text:
Talk:Taxation in the United Kingdom

Perhaps once the content is fleshed out better we could split out the more general tax stuff into a seperate General Tax book?

This would then form part of the series:
 * Accounting
 * Tax
 * Tax in UK
 * Tax in ZA

ps. If you have an accounting background Accounting needs some help too (But flesh out here first, we have at least 2 active contributors in accounts) --Mig77 13:39, 17 February 2006 (UTC)

Just a quick note to say THANK YOU for starting up this wikibook. Tax is something that I'd love to know more about (but I'm afraid I know very little about right now). Best of luck with the book dan_aka_jack 13:25, 27 April 2006 (UTC)

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

title: Talk:Taxation in the United Kingdom


text:
Tagalog/Appendix D

The following appendix contains information about the salawikain, or traditional Filipino proverb. The salawikain, also known as the kasabihan ("saying"), is part of a rich Filipino cultural tradition, many of which are very old and date back to pre-colonial or colonial times.

The salawikain is a very important part of Philippine history, since it is one of the contributions early Filipinos have made to Philippine history and culture.

Mga Salawikain (Proverbs)
Here are some examples of the salawikain:

Ang hindi marunong lumingon sa pinanggalingan ay hindi makararating sa paroroonan. "He who does not look back from where he came will never reach his destination."

''Ang isda ay hinuhuli sa bibig. Ang tao, sa salita.'' "Fish are caught by the mouth. People, by their word."

Ang hindi magmahal sa kaniyang wika ay mahigit pa sa hayop at malansang isda. (José Rizal) "He who doesn't love his language is worse than an animal or a rotten fish."

Nasa Diyos ang awa, nasa tao ang gawa. "God has compassion, man has action."

Magbiro lamang sa lasing, huwag lang sa bagong gising. "Joke around with someone who is drunk, but not with someone newly awoken."

Magsama-sama at malakas, magwatak-watak at babagsak. "United we stand, divided we fall."

Aanhin pa ang damo kung patay na ang kabayo. "What's the use of grass when the horse is already dead."

Habang may buhay, may pag-asa. "While there is life, there is hope."

Ang maaabot nang paupo, huwag nang kunin nang patayo. "What can be reached by sitting down, do not get by standing up."

Ang bahay mo man ay bato kung ang nakatira'y kuwago, mabuti pa ang isang kubo kung nakatira'y tao. "Your house may be stone if an owl lives in it, but good is a hut when a person lives in it."

Anak na di paluin, ina ay paluluhain. "A child that will not be spanked will make the mother cry."

^ Table of Contents ^ | <<Appendix C: Slang | Appendix D: Proverbs

title: Tagalog/Appendix D


text:
XML - Managing Data Exchange/XSLT and Style Sheets/Answers

Stylesheet Chapter => XSLT and Style Sheets

Stylesheet Exercises => Exercises


text:
Statistics/Numerical Methods/Numerics in Excel

The purpose of this paper is to evaluate the accuracy of MS Excel in terms of statistical procedures and to conclude whether the MS Excel should be used for (statistical) scientific purposes or not.

According to the literature, there are three main problematic areas for Excel if it is used for statistical calculations. These are
 * probability distributions,
 * univariate statistics, ANOVA and Estimations (both linear and non-linear)
 * random number generation.

If the results of statistical packages are assessed, one should take into account that the acceptable accuracy of the results should be achieved in double precision (which means that a result is accepted as accurate if it possesses 15 accurate digits) given that the reliable algorithms are capable of delivering correct results in double precision, as well. If the reliable algorithms can not retrieve results in double precision, it is not fair to anticipate that the package (evaluated) should achieve double precision. Thus we can say that the correct way for evaluating the statistical packages is assessing the quality of underlying algorithm of statistical calculations rather than only counting the accurate digits of results. Besides, test problems must be reasonable which means they must be amenable to solution by known reliable algorithms. (McCullough & Wilson, 1999, S. 28)

In further sections, our judgement about the accuracy of MS Excel will base on certified values and tests. As basis we have Knüsel’s ELV software for probability distributions, StRD (Statistical Reference Datasets) for Univariate Statistics, ANOVA and Estimations and finally Marsaglia’s DIEHARD for Random Number Generation. Each of the tests and certified values will be explained in the corresponding sections.

Assessing Excel Results for Statistical Distributions
As we mentioned above our judgement about Excel’s calculations for probability distributions will base on Knüsel’s ELV Program which can compute probabilities and quantiles of some elementary statistical distributions. Using ELV, the upper and lower tail probabilities of all distributions are computed with six significant digits for probabilities as small as 10−100 and upper and lower quantiles are computed for all distributions for tail probabilities P with 10−12 ≤ P ≤ ½. (Knüsel, 2003, S.1)

In our benchmark Excel should display no inaccurate digits. If six digits are displayed, then all six digits should be correct. If the algorithm is only accurate to two digits, then only two digits should be displayed so as not to mislead the user (McCullough & Wilson, 2005, S. 1245)

In the following sub-sections the exact values in the tables are retrieved from Knüsel’s ELV software and the acceptable accuracy is in single precision, because even the best algorithms can not achieve 15 correct digits in most cases, if the probability distributions are issued.

Normal Distribution

 * Excel Function:NORMDIST
 * Parameters:   mean = 0, variance = 1, x (critical value)
 * Computes:     the tail probability Pr X ≤ x, whereas X denotes a random variable with a standard normal distribution (with mean 0 and variance 1)

As we can see in table 1, Excel 97, 2000 and XP encounter problems and computes small probabilities in tail incorrectly (i.e for x = -8,3 or x = -8.2) However, this problem is fixed in Excel 2003 (Knüsel, 2005, S.446).

Inverse Normal Distribution

 * Excel Function: 	NORMINV
 * Parameters:		mean = 0, variance = 1, p (probability for X < x)
 * Computes: 		the x value (quantile)

X denotes a random variable with a standard normal distribution. In contrast to “NORMDIST” function issued in the last section, p is given and quantile is computed.

If used, Excel 97 prints out quantiles with 10 digits although none of these 10 digits may be correct if p is small. In Excel 2000 and XP, Microsoft tried to fix errors, although results are not sufficient (See table 2). However in Excel 2003 the problem is fixed entirely. (Knüsel, 2005, S.446)

Inverse Chi-Square Distribution

 * Excel Function: 	CHIINV
 * Parameters: 		p (probability for X > x), n (degrees of freedom)
 * Computes: 		the x value (quantile)

X denotes a random variable with a chi-square distribution with n degrees of freedom.

Old Excel Versions: Although the old Excel versions show ten significant digits, only very few of them are accurate if p is small (See table 3). Even if p is not small, the accurate digits are not enough to say that Excel is sufficient for this distribution.

Excel 2003: Problem was fixed. (Knüsel, 2005, S.446)

Inverse F Distribution

 * Excel Function: 	FINV
 * Parameters: 	p (probability for X > x), n1, n2 (degrees of freedom)
 * Computes: 		the x value (quantile)

X denotes a random variable with a F distribution with n1 and n2 degrees of freedom.

Old Excel Versions: Excel prints out x values with 7 or more significant digits although only one or two of these many digits are correct if p is small (See table 4).

Excel 2003: Problem fixed. (Knüsel, 2005, S.446)

Inverse t Distribution

 * Excel Function:	TINV
 * Parameters: 		p (probability for |X| > x), n (degree of freedom)
 * Computes: 		the x value (quantile)

X denotes a random variable with a t distribution with n degrees of freedom. Please note that the |X| value causes a 2 tailed computation. (lower tail & high tail)

Old Excel Versions: Excel prints out quantiles with 9 or more significant digits although only one or two of these many digits are correct if p is small (See table 5).

Excel 2003: Problem fixed. (Knüsel, 2005, S.446)

Poisson Distribution

 * Excel Function: Poisson
 * Parameters:     λ (mean),  k (number of cases)
 * Computes:  	    the tail probability Pr X ≤ k

X denotes a random variable with a Poisson distribution with given parameters.

Old Excel Versions: correctly computes very small probabilities but gives no result for central probabilities near the mean (in the range about 0.5). (See table 6)

Excel 2003: The central probabilities are fixed. However, inaccurate results in the tail. (See table 6)

The strange behaviour of Excel can be encountered for values λ150. (Knüsel, 1998, S.375) It fails even for probabilities in the central range between 0.01 and 0.99 and even for parameter values that cannot be judged as too extreme.

Binomial Distribution

 * Excel Function: 	BINOMDIST
 * Parameters: 		n (= number of trials), υ(= probability for a success) , k(number of successes)
 * Computes: 		the tail probability Pr X ≤ k

-X denotes a random variable with a binoamial distribution with given parameters

Old Excel Versions: As we see in table 7, old versions of Excel correctly computes very small probabilities but gives no result for central probabilities near the mean (same problem with Poisson distribuiton on old Excel versions)

Excel 2003: The central probabilities are fixed. However, inaccurate results in the tail. (Knüsel, 2005, S.446). (same problem with Poisson distribuiton on Excel 2003).

This strange behaviour of Excel can be encountered for values n > 1000. (Knüsel, 1998, S.375) It fails even for probabilities in the central range between 0.01 and 0.99 and even for parameter values that cannot be judged as too extreme.

Other problems

 * Excel 97, 2000 and XP includes flaws by computing the hypergeometric distribution (HYPERGEOM). For some values  (N > 1030) no result is retrieved. This is prevented on Excel 2003, but there is still no option to compute tail probabilities. So computation of  Pr {X = k} is possible, but computation of  Pr {X ≤ k} is not. (Knüsel, 2005, S.447)
 * Function GAMMADIST for gamma distribution retrieves incorrect values on Excel 2003. (Knüsel, 2005, S.447-448)
 * Also the function BETAINV for inverse beta distribution computes incorrect values on Excel 2003 (Knüsel, 2005, S. 448)

Assessing Excel Results for Univariate Statistics, ANOVA and Estimation (Linear & Non-Linear)
Our judgement about Excel’s calculations for univariate statistics, ANOVA and Estimation will base on StRD which is designed by Statistical Engineering Division of National Institute of Standards and Technology (NIST) to assist researchers in benchmarking statistical software packages explicitly. StRD has reference datasets (real-world and generated datasets) with certified computational results that enable the objective evaluation of statistical Software. It comprises four suites of numerical benchmarks for statistical software: univariate summary statistics, one way analysis of variance, linear regression and nonlinear regression and it includes several problems for each suite of tests. All problems have a difficulty level:low, average or high.

By assessing Excel results in this section we are going to use LRE (log relative error) which can be used as a score for accuracy of results of statistical packages. The number of correct digits in results can be calculated via log relative error. Please note that for double precision the computed LRE is in the range 0 - 15, because we can have max. 15 correct digits in double precision.

Formula LRE:

$$\lambda = LRE(x) = -log_{10} \left ( \frac{|x - c|}{|x|} \right ) $$

c: the correct answer (certified computational result) for a particular test problem

x: answer of Excel for the same problem

Univariate Statistics

 * Excel Functions: - AVERAGE, STDEV, PEARSON (also CORREL)

 * Computes (respectively): mean, standard deviation, correlation coefficient

Old Excel Versions: an unstable algorithm for calculation of the sample variance and the correlation coefficient is used. Even for the low difficulty problems (datasets with letter “l” in table 8) the old versions of Excel fail.

Excel 2003: Problem was fixed and the performance is acceptable. The accurate digits less than 15 don’t indicate an unsuccessful implementation because even the reliable algorithms can not retrieve 15 correct digits for these average and high difficulty problems (datasets with letters “a” and “h” in table 8) of StRD.

ONEWAY ANOVA

 * Excel Function: Tools – Data Analysis – ANOVA: Single Factor (requires Analysis Toolpak)
 * Computes: df, ss, ms, F-statistic

Since ANOVA produces many numerical results (such as df, ss, ms, F), here only the LRE for the final F-statistic is presented. Before assessing Excel’s performance one should consider that a reliable algorithm for one way Analysis of Variance can deliver 8-10 digits for the average difficulty problems and 4-5 digits for higher difficulty problems.

Old Excel Versions: Considering numerical solutions, delivering only a few digits of accuracy for difficult problems is not an evidence for bad software, but retrieving 0 accurate digits for average difficulty problems indicates bad software when calculating ANOVA. (McCullough & Wilson, 1999, S. 31). For that reason Excel versions prior than Excel 2003 has an acceptable performance only on low-difficulty problems. It retrieves zero accurate digits for difficult problems. Besides, negative results for “within group sum of squares” and “between group sum of squares” are the further indicators of a bad algorithm used for Excel. (See table 9)

Excel 2003: Problem was fixed (See table 9). The zero digits of accuracy for the Simon 9 test is no cause for concern, since this also occurs when reliable algorithms are employed. Therefore the performance is acceptable. (McCullough & Wilson, 2005, S. 1248)

Linear Regression

 * Excel Function: LINEST
 * Computes:  All numerical results required by Linear Regression

Since LINEST produces many numerical results for linear regression, only the LRE for the coefficients and standard errors of coefficients are taken into account. Table 9 shows the lowest LRE values for each dataset as the weakest link in the chain in order to reflect the worst estimations (smallest $$\lambda_\beta$$-LRE and $$\lambda_\sigma$$-LRE) made by Excel for each linear regression function.

Old Excel Versions: either doesn't check for near-singularity of the input matrix or checking it incorrectly, so the results for ill-conditioned Dataset “Filip (h)” include not a single correct digit. Actually, Excel should have refused the solution and commit a warning to user about the near singularity of data matrix. (McCullough & Wilson, 1999, S.32,33). However, in this case, the user is mislead.

Excel 2003: Problem is fixed and Excel 2003 has an acceptable performance. (see table 10)

Non-Linear Regression
When solving nonlinear regression using Excel, it is possible to make choices about:

 * 1) method of derivative calculation: forward (default) or central numerical derivatives
 * 2) convergence tolerance (default=1.E-3)
 * 3) scaling (recentering) the variables
 * 4) method of solution (default – GRG2 quasi-Newton method)

Excel’s default parameters don’t always produce the best solutions always (like all other solvers). Therefore one needs to give different parameters and test the Excel-Solver for non-linear regression. In table 10 the columns A-B-C-D are combinations of different non-linear options. Because changing the 1st and 4th option doesn’t affect the result, only 2nd and 3rd parameters are changed for testing:
 * A: Default estimation
 * B: Convergence Tolerance = 1E -7
 * C: Automatic Scaling
 * D: Convergence Tolerance =	1E -7 & Automatic Scaling

In Table 11, the lowest LRE principle is applied to simplify the assessment. (like in linear reg.)

Results in table 11 are same for each Excel version (Excel 97, 2000, XP, 2003)

As we see in table 11, the non-linear option combination A produces 21 times, B 17 times, C 20 times and D 14 times “0” accurate digits. which indicates that the performance of Excel in this area is inadequate. Expecting to find all exact solutions for all problems with Excel is not fair, but if it is not able to find the result, it is expected to warn user and commit that the solution can not be calculated. Furthermore, one should emphasize that other statistical packages like SPSS, S-PLUS and SAS exhibit zero digit accuracy only few times (0 to 3) in these tests (McCullough & Wilson, 1999, S. 34).

Assessing Random Number Generator of Excel
Many statistical procedures employ random numbers and it is expected that the generated random numbers are really random. Only random number generators should be used that have solid theoretical properties. Additionally, statistical tests should be applied on samples generated and only generators whose output has successfully passed a battery of statistical tests should be used. (Gentle, 2003)

Based on the facts explained above we should assess the quality of Random Number Generation by:
 * analysing the underlying algorithm for Random Number Generation.
 * analysing the generators output stream. There are many alternatives to test the output of a RNG. One can evaluate the generated output using static tests in which the generation order is not important. These tests are goodness of fit tests. The second way of evaluating the output stream is running a dynamic test on generator, whereas the generation order of the numbers is important.

Excel’s RNG – Underlying algorithm
The objective of random number generation is to produce samples any given size that are indistinguishable from samples of the same size from a U(0,1) distribution. (Gentle, 2003) For this purpose there are different algorithms to use. Excel’s algorithm for random number generation is Wichmann–Hill algorithm. Wichmann–Hill is a useful RNG algorithm for common applications, but it is obsolete for modern needs (McCullough & Wilson, 2005, S. 1250). The formula for this random number generator is defined as follows:

$$X_i =  171.X_i - 1  mod  30269 $$

$$Y_i =  172.Y_i - 1  mod  30307 $$

$$Z_i =  170.Z_i - 1  mod  30323$$

$$U_i =  \frac{X_i}{30269} +  \frac{Y_i}{30307} + \frac {Z_i}{30323} mod 1 $$

Wichmann–Hill is a congruential generator which means that it is a recursive aritmethical RNG as we see in the formula above. It is a combination of three other linear congruential generator and requires three seeds: $$X_0 Y_0 Z_0$$.

Period, in terms of random number generation, is the number of calls that can be made to the RNG before it begins to repeat. For that reason, having a long period is a quality measure for random number generators. It is essential that the period of the generator be larger than the number of random numbers to be used. Modern applications are increasingly demanding longer and longer sequences of random numbers (i.e for using in Monte-Carlo simulations) (Gentle, 2003)

The lowest acceptable period for a good RNG is $$2^{60}$$ and the period of Wichmann-Hill RNG is 6.95E+12 (≈ $$2^{43}$$). In addition to this unacceptable performance, Microsoft claims that the period of Wichmann-Hill RNG is 10E+13 Even if Excel’s RNG has a period of 10E+13, it is still not sufficient to be an acceptable random number generator because this value is also less than $$2^{60}$$. (McCullough & Wilson, 2005, S. 1250)

Furthermore it is known that RNG of Excel produces negative values after the RNG executed many times. However a correct implementation of a Wichmann-Hill Random Number Generator should produce only values between 0 and 1. (McCullough & Wilson, 2005, S. 1249)

Excel’s RNG – The Output Stream
As we discussed above, it is not sufficient to discuss only the underlying algorithm of a random number generation. One needs also some tests on output stream of a random number generator while assessing the quality of this random number generator. So a Random Number Generator should produce output which passes some tests for randomness. Such a battery of tests, called DIEHARD, has been prepared by Marsaglia. A good RNG should pass almost all of the tests but as we can see in table 12 Excel can pass only 11 of them (7 failure), although Microsoft has declaired Wichmann–Hill Algorithm is implemented for Excel’s RNG. However, we know that Wichmann-Hill is able to pass 16 tests from DIEHARD (McCullough & Wilson, 1999, S. 35).

Due to reasons explained in previous and this section we can say that Excel’s performance is inadequate (because of period length, incorrect implementation Wichmann Hill Algorithm, which is already obsolete, DIEHARD test results)

Conclusion
Old versions of Excel (Excel 97, 2000, XP) : For those reasons, we can say that use of Excel 97, 2000, XP for (statistical) scientific purposes should be avoided.
 * shows poor performance on following distributions: Normal, F, t, Chi Square, Binomial, Poisson, Hypergeometric
 * retrieves inadequate results on following calculations: Univariate statistics, ANOVA, linear regression, non-linear regression
 * has an unacceptable random number generator

Although several bugs are fixed in Excel 2003, still use of Excel for (statistical) scientific purposes should be avoided because it:
 * has a poor performance on following distributions: Binomial, Poisson, Gamma, Beta
 * retrieves inadequate results for non-linear regression
 * has an obsolete random number generator.


text:
Macedonian

The Macedonian language - North Macedonia - Slavistics

''This book will help you start learning the Macedonian language through everyday phrases and expressions. At the end you will find some simple texts using the new skills and a vocabulary you have learnt. Remember that no book can cover all of the words you will need to communicate fluently in any language, including Macedonian. The best way to improve your vocabulary is to simply read newspapers and magazines, watch Macedonian TV, and most important of all, listen to and translate Macedonian songs.''

title: Macedonian


text: Macedonian/Alphabet

title: Macedonian/Alphabet


text:
Accountancy/Assets

An asset is an economic resource owned and controlled by an entity as a result of past events. and from which future economic benefits are expected to flow in the enterprise.

An asset can classified as: a. Current or non-current (see the definition below) b. Tangible or intangible (whether or not the asset has a physical existence).

The distinction between current and non-current assets is usually required in the presentation of the assets in the balance sheet.

Evidence of the existence of an asset is that an entity is exposed to the risks, as well as the rewards of ownership.

Importantly, the definition does not take into account legal ownership, but instead reflects substance. For example an asset leased under a contract which confers the significant risks and rewards on the lessee, is capitalized by the lessee despite the fact that legal ownership resides with the lessor.

Assets can be disclosed as current or non-current. A current asset is an asset that forms part of the operating cycle of an entity e.g. inventory (stock) or trade receivables (trade debtors). Alternatively, a current asset is an item which is not a component of operating activity, but which has a maturity of less than 12 months. Cash and cash equivalents are always classed as current assets.

Non-current assets are assets that do not fall under the category of current assets.

title: Accountancy/Assets


text:
Template:Macedonian

<< Macedonian | Macedonian Alphabet | Macedonian Grammar | Macedonian Numbers | Macedonian Verbs | Macedonian Words | Common phrases >>

title: Template:Macedonian


text:
Cookbook:Pita

Pita (also known as pita bread, Arabic bread etc.) is a yeasted wheat flatbread originating from North-African, to Middle-Eastern and Mediterranean cuisines. It is usually eaten with sauces, dips and fillings. Dishes that use pita include kebab, gyros, and shwarma.

Ingredients

 * 250 ml (1 cup) warm water
 * 1 tbsp instant yeast
 * 1 tsp salt
 * 2 tbsp vegetable oil
 * 500 ml (2 cups or 17 fl oz) durum wheat flour

Procedure

 * 1) Dissolve the yeast in a bowl of warm water. Add salt, oil and flour, mix it, and knead it into a dough.
 * 2) Let the dough rise to twice its size.
 * 3) Divide the risen dough into 8 pieces and shape them into balls.
 * 4) Then roll the balls into flat and round pita breads (to a diameter of 15 cm or 6 inches).
 * 5) Let the breads rise in a warm place covered with a cloth for about 30 minutes.
 * 6) Bake at 225°C (435°F) for about 12 minutes.

Notes, tips, and variations

 * Cut the bread into two pieces so it can be used like a sandwich filled with salad or meat. Or the exact same, wrapped into a roll.
 * Pita is delicious when dipped in sauces. There are lots of possible ideas and combinations!
 * Because this is a sticky dough, avoid using extra flour to dust your dough when working with it. Grease your hands and work surface with an oil of your choice.

title: Cookbook:Pita


text: User talk:24.123.120.10

title: User talk:24.123.120.10


text:
Business Continuity Management

/Introduction/

/History & Evolution/

/Managing Business Continuity/

/Risks and Vulnerabilities/

/Bibliography/

/Appendices/

/Index/

/FAQs/

/Tools/

title: Business Continuity Management


text:
Talk:AQA Information and Communication Technology

Proposed rename
All other A-level books are preceded by "A-level". In order to be consistent, this book should be renamed A-level Information and Communication Technology. I hope this proposed move is uncontroversial, but note it here in advance to allow others to comment, Jguk 20:20, 17 February 2006 (UTC)


text:
Water Resources Directory/Education/Planning/Philosophy

Model Curriculum, Core Course #1:

 Philosophy of Water Resources Planning

Learning Objectives:

 * To become familiar with the past theories, methods, and techniques of assessment and evaluation in the planning and management of the Nation’s water resources.
 * To develop an appreciation and insight into the complexity and conflicts in multi-objective water resources planning.
 * To become familiar with the classics in water resources planning and management literature.

Subject Areas

 * perspectives on resource use
 * population growth
 * overview of environmental policy and thought
 * planning and evaluation
 * methods for assessment and evaluation
 * demand forecasting
 * corps mission
 * future of water resources planning


text:
Water Resources Directory/Education/Planning/Legal

Model Curriculum, Core Course #2:

 Institutional Considerations in Water Resources Planning

Learning Objectives:

 * 1) To understand the law making process. The course begins with the U.S. Constitution because it is the master plan directing how the legislative, executive and judicial branches of federal and state governments must interact in creating law. The course highlights the constitutional role of each branch of the federal government, and the role of federal agencies in particular.
 * 2) To understand the particular laws governing the activities of the Corps, including enabling statutes and environmental statutes.

Subject Areas

 * sources of law
 * constitution
 * legislative branch
 * judicial branch
 * executive branch

 * legal roles of federal agencies, duties of Army Corps
 * economic evaluation of political proc.
 * water law
 * environmental law - Endangered Species Act
 * environmental law - Clean Water Act
 * public trust doctrine


text:
Windows Programming/Multitasking

Processes and Threads
Current versions of Windows are multitasking operating systems. In this chapter, we will discuss some of the tools and API functions that are involved in multitasking, threading, and synchronization for use in Windows.

First, let's explain a little bit of terminology. A process is a single program, with a single entry point, and a single exit point. A thread is a part of a process. A process has at least 1 thread; but can have more than 1 thread. When created, processes and threads run automatically and are alloted time slices of execution time by the scheduler in a round-robin fashion. The operating system may activate and deactivate any thread or process at any time. For this reason, we will need to control access to program resources such as global memory and output devices.

If multiple processes are working together, the resulting group is known as a job. Jobs can also be managed by Windows.

Managing Processes

 * CreateProcess, etc

Threads
A few functions are used when dealing with threads, like ,  ,  , and.

If the volatility of threads is disconcerting, Windows also provides an execution object known as a fiber that only runs when activated by the parent thread.

CreateThread
The  function accepts a few parameters:
 * a pointer to the function to be executed within the thread.
 * a pointer to a variable to be passed to the thread's function.

The CreateThread function creates a new thread for a process. The creating thread must specify the starting address of the code that the new thread is to execute. Typically, the starting address is the name of a function defined in the program code. This function takes a single parameter and returns a DWORD value. A process can have multiple threads simultaneously executing the same function.

The following example demonstrates how to create a new thread that executes the locally defined function, ThreadFunc.

For simplicity, this example passes a pointer to a DWORD value as an argument to the thread function. This could be a pointer to any type of data or structure, or it could be omitted altogether by passing a NULL pointer and deleting the references to the parameter in ThreadFunc. It is risky to pass the address of a local variable if the creating thread exits before the new thread, because the pointer becomes invalid. Instead, either pass a pointer to dynamically allocated memory or make the creating thread wait for the new thread to terminate. Data can also be passed from the creating thread to the new thread using global variables. With global variables, it is usually necessary to synchronize access by multiple threads.

Thread Local Storage (TLS)
A single process can (usually) spawn 2000 threads. This is because the default stack size allocated by the linker is 1MB per thread. 1MB x 2000 is around 2GB which is the maximum a user-process can access. Following is a sample code which spawn many threads till a limit is reached:

Debugging
The naming of a thread, a special feature, that only works with a few windows debuggers, is extremely useful for debugging threads, especially when debugging programs with a lot of threads. It consists in generating a special runtime exception (0x406D1388) which allows the program to pass the name for the thread to the debugger.

Next Chapter

 * Interprocess Communication

title: Windows Programming/Multitasking


text:
Windows Programming/Interprocess Communication

When we have multiple threads and fibers working together in a single process, or when we have multiple processes working together in a job, we often need to allow processes and threads to communicate with each other. This is done using a series of different tools provided by Windows.

Next Chapter

 * MDI Programs


text:
Water Resources Directory/Education/Planning/Decision-Making

Model Curriculum, Core Course #3:

Learning Objectives:

 * 1) To gain an understanding of the major ways that public policy decisions about water resources are made, together with an appreciation of the strengths and weaknesses of each approach. More specifically,
 * 2) To be able to describe the current water resources decision-making environment in terms of factors such as presence of multiple interest groups with competing values and priorities, multiple sources of power, etc. and the challenges that this environment presents for decision-making.
 * 3) To describe major public decision-making models, and their strengths and weaknesses in the water resources decision-making environment.

Subject Areas

 * current WR decision-making (environ. decision-making)
 * market mechanisms
 * regulation
 * litigation
 * collaborative methods - public participation
 * collaborative methods - alternative dispute resolution
 * collaborative methods - shared visioning


text:
Water Resources Directory/Education/Planning/Ecology

Model Curriculum, Core Course #4:

 Ecology for Water Resources Planning

Learning Objectives:
* To attain a conceptual understanding of the principles of ecology. * To gain an increased self-confidence in one's personal ability regarding ecology as a science. * To gain an ability to critically analyze a problem or issue from an ecological perspective. * Understand the role of ecological knowledge in determining environmental consequences of human activities. * To recognize the complexity and interrelated nature of ecology relative to other sciences. * To develop an understanding of the ecosystem concept, ecosystem services, ecosystem management, and adaptive management.

Subject Areas

 * organization of the natural world
 * ecol. of organisms-temp & water
 * energy & nutrient relations
 * population distribution & abundance
 * population dynamics
 * population growth models
 * interactions - competition, niches, disease
 * interactions - mutualism, predation
 * community ecology
 * food webs, nutrient cycling
 * ecosystems
 * large-scale ecological phenomena
 * ecosystem management
 * adaptive management
 * sustainability, economics, social/legal/political framework

University Courses
List of similar courses offered at universities, with link to course syllabus.


text:
Orthopaedic Surgery/Hip and Femur

This year the first of 78 million baby boomers turn 60. Despite efforts to prevent osteoporosis a rising incidence of osteoporosis related hip fractures can be anticipated ( Mundy 2006). http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=16470007

title: Orthopaedic Surgery/Hip and Femur


text:
German/Level IV

LEVEL IV: ERWEITERTE LEKTIONEN
Section 1 ~ Kiel, Germany
 * Lektion 1
 * Lektion 2: Undeveloped
 * Lektion 3: Undeveloped
 * Wiederholung 1: Undeveloped

Section 2 ~ Schaan, Liechtenstein

 * Lektion 4: Undeveloped
 * Lektion 5: Undeveloped
 * Lektion 6: Undeveloped
 * Wiederholung 2: Undeveloped

Section 3 ~ Schaffhausen, Switzerland

 * Lektion 7: Undeveloped
 * Lektion 8: ~ Undeveloped
 * Lektion 9: ~ Undeveloped
 * Wiederholung 3: Undeveloped

title: German/Level IV


text:
German/Level V

LEVEL V: REVIEW LESSONS

 * Lesson 1 ~ Starting Point
 * Lesson 2 ~ Berlin, Germany
 * Lesson 3 ~ Vienna, Austria
 * Lesson 4 ~ Bern, Switzerland
 * Lesson 5 ~ Salzburg, Austria
 * Lesson 6 ~ Zürich, Switzerland
 * Lesson 7 ~ Hannover, Germany
 * Lesson 8 ~ Bonn, Germany
 * Lesson 9 ~ Innsbruck, Austria
 * Lesson 10 ~ Bavaria, Germany
 * Lesson 11 ~ Kiel, Germany
 * Lesson 12 ~ Schaan, Liechtenstein
 * Lesson 13 ~ Schaffhausen, Switzerland
 * Lesson 14 ~ Deutschland, Österreich, und Schweiz

title: German/Level V


text:
Modern Greek/Credits

Credits, legal, and licensing information for materials used from third-party sources

=FSI= This book uses some materials from Greek: Basic Course, Volume 1, Obolensky, Sapountzis, and Sapountzis, 1967, Foreign Service Institute, Washington, DC. This book is in the public domain because it is a product of the U.S. government; its title page states that the Foreign Service Institute, Department of State, is the publisher, and that "this work was compiled and published with the support of the Office of Education, Department of Health, Education, and Welfare, United States of America." Also, it was published without a copyright notice in 1967, and any work is in the public domain if it was published in the United States before 1989 without a copyright notice.

title: Modern Greek/Credits


text:
Template:Modern Greek/fsi

(adapted from FSI)

title: Template:Modern Greek/fsi


text:
User talk:Cromwellt

Welcome
Hello and welcome to Wikibooks!

Here are some tips to help you get started:
 * You can sign your name on Talk and vote pages using four tildes, like this: ~~~~.
 * Remember to conduct any editing experiments in the sandbox.
 * Check out our help page if you need additional help.
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Take a look at Naming policy
 * See list of templates which can save your time
 * Remember to maintain a Neutral point of view.
 * Explore, be bold in editing pages, and have fun!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page.

Good luck! --Derbeth talk 23:43, 17 February 2006 (UTC)

Welcome response and question
Yes, WB:NP is still not official policy and a bit messy now. But general practice is to name pages like Bookname/Chaptername or sometimes Bookname/Chaptername/Subchaptername. You can fix naming of existing books, but make sure that noone would oppose it. It's better to take old, orphaned books or books where authors were discussing change of naming convention. Currently we have a bot for mass page renaming (User:tsca.bot) so thigs like changing colon to slash or capitalising all first letters in the title of a book is a simple task. Anyway, thanks for your willing to help. --Derbeth talk 08:20, 19 February 2006 (UTC)

 * The problem with discussion page was reported to me also at Polish Wikibooks so it's not an accident. I hope the developers will fix this bug soon.
 * Bot is moving pages and fixing all pages who link to moved articles. --Derbeth talk 22:53, 19 February 2006 (UTC)

Wikiknowledge has moved.
The link to your user page at Wikiknowledge out of date, it is now http://www.wikiknowledge.net/wiki/index.php?title=User:Cromwellt. --kwhitefoot 21:53, 10 June 2006 (UTC)

I see you fixed the link in my user page too. Thanks, I completely forgot about it! --kwhitefoot 13:47, 15 June 2006 (UTC)

title: User talk:Cromwellt


text:
Business Continuity Management/Appendices

/Organizations/

/Legislation/

/Standards/


text:
Business Continuity Management/Bibliography

/Articles/

/Books/

/Sites/

/Reports/

/Others/

/Events/


text:
Business Continuity Management/Managing Business Continuity

/Project Management/

/Business Impact Analysis/

/Risk Evaluation and Control/

/Developing Business Continuity Management Strategies/

/Emergency Response and Operations/

/Developing and Implementing Business Continuity and Crisis Management Plans/

/Awareness and Training Programmes/

/Maintaining and Exercising Business Continuity and Crisis Managements Plans/

/Crisis Communications/

/Co-ordination with External Agencies/


text:
Introduction to Physical Science/1.7

=States Of Matter= This experiment will demonstrate three of the four basic states of matter. The states of matter which an object or substance takes on is based on the temperature of that subject. Matter has four basic states that can occur in nature. The lowest temperatures generate solids which make up the majority of physical objects on earth. The next state, with slightly higher temperature required is liquids. Liquids are also very common on earth. The next state of matter is the gases which require even more heat that liquids. The air we breath is made up entirely of gases The final state of matter is a plasma. Plasmas require temperatures far beyond the norm for earth-bound objects. The only truly common example of a plasma on earth is fire. In this experiment we will deal with gases, liquids, and solids.

The Experiment
For this lab we will use water to demonstrate the states of matter. Remember all the following steps when writing a laboratory report.
 * 1) Read the entire section before writing your own report
 * 2) Follow procedure (see section 1.1)
 * 3) Be very careful to observe all reactions that occur within the experimental system
 * 4) If you have any questions about anything that is asked of you in this text ask your instructor for help

First, gather your materials together
 * 1) One Pyrex™ beaker of at least a 75ml volume
 * 2) Several ice cubes
 * 3) One glass stirring rod
 * 4) One gas/alcohol burner with a stand or a hot plate
 * 5) Thermometer

Ice is the solid state of water, liquid water is the liquid form, and water vapor is the gaseous form. The goal of this experiment will be to find the temperatures at which water changes state from a solid (ice) to a liquid (liquid water) to a gas (water vapor-as liquid water boils). You will use your materials to change the state of the water and to record the temperatures that you measure.

procedure
First take the ice cubes and place them in the Pyrex™ beaker. Place the thermometer in the tube in contact with the ice. Then, placing the beaker on top of the burner stand, light the burner, and begin to heat the cubes. You should start to see the ice melting slowly. Record the temperature displayed on the thermometer every thirty seconds until the water has completely evaporated from the beaker. When you are finished with this experiment; extinguish your beaker and cleanup your lab area.

=Vocabulary and Questions= solids are the common form of physical objects on earth; they are object that are neither in a liquid or gaseous form.

title: Introduction to Physical Science/1.7


text:
Mathematical Proof/Appendix/Glossary

A | B | C | D | E | F | L | M | N | O | R | S | T

This glossary is mostly just for a quick reminder of terms learned in the book and is not meant to be comprehensive or rigorous. Please visit Wikipedia or Wiktionary for more detail.

A
Back to top
 * Arithmetic : The science of addition and multiplication (subtraction and division are included, since they are the inverse operations of addition and multiplication). Proof by Contrapositive

 * Axiom : A self-evident truth. It is the foundation of logical reasoning.  A statement that is accepted as true without proof, which may be assumed in proving that other things are true.Notation

B
Back to top
 * Basis : A collection $$\mathcal B$$ of open sets in a set $$X$$ such that the intersection of any two open sets in $$X$$ contains a set $$B\in \mathcal B.$$ Proof by Contradiction

C
Back to top
 * Closed set : The complement of an open set in a topological space. Proof by Contradiction

 * Conclusion : The result of a given conditional statement. (The "then" clause of a theorem.)  This is also sometimes referred to as the result.  Constructive Proof

 * Conditional statement
 * An "if" or an "only-if" statement. It is conditional because its truth value is determined by the truth value of two other statements.  Logical Reasoning

 * Contrapositive : The converse and negation of a conditional. The contrapositive of $$P\Rightarrow Q$$ is $$\lnot Q \Rightarrow \lnot P$$.  Logical Reasoning

 * Converse : The "reverse" of a conditional statement. The converse of $$P \Rightarrow Q$$ is $$ P \Leftarrow Q $$. Logical Reasoning

 * Corollary : That which follows, usually without any necessary argument, from a given result. Constructive Proof

D
Back to top
 * Divisor : See factor.

 * Divide : An integer n divides an integer m, if n is a factor of m, equivalently, if m is a multiple of n, or, equivalently, if there's a integer k such that $$n *k = m$$. Proof by Contrapositive

E
Back to top
 * Element : One of the objects in a set. Notation

 * Equivalent : See Logically Equivalent.

F
Back to top
 * Factor : An integer that divides a given integer. (e.g. 3 is a factor of 6.)  This is the "opposite" of multiple.  Proof by Contrapositive

L
Back to top
 * Lemma : A result whose proof is fairly simple or one that is used to simplify or break down a larger argument. Constructive Proof

 * Logcially Equivalent
 * Two statements that are simultaneously true or simultaneously false are logically equivalent. Logical Reasoning

M
Back to top
 * Multiple : An integer obtained by multiplying two integers together. (e.g. 4 is a mulitple of 2).  This is the "opposite" of factor.  Proof by Contrapositive

N
Back to top
 * Negation : The opposite of a truth statement. The negation of   is   and vice-versa.  Logical Reasoning

O
Back to top
 * Open set : A set that is an element of a topology $$\tau$$ defined on a set $$X.$$ Proof by Contradiction

R
Back to top
 * Result : A lemma, theorem, or corollary. A statement of "if-then" that has been proven to be true.  Also, the conclusion of such a statement.  Constructive Proof

S
Back to top
 * Set : A collection of items, or elements. Notation

 * Statement : See Truth Statement.

T
Back to top
 * Theorem : A main result. Usually the proof is somewhat involved and the result is interesting and useful.  Constructive Proof

 * Topological Space : A set $$X$$ together with a topology $$\tau$$ that satisfy the topology axioms. Proof by Contradiction

 * Topology : A collection of subsets of a given set that satisfy the topology axioms. Proof by Contradiction

 * Truth Statement
 * A statement whose truth value can be determined. Therefore, it is either true or false. Logical Reasoning

 * Truth Value
 * The assessment of whether a statement is true or false. Logical Reasoning

title: Mathematical Proof/Appendix/Glossary


text:
Talk:Japanese/Lessons/Introduction/Ogenki desu ka/Questions using ka

Particles
The definition of "particle" as "a part of speech that performs some grammatical function" seems iffy to me. Aren't sentence final よ and ね usually considered particles? What is the "grammatical function" of よ or ね? I wouldn't call their role grammatical (indicating the relationships among different words, etc). I'm no expert, but I think this is potentially confusing. Also, do we really need to introduce the jargon "bound word"? It may be a perfectly fine concept but I can't see us ever needing to mention it again.

ToothingLummox 02:43, 18 February 2006 (UTC)

 * Feel free to come up with a better definition. When you do, please throw it into the Japanese/Grammar module as well. We can leave the bound word discussion out for now as well. --Aaronsama 18:51, 21 February 2006 (UTC)

 * I borrowed and modified the definition from Wikipedia's [w:Japanese particles] article. It occurred to me that ね can be used by itself as an interjection. Is it then no longer a particle? I always thought of it as an abbreviation of sou desu ne.
 * ToothingLummox 08:09, 22 February 2006 (UTC)

 * Usually, the interjection is written ねえ, but I'm sure it's abbreviated ね as well. I think grammarians consider this as a separate word from sentence-final ね. ---136.2.1.103 16:12, 21 April 2006 (UTC)


text:
User talk:Rcpage2001~enwikibooks

Rcpage2001 is an amateur writer and speller. He just spelled amateur wrong! (I think!) His hobbies include video games, writing, and reading!

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title: User talk:Rcpage2001~enwikibooks


text:
Mathematical Proof/Methods of Proof

There are many different ways to prove things in mathematics. This chapter will introduce some of those methods.

Introduction
In this chapter, for every method of proof introduced, we will discuss it in this manner: Before introducing the first proof method, let us go through the meanings of some frequently used terms in mathematics books (some are already used in previous chapters actually), which are used more frequently starting from this chapter.
 * 1) Explaining  the method works, by considering its underlying.
 * 2) Introducing  to use the method.
 * 3) Giving examples of using the method (and possibly also some previous method introduced) to prove some results.
 * Definition: an explanation of meaning of a term.
 * Theorem: an important and interesting true mathematical statement.
 * Proposition: a (relatively) less important theorem.

 * Lemma: a true mathematical statement that is useful in establishing the truth of other true statements, and is less important than a theorem.
 * Corollary: a true mathematical statement that can be deduced from a theorem (or proposition) simply.
 * Proof: an explanation of why a statement is true.
 * Axiom: a true mathematical statement whose truth is accepted.
 * Conjecture: a statement that is to be true, but is  proven to be true.

Direct proof
Many mathematical theorems can be expressed in the form of "$$\forall x\in S,P(x)\implies Q(x)$$, in which $$P(x)$$ and $$Q(x)$$ are about elements $$x$$ in a set $$S$$ . This expression means "If $$P(x)$$ then $$Q(x)$$" is true for every $$x\in S$$. However, in practice, we rarely include the phrase "is true", and usually just write the theorem in the form of "For every $$x\in S$$, if $$P(x)$$, then $$Q(x)$$." Sometimes, we write instead "Let $$x\in S$$. If $$P(x)$$, then $$Q(x)$$.", which has the same meaning.

In some situations, the statement is not stated in such a form, but can be in such a form. For example, the statement "The square of an even integer is even." can be expressed as "For every $$x\in\mathbb Z$$, $$x$$ is even $$\implies$$ $$x^2$$ is even.", or "For every $$x\in\mathbb Z$$, if $$x$$ is even, then $$x^2$$ is even.", or "Let $$x\in\mathbb Z$$. If $$x$$ is even, then $$x^2$$ is even.", etc.

Now, we will introduce the first proof method to prove statements in such a form, which is known as. As suggested by its name, this method is quite "direct", and is probably the simplest method among all methods discussed in this chapter.

Consider the statement "$$\forall x\in S, P(x)\to Q(x)$$". We would like to prove that it is true. Suppose $$P(x_0)$$ is for some $$x_0\in S$$. Then, the conditional $$P(x_0)\to Q(x_0)$$ must be true for this $$x_0$$ by definition, regardless of the truth value of $$Q(x_0)$$. Hence, in the proof, we do not need to consider those $$x\in S$$ for which the hypothesis $$P(x)$$ is false.

Because of this, to give a of "$$\forall x\in S,P(x)\implies Q(x)$$", we first  $$P(x)$$ is  (So, we are considering every $$x\in S$$ for which $$P(x)$$ is true.), and then proceed to show that $$Q(x)$$ is true for every such $$x$$ (or else the conditional $$P(x)\to Q(x)$$ will be false).

This shows that $$P(x)\to Q(x)$$ is true for those $$x\in S$$ for which $$P(x)$$ is, and this is enough to prove $$P(x)\to Q(x)$$ is true for $$x\in S$$ (for which $$P(x)$$ may or may not be true), since we have mentioned that the conditional $$P(x)\to Q(x)$$ must be true for those $$x\in S$$ for which $$P(x)$$ is false.

Combining the two results in the previous example and exercise, we get an "if and only if" result:
 * For every $$x\in\mathbb R$$, we have $$x^2-x-2<-2$$ if and only if $$0<x<1$$.

In other words, using set language, $$\{x\in\mathbb R:0<x<1\}=\{x\in\mathbb R:(x-1)^2<1\}.$$ (Both sets represent the curve (strictly) under the horizontal line $$y=-2$$ in the above graph.)

Letting $$P(x)$$ and $$Q(x)$$ be the open statement "$$x^2-x-2<-2$$" and "$$0<x<1$$" respectively, we can express the above result symbolically: $$\forall x\in\mathbb R,P(x)\iff Q(x).$$ Recall that "$$P(x)\iff Q(x)$$" means "$$P(x)\implies Q(x)$$" "$$Q(x)\implies P(x)$$". So, to give a direct proof to the statement in the form of "$$\forall x\in S,P(x)\iff Q(x)$$", a usual way is to break the proof into two parts:
 * 1) proving that $$\forall x\in S,P(x)\implies Q(x)$$ (known as the "" part, or "$$\Rightarrow$$" direction)
 * 2) proving that $$\forall x\in S,Q(x)\implies P(x)$$ (known as the "" part, or "$$\Leftarrow$$" direction)

Proofs related to congruence of integers
After introducing the method of direct proof, let us apply this method on proving some results relating to congruence of integers. Before this, let us introduce the concept of congruence of integers. We begin by a motivation for the definition of congruence of integers.

We know that an integer $$x$$ is either even or odd, i.e., can be expressed as $$2k$$ or $$2k+1$$ for some integer $$k$$, according to whether the remainder is 0 or 1 when $$x$$ is divided by 2. Thus, if two integers $$x$$ and $$y$$ have the same remainder when divided by 2 (i.e., have the same parity), then the difference $$x-y$$ can be proved to be a multiple of 2 (it can be proved that the converse is also true). Similarly, an integer can be expressed as $$3k,3k+1$$ or $$3k+2$$ for some integer $$k$$, according to whether remainder is 0, 1 or 2 when $$x$$ is divided by 3. Hence, if two integers $$x$$ and $$y$$ have the same remainder when divided by 3, then the difference $$x-y$$ can be proved to be a multiple of 3 (the converse is also true).

Hence, "two integers have the same remainder when divided by some integer $$k$$" is equivalent to "their difference is a multiple of $$k$$". This leads us to the following definition.

Now, let us apply direct proof to prove some results related to the congruence of integers.

In the above result, everything is with the same "modulo $$n$$". It is then natural to ask whether there are any results for different "modulo". The answer is yes, and to discuss a result, we need the concept of integers.

Using this result, we can deduce the following result:

The following lemma is quite useful for proving results about congruence of integers.

Proofs related to sets
The proofs related to sets are often in the form of To prove that a set is a subset of another set, we use the definition of subset:
 * 1) A set is a subset of another set.
 * 2) A set equals another set.
 * A set $$X$$ is a subset of another set $$Y$$ if for every element $$x$$, if $$x\in X$$, then $$x\in Y$$.

Thus, we can employ direct proof to prove that a set is a subset of another set.

For the equality of two sets, recall that:
 * A set $$X$$ equals another set $$Y$$ if and only if $$X\subseteq Y$$ and $$Y\subseteq X$$.

Thus, to prove the equality of two sets, we often need to separate the proof into two parts: (i) proving that $$X\subseteq Y$$; (ii) proving that $$Y\subseteq X$$.

We can observe that to prove different laws for the sets, we just simply use the corresponding laws in logic to prove them. Hence, the proofs for other laws, e.g., commutative law and distributive law, are similar.

Proof by contrapositive
Recall that the of a conditional $$P\to Q$$ is the statement $$\sim Q\to\;\sim P$$, which is  to the original conditional $$P\to Q$$. Hence, if we can show that $$\sim Q\implies \sim P$$, then it follows that $$P\implies Q$$. This gives us another method of proof, namely.

A of the statement $$\forall x\in S,P(x)\implies Q(x)$$ is a  of $$\forall x\in S,\sim Q(x)\implies \sim P(x)$$. That is, we first assume that $$\sim Q(x)$$ is true, and proceed to show that $$\sim P(x)$$ is true. In other words, we first assume that $$Q(x)$$ is, and proceed to show that $$P(x)$$ is.

Generally, when we encounter a statement $$\forall x\in S,P(x)\implies Q(x)$$, and observe that the consequent $$Q(x)$$ is "simpler" than the hypothesis $$P(x)$$, using proof by contrapositive is usually more preferable.

Proof by cases
In the previous sections, we have actually employed the method of already. It is natural to use this method when we need to break down a problem into several cases, and then tackle every case individually ("divide and conquer"). Sometimes, we may even need to further divide a case into several subcases. The following are some typical cases:
 * 1) an integer is even or odd;
 * 2) a real number is less than 0, equal to 0, or greater than 0;
 * 3) for two nonzero real numbers $$x$$ and $$y$$, either $$xy>0$$ or $$xy<0$$.
 * For every of the cases, it can be divided into two subcases:
 * $$xy>0$$: ($$x>0$$ and $$y>0$$) or ($$x<0$$ and $$y<0$$)
 * $$xy<0$$: ($$x>0$$ and $$y<0$$) or ($$x<0$$ and $$y>0$$)

But, we should be aware that when we use proof by cases to prove that "$$\forall x\in S,\dotsc$$", the cases should cover all possibilities, i.e., all $$x\in S$$.

The following inequality is quite important in mathematics.

Proof by contradiction
Another important method of proof is.

Let $$P$$ be a statement. Now, suppose we want to prove that $$P$$ is true. If we can show that the conditional $$\sim S\to\mathbf F$$ is true (i.e., $$\sim S\implies\mathbf F$$), then the truth table for conditional tells us that $$\sim S$$ false, and hence $$S$$ must be true, as desired.

The truth table: $$ \begin{array}{c|c|c|c} S&\sim S&\mathbf F&\sim S\to\mathbf F\\ \hline T&F&F&T\\ F&T&F&F\\ \end{array} $$ In words, the method of proof by contradiction is as follows: we first assume that the statement we want to prove to be true is, and then proceed to show that this gives a , and therefore our assumption must be wrong. That is, it is impossible for the statement to be false since it leads to something "absurd". Hence, the statement is true.

The name of comes from the fact that the assumption that the statement is false is later  by some other fact. This is also known as, which means reduction to absurdity.

In general, we often use the method of proof by contradiction when the statement we want to prove is, e.g. "There is no ....", "There does not exist ...", etc.

Also, to indicate to the reader that we are using proof by contradiction, it is recommended that we write "assume to the contrary that ..." (or something similar) instead of just "assume ..." for the assumption that the statement is false.

The following is a typical example in introducing proof by contradiction:

The following example is another typical example for demonstrating proof by contradiction. But before discussing it, we need to introduce the following theorem since it is used in the proof.

We can also use proof by contradiction to prove that a statement "$$\forall x\in S,P(x)\to Q(x)$$" is true. We first assume that the negation of the statement, i.e., "$$\exists x\in S,P(x)\land(\sim Q(x))$$" is true, and proceed to arrive at a contradiction. (Recall that $$P\to Q\iff (\sim P)\lor Q$$. So, $$\sim(P\to Q)\iff P\land(\sim Q)$$.) Thus, to prove that $$\forall x\in S,P(x)\implies Q(x)$$, we assume that there exists $$x\in S$$ such that $$P(x)$$ is true and $$Q(x)$$ is false, and then deduce a contradiction.

Sometimes we can prove a statement using multiple methods:

We can see that when proving a statement "$$\forall x\in S,P(x)\implies Q(x)$$", we may be able to use both proof by contrapositive and proof by contradiction. However, the two proofs are different, and we will compare them in the following table: From this table, we can see that the proof by contradiction is more advantageous in terms of assumption, since it has one more "help" from $$P(x)$$ (when we assume $$P(x)\land\sim Q(x)$$, we can use both $$\sim Q(x)$$ and $$P(x)$$.) However, in terms of goal, the proof by contrapositive is more advantageous since the goal is more clear ($$\sim P(x)$$), while for the proof by contradiction, it is not clear that what the form of the contradiction is. There can be many "ways" for arriving at a contradiction (compare the contradiction in the proof of "$$\sqrt 2$$ is irrational" vs. that of "there are infinitely many primes").

Existence proof
Consider the statement "$$\exists x\in S,P(x)$$". This statement is true if $$P(x)$$ is true for $$x\in S$$. Otherwise, it is false. Thus, to prove such kind of statement, it suffices to find element $$x\in S$$ such that $$P(x)$$ is true, and this is known as an existence proof. In particular, we should verify that the choice of element $$x$$ is actually belonging to $$S$$ and $$P(x)$$ is actually true for that choice.

The following example demonstrates a more advanced version of existence proof:.

Disproof
Consider the statement "$$\forall x\in S,P(x)$$.". Suppose we somehow believe that the statement is. Then, we want to prove that it is false, i.e., prove that "$$\exists x\in S$$ such that $$P(x)$$ is false." An element $$x_0\in S$$ for which $$P(x_0)$$ is false is known as a of the statement, and a process of showing that the statement is (in other words,  the statement) is called a  of the statement. One counterexample is enough to disprove the statement.

To disprove the statement "$$\forall x\in S,P(x)\to Q(x)$$, we show that there exists an element $$x_0\in S$$ such that $$P(x_0)\to Q(x_0)$$ is false, i.e., $$P(x_0)$$ is true and $$Q(x_0)$$ is false.

To disprove the statement "$$\exists x\in S,P(x)$$", we need to show that there is such $$x$$. That is, we need to show that "$$\forall x\in S,\sim P(x)$$" (the negation of statement) is true. In this case, we are just constructing counterexamples. We have to prove that $${\color{darkgreen}\forall}x\in S,\sim P(x)$$, instead of $$\exists x\in S,\sim P(x)$$.

Often, before proving or disproving statements, the truth of falseness of them are known, and we need to decide whether every of them is true or false. After that, we will prove it if it is true and disprove it otherwise. Of course, our decision is not perfectly accurate, and sometimes we make mistake, which leads us to do the opposite thing, i.e., proving a wrong statement or disproving a correct statement. Clearly, when we do such thing, we cannot succeed. However, the of performing such wrong thing may give us some insights about what the correct direction is, and how to prove/disprove the statement.

To decide whether a statement is true or false, you may follow some tips below:
 * Follow your mathematical intuition. As you have learnt more about mathematics, you may have intuition and idea about whether a statement is true or false, before proving/disproving it.
 * Construct some simple examples. Such examples constructed may be a counterexample (for disproving "$$\forall\dotsb$$")/example (for proving "$$\exists\dotsb$$") if you are lucky. Even if they are not counterexamples/examples, you may observe some kind of patterns from it, which may give you some insights about how to prove/disprove the statement.

Proof by mathematical induction
The last proof method discussed in this chapter is. This method is used to prove a statement in the form of "For every integer $$n$$ greater than or equal to an integer $$m$$, $$P(n)$$ is true.". That is, it is used to prove that a of statements $$P(m),P(m+1),P(m+2),\dotsc$$ is true. Of course, we can prove that every of these statements is true one by one, but the principle of mathematical induction gives an alternative and more convenient method.

To the principle of mathematical induction, we need the. Before introducing it, we need the definition of.

Now, one may ask that whether the set $$\mathbb N$$ is well-ordered. It may that it is well-ordered. Here, we will just regard this as an :

The well-ordering principle can then be used to prove (a less general version of) the principle of mathematical induction.

This theorem is quite intuitive: with the conditions satisfied, we have an "infinite chain of implications":
 * $$P(m)$$ is true (by (i)).
 * Thus, $$P(m+1)$$ is true (by (ii)).
 * Thus, $$P(m+2)$$ is true (by (ii)).
 * Thus, $$P(m+3)$$ is true (by (ii)) ...

But strictly speaking, this is not counted as a proof. We will give a formal (partial) proof to this theorem below:

Using the principle of mathematical induction, to prove that the statement $$P(n)$$ is true for every integer $$n\ge m$$, it suffices to prove two things:

(i) $$P(m)$$ is true.

(ii) For every integer $$k\ge m$$, $$P(k)\implies P(k+1)$$.

Thus, proof in this form is a two-step process. We often call the proof of (i) as the or, and the proof of (ii) is called the. In particular, when we prove (ii), we usually use the method of direct proof, and first assume $$P(k)$$ is true for integer $$k$$ with $$k\ge m$$. Such assumption is often called the.

We can use the proof by mathematical induction for proving some inequalities also:

The strong form of induction
In this section, we will discuss a stronger form of mathematical induction. Sometimes, we need this form of mathematical induction since merely assuming "$$P(k)$$ is true ..." in the inductive hypothesis may not be enough to show that $$P(k+1)$$ is true. We may need some more helps, particularly the fact that the statement is true for from the basis step to the given $$k$$. That is, if "$$m$$" is 1, then we assume $$P(1),\dotsc,P(k)$$ are all true.

Since "$$P(1)\land P(2)\land\dotsb\land P(k)$$" is than "$$P(k)$$", we have the name "the  form of induction".

This theorem is also quite intuitive: with the conditions satisfied, we get an "infinite chain of implications":
 * 1) $$P(m)$$ is true by (i).
 * 2) Because of 1., $$P(m+1)$$ is true by (ii).
 * 3) Because of 1. and 2., $$P(m+2)$$ is true by (ii).
 * 4) Because of 1., 2., and 3., $$P(m+3)$$ is true by (ii) ...

Hence, to prove that the statement $$P(n)$$ is true for every $$n\ge m$$ by the strong form of induction, we need to prove

(i) the basis step: $$P(m)$$ is true; and (ii) the inductive step: for every integer $$k\ge m$$, if $$P(m),P(m+1),\dotsc,P(k)$$ are true (the inductive hypothesis), then $$P(k+1)$$ is true.

We usually use the strong form of induction to prove results that are related to.

title: Mathematical Proof/Methods of Proof


text:
Artificial Intelligence/Logic

For a more in-depth book, see the wikibook Formal Logic. This chapter for AI Logic is aimed to be an introduction of specific elements of Logic that pertain to AI only.

What is logic? The philosophical definition is that logic is a description of how one should think. In the context of AI, logic is "formal," which means it resembles math in its clarity and lack of ambiguity.

There are lots of different kinds of logics out there (temporal logics, ethical logics, etc.). But when most people, particularly in AI, refer to logic, they are talking about propositional logic or the more complex predicate logic. So although you can make a logic with any kind of rules and symbols you want, what most people mean when they refer to logic are the widely-accepted rules and notations commonly taught in introductory logic courses.

Next we will discuss the basics of representation of propositions in logic.

Next: Representational perspectives

title: Artificial Intelligence/Logic


text:
Lojban/Selbri

selbri are Lojban words that describe the relationships between the various parts of a sentence. The majority of words used as selbri are gismu, which can be easily recognized as they always follow the pattern CVCCV or CCVCV (where C is a consonant and V is a vowel).

The word order of the sentence gives selbri their meaning. The positions of the other words in the sentence, with respect to the position of the selbri, determine what the meaning of the sentence is. For example:

la djon. pritu la matius. la maikyl.

pritu is the selbri. It means "x is to the right of y, where y is facing z." The variables (called sumti) are replaced with other things. In this case, x is John (djon.), y is Matthew (matius.), and z is Michael (maikyl.) Thus the whole sentence (a sentence like this is called a bridi) means "John is to the right of Matthew, where Matthew is facing Michael."

selbri follow some rules for determining the structure of their possible sumti:

 * 1) The subject of the sentence, the person or thing that is or does, usually comes first.
 * 2) The object of the sentence, the person or thing that is affected by the action, comes second.
 * 3) The destination (to [place]) comes before the origin (from [place])
 * 4) Any other places come near the end; "by standard," "by means of," "made of," etc.

title: Lojban/Selbri


text:
Talk:String Theory

The purpose of this textbook is primarilly for learning, not so much as a reference. Therefore, try not to leave too much work up to the reader.

I have begun this book with a chapter on supersymmetry. This only needs to be an introduction to give the reader a feel for it. To learn most easily, I find that it works best to discuss only one topic at a time rather than to throw many things at the reader at once, and therefore supersymmetry may need its own discussion independant of strings.

Strings are introduced as classical relativistic theories with a 2D conformal invariance. I think this development is more logical and elegant than introducing strings as a generalization of point particles. The former version starts with principles, whereas the latter begins with an ad-hoc idea.Smeapancol 06:44, 18 February 2006 (UTC)

title: Talk:String Theory


text:
Mathematical Proof/Methods of Proof/Direct Proof

A constructive proof is the most basic kind of proof there is. It is a proof that starts with a hypothesis, and a person uses a series of logical steps and a list of axioms, to arrive at a conclusion.

Parts of a Theorem
A theorem is a proven statement that was constructed using previously proven statements, such as theorems, or constructed using axioms. Some theorems are very complicated and involved, so we will discuss their different parts.

The Hypothesis
The hypothesis is the "if" statement of a theorem. In a way, it is similar to an axiom because it is assumed to be true in order to prove a theorem. We will consider a simple example.

 * Theorem 2.1.1. If A and B are sets such that $$A\subseteq B$$ and $$B\subseteq A$$, then A=B.

In this theorem, the hypothesis is everything before the word "then." This is a very simple proof. We need to prove that for every x, $$x\in A \Leftrightarrow x\in B$$. For the purpose of analyzing proofs, we will define $$ P(x) = x\in A$$ and $$ Q(x) = x\in B$$. The most common way to prove an "if and only if" statement is to prove necessity and sufficiency separately

 * So we start by showing that $$ P(x) \Rightarrow Q(x).$$ Therefore, we assume that P(x) is true.  That is, $$x\in A.$$  Since we assumed, by hypothesis, that $$A\subseteq B,$$, we know that $$ x\in B$$, which means that Q(x) is true.

 * Now we show that $$ Q(x) \Rightarrow P(x)$$, so we assume that Q(x) is true. This means that $$x\in B$$.  Since we know that $$B\subseteq A,$$ we know that $$ x\in A,$$ so P(x) is true.

By these two conclusions, we see that $$P(x)\Leftrightarrow Q(x).$$

Now, by axiom 3, A=B, since $$x\in A \Leftrightarrow x\in B.$$ This concludes the proof. This is a very trivial proof, but its point was to show how to use a hypothesis or set of hypotheses in order to reach the desired conclusion. This method here is the most common in proving that two sets are equal. You prove that each set is a subset of the other.

The Conclusion
The part of the theorem after the word "then" is called the conclusion. The proof of a theorem is merely the logical connection between the hypothesis and the conclusion. Once you've seen and proved a few theorems, a conclusion is almost predictable. For example, what conclusion would you naturally draw from the following two statements? These two statements are the hypotheses. To word this as a theorem, we would have "If all Americans are people and all people live on Earth, then all Americans live on Earth." This statement is what most people would call completely obvious and requires no proof. However, to show how this concept is applied in mathematics, we will abstract this theorem and prove it.
 * 1) All Americans are people.
 * 2) All people live on Earth.
 * Theorem 2.1.2. If $$A\subset B$$ and $$B\subset C$$, then $$A\subset C$$.

To see how this relates to our problem, let A be the set of all Americans, B the set of all people, and C the set of all things that live on Earth.

To show that $$A\subset C$$, we need to show that $$\forall x\in A, x\in C.$$ So we suppose $$x\in A.$$  By hypothesis, $$A\subset B,$$ so $$x\in B.$$  Also by hypothesis, $$B\subset C$$, so $$x\in C.$$  Since this was true for any arbitrary $$x\in A,$$ we have shown that $$A\subset C.$$

The Definition
While a definition is not usually part of a theorem, they are commonly introduced immediately before a theorem, in order to help define the symbols you use or to help prove it.
 * Definition 2.1.3. If a set A has only finitely many elements, then the order of A, denoted by |A|, is the number of elements in A.

This definition gives meaning to the following theorem.
 * Theorem 2.1.4. If A and B are finite sets such that A = B, then |A|=|B|.

Here we take advantage of the fact that A is a finite set. Let n be the integer such that |A| = n. Then index the elements of A so that $$A=\{x_1,x_2,\ldots,x_n\}.$$ Now $$\forall i = 1, 2, \ldots, n$$, we have $$ x_i\in B$$. So we see that B has at least n elements, that is $$|B|\ge n.$$ Also, every element of B is in A (by hypothesis), so it follows that there are no more elements in B than there are in A, so $$|B|\le n$$, thus |B| = n = |A|, which concludes the proof.

The Given
Sometimes, the first part of a theorem lists what is required in order for a theorem to be proven. Hence, this list is described in the theorem as what is given. This helps readers understand exactly what is the hypothesis and what is the conclusion in a theorem statement. For example, Theorem 2.1.4 can be rewritten so that it lists what is required beforehand.

Theorem 2.1.4. Given finite sets A and B, if A = B, then |A|=|B|.

This statement clearly highlights what the hypothesis is and what the conclusion is, in this example.

Classifying Theorems
There are different terms that mathematicians like to use in stating mathematical results. Theorem is probably the most common and well-known, especially to non-mathematicians. There are, however, a few other related terms used in mathematics. They are all theorems, but have more specific uses.

Lemma
A lemma is a "small theorem." When a result is less profound, more trivial, or boring, it can be called a lemma. A lemma is also used to make the proof of a theorem shorter. That is, if a chunk of a proof can be pulled off and proved separately, then it is called a lemma and the proof of the theorem will say something to the effect of "as proved in the lemma."

For example, the following lemma will help to make the proof of Theorem 2.1.4 more concise.
 * Lemma 2.1.5. If A and B are finite sets and $$A\subset B$$ then $$|A|\le |B|$$.

As you might guess, this is one motivation for the use of the symbol $$\subset$$, since it is similar in appearance to <.

Let n = |A|. Then number the elements of A, so $$A=\{x_1,x_2,\ldots,x_n\}.$$  Then for each i from 1 to n we see that $$x_i\in B$$, which means that B has at least n different elements, or that $$|B|\ge n =|A|,$$ which is what we were trying to prove.

Now if we use this lemma twice on Theorem 2.1.4, we will get a very brief proof. Since $$A\subset B$$ we know that $$ |A|\le |B|.$$ Also, since $$B\subset A$$, we see that $$|B|\le |A|.$$  Now we use a fact about numbers, that if $$x\le y$$ and $$y\le x$$, it must follow that x = y.

Corollary
A corollary is similar to a lemma in that it is usually a small and not as important as a theorem. However, a corollary is usually a result that follows almost immediately from a theorem. Corollaries tend to use a few well-known or established theorems and the primary theorem to prove. This is why corollaries are often found appended after a big theorem.

For example, Let's suppose we proved the theorem that "All people are pigs." Then a corollary would be "People who have heads are pigs." which clearly follows from the first result since "People have heads" is well-known and true (A person without a head would be rather odd, no?). Another, slightly more interesting, corollary would be "People can be sold for bacon when they die." since "bacon comes from pigs" is well-known and true.

So we see that a corollary is something that follows from a preceding theorem with minimal argument to support it. Note that theorems you declare as a corollary may not be so to others, as corollaries are subjective. However, thinking of a corollary as relying on either "common sense" or "obvious" to the reader that it is a direct consequence of the theorem is the proper thought when figuring out what to assign as a corollary.

Exercises

 * 1) Prove that the following sets are equal. Verify it with a truth table or a Venn Diagram.  You may assume that A, B, and C are nonempty sets.  Also assume that U is the universe.
 * 2) $$A\cup(B\cap C) = (A\cup B)\cap(A\cup C)$$
 * 3) $$A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$$
 * 4) $$A^c - B = (A\cup B)^c$$
 * 5) $$A-B^c = A\cap B$$
 * 6) $$(A-B)\cup(A\cap C)=A-(B-C)$$
 * 7) Prove that if A and B are finite sets then $$|A\cup B|\le|A|+|B|$$ and that equality holds when $$A\cap B =\varnothing.$$

Something to think about

 * We have defined the order or size of a set for a finite set. Would it make sense to define this order for an infinite set?  How would you tell whether two infinite sets are the same size?
 * If you know that $$|A|\le |B|,$$ can you show that $$A\subset B?$$


text:
Talk:Physics Exercises/Trig Tables

Purpose of this page
Why can't they just use a calculator? Brianjd 09:02, 18 February 2006 (UTC)

title: Talk:Physics Exercises/Trig Tables


text:
WebObjects/Project WONDER

If you've ever been working on a WebObjects project and thought to yourself "I wonder if anyone has written this?" or "Gee, this seems like it could be easier," then Project WONDER is the thing for you. Project WONDER is a massive open source Java project designed to ease the development of WebObjects applications. The project contains many subsystems -- reusable components, frameworks, database plugins, TONS of utilities, and a set of sample apps that show off some of Wonder's features.

The power, size, and scope of Project WONDER are both a blessing and a curse. To the initiated, the answer to most WO questions is "Project WONDER." However, many people seem to be turned off by the apparent complexity of using WONDER. With the deployment of the nightly build server, though, these fears are mostly unfounded. Installing and using Project WONDER requires adding a handful of Frameworks to your project and changing your Application, Session, and DirectAction superclasses to be ERXApplication, ERXSession, and ERXDirectAction, and you're done. WONDER will do the rest for you, bringing in patched, enhanced, and just-plain-better versions of various core WO components, and providing a huge number of prebuilt components and technologies to pull from to build your WO applications. It is WELL worth the time to start using it if you do anything beyond the simplest of applications.

For a slightly dated, yet still good, walkthrough you can check out Jonathan Rentzsch's 2002 CAWUG presentation.

The project itself is hosted as SourceForge under the project name Wonder, and has a nightly build server at http://webobjects.mdimension.com/wonder.

The latest javadoc is available at http://webobjects.mdimension.com/wonder/api.

A downloadable version of the latest javadoc is available at http://webobjects.mdimension.com/wonder/Wonder-latest-Documentation.tar.gz

Any questions can be directed to the Project WONDER mailing list.

Wonder Community Tutorials available at: ObjectStyle Confluence Wiki WOnder Tutorials

title: WebObjects/Project WONDER


text:
WebObjects/Noxymo Core/Overview

NoxymoCore is no longer available

title: WebObjects/Noxymo Core/Overview