The United States of America known as the United States or America, is a country composed of 50 states, a federal district, five major self-governing territories, various possessions. At 3.8 million square miles, the United States is the world's third or fourth largest country by total area and is smaller than the entire continent of Europe's 3.9 million square miles. With a population of over 327 million people, the U. S. is the third most populous country. The capital is Washington, D. C. and the largest city by population is New York City. Forty-eight states and the capital's federal district are contiguous in North America between Canada and Mexico; the State of Alaska is in the northwest corner of North America, bordered by Canada to the east and across the Bering Strait from Russia to the west. The State of Hawaii is an archipelago in the mid-Pacific Ocean; the U. S. territories are scattered about the Pacific Ocean and the Caribbean Sea, stretching across nine official time zones. The diverse geography and wildlife of the United States make it one of the world's 17 megadiverse countries.
Paleo-Indians migrated from Siberia to the North American mainland at least 12,000 years ago. European colonization began in the 16th century; the United States emerged from the thirteen British colonies established along the East Coast. Numerous disputes between Great Britain and the colonies following the French and Indian War led to the American Revolution, which began in 1775, the subsequent Declaration of Independence in 1776; the war ended in 1783 with the United States becoming the first country to gain independence from a European power. The current constitution was adopted in 1788, with the first ten amendments, collectively named the Bill of Rights, being ratified in 1791 to guarantee many fundamental civil liberties; the United States embarked on a vigorous expansion across North America throughout the 19th century, acquiring new territories, displacing Native American tribes, admitting new states until it spanned the continent by 1848. During the second half of the 19th century, the Civil War led to the abolition of slavery.
By the end of the century, the United States had extended into the Pacific Ocean, its economy, driven in large part by the Industrial Revolution, began to soar. The Spanish–American War and World War I confirmed the country's status as a global military power; the United States emerged from World War II as a global superpower, the first country to develop nuclear weapons, the only country to use them in warfare, a permanent member of the United Nations Security Council. Sweeping civil rights legislation, notably the Civil Rights Act of 1964, the Voting Rights Act of 1965 and the Fair Housing Act of 1968, outlawed discrimination based on race or color. During the Cold War, the United States and the Soviet Union competed in the Space Race, culminating with the 1969 U. S. Moon landing; the end of the Cold War and the collapse of the Soviet Union in 1991 left the United States as the world's sole superpower. The United States is the world's oldest surviving federation, it is a representative democracy.
The United States is a founding member of the United Nations, World Bank, International Monetary Fund, Organization of American States, other international organizations. The United States is a developed country, with the world's largest economy by nominal GDP and second-largest economy by PPP, accounting for a quarter of global GDP; the U. S. economy is post-industrial, characterized by the dominance of services and knowledge-based activities, although the manufacturing sector remains the second-largest in the world. The United States is the world's largest importer and the second largest exporter of goods, by value. Although its population is only 4.3% of the world total, the U. S. holds 31% of the total wealth in the world, the largest share of global wealth concentrated in a single country. Despite wide income and wealth disparities, the United States continues to rank high in measures of socioeconomic performance, including average wage, human development, per capita GDP, worker productivity.
The United States is the foremost military power in the world, making up a third of global military spending, is a leading political and scientific force internationally. In 1507, the German cartographer Martin Waldseemüller produced a world map on which he named the lands of the Western Hemisphere America in honor of the Italian explorer and cartographer Amerigo Vespucci; the first documentary evidence of the phrase "United States of America" is from a letter dated January 2, 1776, written by Stephen Moylan, Esq. to George Washington's aide-de-camp and Muster-Master General of the Continental Army, Lt. Col. Joseph Reed. Moylan expressed his wish to go "with full and ample powers from the United States of America to Spain" to seek assistance in the revolutionary war effort; the first known publication of the phrase "United States of America" was in an anonymous essay in The Virginia Gazette newspaper in Williamsburg, Virginia, on April 6, 1776. The second draft of the Articles of Confederation, prepared by John Dickinson and completed by June 17, 1776, at the latest, declared "The name of this Confederation shall be the'United States of America'".
The final version of the Articles sent to the states for ratification in late 1777 contains the sentence "The Stile of this Confederacy shall be'The United States of America'". In June 1776, Thomas Jefferson wrote the phrase "UNITED STATES OF AMERICA" in all capitalized letters in the headline of his "original Rough draught" of the Declaration of Independence; this draft of the document did not surface unti
Arizona is a state in the southwestern region of the United States. It is part of the Western and the Mountain states, it is the 14th most populous of the 50 states. Its capital and largest city is Phoenix. Arizona shares the Four Corners region with Utah and New Mexico. Arizona is the 48th state and last of the contiguous states to be admitted to the Union, achieving statehood on February 14, 1912, coinciding with Valentine's Day. Part of the territory of Alta California in New Spain, it became part of independent Mexico in 1821. After being defeated in the Mexican–American War, Mexico ceded much of this territory to the United States in 1848; the southernmost portion of the state was acquired in 1853 through the Gadsden Purchase. Southern Arizona is known for its desert climate, with hot summers and mild winters. Northern Arizona features forests of pine, Douglas fir, spruce trees. There are ski resorts in the areas of Flagstaff and Tucson. In addition to the Grand Canyon National Park, there are several national forests, national parks, national monuments.
About one-quarter of the state is made up of Indian reservations that serve as the home of 27 federally recognized Native American tribes, including the Navajo Nation, the largest in the state and the United States, with more than 300,000 citizens. Although federal law gave all Native Americans the right to vote in 1924, Arizona excluded those living on reservations in the state from voting until the state Supreme Court ruled in favor of Native American plaintiffs in Trujillo v. Garley; the state's name appears to originate from an earlier Spanish name, derived from the O'odham name alĭ ṣonak, meaning "small spring", which applied only to an area near the silver mining camp of Planchas de Plata, Sonora. To the European settlers, their pronunciation sounded like "Arissona"; the area is still known as alĭ ṣonak in the O'odham language. Another possible origin is the Basque phrase haritz ona, as there were numerous Basque sheepherders in the area. A native Mexican of Basque heritage established the ranchería of Arizona between 1734 and 1736 in the current Mexican state of Sonora, which became notable after a significant discovery of silver there, c.
1737. There is a misconception. For thousands of years before the modern era, Arizona was home to numerous Native American tribes. Hohokam and Ancestral Puebloan cultures were among the many that flourished throughout the state. Many of their pueblos, cliffside dwellings, rock paintings and other prehistoric treasures have survived, attracting thousands of tourists each year; the first European contact by native peoples was with Marcos de Niza, a Spanish Franciscan, in 1539. He explored parts of the present state and made contact with native inhabitants the Sobaipuri; the expedition of Spanish explorer Coronado entered the area in 1540–1542 during its search for Cíbola. Few Spanish settlers migrated to Arizona. One of the first settlers in Arizona was José Romo de Vivar. Father Kino was the next European in the region. A member of the Society of Jesus, he led the development of a chain of missions in the region, he converted many of the Indians to Christianity in the Pimería Alta in the 1690s and early 18th century.
Spain founded presidios at Tubac in 1752 and Tucson in 1775. When Mexico achieved its independence from the Kingdom of Spain and its Spanish Empire in 1821, what is now Arizona became part of its Territory of Nueva California known as Alta California. Descendants of ethnic Spanish and mestizo settlers from the colonial years still lived in the area at the time of the arrival of European-American migrants from the United States. During the Mexican–American War, the U. S. Army occupied the national capital of Mexico City and pursued its claim to much of northern Mexico, including what became Arizona Territory in 1863 and the State of Arizona in 1912; the Treaty of Guadalupe Hidalgo specified that, in addition to language and cultural rights of the existing inhabitants of former Mexican citizens being considered as inviolable, the sum of US$15 million dollars in compensation be paid to the Republic of Mexico. In 1853, the U. S. acquired the land south below the Gila River from Mexico in the Gadsden Purchase along the southern border area as encompassing the best future southern route for a transcontinental railway.
What is now known as the state of Arizona was administered by the United States government as part of the Territory of New Mexico until the southern part of that region seceded from the Union to form the Territory of Arizona. This newly established territory was formally organized by the Confederate States government on Saturday, January 18, 1862, when President Jefferson Davis approved and signed An Act to Organize the Territory of Arizona, marking the first official use of the name "Territory of Arizona"; the Southern territory supplied the Confederate government with men and equipment. Formed in 1862, Arizona scout companies served with the Confederate States Army duri
Julia Hall Bowman Robinson was an American mathematician noted for her contributions to the fields of computability theory and computational complexity theory–most notably in decision problems. Her work on Hilbert's 10th problem played a crucial role in its ultimate resolution. In 1936, she entered San Diego State University at the age of 16. Dissatisfied with the mathematics curriculum at San Diego State University, she transferred to University of California, Berkeley in 1939 for her senior year and received her BA degree in 1940. After graduating, Robinson continued in graduate studies at Berkeley; as a graduate student, Robinson was employed as a teaching assistant with the Department of Mathematics and by Jerzy Neyman in the Berkeley Statistical Laboratory, where her work resulted in her first published paper. Robinson received her Ph. D. degree in 1948 under Alfred Tarski with a dissertation on "Definability and Decision Problems in Arithmetic". Julia Bowman Robinson's first paper was published in 1948 titled "A Note on Exact Sequential Analysis", a product of her time as a statistics lab assistant under Jerzy Neyman at Berkeley.
Robinson's Ph. D. thesis, "Definability and Decision Problems in Arithmetic," showed that the theory of the rational numbers was an undecidable problem, by demonstrating that elementary number theory could be defined in terms of the rationals. (Elementary number theory was known to be undecidable by Gödel's first Incompleteness Theorem. Here is an excerpt from her thesis:"This consequence of our discussion is interesting because of a result of Godel which shows that the variety of relations between integers which are arithmetically definable in terms of addition and multiplication of integers is great. For instance from Theorem 3.2 and Godel's result, we can conclude that the relation which holds between three rationals A, B, N if and only if N is a positive integer and A=BN is definable in the arithmetic of rationals." Hilbert's tenth problem asks for an algorithm to determine whether a Diophantine equation has any solutions in integers. Robinson began exploring methods for this problem in 1948 while at the RAND Corporation.
Her work regarding Diophantine representation for exponentiation and her method of using Pell's equation led to the J. R. hypothesis in 1950. Proving this hypothesis would be central in the final solution, her research publications would lead to collaborations with Martin Davis, Hilary Putnam, Yuri Matiyasevich. In 1970, the problem was resolved in the negative. Through the 1970's, Robinson continued working with Matiyasevich on one of their solution's corollaries, which stated that there is a constant N such that, given a Diophantine equation with any number of parameters and in any number of unknowns, one can transform this equation into another with the same parameters but in only N unknowns such that both equations are solvable or unsolvable for the same values of the parameters. At the time the solution was first published, the authors established N = 200. Robinson and Matiyasevich's joint work would produce further reduction to 9 unknowns."Hilbert’s tenth problem asks in essence whether the set D is decidable.
The answer is negative. In contrast, we can show. Before doing so, let’s consider a simpler problem, it is an analog of Hilbert’s tenth problem for polynomials that have only a single variable, such as 4x3 −2x2 + x − 7. Let D1 =. Here is a TM M1 that recognizes D1: M1 ='On input ⟨p⟩: where p is a polynomial over the variable x.'1.' Evaluate p with x set successively to the values 0, 1, −1, 2, −2, 3, −3... If at any point the polynomial evaluates to 0, accept." After Robinson solved Hilbert's tenth problem, Saunders Mac Lane nominated her for the National Academy of Sciences. Alfred Tarski and Jerzy Neyman flew out to Washington, D. C. to further explain to the NAS why her work is so important and how it tremendously contributed to mathematics. In 1975, she was the first female mathematician to be elected to the National Academy of Sciences. Not allowed to teach at in the Mathematics Department at Berkeley as there was a rule which prevented family members from working together in the same department after marrying Raphael M. Robinson in 1941.
Robinson instead stayed in the statistics department despite wanting to teach calculus. Although Raphael retired in 1973, it wasn't until 1976 she was offered a full time professorship position at Berkeley only after the department heard of her nomination to the National Academy of Sciences. During the late 1940s, Robinson spent a year or so at the RAND Corporation in Santa Monica researching game theory which led to her published paper called "An Interactive Method of Solving a Game" in 1951. In her paper, she proved that the fictitious play dynamics converges to the mixed strategy Nash equilibrium in two-player zero-sum games; this was posed by George W. Brown as a prize problem at RAND Corporation. Julia was elected the first female president of the American Mathematical Society, it took time for her to accept the nominated, stated in her autobiography:"In 1982 I was nominated for the presidency of the American Mathematical Society. I realized that I had been chosen because I was a woman and because I had the seal of approval, as it were, of the National Academy.
After discussion with Raphael, who thought I should decline and save my energy for mathematics, other members of my family, who differed with him, I decided that as a woman and a mathematician I had no alternative but to accept. I have always tried to do everything I cou
San Diego is a city in the U. S. state of California. It is in San Diego County, on the coast of the Pacific Ocean in Southern California 120 miles south of Los Angeles and adjacent to the border with Mexico. With an estimated population of 1,419,516 as of July 1, 2017, San Diego is the eighth-largest city in the United States and second-largest in California, it is part of the San Diego–Tijuana conurbation, the second-largest transborder agglomeration between the U. S. and a bordering country after Detroit–Windsor, with a population of 4,922,723 people. The city is known for its mild year-round climate, natural deep-water harbor, extensive beaches, long association with the United States Navy, recent emergence as a healthcare and biotechnology development center. San Diego has been called "the birthplace of California". Home to the Kumeyaay people, it was the first site visited by Europeans on what is now the West Coast of the United States. Upon landing in San Diego Bay in 1542, Juan Rodríguez Cabrillo claimed the area for Spain, forming the basis for the settlement of Alta California 200 years later.
The Presidio and Mission San Diego de Alcalá, founded in 1769, formed the first European settlement in what is now California. In 1821, San Diego became part of the newly independent Mexico, which reformed as the First Mexican Republic two years later. California became part of the United States in 1848 following the Mexican–American War and was admitted to the union as a state in 1850; the city is the seat of San Diego County and is the economic center of the region as well as the San Diego–Tijuana metropolitan area. San Diego's main economic engines are military and defense-related activities, international trade, manufacturing; the presence of the University of California, San Diego, with the affiliated UCSD Medical Center, has helped make the area a center of research in biotechnology. The original inhabitants of the region are now known as the San La Jolla people; the area of San Diego has been inhabited by the Kumeyaay people. The first European to visit the region was explorer Juan Rodríguez Cabrillo, sailing under the flag of Castile but born in Portugal.
Sailing his flagship San Salvador from Navidad, New Spain, Cabrillo claimed the bay for the Spanish Empire in 1542, named the site "San Miguel". In November 1602, Sebastián Vizcaíno was sent to map the California coast. Arriving on his flagship San Diego, Vizcaíno surveyed the harbor and what are now Mission Bay and Point Loma and named the area for the Catholic Saint Didacus, a Spaniard more known as San Diego de Alcalá. On November 12, 1602, the first Christian religious service of record in Alta California was conducted by Friar Antonio de la Ascensión, a member of Vizcaíno's expedition, to celebrate the feast day of San Diego. Permanent colonization of California and of San Diego began in 1769 with the arrival of four contingents of Spaniards from New Spain and the Baja California peninsula. Two seaborne parties reached San Diego Bay: the San Carlos, under Vicente Vila and including as notable members the engineer and cartographer Miguel Costansó and the soldier and future governor Pedro Fages, the San Antonio, under Juan Pérez.
An initial overland expedition to San Diego from the south was led by the soldier Fernando Rivera and included the Franciscan missionary and chronicler Juan Crespí, followed by a second party led by the designated governor Gaspar de Portolà and including the mission president Junípero Serra. In May 1769, Portolà established the Fort Presidio of San Diego on a hill near the San Diego River, it was the first settlement by Europeans in. In July of the same year, Mission San Diego de Alcalá was founded by Franciscan friars under Serra. By 1797, the mission boasted the largest native population in Alta California, with over 1,400 neophytes living in and around the mission proper. Mission San Diego was the southern anchor in Alta California of the historic mission trail El Camino Real. Both the Presidio and the Mission are National Historic Landmarks. In 1821, Mexico won its independence from Spain, San Diego became part of the Mexican territory of Alta California. In 1822, Mexico began its attempt to extend its authority over the coastal territory of Alta California.
The fort on Presidio Hill was abandoned, while the town of San Diego grew up on the level land below Presidio Hill. The Mission was secularized by the Mexican government in 1834, most of the Mission lands were granted to former soldiers; the 432 residents of the town petitioned the governor to form a pueblo, Juan María Osuna was elected the first alcalde, defeating Pío Pico in the vote. However, San Diego had been losing population throughout the 1830s and in 1838 the town lost its pueblo status because its size dropped to an estimated 100 to 150 residents. Beyond town Mexican land grants expanded the number of California ranchos that modestly added to the local economy. Americans gained increased awareness of California, its commercial possibilities, from the writings of two countrymen involved in the officially forbidden, to foreigners, but economically significant hide and tallow trade, where San Diego was a major port and the only one with an adequate harbor: William Shaler's "Journal of a Voyage Between China and the North-Western Coast of America, Made in 1804" and Richard Henry Dana's more substantial and convincing account, of his 1834–36 voyage, the classic Two Years Before the Mast.
In 1846, the United States went to war against Mexico and sent a naval and land expedition to conquer Alta California. At firs
Raphael M. Robinson
Raphael Mitchel Robinson was an American mathematician. Born in National City, Robinson was the youngest of four children of a lawyer and a teacher, he was awarded from the University of California, Berkeley in mathematics: the BA, MA, Ph. D.. His Ph. D. thesis, on complex analysis, was titled Some results in the theory of Schlicht functions. In 1941, Robinson married his former student Julia Bowman, she became his Berkeley colleague and the first woman president of the American Mathematical Society. Robinson worked on mathematical logic, set theory, number theory, combinatorics. In 1937 he set out a simpler and more conventional version of the John von Neumann 1923 axiomatic set theory. Soon after Alfred Tarski joined Berkeley's mathematics department in 1942, Robinson began to do major work on the foundations of mathematics, building on Tarski's concept of "essential undecidability," by proving a number of mathematical theories undecidable. In 1950 Robinson proved that an undecidable theory need not have an infinite number of axioms by coming up with a counterexample: Robinson arithmetic Q.
Q is finitely axiomatizable. Robinson's work on undecidability culminated in his coauthoring Tarski et al. which established, among other things, the undecidability of group theory, lattice theory, abstract projective geometry, closure algebras. Robinson worked in number theory employing early computers to obtain results. For example, he coded the Lucas-Lehmer primality test to determine whether 2n − 1 was prime for all prime n < 2304 on a SWAC. In 1952, he showed that these Mersenne numbers were all composite except for 17 values of n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, he discovered the last five of the largest ones known at the time. Robinson wrote several papers on tilings of the plane, in particular a clear and remarkable 1971 paper Undecidability and nonperiodicity for tilings of the plane simplifying what had been a tangled theory. Robinson became a full professor at Berkeley in 1949, retired in 1973, remained active in his educational interests for the duration of his life having published late in his life: Minsky's small universal Turing machine, describing a universal Turing machine with four symbols and seven states.
Robinson arithmetic Robinson, R. M. "The theory of classes: A modification of Von Neumann's system", Journal of Symbolic Logic, 2: 29–36, doi:10.2307/2268798, JSTOR 2268798. ———, "An Essentially Undecidable Axiom System", Proceedings of the International Congress of Mathematics: 729–730. Alfred Tarski, A. Mostowski, R. M. Robinson, 1953. Undecidable theories. North Holland. Leon Henkin, 1995, "In memoriam: Raphael Mitchell Robinson," Bull. Symbolic Logic 1: 340–43. "In memoriam: Raphael Mitchell Robinson," Modern Logic 5: 329. O'Connor, John J.. The source for much of this entry. Raphael M. Robinson at the Mathematics Genealogy Project
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to