probability theory: Measure theory: …only the weaker axiom of finite additivity, but the absence of interesting models that fail to satisfy the axiom of countable additivity has led to its virtually universal acceptance.

Michael Oser Rabin: …their early joint paper “Finite Automata and Their Decision Problem,” which has had a lasting impact on the field of automata theory, and for their subsequent independent work.

Judaism: Ḥasdai Crescas: …between an infinite being and finite beings. It is through infinitude that God’s essential attributes—wisdom, for instance—differ from the corresponding and otherwise similar attributes found in created beings. In Crescas’s doctrine, as in that of Spinoza, God’s attributes are infinite in number. The central place assigned to the doctrine of…

mechanics of solids: Small-strain tensor: …are instead best approached through finite deformation theory.

numerical analysis: Solving differential and integral equations: …numerical procedures are often called finite difference methods. Most initial value problems for ordinary differential equations and partial differential equations are solved in this way. Numerical methods for solving differential and integral equations often involve both approximation theory and the solution of quite large linear and nonlinear systems of equations.

mechanics of solids: Computational mechanics: …most common version of the finite-element method, the domain to be analyzed is divided into cells, or elements, and the displacement field within each element is interpolated in terms of displacements at a few points around the element boundary (and sometimes within it) called nodes. The interpolation is done so…

combinatorics: BIB (balanced incomplete block) designs: …depend on the use of finite fields, finite geometries, and number theory. Some general methods were given in 1939 by the Indian mathematician Raj Chandra Bose, who has since emigrated to the United States.

game theory: Classification of games: …game is said to be finite when each player has a finite number of options, the number of players is finite, and the game cannot go on indefinitely. Chess, checkers, poker, and most parlour games are finite. Infinite games are more subtle and will only be touched upon in this…

Christianity: 20th-century discussions: The concept of a finite deity developing through time was also proposed (e.g., by Charles Hartshorne) to meet objections to some of these concepts: If God is immutable, how can God be aware of successive events in time? If God has absolute self-existence, how can God respond with sympathy…

Burnside problem: …order must necessarily be a finite group. The problem was formulated by the English mathematician William Burnside in 1902.

numerical analysis: Common perspectives in numerical analysis: …in the effects of using finite precision computer arithmetic. This is especially important in numerical linear algebra, as large problems contain many rounding errors. Numerical analysts are generally interested in measuring the efficiency (or “cost”) of an algorithm. For example, the use of Gaussian elimination to solve a linear system…

Georg Cantor: Set theory: …agreed that a set, whether finite or infinite, is a collection of objects (e.g., the integers, {0, ±1, ±2,…}) that share a particular property while each object retains its own individuality. But when Cantor applied the device of the one-to-one correspondence (e.g., {a, b, c} to {1, 2, 3}) to…

mechanics of solids: Strain and strain-displacement relations: …magnitude; such expressions are called finite strain.

automata theory: Finite transducers: The most important transducers are the finite transducers, or sequential machines, which may be characterized as one-way Turing machines with output. They are the weakest with respect to computing power, while the universal machine is the most powerful. There are also transducers of…

Burnside problem: …problem of determining if a finitely generated periodic group with each element of finite order must necessarily be a finite group. The problem was formulated by the English mathematician William Burnside in 1902.

foundations of mathematics: Intuitionistic logic: …the more extreme position of finitism. According to this view, which goes back to Aristotle, infinite sets do not exist, except potentially. In fact, it is precisely in the presence of infinite sets that intuitionists drop the classical principle of the excluded third.

Silvio Berlusconi: Early life and first term as prime minister: …under the umbrella of the Fininvest holding company, a vast conglomerate that grew to control more than 150 businesses.

Albert Fink: …also the inventor of the Fink truss, used to support bridges and the roofs of buildings.

Albert Fink was a German-born American railroad engineer and executive who was the first to investigate the economics of railroad operation on a systematic basis. He was also the inventor of the Fink truss, used to support bridges and the roofs of buildings. Educated in Germany, Fink immigrated to

Diane Ackerman is an American writer whose works often reflect her interest in natural science. Ackerman was educated at Pennsylvania State University (B.A., 1970) and Cornell University, Ithaca, N.Y. (M.F.A., 1973; M.A., 1976; Ph.D., 1978). From 1980 to 1983 she taught English at the University of

phenomenology: Other developments: Eugen Fink, for several years Husserl’s collaborator, whose essay “Die phänomenologische Philosophie Edmund Husserls in der gegenwärtigen Kritik” (1933) led to a radicalization of Husserl’s philosophical, transcendental idealism, later turned in another direction, one that approached Heidegger’s position and divorced itself at the same time…

Janis Ian is an American singer, songwriter, and musician who wrote multiple popular soft rock songs in the late 1960s and ’70s, most notably “Society’s Child (Baby I’ve Been Thinking),” the story of a white girl pressured by her family and society to relinquish her Black boyfriend. Ian recorded

Kay Thompson was an American entertainer and writer who was best known as the author of the highly popular Eloise books, featuring a comically endearing enfant terrible who bedeviled New York City’s Plaza Hotel. Thompson early displayed a considerable talent for the piano, and at the age of 16 she

Kay Thompson was an American entertainer and writer who was best known as the author of the highly popular Eloise books, featuring a comically endearing enfant terrible who bedeviled New York City’s Plaza Hotel. Thompson early displayed a considerable talent for the piano, and at the age of 16 she

Few people in the financial industry are as well-known as Larry Fink, the cofounder, chair, and chief executive officer of BlackRock (BLK), the world’s largest money manager. With more than $14 trillion in assets under management as of 2025, BlackRock holds stakes in thousands of companies through

Mike Fink was an American keelboatman of the Old West, who became the legendary hero of the American tall tale. As a youth Fink won fame as a marksman and Indian scout around Fort Pitt. Later, when keelboats became the chief vessels of commerce on the Ohio and Mississippi rivers, he became “the

Theodore Fink was an Australian politician and publisher, noted for his interests in education. Fink was brought to Australia as a child (1861), studied at local schools in Melbourne, and then read law at Melbourne University, becoming a solicitor in 1877. He became prosperous, sat on the Victorian

Finke River, major but intermittent river of central Australia that rises south of Mount Ziel in the MacDonnell Ranges of south-central Northern Territory. The Finke passes through Glen Helen Gorge and Palm Valley and then meanders generally southeast over the Missionary Plain. Entering a 40-mile

Cyd Charisse was an American dancer and actress known for her glamorous looks and sensual technically flawless dancing in 1950s movie musicals. Most notable were her appearances with actor Gene Kelly in the highly acclaimed musical film Singin’ in the Rain (1952) and with actor Fred Astaire in the

William Zorach was a traditionalist sculptor of simple, figurative subjects who was a leading figure in the early 20th-century revival of direct carving, whereby the sculptor seeks an image directly from the material to be carved, relying on neither the inspiration of models nor the aid of

Finland, country located in northern Europe. Finland is one of the world’s most northern and geographically remote countries and is subject to a severe climate. Nearly two-thirds of Finland is blanketed by thick woodlands, making it the most densely forested country in Europe. Finland forms a

St. Petersburg: Vyborg Side: …most famous features is the Finland Railway Station, which faces the Admiralty Side across the Neva. Lenin returned to Russia in April 1917 via this station, and there he made his initial pronouncement of a new course that would bring the Bolsheviks to power. A major street of the Vyborg…

Finland: Finance: The Bank of Finland (Suomen Pankki), established in 1811 and guaranteed and supervised by the parliament since 1868, is the country’s central bank and a member of the European System of Central Banks. In 2002, the EU’s common currency, the euro, replaced the markka, which had…

Church of Finland, national church of Finland, which changed from the Roman Catholic to the Lutheran faith during the Protestant Reformation in the 16th century. Christianity was known in Finland as early as the 11th century, and in the 12th century Henry, bishop of Uppsala (Sweden), began

In the 16th century the grand duchy of Finland acquired a coat of arms of its own. Its red shield bore a rampant yellow lion with an armoured human arm holding a sword; white roses were scattered around the field. When Finland passed from Swedish rule to Russian in 1809, it lost much of its

Gulf of Finland, easternmost arm of the Baltic Sea, between Finland (north) and Russia and Estonia (east and south, respectively). Covering an area of 11,600 square miles (30,000 square km), the gulf extends for 250 miles (400 km) from east to west but only 12 to 80 miles (19 to 130 km) from north

history of Finland, a survey of important events and people in the history of Finland from the time of its settlement. One of the world’s most northern and geographically remote countries, Finland forms a symbolic northern border between western and eastern Europe: dense wilderness and Russia to

Orthodox Church of Finland, Eastern Orthodox church, recognized as the second state church of Finland. Most of the Orthodox Finns were originally from Karelia, the southeastern part of Finland that was annexed by the Soviet Union in 1940, which was Christianized by Russian monks in the 12th

Finland, country located in northern Europe. Finland is one of the world’s most northern and geographically remote countries and is subject to a severe climate. Nearly two-thirds of Finland is blanketed by thick woodlands, making it the most densely forested country in Europe. Finland forms a

Finland, country located in northern Europe. Finland is one of the world’s most northern and geographically remote countries and is subject to a severe climate. Nearly two-thirds of Finland is blanketed by thick woodlands, making it the most densely forested country in Europe. Finland forms a

Finlandia, tone poem for orchestra by Jean Sibelius, the best-known of his works. It was composed in 1899 and premiered in the composer’s native Finland, reaching an international audience the following year. The central melody is sometimes sung—with words not original to Sibelius—as the hymn “Be

Mackenzie River: From the headwaters of the Finlay River, which flows into Williston Lake (the impounded waters of the Peace River) west of the Rocky Mountains, the entire river system runs for 2,635 miles (4,241 km) through the lake-strewn Canadian north to empty into the cold and often-frozen waters of the Beaufort…

Carlos J. Finlay was a Cuban epidemiologist who discovered that yellow fever is transmitted from infected to healthy humans by a mosquito. Although he published experimental evidence of this discovery in 1886, his ideas were ignored for 20 years. A graduate of Jefferson Medical College,

Carlos J. Finlay was a Cuban epidemiologist who discovered that yellow fever is transmitted from infected to healthy humans by a mosquito. Although he published experimental evidence of this discovery in 1886, his ideas were ignored for 20 years. A graduate of Jefferson Medical College,

George Finlay was a British historian and participant in the War of Greek Independence (1821–32) who is known principally for his histories of Greece and the Byzantine Empire. After attending the University of Glasgow, Finlay spent two years studying Roman law at the University of Göttingen but

porpoise: The finless porpoises (Neophocaena phocaenoides and N. asiaeorientalis) are small slow-moving inhabitants of coastal waters and rivers along the Indian Ocean and western Pacific Ocean. Black above and white below, finless porpoises have a rounded head. Unlike other porpoises, they lack a dorsal fin entirely. Finless…

Thomas K. Finletter was an American lawyer and government official whose policy recommendations reshaped the United States military during the Cold War. A corporate lawyer by profession, Finletter frequently interrupted his practice to hold government posts. Before the U.S. entry into World War II,

Thomas K. Finletter was an American lawyer and government official whose policy recommendations reshaped the United States military during the Cold War. A corporate lawyer by profession, Finletter frequently interrupted his practice to hold government posts. Before the U.S. entry into World War II,

Charlie Finley was an American insurance executive and professional baseball club owner who was frequently involved in controversy with the commissioner of baseball, the American League, managers, and players. His Oakland Athletics won three consecutive World Series (1972–74). Finley was a farm boy

Charlie Finley was an American insurance executive and professional baseball club owner who was frequently involved in controversy with the commissioner of baseball, the American League, managers, and players. His Oakland Athletics won three consecutive World Series (1972–74). Finley was a farm boy

Martha Finley was a prolific and, in her day, immensely popular American writer of children’s books about pious youngsters rewarded for their virtue. In 1853, after the deaths of both her parents, Finley moved to New York City; later she moved to Philadelphia and then to Phoenixville, Pennsylvania.

Dallas Mavericks: The Dirk Nowitzki era: …by Nash, Nowitzki, and sharpshooter Michael Finley.

Finn, legendary Irish hero, leader of the group of warriors known as the Fianna Éireann. See Fenian

Finland: Ethnic groups: As other groups began to enter Finland some 3,000 years later, the proto-Sami probably retreated northward. Archaeological remains suggest that this second wave of settlers came from or had contact with what was to become Russia and also Scandinavia and central Europe. Peoples of Uralic (specifically Finno-Ugric)…

Finn, legendary Irish hero, leader of the group of warriors known as the Fianna Éireann. See Fenian

Finn, legendary Irish hero, leader of the group of warriors known as the Fianna Éireann. See Fenian

Huckleberry Finn, one of the enduring characters in American fiction, the protagonist of Mark Twain’s Huckleberry Finn (1884), who was introduced in Tom Sawyer (1876). Huck, as he is best known, is an uneducated, superstitious boy, the son of the town drunkard. Although he sometimes is deceived by

Huckleberry Finn, one of the enduring characters in American fiction, the protagonist of Mark Twain’s Huckleberry Finn (1884), who was introduced in Tom Sawyer (1876). Huck, as he is best known, is an uneducated, superstitious boy, the son of the town drunkard. Although he sometimes is deceived by

T. Rex: Formation and early albums: …band, replacing him with percussionist Mickey Finn. The new duo released A Beard of Stars (1970) as Tyrannosaurus Rex. The album reflected Bolan’s transition toward an electric sound.

River Finn, river in County Donegal, Ireland, rising at Lough (lake) Finn and flowing eastward across the county for about 30 miles (48 km) before uniting with the River Mourne at Strabane to form a ribbon of continuous agricultural settlement in the mountain heartland of Donegal. The main

Icelandic literature: Prose: Among them are the Finnboga saga ramma (“Saga of Finnbogi the Strong”), about a 10th-century hero, and a saga that tells the love story of its hero Víglundr. Sagas about bishops, already a theme in the 13th century, became more numerous, as did lives of foreign saints. A large…

Vigdís Finnbogadóttir is an Icelandic teacher, cultural figure, and politician who served as president of Iceland from 1980 to 1996. She was the first woman in the world to be elected head of state in a national election. Finnbogadóttir was born into a wealthy and well-connected family. Her mother

Billie Eilish: Early life: …who performs under the moniker FINNEAS, enlisted her to sing “Ocean Eyes,” a song he had written and produced for his band at the time and later put online. The song quickly gained success, and Eilish signed with the boutique label Darkroom in the summer of 2016.

cephalopod: Annotated classification: Suborder Palaeoctopoda (finned octopod) Cretaceous, some living. Suborder Cirrata (Cirromorpha) Holocene; soft-bodied, deep-webbed forms with cirri on arms and small to large paddle-shaped fins; primarily deep-sea. Suborder Incirrata (common octopus)

Finnegans Wake, experimental novel by James Joyce. Extracts of the work appeared as Work in Progress from 1928 to 1937, and it was published in its entirety as Finnegans Wake in 1939. Finnegans Wake is a complex novel that blends the reality of life with a dream world. The motive idea of the novel,

Albert Finney was an English actor noted for his versatility. Finney established himself as a Shakespearean actor in the late 1950s. In 1960 he won praise in the roles of working-class rebels in the play Billy Liar and the film Saturday Night and Sunday Morning. Taking on additional leading parts,

Charles Grandison Finney was an American lawyer, president of Oberlin College, and a central figure in the religious revival movement of the early 19th century; he is sometimes called the first of the professional evangelists. After teaching school briefly, Finney studied law privately and entered

Finno-Ugric languages: …classified together as the Volga-Finnic group of languages. Also, because the dialects of Sami are almost mutually unintelligible, they are often classified as separate languages.

Finnic peoples, descendants of a collection of tribal peoples speaking closely related languages of the Finno-Ugric family who migrated to the area of the eastern Baltic, Finland, and Karelia before ad 400—probably between 100 bc and ad 100, though some authorities place the migration many

shark: Shark finning: Among the threats from humans that sharks face is finning, the practice of harvesting the lateral and dorsal fins and the lower tail fin from a shark by commercial fishing operations and others worldwide. After the shark has been captured and its fins have…

philosophy of law: John Finnis: John Finnis took a more-ambitious philosophical tack against positivism than Dworkin did. He argued that any theory of a social phenomenon, including law, must identify its “central” cases, since the goal of any theory is to describe the central or important features of…

Finland: Ethnic groups: As other groups began to enter Finland some 3,000 years later, the proto-Sami probably retreated northward. Archaeological remains suggest that this second wave of settlers came from or had contact with what was to become Russia and also Scandinavia and central Europe. Peoples of Uralic (specifically Finno-Ugric)…

Finland: Media and publishing: The state-run Finnish Broadcasting Company (Yleisradio Oy [YLE]; established 1926) operates a number of nationwide television networks—both public service and commercial—along with several digital channels and offers programming in Swedish. YLE also owns Radio Finland, which broadcasts in Finnish, Swedish, English, and Russian. Jointly owned by Finland,…

Finland: Agrarian reform: …the Agrarian Party (now the Centre Party), have been a major factor in Finnish politics.

Finland: Early independence: …a small contingent founded the Finnish Communist Party in Moscow; others continued their flight to the United States and western Europe, some gradually returning to Finland.

Finnish language, member of the Finno-Ugric group of the Uralic language family, spoken in Finland. At the beginning of the 19th century, Finnish had no official status, with Swedish being used in Finnish education, government, and literature. The publication in 1835 of the Kalevala, a national

Finnish literature, the oral and written literature produced in Finland in the Finnish, Swedish, and, during the Middle Ages, Latin languages. The history of Finnish literature and that of Swedish literature are intertwined. From the mid-12th century until 1809, Finland was ruled by Sweden, and

Kaarlo Bergbom: …first stable Finnish-language theatre, the Finnish National Theatre. Bergbom, himself the author of a romantic tragedy, directed the first performance of Aleksis Kivi’s one-act biblical drama Lea (1869), the event cited as the beginning of professional theatre in the Finnish language.

Orthodox Church of Finland, Eastern Orthodox church, recognized as the second state church of Finland. Most of the Orthodox Finns were originally from Karelia, the southeastern part of Finland that was annexed by the Soviet Union in 1940, which was Christianized by Russian monks in the 12th

Finlandia: It was written for the Finnish Press Pension Celebration of 1899, a thinly veiled rally in support of freedom of the Finnish press, then largely controlled by tsarist Russia. Sibelius’s contribution to the three-day pageant was a set of nationalistic musical tableaux. Several of these pieces he later recycled into…

Finnish spitz, breed of dog native to Finland, where a breed standard has existed since 1812. It is nicknamed the “barking bird dog” for its habit of “yodeling,” or barking continuously, to alert the hunter to the location of game birds. The breed continues to be a sporting dog in Finland but

Daly River: …Daly, explored in 1865 by Boyle Finniss, first governor of a proposed settlement in the territory, and named after Sir Dominick Daly, then governor of South Australia, is navigable for 70 miles (115 km) above its tidal mouth. The river is the eastern boundary of an Aboriginal reserve that extends…

Treaty of Novgorod: …was then generally known as Finnmark (including the present Norwegian province of Finnmark and Russia’s Kola Peninsula). The treaty, rather than delimiting a clear frontier between Norway and Novgorod, created a buffer zone, the “common districts.” The buffer zone offered Norway and Novgorod taxing rights over the indigenous Sami and…

Norway: Constitutional framework: The Finnmark Act, adopted by the Storting in 2005, transferred some 95 percent of the fylke (county) of Finnmark from state ownership to its residents through the establishment of the Finnmark Estate. The act recognized in particular that the Sami people, through protracted traditional use of…

Norway: Relief: …in southern Norway; and the Finnmark Plateau (1,000 feet [300 metres] above sea level), occupying most of Finnmark, the northernmost and largest county of Norway.

Finnmarksvidda, swampy plain, northern Norway. Though it has no exact natural boundaries, the plain’s principal section is about 60 miles (100 km) from east to west and 50 miles from north to south. The Finnmarksvidda, made up of ancient crystalline rock, is characterized by numerous small lakes

Finland: Ethnic groups: Peoples of Uralic (specifically Finno-Ugric) stock dominated two settlement areas. Those who entered southwestern Finland across the Gulf of Finland were the ancestors of the Hämäläiset (Tavastians, or Tavastlanders), the people of southern and western Finland (especially the historic region of Häme); those who entered from the southeast were…

Finno-Ugric languages, group of languages constituting much the larger of the two branches of a more comprehensive grouping, the Uralic languages (q.v.). The Finno-Ugric languages are spoken by several million people distributed discontinuously over an area extending from Norway in the west to the

Finno-Ugric religion, pre-Christian and pre-Islamic religious beliefs and practices of the Finno-Ugric peoples, who inhabit regions of northern Scandinavia, Siberia, the Baltic area, and central Europe. In modern times the religion of many of these peoples has been an admixture of agrarian and

canals and inland waterways: Germany: The 40-km (25-mile) Finow Canal along the Havel to the Liepe, a tributary of the Oder, had been built earlier but fell into decay because of flooding and neglect and was not rebuilt until 1751. Thus, in the late 17th and early 18th centuries, under Frederick William, the…

Financial markets move fast and often involve complicated financial products, which makes oversight and investor protection an important part of the financial system. The Financial Industry Regulatory Authority (FINRA) plays a central role in regulating brokerage firms and the brokers who work for

Finschhafen, town and port at the tip of Huon Peninsula, eastern Papua New Guinea, southwestern Pacific Ocean. The three-basin harbour, an inlet of the Solomon Sea, was charted by the British navigator Capt. John Moresby in 1873–74. Named for German explorer Otto Finsch, the town was claimed by

creeper: The brown creeper (Mohoua novaeseelandiae, or Finschia novaeseelandiae) of New Zealand belongs to the family Pachycephalidae. It is about 13 cm long, with a rather long tail and a tiny bill. Flocks or pairs call constantly in forests of South Island.

Niels Ryberg Finsen was a Danish physician and the founder of modern phototherapy (the treatment of disease by the influence of light). He received the 1903 Nobel Prize for Physiology or Medicine for the application of light in the treatment of skin diseases. Finsen was born into a prominent

Gulf of Finland, easternmost arm of the Baltic Sea, between Finland (north) and Russia and Estonia (east and south, respectively). Covering an area of 11,600 square miles (30,000 square km), the gulf extends for 250 miles (400 km) from east to west but only 12 to 80 miles (19 to 130 km) from north

Finsteraarhorn, highest peak (14,022 feet [4,274 metres]) of the Bernese Alps in south-central Switzerland, it lies between the cantons of Bern and Valais south-southeast of the mountain resort of Grindelwald. First ascended in 1812 by three Swiss guides (though this is disputed, and the first

bridge: Ulrich Finsterwalder: During the years after World War II, a German engineer and builder, Ulrich Finsterwalder, developed the cantilever method of construction with prestressed concrete. Finsterwalder’s Bendorf Bridge over the Rhine at Koblenz, Germany, was completed in 1962 with thin piers and a center

Wolfgang Amadeus Mozart: Early maturity: …to write an opera buffa, La finta giardiniera (“The Feigned Gardener Girl”), for the Munich carnival season, where it was duly successful. It shows Mozart, in his first comic opera since his childhood, finding ways of using the orchestra more expressively and of giving real personality to the pasteboard figures…

Wolfgang Amadeus Mozart: Early life and works: …having an Italian opera buffa, La finta semplice (“The Feigned Simpleton”), done at the court theater—hopes that were, however, frustrated, much to Leopold’s indignation. But a substantial, festal mass setting (probably K 139/47a) was successfully given before the court at the dedication of the Orphanage Church. La finta semplice was…

Puerto Rico: Hurricanes Maria and Fiona: …not as strong as Maria, Hurricane Fiona caused flooding and landslides, and it knocked out the island’s electricity.