest, most concise manner possible. Your clear, concise delivery should resonate with your audience in a way that leads them to not only believe what you are saying, but also to be persuaded to your causes.
Here is a basic primer on Rhetoric.

## The Three Means of Persuasion

The systematical core of Aristotle's Rhetoric is the doctrine that there are three technical means of persuasion.
* Ethos - an appeal based on the character of the speaker.
* Pathos - an appeal based on emotion.
* Logos - an appeal based on logic or reason.

## Key Scholars

* Classical (Greek)
** Gorgias
** Isocrates
** Plato
** Aristotle
* Classical (Roman)
** Cicero
** Quintilian
* Medieval
** Augustine of Hippo
** Desiderius Erasmus
** Petrus Ramus
* Modern
** I.A. Richards
** Wayne C. Booth
* Contemporary
** Stephen Toulmin
** Kenneth Burke

## Related Wikibooks

* Visual Rhetoric
# Rhetoric by Aristotle (Chapter 14, Book 3)

(The inscription from the book is by Electronic Home Library (TM), World Library, Inc., 1991. Translated by W. Rhys Roberts. This translation is already not copyrighted.)
Aristotle included the introductions to forensic speeches among the introductions to speeches of display. He did it, pleading the sentence denoted in extra bold type:
"The Introduction is the beginning of a speech, corresponding to the prologue in poetry and the prelude in flute-music; they are all beginnings, paving the way, as it were, for what is to follow. The musical prelude resembles the introduction to speeches of display; as flute players play first some brilliant passage they know well and then fit it on to the opening notes of the piece itself, so in speeches of display the writer should proceed in the same way; he should begin with what best takes his fancy, and then strike up his theme and lead into it; which is indeed what is always done. (Take as an example the introduction to the Helen of Isocrates-there is nothing in common between the 'eristics' and Helen.) And here, even if you travel far from your subject, it is fitting, rather than that there should be sameness in the entire speech.
The usual subject for the introductions to speeches of display is some piece of praise or censure. Thus Gorgias writes in his Olympic Speech, 'You deserve widespread admiration, men of Greece', praising thus those who start,ed the festival gatherings.' Isocrates, on the other hand, censures them for awarding distinctions to fine athletes but giving no prize for intellectual ability. Or one may begin with a piece of advice, thus: 'We ought to honour good men and so I myself am praising Aristeides' or 'We ought to honour those who are unpopular but not bad men, men whose good qualities have never been noticed, like Alexander son of Priam.' Here the orator gives advice. Or we may begin as speakers do in the law-courts; that is to say, with appeals to the audience to excuse us if our speech is about something paradoxical, difficult, or hackneyed; like Choerilus in the lines-
Introductions to speeches of display, then, may be composed of some piece of praise or censure, of advice to do or not to do something, or of appeals to the audience; and you must choose between making these preliminary passages connected or disconnected with the speech itself.
Introductions to forensic speeches, it must be observed, have the same value as the prologues of dramas and the introductions to epic poems; the dithyrambic prelude resembling the introduction to a speech of display, as
In prologues, and in epic poetry, a foretaste of the theme is given, intended to inform the hearers of it in advance instead of keeping their minds in suspense. Anything vague puzzles them: so give them a grasp of the beginning, and they can hold fast to it and follow the argument."

## Footnotes

* ↑ The explaining remark concerning this problem was published on page 430 of The "Art" of Rhetoric by John Henry Freese.