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In abstract algebra and formal logic, the **distributive property** of binary operations generalizes the **distributive law** from elementary algebra. In propositional logic, **distribution** refers to two valid rules of replacement. The rules allow one to reformulate conjunctions and disjunctions within logical proofs.

For example, in arithmetic:

In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards.
Because these give the same final answer (8), it is said that multiplication by 2 *distributes* over addition of 1 and 3.
Since one could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers *distributes* over addition of real numbers.

Given a set *S* and two binary operators ∗ and + on *S*, we say that the operation:

∗ is *left-distributive* over + if, given any elements *x*, *y*, and *z* of *S*,

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In probability and statistics, a **probability distribution** assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

In applied probability, a probability distribution can be specified in a number of different ways, often chosen for mathematical convenience:

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In mathematics and computer science, a **history monoid** is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thread joins) for providing rendezvous points between a set of independently executing processes or threads.

History monoids occur in the theory of concurrent computation, and provide a low-level mathematical foundation for process calculi, such as CSP the language of communicating sequential processes, or CCS, the calculus of communicating systems. History monoids were first presented by M.W. Shields.

History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.

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**Omaha** (/ˈoʊməhɑː/ * OH-mə-hah*) is the largest city in the state of Nebraska, United States, and is the county seat of Douglas County. It is located in the Midwestern United States on the Missouri River, about 10 miles (15 km) north of the mouth of the Platte River. Omaha is the anchor of the Omaha-Council Bluffs metropolitan area, which includes Council Bluffs, Iowa, across the Missouri River from Omaha. According to the 2010 census, Omaha's population was 408,958, making it the nation's 41st-largest city. According to the 2014 Population Estimates, Omaha's population was 446,599. Including its suburbs, Omaha formed the 60th-largest metropolitan area in the United States in 2013 with an estimated population of 895,151 residing in eight counties. The Omaha-Council Bluffs-Fremont, Nebraska-IA Combined Statistical Area is 931,667, according to the U.S. Census Bureau's 2013 estimate. There are nearly 1.3 million residents within a 50-mile (80 km) radius of the city's center, forming the Greater Omaha area.

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**Omaha** is a city in Nebraska, U.S.

**Omaha** may also refer to:

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**Omaha** (March 24, 1932 – April 24, 1959) was a United States Thoroughbred horse racing champion. In a racing career which lasted from 1934 through 1936, he ran twenty-two times and won nine races. He had his greatest success as a three-year-old in 1935, when he won the Triple Crown. As a four-year-old, he had success running in England, where he narrowly lost the Ascot Gold Cup.

Foaled at Claiborne Farm in Paris, Kentucky, Omaha was a chestnut horse with a white blaze who stood 16.3 hands high. He was the son of 1930 U.S. Triple Crown winner Gallant Fox and the mare Flambino. Omaha was the third horse to ever win the Triple Crown, which he did in 1935. Flambino also produced the Ascot Gold Cup winner Flares and was the sister of La France, the direct female ancestor of many notable thoroughbreds including Danzig Connection, Decidedly, and Johnstown.

The horse was owned by and bred William Woodward, Sr.'s famous Belair Stud in Bowie, Maryland. He was trained by Sunny Jim Fitzsimmons, who also trained Omaha's sire to the Triple Crown. As a yearling, Omaha was leggy and awkward-looking but a favorite of Woodward, who reportedly considered sending the horse to England to be trained for the Epsom Derby. In the event, Omaha's move to England was postponed until 1936. He was ridden to his biggest wins by Canadian jockey Smokey Saunders.

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