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Degenerate distribution

In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter definition, it is a deterministic distribution and takes only a single value. Examples include a two-headed coin and rolling a die whose sides all show the same number. This distribution satisfies the definition of "random variable" even though it does not appear random in the everyday sense of the word; hence it is considered degenerate.

In the case of a real-valued random variable, the degenerate distribution is a one-point distribution, localized at a point k0 on the real line. The probability mass function equals 1 at this point and 0 elsewhere.

The degenerate univariate distribution can be viewed as the limiting case of a continuous distribution whose variance goes to 0 causing the probability density function to be a delta function at k0, with infinite height there but area equal to 1.

The cumulative distribution function of the univariate degenerate distribution is:

$$F_{k_0}(x)=\left\{\begin{matrix} 1, & \mbox{if }x\ge k_0 \\ 0, & \mbox{if }x<k_0 \end{matrix}\right.$$

Constant random variable
In probability theory, a constant random variable is a discrete random variable that takes a constant value, regardless of any event that occurs. This is technically different from an almost surely constant random variable, which may take other values, but only on events with probability zero. Constant and almost surely constant random variables, which have a degenerate distribution, provide a way to deal with constant values in a probabilistic framework.

Let  X: Ω → R  be a random variable defined on a probability space  (Ω, P). Then X  is an almost surely constant random variable if there exists $$ k_0 \in \mathbb{R} $$ such that
 * $$\Pr(X = k_0) = 1,$$

and is furthermore a constant random variable if
 * $$X(\omega) = k_0, \quad \forall\omega \in \Omega.$$

A constant random variable is almost surely constant, but not necessarily vice versa, since if X  is almost surely constant then there may exist  γ ∈ Ω  such that  X(γ) ≠ k0  (but then necessarily Pr({γ}) = 0, in fact Pr(X ≠ k0) = 0).

For practical purposes, the distinction between X  being constant or almost surely constant is unimportant, since the cumulative distribution function  F(x)  of  X  does not depend on whether  X  is constant or 'merely' almost surely constant. In either case,
 * $$F(x) = \begin{cases}1, &x \geq k_0,\\0, &x < k_0.\end{cases}$$

The function F(x)  is a step function; in particular it is a translation of the Heaviside step function.

Higher dimensions
Degeneracy of a multivariate distribution in n random variables arises when the support lies in a space of dimension less than n. This occurs when at least one of the variables is a deterministic function of the others. For example, in the 2-variable case suppose that Y = aX + b for scalar random variables X and Y and scalar constants a ≠ 0 and b; here knowing the value of one of X or Y gives exact knowledge of the value of the other. All the possible points (x, y) fall on the one-dimensional line y = ax + b.

In general when one or more of n random variables are exactly linearly determined by the others, if the covariance matrix exists its rank is less than n  and its determinant is 0, so it is positive semi-definite but not positive definite, and the joint probability distribution is degenerate.

Degeneracy can also occur even with non-zero covariance. For example, when scalar X is symmetrically distributed about 0 and Y is exactly given by Y = X 2, all possible points (x, y) fall on the parabola y = x 2, which is a one-dimensional subset of the two-dimensional space.

title: Degenerate distribution


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Icelandic

Icelandic refers to anything of, from, or related to Iceland and may refer to:
 * Icelandic people
 * Icelandic language
 * Icelandic orthography
 * Icelandic cuisine

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

title: Icelandic


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Canadian dollar

The Canadian dollar (symbol: $; code: CAD; dollar canadien) is the currency of Canada. It is abbreviated with the dollar sign $. There is no standard disambiguating form, but the abbreviations Can$, CA$ and C$ are frequently used for distinction from other dollar-denominated currencies (though C$ remains ambiguous with the Nicaraguan córdoba). It is divided into 100 cents (¢).

Owing to the image of a common loon on its reverse, the dollar coin, and sometimes the unit of currency itself, may be referred to as the loonie by English-speaking Canadians and foreign exchange traders and analysts.

Accounting for approximately 2% of all global reserves, the Canadian dollar is the sixth-most held reserve currency in the world, behind the U.S. dollar, euro, yen, sterling, and renminbi. The Canadian dollar is popular with central banks because of Canada's relative economic soundness, the Canadian government's strong sovereign position, and the stability of the country's legal and political systems.

Colonial currencies
The 1850s in Canada were a decade of debate over whether to adopt a £sd-based monetary system or a decimal monetary system based on the US dollar. The British North American provinces, for reasons of practicality in relation to the increasing trade with the neighbouring United States, had a desire to assimilate their currencies with the American unit, but the imperial authorities in London still preferred sterling as the sole currency throughout the British Empire. The British North American provinces nonetheless gradually adopted currencies tied to the American dollar.

Province of Canada

In 1841, the Province of Canada adopted a new system based on the Halifax rating. The new Canadian pound was equal to four US dollars (92.88 grains gold), making £1 sterling equal to £1.4s.4d. Canadian. Thus, the new Canadian pound was worth 16 shillings and 5.3 pence sterling.

In 1851, the Parliament of the Province of Canada passed an act for the purposes of introducing a sterling-based unit, with decimal fractional coinage. The idea was that the decimal coins would correspond to exact amounts in relation to the U.S. dollar fractional coinage.

In response to British concerns, in 1853, an act of the Parliament of the Province of Canada introduced the gold standard into the colony, based on both the British gold sovereign and the American gold eagle coins. This gold standard was introduced with the gold sovereign being legal tender at £1 = US$$1/100$. No coinage was provided for under the 1853 act. Sterling coinage was made legal tender and all other silver coins were demonetized. The British government in principle allowed for a decimal coinage but nevertheless held out the hope that a sterling unit would be chosen under the name of "royal". However, in 1857, the decision was made to introduce a decimal coinage into the Province of Canada in conjunction with the U.S. dollar unit. Hence, when the new decimal coins were introduced in 1858, the colony's currency became aligned with the U.S. currency, although the British gold sovereign continued to remain legal tender at the rate of £1 = Can$$4.86 2/3$ right up until the 1990s. In 1859, Canadian colonial postage stamps were issued with decimal denominations for the first time. In 1861, Canadian postage stamps were issued with the denominations shown in dollars and cents.

Maritime colonies
In 1860, the colonies of New Brunswick and Nova Scotia followed the Province of Canada in adopting a decimal system based on the U.S. dollar unit.

In 1871, Prince Edward Island went decimal within the U.S. dollar unit and introduced coins in the denomination of 1 cent. However, the currency of Prince Edward Island was absorbed into the Canadian system shortly afterwards, when Prince Edward Island joined the Dominion of Canada in 1873.

Newfoundland
Newfoundland went decimal in 1865, but unlike the Province of Canada, New Brunswick, and Nova Scotia, it decided to adopt a unit based on the Spanish dollar rather than on the U.S. dollar, and there was a slight difference between these two units. The U.S. dollar was created in 1792 on the basis of the average weight of a selection of worn Spanish dollars. As such, the Spanish dollar was worth slightly more than the U.S. dollar, and likewise, the Newfoundland dollar, until 1895, was worth slightly more than the Canadian dollar.

British Columbia
The Colony of British Columbia adopted the British Columbia dollar as its currency in 1865, at par with the Canadian dollar. When British Columbia joined Canada as its sixth province in 1871, the Canadian dollar replaced the British Columbia dollar.

Post-Confederation history
In 1867, the Province of Canada, New Brunswick, and Nova Scotia united into a federation named Canada. As a result, their respective currencies were merged into a singular Canadian dollar. The Canadian Parliament passed the Uniform Currency Act in April 1871, tying up loose ends as to the currencies of the various provinces and replacing them with a common Canadian dollar.

The gold standard was temporarily abandoned during World War I and definitively abolished on April 10, 1933. At the outbreak of World War II, the exchange rate to the U.S. dollar was fixed at Can$1.10 = US$1.00. This was changed to parity in 1946. In 1949, sterling was devalued and Canada followed, returning to a peg of Can$1.10 = US$1.00. However, Canada allowed its dollar to float in 1950, whereupon the currency rose to a slight premium over the U.S. dollar for the next decade. But the Canadian dollar fell sharply after 1960 before it was again pegged in 1962 at Can$1.00 = US$0.925. This was sometimes pejoratively referred to as the "Diefenbuck" or the "Diefendollar", after the then Prime Minister, John Diefenbaker. This peg lasted until 1970, with the currency's value being floated since then.

Terminology
Canadian English, similar to American English, used the slang term "buck" for a former paper dollar. The Canadian origin of this term derives from a coin struck by the Hudson's Bay Company during the 17th century with a value equal to the pelt of a male beaver – a "buck". Because of the appearance of the common loon on the back of the $1 coin that replaced the dollar bill in 1987, the word loonie was adopted in Canadian parlance to distinguish the Canadian dollar coin from the dollar bill. When the two-dollar coin was introduced in 1996, the derivative word toonie ("two loonies") became the common word for it in Canadian English slang.

In French, the currency is also called le dollar; Canadian French slang terms include piastre or piasse (the original word used in 18th-century French to translate "dollar") and huard (equivalent to loonie, since huard is French for "loon," the bird appearing on the coin). The French pronunciation of cent (pronounced similarly to English as or, not like the word for hundred,  or ) is generally used for the subdivision; sou is another, informal, term for 1¢. 25¢ coins in Quebec French are often called trente sous ("thirty cents") because of a series of changes in terminology, currencies, and exchange rates. After the British conquest of Canada in 1760, French coins gradually went out of use, and sou became a nickname for the halfpenny, which was similar in value to the French sou. Spanish dollars and U.S. dollars were also in use, and from 1841 to 1858, the exchange rate was fixed at $4 = £1 (or 400¢ = 240d). This made 25¢ equal to 15d, or 30 halfpence (trente sous). After decimalization and the withdrawal of halfpenny coins, the nickname sou began to be used for the 1¢ coin, but the idiom trente sous for 25¢ endured.

Coins
Coins are produced by the Royal Canadian Mint's facilities in Winnipeg, Manitoba, and Ottawa, Ontario, in denominations of 5¢ (nickel), 10¢ (dime), 25¢ (quarter), 50¢ (50¢ piece) (though the 50¢ piece is no longer distributed to banks and is only available directly from the mint, therefore seeing very little circulation), $1 (loonie), and $2 (toonie). The last 1¢ coin (penny) to be minted in Canada was struck on May 4, 2012, and distribution of the penny ceased on February 4, 2013. Ever since, the price for a cash transaction is rounded to the nearest five cents. The penny continues to be legal tender, although it is only accepted as payment and is not given back as change.

The standard set of designs has Canadian symbols, usually wildlife, on the reverse, and an effigy of Charles III on the obverse. A large number of pennies, nickels, and dimes are in circulation bearing the effigy of Elizabeth II, and occasionally some depicting George VI can be found. It is also common for American coins to be found among circulation due to the close proximity to the United States and the fact that the sizes and colours of the coins are similar. Commemorative coins with differing reverses are also issued on an irregular basis, most often quarters. 50¢ coins are rarely found in circulation; they are often collected and not regularly used in day-to-day transactions in most provinces.

Coin history
In 1858, bronze 1¢ and 0.925 silver 5¢, 10¢ and 20¢ coins were issued by the Province of Canada. Except for 1¢ coins struck in 1859, no more coins were issued until 1870, when production of the 5¢ and 10¢ was resumed and silver 25¢ and 50¢ were introduced. Between 1908 and 1919, sovereigns (legal tender in Canada for $$4.86 2/3$) were struck in Ottawa with a "C" mintmark.

Canada produced its first gold dollar coins in 1912 in the form of $5 and $10. These coins were produced from 1912 to 1914. The obverse carries an image of King George V and on the reverse is a shield with the arms of the Dominion of Canada. Gold from the Klondike River valley in the Yukon accounts for much of the gold in the coins.

Two years into the coin's production World War I began and production of the coins stopped in favour of tighter control over Canadian gold reserves. Most of the 1914 coins produced never reached circulation at the time and some were stored for more than 75 years until being sold off in 2012. The high quality specimens were sold to the public and the visually unappealing ones were melted.

In 1920, the size of the 1¢ was reduced and the silver fineness of the 5¢, 10¢, 25¢ and 50¢ coins was reduced to 0.800 silver/.200 copper. This composition was maintained for the 10¢, 25¢ and 50¢ piece through 1966, but the debasement of the 5¢ piece continued in 1922 with the silver 5¢ being entirely replaced by a larger nickel coin. In 1942, as a wartime measure, nickel was replaced by tombac in the 5¢ coin, which was changed in shape from round to dodecagonal. Chromium-plated steel was used for the 5¢ in 1944 and 1945 and between 1951 and 1954, after which nickel was readopted. The 5¢ returned to a round shape in 1963.

In 1935, the 0.800 silver voyageur dollar was introduced. Production was maintained through 1967 with the exception of the war years between 1939 and 1945.

In 1967 both 0.800 silver/0.200 copper and, later that year, 0.500 silver/.500 copper 10¢ and 25¢ coins were issued. 1968 saw further debasement: the 0.500 fine silver dimes and quarters were completely replaced by nickel ones mid-year. All 1968 50¢ and $1 coins were reduced in size and coined only in pure nickel. Thus, 1968 marked the last year in which any circulating silver coinage was issued in Canada.

In 1982, the 1¢ coin was changed to dodecagonal, and the 5¢ was further debased to a cupro-nickel alloy. In 1987 a $1 coin struck in aureate-plated nickel was introduced. A bimetallic $2 coin followed in 1996. In 1997, copper-plated zinc replaced bronze in the 1¢, and it returned to a round shape. This was followed, in 2000, by the introduction of even cheaper plated-steel 1¢, 5¢, 10¢, 25¢ and 50¢ coins, with the 1¢ plated in copper and the others plated in cupro-nickel. In 2012, the multi-ply plated-steel technology was introduced for $1 and $2 coins as well. Also in that year mintage of the 1¢ coin ceased and its withdrawal from circulation began in 2013.

Banknotes
The first paper money issued in Canada denominated in dollars were British Army bills, issued between 1813 and 1815. Canadian dollar banknotes were later issued by the chartered banks starting in the 1830s, by several pre-Confederation colonial governments (most notably the Province of Canada in 1866), and after confederation, by the Canadian government starting in 1870. Some municipalities also issued notes, most notably depression scrip during the 1930s.

On July 3, 1934, with only 10 chartered banks still issuing notes, the Bank of Canada was founded. This new government agency became the sole issuer of all federal notes. In 1935, it issued its first series of notes in denominations of $1, $2, $5, $10, $20, $25, $50, $100, $500 and $1000. The $25 note was a commemorative issue, released to mark the Silver Jubilee of King George V. In 1944, the chartered banks were prohibited from issuing their own currency, with the Royal Bank of Canada and the Bank of Montreal among the last to issue notes. Significant design changes to the notes have occurred since 1935, with new series introduced in 1937, 1954, 1970, 1986, and 2001. In June 2011, newly designed notes printed on a polymer substrate, as opposed to cotton fibre, were announced; the first of these polymer notes, the $100 bill, began circulation on November 14, 2011, the $50 bill began circulation on March 26, 2012, the $20 denomination began circulation on November 7, 2012, and the $5 and $10 denominations began circulation on November 12, 2013.

Since 1935, all banknotes are printed by the Ottawa-based Canadian Bank Note Company under contract to the Bank of Canada. Previously, a second company, BA International (founded in 1866 as the British American Bank Note Company), shared printing duties. In 2011, BA International announced it would close its banknote printing business and cease printing banknotes at the end of 2012; since then, the Canadian Bank Note Company has been the sole printer of Canadian banknotes.

All banknotes from series prior to the current polymer series are now considered unfit for circulation due to their lacking of any modern security features, such as a metallic stripe. Financial institutions must return the banknotes to the Bank of Canada, which will then destroy them. Individuals may keep the banknotes indefinitely.

Legal tender
As of January 1, 2021, the $1, $2, $25, $500 and $1000 notes issued by the Bank of Canada are no longer legal tender. All other current and prior Canadian dollar banknotes issued by the Bank of Canada remain as legal tender in Canada. However, commercial transactions may legally be settled in any manner agreed by the parties involved.

Legal tender of Canadian coinage is governed by the Currency Act, which sets out limits of:
 * $40 if the denomination is $2 or greater but does not exceed $10;
 * $25 if the denomination is $1;
 * $10 if the denomination is 10¢ or greater but less than $1;
 * $5 if the denomination is 5¢;
 * 25¢ if the denomination is 1¢.

Retailers in Canada may refuse bank notes without breaking the law. According to legal guidelines, the method of payment has to be mutually agreed upon by the parties involved with the transactions. For example, stores may refuse $100 banknotes if they feel that would put them at risk of being counterfeit victims; however, official policy suggests that the retailers should evaluate the impact of that approach. In the case that no mutually acceptable form of payment can be found for the tender, the parties involved should seek legal advice.

Canadian dollars, especially coins, are accepted by some businesses in the northernmost cities of the United States and in many Canadian snowbird enclaves, just as U.S. dollars are accepted by some Canadian businesses.

In 2012, Iceland considered adopting the Canadian dollar as a stable alternative to the Icelandic króna. Canada was favoured due to its northern geography and similar resource-based economy, in addition to its relative economic stability. The Canadian ambassador to Iceland said that Iceland could adopt the currency; although Iceland ultimately decided not to move on with the proposal.

Value

Since 76.7% of Canada's exports go to the U.S., and 53.3% of imports into Canada come from the U.S., Canadians are interested in the value of their currency mainly against the U.S. dollar. Although domestic concerns arise when the dollar trades much lower than its U.S. counterpart, there is also concern among exporters when the dollar appreciates quickly. A rise in the value of the dollar increases the price of Canadian exports to the U.S. On the other hand, there are advantages to a rising dollar, in that it is cheaper for Canadian industries to purchase foreign material and businesses.

The Bank of Canada currently has no specific target value for the Canadian dollar and has not intervened in foreign exchange markets since 1998. The Bank's official position is that market conditions should determine the worth of the Canadian dollar, although it occasionally makes minor attempts to influence its value.

On world markets, the Canadian dollar historically tended to move in tandem with the U.S. dollar. An apparently rising Canadian dollar (against the U.S. dollar) was decreasing against other international currencies; however, during the rise of the Canadian dollar between 2002 and 2013, it gained value against the U.S. dollar as well as other international currencies. In recent years, dramatic fluctuations in the value of the Canadian dollar have tended to correlate with shifts in oil prices, reflecting the Canadian dollar's status as a petrocurrency owing to Canada's significant oil exports.

The Canadian dollar traded at a record high of US$2.78 in terms of American greenbacks on July 11, 1864, since the latter was inconvertible paper currency. However, the Canadian dollar remained close to par or 1:1 versus the gold or silver US dollar of the time.

Unlike other currencies in the Bretton Woods system, whose values were fixed, the Canadian dollar was allowed to float from 1950 to 1962. Between 1952 and 1960, the Canadian dollar traded at a slight premium over the U.S. dollar, reaching a high of US$1.0614 on August 20, 1957.

The Canadian dollar fell considerably after 1960, and this contributed to Prime Minister John Diefenbaker's defeat in the 1963 election. The Canadian dollar returned to a fixed exchange rate regime in 1962 when its value was set at US$0.925, where it remained until 1970.

As an inflation-fighting measure, the Canadian dollar was allowed to float in 1970. Its value appreciated and it was worth more than the U.S. dollar for part of the 1970s. The high point was on April 25, 1974, when it reached US$1.0443.

The Canadian dollar fell in value against its American counterpart during the technological boom of the 1990s that was centred in the United States, and was traded for as little as US$0.6179 on January 21, 2002, which was an all-time low. Since then, its value against all major currencies rose until 2013, due in part to high prices for commodities (especially oil) that Canada exports.

The Canadian dollar's value against the U.S. dollar rose sharply in 2007 because of the continued strength of the Canadian economy and the U.S. currency's weakness on world markets. During trading on September 20, 2007, it met the U.S. dollar at parity for the first time since November 25, 1976.

Inflation in the value of the Canadian dollar has been fairly low since the 1990s. In 2007 the Canadian dollar rebounded, soaring 23% in value.

On September 28, 2007, the Canadian dollar closed above the U.S. dollar for the first time in 30 years, at US$1.0052. On November 7, 2007, it hit US$1.1024 during trading, a modern-day high after China announced it would diversify its US$1.43trillion foreign exchange reserve away from the U.S. dollar. By November 30, however, the Canadian dollar was once again at par with the U.S. dollar, and on December 4, the dollar had retreated back to US$0.98, through a cut in interest rates made by the Bank of Canada due to concerns about exports to the U.S.

Due to its soaring value and new record highs at the time, the Canadian dollar was named the Canadian Newsmaker of the Year for 2007 by the Canadian edition of Time magazine.

Since the late 2000s, the Canadian dollar has been valued at levels comparable to the years before its swift rise in 2007. For most of the 2010s, the exchange rate of Canadian to US dollars was approximately US$0.70 to Can$1.00.

Reserve currency
A number of central banks (and commercial banks) keep Canadian dollars as a reserve currency. The Canadian dollar is considered to be a benchmark currency.

In the economy of the Americas, the Canadian dollar plays a similar role to that of the Australian dollar (AUD) in the Asia-Pacific region. The Canadian dollar (as a regional reserve currency for banking) has been an important part of the British, French and Dutch Caribbean states' economies and finance systems since the 1950s. The Canadian dollar is held by many central banks in Central and South America as well.

By observing how the Canadian dollar behaves against the U.S. dollar, foreign exchange economists can indirectly observe internal behaviours and patterns in the U.S. economy that could not be seen by direct observation. The Canadian dollar has fully evolved into a global reserve currency only since the 1970s, when it was floated against all other world currencies. Some economists have attributed the rise of importance of the Canadian dollar to the long-term effects of the Nixon Shock that effectively ended the Bretton Woods system of global finance.

title: Canadian dollar


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ALOHAnet

ALOHAnet, also known as the ALOHA System, or simply ALOHA, was a pioneering computer networking system developed at the University of Hawaii. ALOHAnet became operational in June 1971, providing the first public demonstration of a wireless packet data network.

The ALOHAnet used a new method of medium access, called ALOHA random access, and experimental ultra high frequency (UHF) for its operation. In its simplest form, later known as Pure ALOHA, remote units communicated with a base station (Menehune) over two separate radio frequencies (for inbound and outbound respectively). Nodes did not wait for the channel to be clear before sending, but instead waited for acknowledgement of successful receipt of a message, and re-sent it if this was not received. Nodes would also stop and re-transmit data if they detected any other messages while transmitting. While simple to implement, this results in an efficiency of only 18.4%. A later advancement, Slotted ALOHA, improved the efficiency of the protocol by reducing the chance of collision, improving throughput to 36.8%.

ALOHA was subsequently employed in the Ethernet cable based network in the 1970s, and following regulatory developments in the early 1980s it became possible to use the ALOHA random-access techniques in both Wi-Fi and in mobile telephone networks. ALOHA channels were used in a limited way in the 1980s in 1G mobile phones for signaling and control purposes. In the late 1980s, the European standardization group GSM who worked on the Pan-European Digital mobile communication system GSM greatly expanded the use of ALOHA channels for access to radio channels in mobile telephony. In the early 2000s additional ALOHA channels were added to 2.5G and 3G mobile phones with the widespread introduction of General Packet Radio Service (GPRS), using a slotted-ALOHA random-access channel combined with a version of the Reservation ALOHA scheme first analyzed by a group at BBN Technologies.

History
One of the early computer networking designs, development of the ALOHA network was begun in September 1968 at the University of Hawaii under the leadership of Norman Abramson and Franklin Kuo, along with Thomas Gaarder, Shu Lin, Wesley Peterson and Edward ("Ned") Weldon. The goal was to use low-cost commercial radio equipment to connect users on Oahu and the other Hawaiian islands with a central time-sharing computer on the main Oahu campus. The first packet broadcasting unit went into operation in June 1971. Terminals were connected to a special purpose terminal connection unit using RS-232 at 9600 bit/s.

ALOHA was originally a contrived acronym standing for Additive Links On-line Hawaii Area.

The original version of ALOHA used two distinct frequencies in a hub configuration, with the hub machine broadcasting packets to everyone on the outbound channel, and the various client machines sending data packets to the hub on the inbound channel. If data was received correctly at the hub, a short acknowledgment packet was sent to the client; if an acknowledgment was not received by a client machine after a short wait time, it would automatically retransmit the data packet after waiting a randomly selected time interval. This acknowledgment mechanism was used to detect and correct for collisions created when two client machines both attempted to send a packet at the same time.

ALOHAnet's primary importance was its use of a shared medium for client transmissions. Unlike the ARPANET where each node could only talk to a single node at the other end of a wire or satellite circuit, in ALOHAnet all client nodes communicated with the hub on the same frequency. This meant that some sort of mechanism was needed to control who could talk at what time. The ALOHAnet solution was to allow each client to send its data without controlling when it was sent, and implementing an acknowledgment/retransmission scheme to deal with collisions. This approach radically reduced the complexity of the protocol and the networking hardware, since nodes do not need to negotiate who is allowed to speak.

This solution became known as a pure ALOHA, or random-access channel, and was the basis for subsequent Ethernet development and later Wi-Fi networks. Various versions of the ALOHA protocol (such as Slotted ALOHA) also appeared later in satellite communications, and were used in wireless data networks such as ARDIS, Mobitex, CDPD, and GSM.

The Aloha network introduced the mechanism of randomized multiple access, which resolved device transmission collisions by transmitting a packet immediately if no acknowledgement is present, and if no acknowledgment was received, the transmission was repeated after a random waiting time. The probability distribution of this random waiting time for retransmission of a packet that has not been acknowledged as received is critically important for the stability of Aloha-type communication systems. The average waiting time for retransmission is typically shorter than the average time for generation of a new packet from the same client node, but it should not be allowed to be so short as to compromise the stability of the network, causing a collapse in its overall throughput.

Also important was ALOHAnet's use of the outgoing hub channel to broadcast packets directly to all clients on a second shared frequency and using an address in each packet to allow selective receipt at each client node. Separate frequencies were used for incoming and outgoing communications to the hub so that devices could receive acknowledgments regardless of transmissions.

Pure ALOHA

The original version of the protocol (now called Pure ALOHA, and the one implemented in ALOHAnet) was quite simple:

 * If you have data to send, send the data
 * If, while you are transmitting data, you receive any data from another station, there has been a message collision. All transmitting stations will need to try resending later.

Pure ALOHA does not check whether the channel is busy before transmitting. Since collisions can occur and data may have to be sent again, ALOHA cannot efficiently use 100% of the capacity of the communications channel. How long a station waits until it retransmits, and the likelihood a collision occurs are interrelated, and both affect how efficiently the channel can be used. This means that the concept of retransmit later is a critical aspect; The quality of the backoff scheme chosen significantly influences the efficiency of the protocol, the ultimate channel capacity, and the predictability of its behavior.

To assess Pure ALOHA, there is a need to predict its throughput, the rate of (successful) transmission of frames. First make a few simplifying assumptions:

 * All frames have the same length.
 * Stations cannot generate a frame while transmitting or trying to transmit. That is, while a station is sending or trying to resend a frame, it cannot be allowed to generate more frames to send.
 * The population of stations attempting to transmit (both new transmission and retransmissions) follows a Poisson distribution.

Let $T$ refer to the time needed to transmit one frame on the channel, and define frame-time as a unit of time equal to $T$. Let $G$ refer to the mean used in the Poisson distribution over transmission-attempt amounts. That is, on average, there are $G$ transmission attempts per frame-time.

Consider what needs to happen for a frame to be transmitted successfully. Let $t$ refer to the time at which it is intended to send a frame. It is preferable to use the channel for one frame-time beginning at $t$, and all other stations to refrain from transmitting during this time.

For any frame-time, the probability of there being $k$ transmission-attempts during that frame-time is:

$$\frac{G^k e^{-G}}{k!}$$

The average number of transmission-attempts for two consecutive frame-times is $2G$. Hence, for any pair of consecutive frame-times, the probability of there being $k$ transmission attempts during those two frame-times is:

 * $$\frac{(2G)^k e^{-2G}}{k!}$$

Therefore, the probability ($$Prob_{pure}$$) of there being zero transmission-attempts between $t-T$ and $t+T$ (and thus of a successful transmission for us) is:

 * $$Prob_{pure}=e^{-2G}$$

The throughput can be calculated as the rate of transmission attempts multiplied by the probability of success, and it can be concluded that the throughput ($$S_{pure}$$) is:

 * $$S_{pure}=Ge^{-2G}$$

The maximum throughput is $0.5/e$ frames per frame-time (reached when $$G=0.5$$), which is approximately 0.184 frames per frame-time. This means that, in Pure ALOHA, only about 18.4% of the time is used for successful transmissions.

Slotted ALOHA

An improvement to the original ALOHA protocol was Slotted ALOHA, which introduced discrete time slots and increased the maximum throughput. A station can start a transmission only at the beginning of a time slot, and thus collisions are reduced. In this case, only transmission-attempts within 1 frame-time and not 2 consecutive frame-times need to be considered, since collisions can only occur during each time slot. Thus, the probability of there being zero transmission attempts by other stations in a single time slot is:

$$Prob_{slotted} = e^{-G}$$

the probability of a transmission requiring exactly k attempts is (k-1 collisions and 1 success):

$$Prob_{slotted} k = e^{-G} ( 1 - e^{-G} )^{k-1}$$

The throughput is:

$$S_{slotted}=Ge^{-G}$$

The maximum throughput is 1/e frames per frame-time (reached when G = 1), which is approximately 0.368 frames per frame-time, or 36.8%.

Slotted ALOHA is used in low-data-rate tactical satellite communications networks by military forces, in subscriber-based satellite communications networks, mobile telephony call setup, set-top box communications and in the contactless RFID technologies.

Reservation ALOHA
Reservation ALOHA, or R-ALOHA, is an effort to improve the efficiency of Slotted ALOHA. The improvements with Reservation ALOHA are markedly shorter delays and ability to efficiently support higher levels of utilization. As a contrast of efficiency, simulations have shown that Reservation ALOHA exhibits less delay at 80% utilization than Slotted ALOHA at 20–36% utilization.

The chief difference between Slotted and Reservation ALOHA is that with Slotted ALOHA, any slot is available for utilization without regards to prior usage. Under Reservation ALOHA's contention-based reservation schema, the slot is temporarily considered "owned" by the station that successfully used it. Additionally, Reservation ALOHA simply stops sending data once the station has completed its transmission. As a rule, idle slots are considered available to all stations that may then implicitly reserve (utilize) the slot on a contention basis.

Other protocols
The use of a random-access channel in ALOHAnet led to the development of carrier-sense multiple access (CSMA), a listen before send random-access protocol that can be used when all nodes send and receive on the same channel. CSMA in radio channels was extensively modeled. The AX.25 packet radio protocol is based on the CSMA approach with collision recovery, based on the experience gained from ALOHAnet. A variation of CSMA, CSMA/CD is used in early versions of Ethernet.

ALOHA and the other random-access protocols have an inherent variability in their throughput and delay performance characteristics. For this reason, applications that need highly deterministic load behavior may use master/slave or token-passing schemes (such as Token Ring or ARCNET) instead of contention systems.

Hardware
The central node communications processor was an HP 2100 minicomputer called the Menehune, which is the Hawaiian language word for dwarf people, and was named for its similar role to the original ARPANET Interface Message Processor (IMP) which was being deployed at about the same time. In the original system, the Menehune forwarded correctly received user data to the UH central computer, an IBM System 360/65 time-sharing system. Outgoing messages from the 360 were converted into packets by the Menehune, which were queued and broadcast to the remote users at a data rate of 9600 bit/s. Unlike the half-duplex radios at the user TCUs, the Menehune was interfaced to the radio channels with full-duplex radio equipment.

The original user interface developed for the system was an all-hardware unit called an ALOHAnet Terminal Control Unit (TCU) and was the sole piece of equipment necessary to connect a terminal into the ALOHA channel. The TCU was composed of a UHF antenna, transceiver, modem, buffer and control unit. The buffer was designed for a full line length of 80 characters, which allowed handling of both the 40- and 80-character fixed-length packets defined for the system. The typical user terminal in the original system consisted of a Teletype Model 33 or a dumb CRT user terminal connected to the TCU using a standard RS-232 interface. Shortly after the original ALOHA network went into operation, the TCU was redesigned with one of the first Intel microprocessors, and the resulting upgrade was called a Programmable Control Unit (PCU).

Additional basic functions performed by the TCUs and PCUs were generation of a cyclic-parity-check code vector and decoding of received packets for packet error detection purposes, and generation of packet retransmissions using a simple random interval generator. If an acknowledgment was not received from the Menehune after the prescribed number of automatic retransmissions, a flashing light was used as an indicator to the human user. Also, since the TCUs and PCUs did not send acknowledgments to the Menehune, a steady warning light was displayed to the human user when an error was detected in a received packet. Considerable simplification was incorporated into the initial design of the TCU as well as the PCU for interfacing a human user into the network.

In later versions of the system, simple radio relays were placed in operation to connect the main network on the island of Oahu to other islands in Hawaii, and Menehune routing capabilities were expanded to allow user nodes to exchange packets with other user nodes, the ARPANET, and an experimental satellite network.

Network architecture
Two fundamental choices which dictated much of the ALOHAnet design were the two-channel star configuration of the network and the use of random access for user transmissions.

The two-channel configuration was primarily chosen to allow for efficient transmission of the relatively dense total traffic stream being returned to users by the central time-sharing computer. An additional reason for the star configuration was the desire to centralize as many communication functions as possible at the central network node (the Menehune) to minimize the cost of the original all-hardware terminal control unit (TCU) at each user node.

The random-access channel for communication between users and the Menehune was designed specifically for the traffic characteristics of interactive computing. In a conventional communication system, a user might be assigned a portion of the channel on either a frequency-division multiple access or time-division multiple access basis. Since it was well known that in time-sharing systems (circa 1970), computer and user data are bursty, such fixed assignments are generally wasteful of bandwidth because of the high peak-to-average data rates that characterize the traffic.

To achieve a more efficient use of bandwidth for bursty traffic, ALOHAnet developed the random-access packet switching method that has come to be known as a pure ALOHA channel. This approach effectively dynamically allocates bandwidth immediately to a user who has data to send, using the acknowledgment and retransmission mechanism described earlier to deal with occasional access collisions. While the average channel loading must be kept below about 10% to maintain a low collision rate, this still results in better bandwidth efficiency than when fixed allocations are used in a bursty traffic context.

Two 100 kHz channels in the experimental UHF band were used in the implemented system, one for the user-to-computer random-access channel and one for the computer-to-user broadcast channel. The system was configured as a star network, allowing only the central node to receive transmissions in the random-access channel. All user TCUs received each transmission made by the central node in the broadcast channel. All transmissions were made in bursts at $9,600 bit/s$, with data and control information encapsulated in packets.

Each packet consisted of a 32-bit header and a 16-bit header parity check word, followed by up to 80 bytes of data and a 16-bit parity check word for the data. The header contained address information identifying a particular user so that when the Menehune broadcast a packet, only the intended user's node would accept it.

Legacy
In the 1970s ALOHA random access was employed in the nascent Ethernet cable based network and then in the Marisat (now Inmarsat) satellite network.

In the early 1980s frequencies for mobile networks became available, and in 1985 frequencies suitable for what became known as Wi-Fi were allocated in the US. These regulatory developments made it possible to use the ALOHA random-access techniques in both Wi-Fi and in mobile telephone networks.

ALOHA channels were used in a limited way in the 1980s in 1G mobile phones for signaling and control purposes. In the late 1980s, the European standardization group GSM who worked on the Pan-European Digital mobile communication system GSM greatly expanded the use of ALOHA channels for access to radio channels in mobile telephony. In addition, SMS message texting was implemented in 2G mobile phones. In the early 2000s additional ALOHA channels were added to 2.5G and 3G mobile phones with the widespread introduction of General Packet Radio Service (GPRS), using a slotted-ALOHA random-access channel combined with a version of the Reservation ALOHA scheme first analyzed by a group at BBN Technologies.

title: ALOHAnet


text:
National nature reserves in Norfolk

National nature reserves in Norfolk, England are established by Natural England and managed by it or by non-governmental organisations such as the Norfolk Wildlife Trust, the Royal Society for the Protection of Birds, and the National Trust.

List of reserves
A list of national nature reserves in Norfolk:
 * Ant Broads & Marshes NNR (in The Broads National Park)
 * Blakeney NNR, including Blakeney Point
 * Brettenham Heath NNR
 * Bure Marshes NNR (in The Broads National Park)
 * Calthorpe Broad NNR (in The Broads National Park)
 * Dersingham Bog NNR
 * Foxley Wood NNR
 * Heigham Holmes
 * Hickling Broad NNR (in The Broads National Park)
 * Holkham NNR
 * Holme Dunes NNR
 * Ludham - Potter Heigham NNR (in The Broads National Park)
 * Martham Broad NNR (in The Broads National Park)
 * Mid-Yare NNR (in The Broads National Park)
 * Redgrave and Lopham Fen
 * Roydon Common NNR
 * Scolt Head Island NNR
 * Swanton Novers NNR
 * The Wash NNR
 * Weeting Heath NNR
 * Winterton Dunes NNR

title: National nature reserves in Norfolk


text:
Hilbert's tenth problem

Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.

For example, the Diophantine equation $$3x^2-2xy-y^2z-7=0$$ has an integer solution: $$x=1,\ y=2,\ z=-2$$. By contrast, the Diophantine equation $$x^2+y^2+1=0$$ has no such solution.

Hilbert's tenth problem has been solved, and it has a negative answer: such a general algorithm cannot exist. This is the result of combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson that spans 21 years, with Matiyasevich completing the theorem in 1970. The theorem is now known as Matiyasevich's theorem or the MRDP theorem (an initialism for the surnames of the four principal contributors to its solution).

When all coefficients and variables are restricted to be positive integers, the related problem of polynomial identity testing becomes a decidable (exponentiation-free) variation of Tarski's high school algebra problem, sometimes denoted $$\overline{HSI}.$$

Original formulation
Hilbert formulated the problem as follows:

"Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers."

The words "process" and "finite number of operations" have been taken to mean that Hilbert was asking for an algorithm. The term "rational integral" simply refers to the integers, positive, negative or zero: 0, ±1, ±2, ... . So Hilbert was asking for a general algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution in integers.

Hilbert's problem is not concerned with finding the solutions. It only asks whether, in general, we can decide whether one or more solutions exist. The answer to this question is negative, in the sense that no "process can be devised" for answering that question. In modern terms, Hilbert's 10th problem is an undecidable problem.

Diophantine sets
In a Diophantine equation, there are two kinds of variables: the parameters and the unknowns. The Diophantine set consists of the parameter assignments for which the Diophantine equation is solvable. A typical example is the linear Diophantine equation in two unknowns,

 * $$a_1x + a_2y = a_3$$,

where the equation is solvable if and only if the greatest common divisor $$\gcd(a_1, a_2)$$ evenly divides $$a_3$$. The set of all ordered triples $$(a_1, a_2, a_3)$$ satisfying this restriction is called the Diophantine set defined by $$a_1x + a_2y = a_3$$. In these terms, Hilbert's tenth problem asks whether there is an algorithm to determine if the Diophantine set corresponding to an arbitrary polynomial is non-empty.

The problem is generally understood in terms of the natural numbers (that is, the non-negative integers) rather than arbitrary integers. However, the two problems are equivalent: any general algorithm that can decide whether a given Diophantine equation has an integer solution could be modified into an algorithm that decides whether a given Diophantine equation has a natural-number solution, and vice versa. By Lagrange's four-square theorem, every natural number is the sum of the squares of four integers, so we could rewrite every natural-valued parameter in terms of the sum of the squares of four new integer-valued parameters. Similarly, since every integer is the difference of two natural numbers, we could rewrite every integer parameter as the difference of two natural parameters. Furthermore, we can always rewrite a system of simultaneous equations $$p_1=0,\ldots,p_k=0$$ (where each $$p_i$$ is a polynomial) as a single equation $$p_1^{\,2}+\cdots+p_k^{\,2}=0$$.

Recursively enumerable sets
A recursively enumerable set can be characterized as one for which there exists an algorithm that will ultimately halt when a member of the set is provided as input, but may continue indefinitely when the input is a non-member. It was the development of computability theory (also known as recursion theory) that provided a precise explication of the intuitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable (also known as semi-decidable). This is because one can arrange all possible tuples of values of the unknowns in a sequence and then, for a given value of the parameter(s), test these tuples, one after another, to see whether they are solutions of the corresponding equation. The unsolvability of Hilbert's tenth problem is a consequence of the surprising fact that the converse is true: "Every recursively enumerable set is Diophantine." This result is variously known as Matiyasevich's theorem (because he provided the crucial step that completed the proof) and the MRDP theorem (for Yuri Matiyasevich, Julia Robinson, Martin Davis, and Hilary Putnam). Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence. In fact, more can be said: there is a polynomial
 * $$p(a,x_1,\ldots,x_n)$$

with integer coefficients such that the set of values of $$a$$ for which the equation
 * $$p(a,x_1,\ldots,x_n)=0$$

has solutions in natural numbers is not computable. So, not only is there no general algorithm for testing Diophantine equations for solvability, even for this one parameter family of equations, there is no algorithm.

Applications
The Matiyasevich/MRDP theorem relates two notions—one from computability theory, the other from number theory—and has some surprising consequences. Perhaps the most surprising is the existence of a universal Diophantine equation:

 * There exists a polynomial $$p(a,n,x_1,\ldots,x_k)$$ such that, given any Diophantine set $$S$$ there is a number $$n_0$$ such that
 * $$ S = \{\,a \mid \exists x_1, \ldots, x_k[p(a,n_0,x_1,\ldots,x_k)=0]\,\}.$$

This is true simply because Diophantine sets, being equal to recursively enumerable sets, are also equal to Turing machines. It is a well known property of Turing machines that there exist universal Turing machines, capable of executing any algorithm.

Hilary Putnam has pointed out that for any Diophantine set $$S$$ of positive integers, there is a polynomial

 * $$q(x_0,x_1,\ldots,x_n)$$

such that $$S$$ consists of exactly the positive numbers among the values assumed by $$q$$ as the variables

 * $$x_0,x_1,\ldots,x_n$$

range over all natural numbers. This can be seen as follows: If

 * $$p(a,y_1,\ldots,y_n)=0$$

provides a Diophantine definition of $$S$$, then it suffices to set

 * $$q(x_0,x_1,\ldots,x_n)= x_0[1- p(x_0,x_1,\ldots,x_n)^2].$$

So, for example, there is a polynomial for which the positive part of its range is exactly the prime numbers. (On the other hand, no polynomial can only take on prime values.) The same holds for other recursively enumerable sets of natural numbers: the factorial, the binomial coefficients, the fibonacci numbers, etc.

Other applications concern what logicians refer to as $$\Pi^{0}_1$$ propositions, sometimes also called propositions of Goldbach type. These are like Goldbach's conjecture, in stating that all natural numbers possess a certain property that is algorithmically checkable for each particular number. The Matiyasevich/MRDP theorem implies that each such proposition is equivalent to a statement that asserts that some particular Diophantine equation has no solutions in natural numbers. A number of important and celebrated problems are of this form: in particular, Fermat's Last Theorem, the Riemann hypothesis, and the four color theorem. In addition the assertion that particular formal systems such as Peano arithmetic or ZFC are consistent can be expressed as $$\Pi^{0}_1$$ sentences. The idea is to follow Kurt Gödel in coding proofs by natural numbers in such a way that the property of being the number representing a proof is algorithmically checkable.

$$\Pi^{0}_1$$ sentences have the special property that if they are false, that fact will be provable in any of the usual formal systems. This is because the falsity amounts to the existence of a counter-example that can be verified by simple arithmetic. So if a $$\Pi^{0}_1$$ sentence is such that neither it nor its negation is provable in one of these systems, that sentence must be true.

A particularly striking form of Gödel's incompleteness theorem is also a consequence of the Matiyasevich/MRDP theorem:

Let

 * $$p(a,x_1,\ldots,x_k)=0$$

provide a Diophantine definition of a non-computable set. Let $$A$$ be an algorithm that outputs a sequence of natural numbers $$n$$ such that the corresponding equation

 * $$p(n,x_1,\ldots,x_k)=0$$

has no solutions in natural numbers. Then there is a number $$n_0$$ that is not output by $$A$$ while in fact the equation

 * $$p(n_0,x_1,\ldots,x_k)=0$$

has no solutions in natural numbers.

To see that the theorem is true, it suffices to notice that if there were no such number $$n_0$$, one could algorithmically test membership of a number $$n$$ in this non-computable set by simultaneously running the algorithm $$A$$ to see whether $$n$$ is output while also checking all possible $$k$$-tuples of natural numbers seeking a solution of the equation

 * $$p(n,x_1,\ldots,x_k)=0$$

and we may associate an algorithm $$A$$ with any of the usual formal systems such as Peano arithmetic or ZFC by letting it systematically generate consequences of the axioms and then output a number $$n$$ whenever a sentence of the form

 * $$\neg \exists x_1,\ldots, x_k [p(n,x_1,\ldots,x_k)=0]$$

is generated. Then the theorem tells us that either a false statement of this form is proved or a true one remains unproved in the system in question.

Further results
We may speak of the degree of a Diophantine set as being the least degree of a polynomial in an equation defining that set. Similarly, we can call the dimension of such a set the fewest unknowns in a defining equation. Because of the existence of a universal Diophantine equation, it is clear that there are absolute upper bounds to both of these quantities, and there has been much interest in determining these bounds.

Already in the 1920s Thoralf Skolem showed that any Diophantine equation is equivalent to one of degree 4 or less. His trick was to introduce new unknowns by equations setting them equal to the square of an unknown or the product of two unknowns. Repetition of this process results in a system of second degree equations; then an equation of degree 4 is obtained by summing the squares. So every Diophantine set is trivially of degree 4 or less. It is not known whether this result is best possible.

Julia Robinson and Yuri Matiyasevich showed that every Diophantine set has dimension no greater than 13. Later, Matiyasevich sharpened their methods to show that 9 unknowns suffice. Although it may well be that this result is not the best possible, there has been no further progress. So, in particular, there is no algorithm for testing Diophantine equations with 9 or fewer unknowns for solvability in natural numbers. For the case of rational integer solutions (as Hilbert had originally posed it), the 4-squares trick shows that there is no algorithm for equations with no more than 36 unknowns. But Zhi Wei Sun showed that the problem for integers is unsolvable even for equations with no more than 11 unknowns.

Martin Davis studied algorithmic questions involving the number of solutions of a Diophantine equation. Hilbert's tenth problem asks whether or not that number is 0. Let $$A=\{0,1,2,3,\ldots,\aleph_0\}$$ and let $$C$$ be a proper non-empty subset of $$A$$. Davis proved that there is no algorithm to test a given Diophantine equation to determine whether the number of its solutions is a member of the set $$C$$. Thus there is no algorithm to determine whether the number of solutions of a Diophantine equation is finite, odd, a perfect square, a prime, etc.

The proof of the MRDP theorem has been formalized in Coq.

Extensions of Hilbert's tenth problem
Although Hilbert posed the problem for the rational integers, it can be just as well asked for many rings (in particular, for any ring whose number of elements is countable). Obvious examples are the rings of integers of algebraic number fields as well as the rational numbers.

There has been much work on Hilbert's tenth problem for the rings of integers of algebraic number fields. Basing themselves on earlier work by Jan Denef and Leonard Lipschitz and using class field theory, Harold N. Shapiro and Alexandra Shlapentokh were able to prove: "Hilbert's tenth problem is unsolvable for the ring of integers of any algebraic number field whose Galois group over the rationals is abelian."

Shlapentokh and Thanases Pheidas (independently of one another) obtained the same result for algebraic number fields admitting exactly one pair of complex conjugate embeddings.

The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open. Likewise, despite much interest, the problem for equations over the rationals remains open. Barry Mazur has conjectured that for any variety over the rationals, the topological closure over the reals of the set of solutions has only finitely many components. This conjecture implies that the integers are not Diophantine over the rationals and so if this conjecture is true a negative answer to Hilbert's Tenth Problem would require a different approach than that used for other rings.

title: Hilbert's tenth problem


text:
Talk:Leleti Khumalo

Untitled
I've just changed the article to read from a NPOV. I've kept this sentence here because it's not NPOV, but it could be developed into useful info by someone who knows something about this film. -- Sam 21:58 Oct 9, 2002 (UTC)
 * Sarafina! showed the world the tragedy and misfortune that a misguided political propaganda like Apartheid caused.

Fair use rationale for Image:Khumalo yesterday.jpg
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Please go to the image description page and edit it to include a fair use rationale. Using one of the templates at Fair use rationale guideline is an easy way to insure that your image is in compliance with Wikipedia policy, but remember that you must complete the template. Do not simply insert a blank template on an image page.

If there is other fair use media, consider checking that you have specified the fair use rationale on the other images used on this page. Note that any fair use images lacking such an explanation can be deleted one week after being tagged, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.

BetacommandBot (talk) 16:45, 2 January 2008 (UTC)

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title: Talk:Leleti Khumalo


text:
Talk:Canadian dollar

Restoration to original form
I have restored the article to its original form. Further to points above, especially on sourcing WP:QUESTIONABLE and bold editing WP:BRD issues, the ‘Declining Value’ subsection is superfluous (i.e. the ‘Value’ section already mentions historical fluctuations in forex value). A 'Declining Value' subsection also presents the need for an ‘Increasing Value’ subsection, as forex values are relative and constantly fluctuating. This means that even if a currency hits a 15-year low one day, but starts to rise again the next, the currency is rightly said to be ‘increasing’ in value. A ‘Declining Value’ subsection is therefore no longer relevant. Other editors have also tried in vain to caution Peter K Burian with respect to his similar line of reasoning on other articles, yet he continues to ignore the guidelines and attempts to impose his way or the highway. I feel mentioning the Canadian dollar’s all-time highs and lows is both sufficient and appropriate given the precedent of other Wikipedia currency pages. There’s no need to mention daily forex rates as Wikipedia isn’t a newspaper WP:NOTNP & WP:RECENTISM. Besides, forex rates can already be seen in the context of the ‘current exchange rates’ section. The Canadian dollar’s value is not the article’s central theme! If other editors disagree or have anything more to add to this discussion, please discuss here before boldly changing the article from its original form. Thank you. NorthernFactoid (talk) 01:04, 19 January 2016 (UTC)

File:CAN-S559-Bank of Montreal-10 Dollars (1935).jpg to appear as POTD soon
Hello! This is a note to let the editors of this article know that File:CAN-S559-Bank of Montreal-10 Dollars (1935).jpg will be appearing as picture of the day on March 5, 2018. You can view and edit the POTD blurb at Template:POTD/2018-03-05. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page. — Chris Woodrich (talk) 07:03, 20 February 2018 (UTC)

"Absurd" is not civil
I have a request. Please don't characterize previous drafts of articles as "absurd" because they didn't include something you know about. Perhaps other editors didn't know about it. Why characterize someone with less knowledge than yourself as writing an "absurd" passage in an article? Please remember WP:Civility. Mr Serjeant Buzfuz (talk) 04:48, 17 September 2019 (UTC)
 * I used that word only in reference to the content of the article, without any mention of the behaviour of other editors. I was not characterizing anyone. Perhaps it was just an unusual oversight that the (body of the) article did not mention that "CA$" is sometimes used to refer to the Canadian dollar. That seemed like a very common abbreviation to me – e.g., since it is used by default by the CAD template on Wikipedia and is based on the standard two-letter abbreviation for Canada and since that template results in links directly to this article as illustrated here: " CA$123.45 " produces "CA$123.45". The omission seemed very strange for an article about such an important topic as the currency of one of the largest countries in the world (an English-speaking country being discussed on the English Wikipedia). I have never edited the article before and was not familiar with any possible history that might exist about that question (and I don't consider myself as having any special expertise on the subject). I looked in the talk page archives a bit, but did not notice any previous commentary. Perhaps it would have been better if I had said the omission "seemed very strange", rather than saying it was "absurd". I suppose I was using a defensive tone in anticipation that someone might actually think the abbreviation shouldn't be mentioned. It seems that is not the case, which I am glad to discover. —BarrelProof (talk) 17:13, 17 September 2019 (UTC)

currency template
There is a discussion at Template talk:Currency about whether the Canadian currency should be displayed as CA$, Can$, CAD or something else when used in an international context (ie, when it could be confused with the US dollar, Australian dollar, etc). Please comment there, not here.  Stepho  talk 22:39, 1 December 2023 (UTC)

title: Talk:Canadian dollar


text:
Talk:Second Great Awakening

Heyman reference
The article references a piece by Christine Leigh Heyrman to support the claim that the First Awakening was part of a Romantic movement in England, Scotland, and Germany. The Heyman article does not use the word romantic, but evangelical. It's not clear why this reference about the First Awaking is here. The article is about the Second rather than the First Awakening. There might be an argument that the Second Awakening was related to the Romantic movement of the early 1800s, but the Heyman article does not discuss that. BTWaters (talk) 22:36, 2 June 2023 (UTC)

 * I agree. I went ahead and changed "Romantic" to "evangelical" .Ltwin (talk) 16:20, 5 June 2023 (UTC)
 * @BTWaters Ltwin (talk) 16:20, 5 June 2023 (UTC)

title: Talk:Second Great Awakening


text:
User:Tubby

My websites: Organic Games Bela's Handstand

In my utopia, there will be no wire coat-hangers. In my utopia, there will be an abundance of comfortable, practical footwear. In my utopia, there will be no air-conditioners. In my utopia, I will have fresh sheets every night. In my utopia, I will have a new razor blade every morning. In my utopia, everything will be within reach. In my utopia, I will be unemployed. In my utopia, windshield wipers will work as they are supposed to. In my utopia, the hot water will be hot right away, like in a hotel. In my utopia, I will surround myself with things of high quality. In my utopia, socks will not be the hassle they are now. In my utopia, all the knives will be sharp. In my utopia, everyone will be able to leave. In my utopia, there will be no melodrama. In my utopia, where have all the counters gone?

If there are no air-conditioners, I'll have to create my own utopia. I agree about the coat hangers and footwear, though. ;P

BTW, Welcome to Wikipedia. Enjoy. Stormwriter

But those loud big ugly ones? Yuck. There must be another way. -Tubby

ouch! ... I've just edited Tubby's user entry... I'm not sure this acceptable here, but I acted in good will and I'm new here so please excuse me if I did wrong. BlueShade

Socks are a hassle? CircleSky

In my Utopia, I would have you around! Fly-By

title: User:Tubby


text:
Ant Broads & Marshes National Nature Reserve

Ant Broads & Marshes NNR is a national nature reserve in Norfolk, England established by English Nature. It is named after the River Ant.

It is part of the "Ant Broads and Marshes" Site of Special Scientific Interest and is within The Broads National Park.

The NNR includes:
 * Barton Broad and Catfield Fen, owned and managed by Norfolk Wildlife Trust
 * Catfield Fen Reserve, owned and managed by the British Butterfly Conservation Society.

The fen is the best example of unpolluted valley fen in Western Europe.


text:
Evanston

Evanston may refer to locations:

in the United States: in Canada:
 * Evanston, Cincinnati, a neighborhood of Cincinnati, Ohio
 * Evanston, Illinois, a city on Lake Michigan
 * Evanston, Indiana, an unincorporated community
 * Evanston, Kentucky
 * Evanston, Wyoming
 * Evanston, Calgary, a neighbourhood in Calgary, Alberta
 * Evanston, Nova Scotia

in Australia:
 * Evanston, South Australia

title: Evanston


text:
Linear independence

In the theory of vector spaces, a set of vectors is said to be  if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension.

A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

Definition
A sequence of vectors $$\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$$ from a vector space $V$ is said to be linearly dependent, if there exist scalars $$a_1, a_2, \dots, a_k,$$ not all zero, such that
 * $$a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \cdots + a_k\mathbf{v}_k = \mathbf{0},$$

where $$\mathbf{0}$$ denotes the zero vector.

This implies that at least one of the scalars is nonzero, say $$a_1\ne 0$$, and the above equation is able to be written as
 * $$\mathbf{v}_1 = \frac{-a_2}{a_1}\mathbf{v}_2 + \cdots + \frac{-a_k}{a_1} \mathbf{v}_k,$$

if $$k>1,$$ and $$\mathbf{v}_1 = \mathbf{0}$$ if $$k=1.$$

Thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others.

A sequence of vectors $$\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$$ is said to be linearly independent if it is not linearly dependent, that is, if the equation
 * $$a_1\mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n\mathbf{v}_n = \mathbf{0},$$

can only be satisfied by $$a_i=0$$ for $$i=1,\dots,n.$$ This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence. In other words, a sequence of vectors is linearly independent if the only representation of $$\mathbf 0$$ as a linear combination of its vectors is the trivial representation in which all the scalars $a_i$ are zero. Even more concisely, a sequence of vectors is linearly independent if and only if $$\mathbf 0$$ can be represented as a linear combination of its vectors in a unique way.

If a sequence of vectors contains the same vector twice, it is necessarily dependent. The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is linearly independent if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is often useful.

A sequence of vectors is linearly independent if and only if it does not contain the same vector twice and the set of its vectors is linearly independent.

Infinite case
An infinite set of vectors is linearly independent if every nonempty finite subset is linearly independent. Conversely, an infinite set of vectors is linearly dependent if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.

An indexed family of vectors is linearly independent if it does not contain the same vector twice, and if the set of its vectors is linearly independent. Otherwise, the family is said to be linearly dependent.

A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space. For example, the vector space of all polynomials in $x$ over the reals has the (infinite) subset ${1, x, x^{2}, ...}$ as a basis.

Geometric examples

 * $$\vec u$$ and $$\vec v$$ are independent and define the plane P.
 * $$\vec u$$, $$\vec v$$ and $$\vec w$$ are dependent because all three are contained in the same plane.
 * $$\vec u$$ and $$\vec j$$ are dependent because they are parallel to each other.
 * $$\vec u$$, $$\vec v$$ and $$\vec k$$ are independent because $$\vec u$$ and $$\vec v$$ are independent of each other and $$\vec k$$ is not a linear combination of them or, equivalently, because they do not belong to a common plane. The three vectors define a three-dimensional space.
 * The vectors $$\vec o$$ (null vector, whose components are equal to zero) and $$\vec k$$ are dependent since $$\vec o = 0 \vec k$$

Geographic location
A person describing the location of a certain place might say, "It is 3 miles north and 4 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of the Earth's surface). The person might add, "The place is 5 miles northeast of here." This last statement is true, but it is not necessary to find the location. In this example the "3 miles north" vector and the "4 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "5 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary to define a specific location on a plane.

Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, $n$ linearly independent vectors are required to describe all locations in $n$-dimensional space.

The zero vector
If one or more vectors from a given sequence of vectors $$\mathbf{v}_1, \dots, \mathbf{v}_k$$ is the zero vector $$\mathbf{0}$$ then the vector $$\mathbf{v}_1, \dots, \mathbf{v}_k$$ are necessarily linearly dependent (and consequently, they are not linearly independent). To see why, suppose that $$i$$ is an index (i.e. an element of $$\{ 1, \ldots, k \}$$) such that $$\mathbf{v}_i = \mathbf{0}.$$ Then let $$a_{i} := 1$$ (alternatively, letting $$a_{i}$$ be equal any other non-zero scalar will also work) and then let all other scalars be $$0$$ (explicitly, this means that for any index $$j$$ other than $$i$$ (i.e. for $$j \neq i$$), let $$a_{j} := 0$$ so that consequently $$a_{j} \mathbf{v}_j = 0 \mathbf{v}_j = \mathbf{0}$$). Simplifying $$a_1 \mathbf{v}_1 + \cdots + a_k\mathbf{v}_k$$ gives:
 * $$a_1 \mathbf{v}_1 + \cdots + a_k\mathbf{v}_k = \mathbf{0} + \cdots + \mathbf{0} + a_i \mathbf{v}_i + \mathbf{0} + \cdots + \mathbf{0} = a_i \mathbf{v}_i = a_i \mathbf{0} = \mathbf{0}.$$

Because not all scalars are zero (in particular, $$a_{i} \neq 0$$), this proves that the vectors $$\mathbf{v}_1, \dots, \mathbf{v}_k$$ are linearly dependent.

As a consequence, the zero vector can not possibly belong to any collection of vectors that is linearly independent.

Now consider the special case where the sequence of $$\mathbf{v}_1, \dots, \mathbf{v}_k$$ has length $$1$$ (i.e. the case where $$k = 1$$). A collection of vectors that consists of exactly one vector is linearly dependent if and only if that vector is zero. Explicitly, if $$\mathbf{v}_1$$ is any vector then the sequence $$\mathbf{v}_1$$ (which is a sequence of length $$1$$) is linearly dependent if and only if $\mathbf{v}_1 = \mathbf{0}$; alternatively, the collection $$\mathbf{v}_1$$ is linearly independent if and only if $$\mathbf{v}_1 \neq \mathbf{0}.$$

Linear dependence and independence of two vectors
This example considers the special case where there are exactly two vector $$\mathbf{u}$$ and $$\mathbf{v}$$ from some real or complex vector space. The vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ are linearly dependent if and only if at least one of the following is true: If $$\mathbf{u} = \mathbf{0}$$ then by setting $$c := 0$$ we have $$c \mathbf{v} = 0 \mathbf{v} = \mathbf{0} = \mathbf{u}$$ (this equality holds no matter what the value of $$\mathbf{v}$$ is), which shows that (1) is true in this particular case. Similarly, if $$\mathbf{v} = \mathbf{0}$$ then (2) is true because $$\mathbf{v} = 0 \mathbf{u}.$$ If $$\mathbf{u} = \mathbf{v}$$ (for instance, if they are both equal to the zero vector $$\mathbf{0}$$) then both (1) and (2) are true (by using $$c := 1$$ for both).
 * 1) $$\mathbf{u}$$ is a scalar multiple of $$\mathbf{v}$$ (explicitly, this means that there exists a scalar $$c$$ such that $$\mathbf{u} = c \mathbf{v}$$) or
 * 2) $$\mathbf{v}$$ is a scalar multiple of $$\mathbf{u}$$ (explicitly, this means that there exists a scalar $$c$$ such that $$\mathbf{v} = c \mathbf{u}$$).

If $$\mathbf{u} = c \mathbf{v}$$ then $$\mathbf{u} \neq \mathbf{0}$$ is only possible if $$c \neq 0$$ and $$\mathbf{v} \neq \mathbf{0}$$; in this case, it is possible to multiply both sides by $\frac{1}{c}$ to conclude $\mathbf{v} = \frac{1}{c} \mathbf{u}.$ This shows that if $$\mathbf{u} \neq \mathbf{0}$$ and $$\mathbf{v} \neq \mathbf{0}$$ then (1) is true if and only if (2) is true; that is, in this particular case either both (1) and (2) are true (and the vectors are linearly dependent) or else both (1) and (2) are false (and the vectors are linearly independent). If $$\mathbf{u} = c \mathbf{v}$$ but instead $$\mathbf{u} = \mathbf{0}$$ then at least one of $$c$$ and $$\mathbf{v}$$ must be zero. Moreover, if exactly one of $$\mathbf{u}$$ and $$\mathbf{v}$$ is $$\mathbf{0}$$ (while the other is non-zero) then exactly one of (1) and (2) is true (with the other being false).

The vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ are linearly independent if and only if $$\mathbf{u}$$ is not a scalar multiple of $$\mathbf{v}$$ and $$\mathbf{v}$$ is not a scalar multiple of $$\mathbf{u}$$.

Vectors in R2
Three vectors: Consider the set of vectors $$\mathbf{v}_1 = (1, 1),$$ $$\mathbf{v}_2 = (-3, 2),$$ and $$\mathbf{v}_3 = (2, 4),$$ then the condition for linear dependence seeks a set of non-zero scalars, such that
 * $$a_1 \begin{bmatrix} 1\\1\end{bmatrix} + a_2 \begin{bmatrix} -3\\2\end{bmatrix} + a_3 \begin{bmatrix} 2\\4\end{bmatrix} =\begin{bmatrix} 0\\0\end{bmatrix},$$

or
 * $$\begin{bmatrix} 1 & -3 & 2 \\ 1 & 2 & 4 \end{bmatrix}\begin{bmatrix} a_1\\ a_2 \\ a_3 \end{bmatrix}= \begin{bmatrix} 0\\0\end{bmatrix}.$$

Row reduce this matrix equation by subtracting the first row from the second to obtain,
 * $$\begin{bmatrix} 1 & -3 & 2 \\ 0 & 5 & 2 \end{bmatrix}\begin{bmatrix} a_1\\ a_2 \\ a_3 \end{bmatrix}= \begin{bmatrix} 0\\0\end{bmatrix}.$$

Continue the row reduction by (i) dividing the second row by 5, and then (ii) multiplying by 3 and adding to the first row, that is
 * $$\begin{bmatrix} 1 & 0 & 16/5 \\ 0 & 1 & 2/5 \end{bmatrix}\begin{bmatrix} a_1\\ a_2 \\ a_3 \end{bmatrix}= \begin{bmatrix} 0\\0\end{bmatrix}.$$

Rearranging this equation allows us to obtain
 * $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} a_1\\ a_2 \end{bmatrix}= \begin{bmatrix} a_1\\ a_2 \end{bmatrix}=-a_3\begin{bmatrix} 16/5\\2/5\end{bmatrix}.$$

which shows that non-zero ai exist such that $$\mathbf{v}_3 = (2, 4)$$ can be defined in terms of $$\mathbf{v}_1 = (1, 1)$$ and $$\mathbf{v}_2 = (-3, 2).$$ Thus, the three vectors are linearly dependent.

Two vectors: Now consider the linear dependence of the two vectors $$\mathbf{v}_1 = (1, 1)$$ and $$\mathbf{v}_2 = (-3, 2),$$ and check,
 * $$a_1 \begin{bmatrix} 1\\1\end{bmatrix} + a_2 \begin{bmatrix} -3\\2\end{bmatrix} =\begin{bmatrix} 0\\0\end{bmatrix},$$

or
 * $$\begin{bmatrix} 1 & -3 \\ 1 & 2  \end{bmatrix}\begin{bmatrix} a_1\\ a_2 \end{bmatrix}= \begin{bmatrix} 0\\0\end{bmatrix}.$$

The same row reduction presented above yields,
 * $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} a_1\\ a_2 \end{bmatrix}= \begin{bmatrix} 0\\0\end{bmatrix}.$$

This shows that $$a_i = 0,$$ which means that the vectors $$\mathbf{v}_1 = (1, 1)$$ and $$\mathbf{v}_2 = (-3, 2)$$ are linearly independent.

Vectors in R4
In order to determine if the three vectors in $$\mathbb{R}^4,$$
 * $$\mathbf{v}_1= \begin{bmatrix}1\\4\\2\\-3\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}7\\10\\-4\\-1\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}-2\\1\\5\\-4\end{bmatrix}.$$

are linearly dependent, form the matrix equation,

 * $$\begin{bmatrix}1&7&-2\\4& 10& 1\\2&-4&5\\-3&-1&-4\end{bmatrix}\begin{bmatrix} a_1\\ a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix}0\\0\\0\\0\end{bmatrix}.$$

Row reduce this equation to obtain,
 * $$\begin{bmatrix} 1& 7 & -2 \\ 0& -18& 9\\ 0 & 0 & 0\\ 0& 0& 0\end{bmatrix} \begin{bmatrix} a_1\\ a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix}0\\0\\0\\0\end{bmatrix}.$$

Rearrange to solve for v3 and obtain,
 * $$\begin{bmatrix} 1& 7 \\ 0& -18 \end{bmatrix} \begin{bmatrix} a_1\\ a_2 \end{bmatrix} =  -a_3\begin{bmatrix}-2\\9\end{bmatrix}.$$

This equation is easily solved to define non-zero ai,
 * $$a_1 = -3 a_3 /2, a_2 = a_3/2,$$

where $$a_3$$ can be chosen arbitrarily. Thus, the vectors $$\mathbf{v}_1, \mathbf{v}_2,$$ and $$\mathbf{v}_3$$ are linearly dependent.

Alternative method using determinants
An alternative method relies on the fact that $$n$$ vectors in $$\mathbb{R}^n$$ are linearly independent if and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero.

In this case, the matrix formed by the vectors is
 * $$A = \begin{bmatrix}1&-3\\1&2\end{bmatrix} .$$

We may write a linear combination of the columns as
 * $$A \Lambda = \begin{bmatrix}1&-3\\1&2\end{bmatrix} \begin{bmatrix}\lambda_1 \\ \lambda_2 \end{bmatrix} .$$

We are interested in whether $AΛ = 0$ for some nonzero vector Λ. This depends on the determinant of $$A$$, which is
 * $$\det A = 1\cdot2 - 1\cdot(-3) = 5 \ne 0.$$

Since the determinant is non-zero, the vectors $$(1, 1)$$ and $$(-3, 2)$$ are linearly independent.

Otherwise, suppose we have $$m$$ vectors of $$n$$ coordinates, with $$m < n.$$ Then A is an n×m matrix and Λ is a column vector with $$m$$ entries, and we are again interested in AΛ = 0. As we saw previously, this is equivalent to a list of $$n$$ equations. Consider the first $$m$$ rows of $$A$$, the first $$m$$ equations; any solution of the full list of equations must also be true of the reduced list. In fact, if $⟨i_{1},...,i_{m}⟩$ is any list of $$m$$ rows, then the equation must be true for those rows.
 * $$A_{\lang i_1,\dots,i_m \rang} \Lambda = \mathbf{0} .$$

Furthermore, the reverse is true. That is, we can test whether the $$m$$ vectors are linearly dependent by testing whether
 * $$\det A_{\lang i_1,\dots,i_m \rang} = 0$$

for all possible lists of $$m$$ rows. (In case $$m = n$$, this requires only one determinant, as above. If $$m > n$$, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.

More vectors than dimensions
If there are more vectors than dimensions, the vectors are linearly dependent. This is illustrated in the example above of three vectors in $$\R^2.$$

Natural basis vectors
Let $$V = \R^n$$ and consider the following elements in $$V$$, known as the natural basis vectors:

 * $$\begin{matrix}

\mathbf{e}_1 & = & (1,0,0,\ldots,0) \\ \mathbf{e}_2 & = & (0,1,0,\ldots,0) \\ & \vdots \\ \mathbf{e}_n & = & (0,0,0,\ldots,1).\end{matrix}$$

Then $$\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n$$ are linearly independent.

$$

Linear independence of functions
Let $$V$$ be the vector space of all differentiable functions of a real variable $$t$$. Then the functions $$e^t$$ and $$e^{2t}$$ in $$V$$ are linearly independent.

Proof
Suppose $$a$$ and $$b$$ are two real numbers such that

 * $$ae ^ t + be ^ {2t} = 0$$

Take the first derivative of the above equation:

 * $$ae ^ t + 2be ^ {2t} = 0$$

for values of $$t.$$ We need to show that $$a = 0$$ and $$b = 0.$$ In order to do this, we subtract the first equation from the second, giving $$be^{2t} = 0$$. Since $$e^{2t}$$ is not zero for some $$t$$, $$b=0.$$ It follows that $$a = 0$$ too. Therefore, according to the definition of linear independence, $$e^{t}$$ and $$e^{2t}$$ are linearly independent.

Space of linear dependencies
A linear dependency or linear relation among vectors $v_{1}, ..., v_{n}$ is a tuple $(a_{1}, ..., a_{n})$ with $n$ scalar components such that

 * $$a_1 \mathbf{v}_1 + \cdots + a_n \mathbf{v}_n= \mathbf{0}.$$

If such a linear dependence exists with at least a nonzero component, then the $n$ vectors are linearly dependent. Linear dependencies among $v_{1}, ..., v_{n}$ form a vector space.

If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous system of linear equations, with the coordinates of the vectors as coefficients. A basis of the vector space of linear dependencies can therefore be computed by Gaussian elimination.

Affine independence
A set of vectors is said to be affinely dependent if at least one of the vectors in the set can be defined as an affine combination of the others. Otherwise, the set is called affinely independent. Any affine combination is a linear combination; therefore every affinely dependent set is linearly dependent. Conversely, every linearly independent set is affinely independent.

Consider a set of $$m$$ vectors $$\mathbf{v}_1, \ldots, \mathbf{v}_m$$ of size $$n$$ each, and consider the set of $$m$$ augmented vectors $\left(\left[\begin{smallmatrix} 1 \\ \mathbf{v}_1\end{smallmatrix}\right], \ldots, \left[\begin{smallmatrix}1 \\ \mathbf{v}_m\end{smallmatrix}\right]\right)$ of size $$n + 1$$ each. The original vectors are affinely independent if and only if the augmented vectors are linearly independent.

Linearly independent vector subspaces
Two vector subspaces $$M$$ and $$N$$ of a vector space $$X$$ are said to be if $$M \cap N = \{0\}.$$ More generally, a collection $$M_1, \ldots, M_d$$ of subspaces of $$X$$ are said to be if $M_i \cap \sum_{k \neq i} M_k = \{0\}$  for every index $$i,$$ where $\sum_{k \neq i} M_k = \Big\{m_1 + \cdots + m_{i-1} + m_{i+1} + \cdots + m_d : m_k \in M_k \text{ for all } k\Big\} = \operatorname{span} \bigcup_{k \in \{1,\ldots,i-1,i+1,\ldots,d\}} M_k.$ The vector space $$X$$ is said to be a of $$M_1, \ldots, M_d$$ if these subspaces are linearly independent and $$M_1 + \cdots + M_d = X.$$

title: Linear independence


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List of cities in North Korea

The important cities of North Korea have self-governing status equivalent to that of provinces. Pyongyang, the largest city and capital, is classified as a chikhalsi (capital city), while three cities (see the list below) are classified as t'ŭkpyŏlsi (special city). Other cities are classified as si (city) and are under provincial jurisdiction, at the same level as counties (see Administrative divisions of North Korea).

List

 * Notes:
 * All population figures come from the 2008 North Korean census.
 * Several former special cities have been re-merged with their provinces, including Chongjin, Hamhung and Kaesong.
 * Rason was annexed into North Hamgyong Province in 2004, but was later promoted back to special city in 2010 to help manage it for foreign investment.
 * Chosŏn'gŭl has replaced Hancha; Hancha has not been officially used in North Korea since the 1950s.

(Note: foundation dates are the dates the cities were legally founded as their current status by the North Korean government. They all existed as prior settlements before these dates.)

title: List of cities in North Korea


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Jacques Prévert

Jacques Prévert (4 February 1900 – 11 April 1977) was a French poet and screenwriter. His poems became and remain popular in the French-speaking world, particularly in schools. His best-regarded films formed part of the poetic realist movement, and include Les Enfants du Paradis (1945). He published his first book in 1946.

Life and education
Prévert was born in Neuilly-sur-Seine and grew up in Paris. After receiving his Certificat d'études upon completing his primary education, he quit school and went to work in Le Bon Marché, a major department store in Paris. In 1918, he was called up for military service in the First World War. After this, he was sent to the Near East to defend French interests there.

He died of lung cancer in Omonville-la-Petite, on 11 April 1977. He had been working on the last scene of the animated movie Le Roi et l'Oiseau (The King and the Mockingbird) with his friend and collaborator Paul Grimault. When the film was released in 1980, it was dedicated to Prévert's memory, and on opening night, Grimault kept the seat next to him empty. His dog Auto was given to a family friend after his death.

Poetry
When Prévert was attending primary school, he at first hated writing. Later, he participated actively in the Surrealist movement. Together with the writers Raymond Queneau and Marcel Duhamel, he was a member of the Rue du Château group. He was also a member of the agitprop theater company Groupe Octobre where he helped craft a left-wing cinema in support of the causes of the Popular Front. Prévert remained supportive of left-wing causes throughout his life. In 1971, he wrote a poem in support of the communist Angela Davis after her arrest.

Prévert's poems were collected and published in his books: Paroles (Words) (1946), Spectacle (1951), La Pluie et le beau temps (Rain and Good Weather) (1955), Histoires (Stories) (1963), Fatras (1971) and Choses et autres (Things and Others) (1973). His poems are often about life in Paris and life after the Second World War. They are widely taught in schools in France, and frequently appear in French language textbooks published worldwide. Some, such as "Déjeuner du Matin", are also often taught in American upper-level French classes, for the students to learn basics.

Some of Prévert's poems, such as "Les feuilles mortes" ("Autumn Leaves"), "L'Addition", "La grasse matinée" ("Sleeping in"), "Les bruits de la nuit" ("The sounds of the night") and "Chasse à l'enfant" ("The hunt for the child") were set to music by Joseph Kosma—and in some cases by Germaine Tailleferre of Les Six, Christiane Verger, Marjo Tal, and Hanns Eisler. They have been sung by prominent French vocalists, including Marianne Oswald, Yves Montand, and Édith Piaf, as well as by the later American singers Joan Baez and Nat King Cole. In 1961, French singer-songwriter Serge Gainsbourg paid tribute to "Les feuilles mortes" in his own song "La chanson de Prévert". "Les feuilles mortes" was also translated into German by the German poet and Liedermacher (singer-songwriter) Wolf Biermann, titled "Welke Blätter", and was performed by him and others.

The British remix DJs Coldcut released their own version in 1993. Another German version has been published and covered by Didier Caesar (alias Dieter Kaiser), which he named "Das welke Laub". "Les feuilles mortes" also bookends Iggy Pop's 2009 album Préliminaires.

Prévert's poems are translated into many languages worldwide. Many translators have translated his poems into English. The poet and translator Suman Pokhrel has translated some of his poems into Nepali.

Films
Prévert wrote a number of screenplays for the film director Marcel Carné. Among them were the scripts for Drôle de drame (Bizarre, Bizarre, 1937), Quai des brumes (Port of Shadows, 1938), Le Jour se lève (Daybreak, 1939), Les Visiteurs du soir (The Night Visitors, 1942) and Children of Paradise (Les Enfants du Paradis, 1945). The last of these regularly gains a high placing in lists of best films ever and earned him an Oscar nomination for best original screenplay.

His poems were the basis for a film by the director and documentarian Joris Ivens, The Seine Meets Paris (La Seine a rencontré Paris, 1957), about the River Seine. The poem was read as narration during the film by singer Serge Reggiani. In 2007, a filmed adaptation of Prévert's poem "To Paint the Portrait of a Bird" was directed by Seamus McNally, featuring T.D. White and Antoine Ray- English translation by Lawrence Ferlinghetti.

Prévert had a long working relationship with Paul Grimault, also a member of Groupe Octobre. Together they wrote the screenplays of a number of animated movies, starting with the short "The Little Soldier" ("Le Petit Soldat", 1947). They worked together until his death in 1977, when he was finishing The King and the Mocking Bird (Le Roi et l'Oiseau), a second version of which was released in 1980. Prévert adapted several Hans Christian Andersen tales into animated or mixed live-action/animated movies, often in versions loosely connected to the original. Two of these were with Grimault, including The King and the Mocking Bird, while another was with his brother Pierre Prévert (fr).

Selected filmography
Prévert wrote the scenarios and sometimes the dialogue in the following films:
 * Baleydier (1932)
 * Ciboulette (1933)
 * Hotel Free Exchange (1934)
 * If I Were Boss (1934)
 * Le Crime de monsieur Lange (1936)
 * 27 Rue de la Paix (1936)
 * Moutonnet (1936)
 * Drôle de drame (1937)
 * Quai des brumes (1938)
 * Ernest the Rebel (1938)
 * Les Disparus de Saint-Agil (1938) (fr)
 * Le Jour se lève (1939)
 * The Mysterious Mr. Davis (1939)
 * Remorques (1941)
 * Les Visiteurs du soir (1942)
 * A Woman in the Night (1943)
 * Summer Light (1943)
 * Goodbye Leonard (1943)
 * Les Enfants du paradis (1945)
 * Les Portes de la nuit (1945)
 * The Bellman (1945)
 * Le Petit Soldat (The Little Soldier) (short animated film, 1947), with Paul Grimault, after The Steadfast Tin Soldier by Hans Christian Andersen
 * Mystery Trip (1947)
 * The Lovers of Verona (1949)
 * La Bergère et le ramoneur (The Shepherdess and the Chimney Sweep) (animated film, 1953), with Paul Grimault after tale by Hans Christian Andersen, later revised and finished as Le Roi et l'oiseau
 * The Hunchback of Notre Dame (1956)
 * Le Petit Claus et le Grand Claus (fr), by Pierre Prévert, after the tale Little Claus and Big Claus by Hans Christian Andersen (live action and animation, 1964)
 * Le diamant (The diamond) (short animated film, 1970), with Paul Grimault, complement to L'Aveu of Costa-Gavras
 * Le Chien mélomane (The Music-Loving Dog) (short animated film, 1973), with Paul Grimault
 * Le Roi et l'oiseau (animated film, 1980), with Paul Grimault

title: Jacques Prévert


text:
Talk:Bernoulli experiment

I don't feel like I'm an authority on the matter here; I just inferred that this must be right from the context of other articles, and some basic knowledge of probability. Can anyone vouch for correctness here?

-- Ryguasu

title: Talk:Bernoulli experiment