Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. The result of a subtraction is called a difference. Subtraction is signified by the minus sign. For example, in the adjacent picture, there are 5 − 2 apples—meaning 5 apples with 2 taken away, a total of 3 apples. Therefore, the difference of 5 and 2 is 3, that is, 5 − 2 = 3. Subtraction represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, irrational numbers, decimals and matrices. Subtraction follows several important patterns, it is anticommutative. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters; because 0 is the additive identity, subtraction of it does not change a number. Subtraction obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond.

General binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks. Subtraction of small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. In advanced algebra and in computer algebra, an expression involving subtraction like A − B is treated as a shorthand notation for the addition A +. Thus, A − B contains two terms, namely A and −B; this allows an easier use of commutativity. Subtraction is written using the minus sign "−" between the terms; the result is expressed with an equals sign. For example, 2 − 1 = 1 4 − 2 = 2 6 − 3 = 3 4 − 6 = − 2 There are situations where subtraction is "understood" though no symbol appears: A column of two numbers, with the lower number in red indicates that the lower number in the column is to be subtracted, with the difference written below, under a line.

This is most common in accounting. Formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend; the result is the difference. All of this terminology derives from Latin. "Subtraction" is an English word derived from the Latin verb subtrahere, in turn a compound of sub "from under" and trahere "to pull". Using the gerundive suffix -nd results in "subtrahend", "thing to be subtracted". From minuere "to reduce or diminish", one gets "minuend", "thing to be diminished". Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c; this movement to the right is modeled mathematically by addition: a + b = c. From c, it takes b steps to the left to get back to a; this movement to the left is modeled by subtraction: c − b = a. Now, a line segment labeled with the numbers 1, 2, 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3, it takes 2 steps to the left to get to position 1, so 3 − 2 = 1.

This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number. From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0, but 3 − 4 is still invalid. The natural numbers are not a useful context for subtraction; the solution is to consider the integer number line. From 3, it takes 4 steps to the left to get to −1: 3 − 4 = −1. Subtraction of natural numbers is not closed; the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number; such a case uses one of two approaches: Say that 26 cannot be subtracted from 11. Give the answer as an integer representing a negative number, so the result of subtracting 26 from 11 is −15. Subtraction of real numbers is defined as addition of signed numbers. A number is subtracted by adding its additive inverse.

We have 3 − π = 3 +. This helps to keep the ring of real numbers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it. A ring has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything. Subtraction is anti-commutative, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, if a and b are any two numbers a − b = −. Subtraction is non-associative. Should the expres

A credal set is a set of probability distributions or, more a set of probability measures. A credal set is assumed or constructed to be a closed convex set, it is intended to express uncertainty or doubt about the probability model that should be used, or to convey the beliefs of a Bayesian agent about the possible states of the world. If a credal set K is closed and convex by the Krein–Milman theorem, it can be equivalently described by its extreme points e x t. In that case, the expectation for a function f of X with respect to the credal set K forms a closed interval, whose lower bound is called the lower prevision of f, whose upper bound is called the upper prevision of f: E _ = min μ ∈ K ∫ f d μ = min μ ∈ e x t ∫ f d μ where μ denotes a probability measure, with a similar expression for E ¯. If X is a categorical variable the credal set K can be considered as a set of probability mass functions over X. If additionally K is closed and convex the lower prevision of a function f of X can be evaluated as: E _ = min p ∈ e x t ∑ x f p where p denotes a probability mass function.

It is easy to see that a credal set over a Boolean variable X cannot have more than two extreme points, while credal sets over variables X that can take three or more values can have any arbitrary number of extreme points. Imprecise probability Dempster–Shafer theory Probability box Robust Bayes analysis Upper and lower probabilities Abellán, J. N.. "Upper entropy of credal sets. Applications to credal classification". International Journal of Approximate Reasoning. 39: 235. Doi:10.1016/j.ijar.2004.10.001

The 2012 Arizona Cardinals season was the franchise's 93rd season in the National Football League and the 25th in Arizona. After a surprising 4–0 start, which included a major upset of the New England Patriots in week 2, the Cardinals lost eleven of their final twelve games, missed the playoffs for a third consecutive season, resulting in the firing of head coach Ken Whisenhunt after six seasons; this was Rod Graves last season as General Manager of the Cardinals. A day after the final game of the regular season, he was fired after 16 seasons. Note: Intra-division opponents are in bold text; the Cardinals began their season at home against their divisional rival Seahawks with QB John Skelton at the starting helm. The team started their season 1–0 despite Skelton finishing the game 14/28 for 149 yards and an interception. Kevin Kolb on made a appearance in relief of Skelton going 6/8 for 66 yards and a touchdown pass. In a huge upset at Gillette Stadium, the Arizona Cardinals beat the New England Patriots and become 2–0.

This was the Cardinals' first win against the Patriots since 1991. This was the Cardinals 700th loss, they are the first team in NFL history to do this. With the tough loss, the Cardinals fell to 4–2 and with the Seahawks' win over the Patriots and the 49ers' loss to the Giants, the team remains in a tie for first place in the NFC West. Losing their fifth straight game of the season by remaining winless at Lambeau Field, the Cardinals head into their bye week at 4–5. With that loss, the Cardinals were not only eliminated from postseason contention and falling to 4–9, but the 58-point loss was the worst in the Cardinals' franchise history, they had 8 turnovers, the most that the Cardinals have committed in a game. With this loss, the Cardinals surpassed the 1989 Chicago Bears for the worst record by an NFL team starting 4–0. Official website Media related to 2012 Arizona Cardinals season at Wikimedia Commons

Milk churn stands were once a common roadside sight in Britain in areas which carried out dairy farming. They were standard-height platforms on which milk churns would be placed for collection by cart or lorry; some were simple and made of wood. Collection of milk churns from stands ceased in Britain in 1979. Many have survived, some being renovated to memorialise the practice, while others have been dismantled or left to decay. Milk churn stands could be made of wood, or were more permanent structures built from concrete or stone blocks. Many were simple cubic structures; some had steps leading up to them, or just a foothole to reach the platform while some could be more elaborate. The simple purpose of the stand was to facilitate collection of milk churns by cart or lorry and so were built at a convenient height for easy transfer. A conical 15 imperial gallons churn weighing 20 pounds. A standard, lighter churn might contain 11 imperial gallons, of milk, weighing about 120 pounds full. Once the full churns had been removed they were replaced by the haulier with empty ones for refilling by the next collection time.

The full churns would be transported directly by road to the dairy, or indirectly by rail. The origin of the milk churn stand dates back at least into the 19th century when commercial trade in milk became widespread, dairies became larger enterprises and widespread distribution was facilitated by rail and improving road networks; when in 1979 churn collection ceased and all milk was collected by tanker, the stands were no longer needed. Many milk churn stands would have been lost during road improvement schemes owing to their proximity to the roadside but many were left in situ to decay. However, many made from more durable materials such as concrete or stone have survived and can be seen throughout the country and, indeed, in other countries; some have been renovated as reminders of the former widespread practice, while some replica stands have been erected for the same reason in stone and the reinstatement or removal of some has been the subject of planning application. Some milk churn stands have been recorded as historical monuments by regional bodies and the National Archives.

Milk churn Dairy Dairy farming

Amīr Sūrī was the king of the Ghurid dynasty from the 9th-century to the 10th-century. He was a descendant of the Ghurid king Amir Banji, whose rule was legitimized by the Abbasid caliph Harun al-Rashid. Amir Suri is known to have fought the Saffarid ruler Ya'qub ibn al-Layth al-Saffar, who managed to conquer much of Khurasan except Ghur. Amir Suri was succeeded by his son Muhammad ibn Suri. Although Amir Suri bore an Arabic title and his son had an Islamic name, they were both Buddhists and were considered pagans by the surrounding Muslim people, it was only during the reign of Muhammad's son Abu Ali ibn Muhammad that the Ghurid dynasty became an Islamic dynasty; the Ghorids were tribal people of Ghoristan mountains, divided into numerous tribes. Among the numerous Ghorid chiefs, the Shansabani tribe had the most authority over all the other Ghorid tribes; the Shansabani were a tribe and the Ghoris were structured as a tribal society. Abu'l-Fadl Bayhaqi, the famous historian of the Ghaznavid era, wrote on page 117 in his book Tarikh-i Bayhaqi: "Sultan Mas'ud left for Ghoristan and sent his learned companion with two people from Ghor as interpreters between this person and the people of that region."

C. Edmund, Bosworth. "GHURIDS". Encyclopaedia Iranica, Online Edition. Retrieved 3 May 2014. Bosworth, C. E.. "The Political and Dynastic History of the Iranian World". In Frye, R. N.. The Cambridge History of Iran, Volume 5: The Saljuq and Mongol periods. Cambridge: Cambridge University Press. Pp. 1–202. ISBN 0-521-06936-X.- Edward Balfour - Google Books

Sergiu Natra is an Israeli composer. Natra compositions include, among others, "Symphony in Red, Blue and Green for symphony orchestra", "Horizons Symphony for symphony string orchestra", "Invincible Symphony for symphony orchestra", "Memories Symphony for symphony string orchestra", "Earth and Water symphony for symphony orchestra", "Spacetime symphony for string orchestra", "March and Choral for symphonic orchestra", "Divertimento in Ancient Style for string orchestra with piano", "Festive Overture - Toccata and Fuge for orchestra", "Variations for Piano and Orchestra", he is known worldwide for his compositions for the harp, including "Music for Violin and Harp", "Sonatina for Harp", "Prayer for Harp", "Divertimento for Harp flute and Strings orchestra", "Music for Nicanor", "Commentaires Sentimentaux", "Ode To The Harp" and "Trio in One Movement no. 3". Sergiu Natra is a Romanian-born in a family originating in Austria and the Czech Republic; as a child he studied piano and music and began particular music studies in 1932, continued at the Jewish conservatory and graduated from the Music Academy of Bucharest.

He studied, among others, theory and orchestration with Leon Klepper and modern music with Michael Andricu. He began composing at an early age and his work titled "March and Choral" for symphony orchestra, earned him the status of a modernist in Romania; the Israel Philharmonic Orchestra performed this work in 1947 under the direction of Edward Lindenberg. NATRA had received many composition awards in for his creations, among which, for "March and Choral" for symphony orchestra and "Divertimento in ancient style" for symphony string orchestra, the George Enescu award for composition in 1945 and for "Suite for symphony orchestra" he received in 1951 the National prize for composition. In 1961, Natra and his wife, Sonia, a sculptor and multidisciplinary artist, emigrated to Israel. A year conducted by Sergiu Comissiona, the Israel Philharmonic Orchestra performed the "Horizons Symphony for symphony string orchestra", the last piece he had written in Romania, the "Music for violin and harp", performed by the violinist Miriam Fried and the French harpist Françoise Netter.

Besides composing music, Natra taught music, including at Tel-Aviv University, where he taught music of the 20th century and analysis of forms. He was a professor at the Music Academy in the Tel-Aviv University until 1985. Among his students were Lior Shambadal, Rafi Kadishson, Erel Paz, Ruben Seroussi, Deborah Rothstein Schramm, Dror Elimelech, Yehonatan Berick, Sally Pinkas, Eugene Alcalay, Sivan Silver and Gil Garburg, Dr. Eran Lupu, Yoni Farhi. See: List of music students by teacher: N to Q#Sergiu Natra. Natra and his wife Sonia, have two sons and Gabi Natra is a composer with a clear European orientation, who has a clear personal stamp and a particular writing style with melodic flow, atonal language, polyphonic idea, gradual development and shaping of motive material, he makes use of a rich palette of sound-colors, unusual instrumental combinations, central registers of instruments, playing techniques which are natural and comfortable and succeed in producing optimal sound, texts in a new language, with its fresh rhythms and sonorities.

An extended list of works and their respective audio and video recordings is to be found at Natra's site Divertimento in Ancient Style for symphony string orchestra.