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140-653: The Geographic Names Information System ( GNIS ) is a database of name and location information about more than two million physical and cultural features throughout the United States and its territories; the associated states of the Marshall Islands , Federated States of Micronesia , and Palau ; and Antarctica . It is a type of gazetteer . It was developed by the United States Geological Survey (USGS) in cooperation with

280-418: A ) {\displaystyle (x_{a},y_{a})} and ( x b , y b ) {\displaystyle (x_{b},y_{b})} Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point x k . The following error estimate shows that linear interpolation is not very precise. Denote

420-414: A , y a ) and ( x b , y b ), and the interpolant is given by: This previous equation states that the slope of the new line between ( x a , y a ) {\displaystyle (x_{a},y_{a})} and ( x , y ) {\displaystyle (x,y)} is the same as the slope of the line between ( x a , y

560-580: A " Special Relationship " with the United Kingdom and strong ties with Canada , Australia , New Zealand , the Philippines , Japan , South Korea , Israel , and several European Union countries ( France , Italy , Germany , Spain , and Poland ). The U.S. works closely with its NATO allies on military and national security issues, and with countries in the Americas through

700-492: A "Concise" subset of the NGNDB that listed "major features", and a "Historical" subset that included the features that no longer exist. There is no differentiation amongst different types of populated places. In the words of the aforementioned 1986 USACE report, "[a] subdivision having one inhabitant is as significant as a major metropolitan center such as New York City ". In comparing GNIS populated place records with data from

840-585: A 1962 replacement of the "Nigger" racial pejorative for African Americans with "Negro" and a 1974 replacement of the "Jap" racial pejorative for Japanese Americans with "Japanese". In 2015, a cross-reference of the GNIS database against the Racial Slur Database had found 1441 racial slur placenames, every state of the United States having them, with California having 159 and the state with

980-692: A 2008 book on ethnic slurs in U.S. placenames Mark Monmonier of Syracuse University discovered "Niger Hill" in Potter County, Pennsylvania , an erroneous transcription of "Nigger Hill" from a 1938 map, and persuaded the USBGN to change it to "Negro Hill". In November 2021, the United States Secretary of the Interior issued an order instructing that "Squaw" be removed from usage by the U.S. federal government. Prior efforts had included

1120-542: A coordinated, continent-wide military effort. Finally, deteriorating relations between the British military establishment and the colonists, relations that were already less than positive, set the stage for further distrust and dislike of British troops. Following their victory in the French and Indian War, Britain began to assert greater control over local colonial affairs, resulting in colonial political resistance ; one of

1260-534: A decentralized government that operated until 1789. After the British surrender at the siege of Yorktown in 1781 American sovereignty was internationally recognized by the Treaty of Paris (1783), through which the U.S. gained territory stretching west to the Mississippi River, north to present-day Canada, and south to Spanish Florida . The Northwest Ordinance (1787) established the precedent by which

1400-407: A few prominent industrialists , largely by their formation of trusts and monopolies to prevent competition. Tycoons led the nation's expansion in the railroad , petroleum , and steel industries. The United States emerged as a pioneer of the automotive industry . These changes were accompanied by significant increases in economic inequality , slum conditions , and social unrest , creating

1540-556: A function s : [ a , b ] → R {\displaystyle s:[a,b]\to \mathbb {R} } such that f ( x i ) = s ( x i ) {\displaystyle f(x_{i})=s(x_{i})} for i = 1 , 2 , … , n {\displaystyle i=1,2,\dots ,n} (that is, that s {\displaystyle s} interpolates f {\displaystyle f} at these points). In general, an interpolant need not be

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1680-430: A functional that represents the entire family of interpolants satisfying those constraints, including those that are discontinuous or partially defined. These functionals identify the subspace of functions where the solution to a constrained optimization problem resides. Consequently, TFC transforms constrained optimization problems into equivalent unconstrained formulations. This transformation has proven highly effective in

1820-416: A given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function. If we consider x {\displaystyle x} as a variable in a topological space , and the function f ( x ) {\displaystyle f(x)} mapping to a Banach space , then

1960-666: A global reach. The U.S. is a founding member of the World Bank , Organization of American States , NATO , and the United Nations , as well as a permanent member of the UN Security Council . The first documented use of the phrase "United States of America" is a letter from January 2, 1776. Stephen Moylan , a Continental Army aide to General George Washington , wrote to Joseph Reed , Washington's aide-de-camp , seeking to go "with full and ample powers from

2100-962: A good approximation, but there are well known and often reasonable conditions where it will. For example, if f ∈ C 4 ( [ a , b ] ) {\displaystyle f\in C^{4}([a,b])} (four times continuously differentiable) then cubic spline interpolation has an error bound given by ‖ f − s ‖ ∞ ≤ C ‖ f ( 4 ) ‖ ∞ h 4 {\displaystyle \|f-s\|_{\infty }\leq C\|f^{(4)}\|_{\infty }h^{4}} where h max i = 1 , 2 , … , n − 1 | x i + 1 − x i | {\displaystyle h\max _{i=1,2,\dots ,n-1}|x_{i+1}-x_{i}|} and C {\displaystyle C}

2240-420: A later phase). Generic designations were given after specific names, so (for examples) Mount Saint Helens was recorded as "Saint Helens, Mount", although cities named Mount Olive , not actually being mountains, would not take "Mount" to be a generic part and would retain their order "Mount Olive". The primary geographic coordinates of features which occupy an area, rather than being a single point feature, were

2380-598: A period of heightened, overt racism following Reconstruction, a time often called the nadir of American race relations . A series of Supreme Court decisions, including Plessy v. Ferguson , emptied the Fourteenth and Fifteenth Amendments of their force, allowing Jim Crow laws in the South to remain unchecked, sundown towns in the Midwest, and segregation in communities across the country , which would be reinforced by

2520-488: A prominent leader in the early 1960s. The Great Society plan of President Lyndon Johnson 's administration resulted in groundbreaking and broad-reaching laws, policies and a constitutional amendment to counteract some of the worst effects of lingering institutional racism . The counterculture movement in the U.S. brought significant social changes, including the liberalization of attitudes toward recreational drug use and sexuality . It also encouraged open defiance of

2660-409: A second Alaska file) data from the 1:100000 and 1:250000 scale USGS maps. Map names were recorded exactly as on the maps themselves, with the exceptions for diacritics as with the NGNDB. Unlike the NGNDB, locations were the geographic coördinates of the south-east corner of the given map, except for American Samoa and Guam maps where they were of the north-east cornder. The TMNDB was later renamed

2800-526: A sense of manifest destiny . The Missouri Compromise attempted to balance the desire of northern states to prevent the expansion of slavery into new territories with that of southern states to extend it, admitting Missouri as a slave state and Maine as a free state. It further prohibited slavery in all other lands of the Louisiana Purchase north of the 36°30′ parallel . As Americans expanded further into land inhabited by Native Americans,

2940-614: A sense of American identity as well. Men who otherwise never left their own colony now traveled across the continent to fight alongside men from decidedly different backgrounds but who were no less "American". British officers trained American officers for battle, most notably George Washington ; these officers would lend their skills and expertise to the colonists' cause during the American Revolutionary War to come. In addition, colonial legislatures and officials found it necessary to cooperate intensively in pursuit of

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3080-691: A smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress. Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates

3220-401: A year). Juvenile correctional facilities are operated by local or state governments and serve as longer-term placements for any minor adjudicated as delinquent and ordered by a judge to be confined. As of January 2023, the United States has the sixth-highest per capita incarceration rate in the world—531 people per 100,000 inhabitants—and the largest prison and jail population in

3360-405: Is a common way to approximate functions. Given a function f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } with a set of points x 1 , x 2 , … , x n ∈ [ a , b ] {\displaystyle x_{1},x_{2},\dots ,x_{n}\in [a,b]} one can form

3500-422: Is a constant. Gaussian process is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a curve through noisy data. In the geostatistics community Gaussian process regression

3640-440: Is also known as Kriging . Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions using Padé approximant , and trigonometric interpolation is interpolation by trigonometric polynomials using Fourier series . Another possibility is to use wavelets . The Whittaker–Shannon interpolation formula can be used if

3780-591: Is believed to be the first widespread culture in the Americas. Over time, indigenous North American cultures grew increasingly sophisticated, and some, such as the Mississippian culture , developed agriculture , architecture , and complex societies . In the post-archaic period , the Mississippian cultures were located in the midwestern , eastern , and southern regions, and the Algonquian in

3920-498: Is known as the presidential system , in contrast to the parliamentary system , where the executive is part of the legislative body. Many countries around the world imitated this aspect of the 1789 Constitution of the United States , especially in the Americas. The Constitution is silent on political parties. However, they developed independently in the 18th century with the Federalist and Anti-Federalist parties. Since then,

4060-438: Is now the United States experienced conflicts with Native Americans, they also engaged in trade, exchanging European tools for food and animal pelts. Relations ranged from close cooperation to warfare and massacres. The colonial authorities often pursued policies that forced Native Americans to adopt European lifestyles, including conversion to Christianity. Along the eastern seaboard, settlers trafficked African slaves through

4200-440: Is or was human activity" not covered by a more specific feature class), "populated place" (a "place or area with clustered or scattered buildings"), "spring" (a spring ), "lava" (a lava flow , kepula , or other such feature), and "well" (a well ). Mountain features would fall into "ridge", "range", or "summit" classes. A feature class "tank" was sometimes used for lakes, which was problematic in several ways. This feature class

4340-421: Is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants. Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linear function . We now replace this interpolant with a polynomial of higher degree . Consider again the problem given above. The following sixth degree polynomial goes through all

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4480-552: Is the federal agency charged with addressing most environmental-related issues . The idea of wilderness has shaped the management of public lands since 1964, with the Wilderness Act . The Endangered Species Act of 1973 provides a way to protect threatened and endangered species and their habitats. The United States Fish and Wildlife Service implements and enforces the Act. In 2024, the U.S. ranked 34th among 180 countries in

4620-423: Is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed. The Theory of Functional Connections (TFC) is a mathematical framework specifically developed for functional interpolation . Given any interpolant that satisfies a set of constraints, TFC derives

4760-541: Is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation , this could be a favourable choice for its speed and simplicity. One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating f (2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f (2.5) midway between f (2) = 0.9093 and f (3) = 0.1411, which yields 0.5252. Generally, linear interpolation takes two data points, say ( x

4900-603: The Allies of World War I , helping to turn the tide against the Central Powers . In 1920, a constitutional amendment granted nationwide women's suffrage . During the 1920s and '30s, radio for mass communication and the invention of early television transformed communications nationwide. The Wall Street Crash of 1929 triggered the Great Depression , which President Franklin D. Roosevelt responded to with

5040-553: The Atlantic slave trade . The original Thirteen Colonies that would later found the United States were administered as possessions of Great Britain , and had local governments with elections open to most white male property owners . The colonial population grew rapidly, eclipsing Native American populations; by the 1770s, the natural increase of the population was such that only a small minority of Americans had been born overseas. The colonies' distance from Britain allowed for

5180-817: The Bering land bridge more than 12,000 years ago, and formed various civilizations and societies . British colonization led to the first settlement of the Thirteen Colonies in Virginia in 1607. Clashes with the British Crown over taxation and political representation sparked the American Revolution , with the Second Continental Congress formally declaring independence on July 4, 1776. Following its victory in

5320-549: The Environmental Performance Index . The country joined the Paris Agreement on climate change in 2016 and has many other environmental commitments. The United States is a federal republic of 50 states and a separate federal capital district, Washington, D.C. It also asserts sovereignty over five unincorporated territories and several uninhabited island possessions . The U.S. is

5460-500: The Federal Bureau of Investigation (FBI) and the U.S. Marshals Service have national jurisdiction and specialized duties, such as protecting civil rights , national security and enforcing U.S. federal courts ' rulings and federal laws. State courts conduct most civil and criminal trials, and federal courts handle designated crimes and appeals of state court decisions . There is no unified "criminal justice system" in

5600-558: The Geographic Cell Names database (GCNDB hereafter) in the 1990s. The Generic database was in essence a machine-readable glossary of terms and abbreviations taken from the map sources, with their definitions, grouped into collections of related terms. The National Atlas database was an abridged version of the NGNDB that contained only those entries that were in the index to the USGS National Atlas of

5740-720: The Great Lakes , Mississippi River and Gulf of Mexico . British colonization of the East Coast began with the Virginia Colony (1607) and Plymouth Colony (1620). The Mayflower Compact and the Fundamental Orders of Connecticut established precedents for representative self-governance and constitutionalism that would develop throughout the American colonies. While European settlers in what

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5880-719: The Great Lakes region and along the Eastern Seaboard , while the Hohokam culture and Ancestral Puebloans inhabited the southwest . Native population estimates of what is now the United States before the arrival of European immigrants range from around 500,000 to nearly 10 million. Christopher Columbus began exploring the Caribbean for Spain in 1492, leading to Spanish-speaking settlements and missions from Puerto Rico and Florida to New Mexico and California . France established its own settlements along

6020-524: The House of Representatives , a lower house based on population; and the Senate , an upper house based on equal representation for each state. Federalism provides substantial autonomy to the 50 states, while the country's political culture promotes liberty , equality , individualism , personal autonomy , and limited government . One of the world's most developed countries , the United States has had

6160-596: The Midwest . At the same time, about one million French Canadians migrated from Quebec to New England . During the Great Migration , millions of African Americans left the rural South for urban areas in the North. Alaska was purchased from Russia in 1867. The Compromise of 1877 effectively ended Reconstruction and white supremacists took local control of Southern politics . African Americans endured

6300-688: The National Park Service and other agencies. About 28% of the country's land is publicly owned and federally managed, primarily in the Western States . Most of this land is protected , though some is leased for commercial use, and less than one percent is used for military purposes. Environmental issues in the United States include debates on non-renewable resources and nuclear energy , air and water pollution , biodiversity , logging and deforestation , and climate change . The U.S. Environmental Protection Agency (EPA)

6440-652: The New Deal , a series of sweeping programs and public works projects combined with financial reforms and regulations . All were intended to protect against future economic depressions. Initially neutral during World War II , the U.S. began supplying war materiel to the Allies of World War II in March 1941 and entered the war in December after the Empire of Japan 's attack on Pearl Harbor . The U.S. developed

6580-712: The Organization of American States and the United States–Mexico–Canada Free Trade Agreement . In South America, Colombia is traditionally considered to be the closest ally of the United States. The U.S. exercises full international defense authority and responsibility for Micronesia , the Marshall Islands , and Palau through the Compact of Free Association . It has increasingly conducted strategic cooperation with India , but its ties with China have steadily deteriorated. Since 2014,

6720-648: The Pentium microprocessor in accordance with Moore's law , rechargeable lithium-ion batteries , the first gene therapy trial, and cloning either emerged in the U.S. or were improved upon there. The Human Genome Project was formally launched in 1990, while Nasdaq became the first stock market in the United States to trade online in 1998. In the Gulf War of 1991, an American-led international coalition of states expelled an Iraqi invasion force that had occupied neighboring Kuwait . The September 11 attacks on

6860-534: The Soviet Union's collapse and the end of the Cold War in 1991, the U.S. emerged as the world's sole superpower , wielding significant geopolitical influence globally . The U.S. national government is a presidential constitutional federal republic and liberal democracy with three separate branches : legislative , executive , and judicial . It has a bicameral national legislature composed of

7000-591: The Spanish–American War . (The Philippines was granted full independence from the U.S. on July 4, 1946, following World War II. Puerto Rico and Guam have remained U.S. territories.) American Samoa was acquired by the United States in 1900 after the Second Samoan Civil War . The U.S. Virgin Islands were purchased from Denmark in 1917. The United States entered World War I alongside

7140-712: The Thematic Mapper of the Landsat program , researchers from the University of Connecticut in 2001 discovered that "a significant number" of populated places in Connecticut had no identifiable human settlement in the land use data and were at road intersections. They found that such populated places with no actual settlement often had "Corner" in their names, and hypothesized that either these were historical records or were "cartographic locators". In surveying in

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7280-494: The United States Board on Geographic Names (BGN) to promote the standardization of feature names. Data were collected in two phases. Although a third phase was considered, which would have handled name changes where local usages differed from maps, it was never begun. The database is part of a system that includes topographic map names and bibliographic references. The names of books and historic maps that confirm

7420-700: The cotton gin made the institution increasingly profitable for Southern elites . This sectional conflict regarding slavery culminated in the American Civil War (1861–1865). Eleven slave states seceded and formed the Confederate States of America , while the other states remained in the Union . War broke out in April 1861 after the Confederates bombarded Fort Sumter . After

7560-567: The displacement interpolation problem used in transportation theory . Multivariate interpolation is the interpolation of functions of more than one variable. Methods include nearest-neighbor interpolation , bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions. They can be applied to gridded or scattered data. Mimetic interpolation generalizes to n {\displaystyle n} dimensional spaces where n > 3 {\displaystyle n>3} . In

7700-494: The electric field , for instance, since the line integral gives the electric potential difference at the endpoints of the integration path. Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the integration path. Linear , bilinear and trilinear interpolation are also considered mimetic, even if it

7840-537: The fall of communism and the collapse of the Soviet Union , which marked the end of the Cold War and left the United States as the world's sole superpower . The 1990s saw the longest recorded economic expansion in American history , a dramatic decline in U.S. crime rates , and advances in technology . Throughout this decade, technological innovations such as the World Wide Web , the evolution of

7980-445: The largest nominal GDP since about 1890 and accounted for over 15% of the global economy in 2023. It possesses by far the largest amount of wealth of any country and has the highest disposable household income per capita among OECD countries. The U.S. ranks among the world's highest in economic competitiveness , productivity , innovation , human rights , and higher education . Its hard power and cultural influence have

8120-733: The 1775 Battles of Lexington and Concord , igniting the American Revolutionary War . At the Second Continental Congress , the colonies appointed George Washington commander-in-chief of the Continental Army , and created a committee that named Thomas Jefferson to draft the Declaration of Independence . Two days after passing the Lee Resolution to create an independent nation the Declaration

8260-408: The 1775–1783 Revolutionary War , the country continued to expand westward across North America , resulting in the dispossession of native inhabitants . As more states were admitted , a North–South division over slavery led to the secession of the Confederate States of America , which fought states remaining in the Union in the 1861–1865 American Civil War . With the victory and preservation of

8400-546: The 1990s (still including tape and paper) to floppy disc , over FTP , and on CD-ROM . The CD-ROM edition only included the NGNDB, the AGNDB, the GCNDB, and a bibliographic reference database (RDB); but came with database search software that ran on PC DOS (or compatible) version 3.0 or later. The FTP site included extra topical databases: a subset of the NGNDB that only included the records with feature classes for populated places,

8540-520: The 2010s, political polarization in the country increased between liberal and conservative factions. This polarization was capitalized upon in the January 2021 Capitol attack , when a mob of insurrectionists entered the U.S. Capitol and sought to prevent the peaceful transfer of power in an attempted self-coup d'état . The United States is the world's third-largest country by total area behind Russia and Canada. The 48 contiguous states and

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8680-596: The American West have an alpine climate . The climate is arid in the Southwest, Mediterranean in coastal California , and oceanic in coastal Oregon , Washington , and southern Alaska . Most of Alaska is subarctic or polar . Hawaii , the southern tip of Florida and U.S. territories in the Caribbean and Pacific are tropical . States bordering the Gulf of Mexico are prone to hurricanes, and most of

8820-420: The Declaration of Independence on July 4, 1776. The term "United States" and the initialism "U.S.", used as nouns or as adjectives in English, are common short names for the country. The initialism "USA", a noun, is also common. "United States" and "U.S." are the established terms throughout the U.S. federal government , with prescribed rules. In English, the term "America" rarely refers to topics unrelated to

8960-404: The District of Columbia occupy a combined area of 3,119,885 square miles (8,080,470 km ). The coastal plain of the Atlantic seaboard gives way to inland forests and rolling hills in the Piedmont plateau region. The Appalachian Mountains and the Adirondack massif separate the East Coast from the Great Lakes and the grasslands of the Midwest . The Mississippi River System ,

9100-458: The Elder concluded that major military resources needed to be devoted to North America to win the war against France. For the first time, the continent became one of the main theaters of what could be termed a " world war ". The British colonies' position as an integral part of the British Empire became more apparent during the war, with British military and civilian officials becoming a more significant presence in American life. The war increased

9240-453: The GNIS web site and can review the justifications and supporters of the proposals. The usual sources of name change requests are an individual state's board on geographic names, or a county board of governors. This does not always succeed, the State Library of Montana having submitted three large sets of name changes that have not been incorporated into the GNIS database. Conversely, a group of middle school students in Alaska succeeded, with

9380-465: The January 1863 Emancipation Proclamation , many freed slaves joined the Union army . The war began to turn in the Union's favor following the 1863 Siege of Vicksburg and Battle of Gettysburg , and the Confederacy surrendered in 1865 after the Union's victory in the Battle of Appomattox Court House . The Reconstruction era followed the war. After the assassination of President Abraham Lincoln , Reconstruction Amendments were passed to protect

9520-476: The Rocky Mountains, the Yellowstone Caldera , is the continent's largest volcanic feature. In 2021, the United States had 8% of global permanent meadows and pastures and 10% of cropland. With its large size and geographic variety, the United States includes most climate types. East of the 100th meridian , the climate ranges from humid continental in the north to humid subtropical in the south. The western Great Plains are semi-arid . Many mountainous areas of

9660-451: The U.S. has become a key ally of Ukraine ; it has also provided the country with significant military equipment and other support in response to Russia's 2022 invasion . The president is the commander-in-chief of the United States Armed Forces and appoints its leaders, the secretary of defense and the Joint Chiefs of Staff . The Department of Defense , which is headquartered at the Pentagon near Washington, D.C., administers five of

9800-661: The U.S. territories." Despite disparate systems of confinement, four main institutions dominate: federal prisons , state prisons , local jails, and juvenile correctional facilities . Federal prisons are run by the Federal Bureau of Prisons and hold people who have been convicted of federal crimes, including pretrial detainees. State prisons, run by the official department of correction of each state, hold sentenced people serving prison time (usually longer than one year) for felony offenses. Local jails are county or municipal facilities that incarcerate defendants prior to trial; they also hold those serving short sentences (typically under

9940-405: The United States , with the coördinates published in the latter substituted for the coördinates from the former. The Board on Geographic Names database was a record of investigative work of the USGS Board on Geographic Names ' Domestic Names Committee, and decisions that it had made from 1890 onwards, as well as names that were enshrined by Acts of Congress . Elevation and location data followed

10080-464: The United States and the peaceful transfer of power , respectively. The Louisiana Purchase of 1803 from France nearly doubled the territory of the United States. Lingering issues with Britain remained , leading to the War of 1812 , which was fought to a draw. Spain ceded Florida and its Gulf Coast territory in 1819. In the late 18th century, American settlers began to expand westward , many with

10220-714: The United States entered the Cold War, where geopolitical tensions between the U.S. and the Soviet Union led the two countries to dominate world affairs . The U.S. utilized the policy of containment to limit the USSR's sphere of influence, and prevailed in the Space Race , which culminated with the first crewed Moon landing in 1969. Domestically, the U.S. experienced economic growth , urbanization , and population growth following World War II . The civil rights movement emerged, with Martin Luther King Jr. becoming

10360-666: The United States has operated as a de facto two-party system , though the parties in that system have been different at different times. The two main national parties are presently the Democratic and the Republican . The former is perceived as relatively liberal in its political platform while the latter is perceived as relatively conservative . In the American federal system , sovereign powers are shared between two levels of elected government: national and state. People in

10500-593: The United States in 2001 by the pan-Islamist militant organization al-Qaeda led to the war on terror , and subsequent military interventions in Afghanistan and Iraq . The cultural impact of the attacks was profound and long-lasting. The U.S. housing bubble culminated in 2007 with the Great Recession , the largest economic contraction since the Great Depression. Coming to a head in

10640-597: The United States of America to Spain" to seek assistance in the Revolutionary War effort. The first known public usage is an anonymous essay published in the Williamsburg newspaper, The Virginia Gazette , on April 6, 1776. By June 1776, the "United States of America" appeared in the Articles of Confederation and the Declaration of Independence . The Second Continental Congress adopted

10780-601: The United States, slavery was abolished nationally . By 1900, the country had established itself as a great power , a status solidified after its involvement in World War I . After Japan 's attack on Pearl Harbor in December 1941, the U.S. entered World War II . Its aftermath left the U.S. and the Soviet Union as the world's two superpowers and led to the Cold War , during which both countries struggled for ideological dominance and international influence . Following

10920-898: The United States, a "Corner" is a corner of the surveyed polygon enclosing an area of land, whose location is, or was (since corners can become "lost" or "obliterated"), marked in various ways including with trees known as "bearing trees" ("witness trees" in older terminology) or "corner monuments". From analysing Native American names in the database in order to compile a dictionary, professor William Bright of UCLA observed in 2004 that some GNIS entries are "erroneous; or refer to long-vanished railroad sidings where no one ever lived". Such false classifications have propagated to other geographical information sources, such as incorrectly classified train stations appearing as towns or neighborhoods on Google Maps. The GNIS accepts proposals for new or changed names for U.S. geographical features through The National Map Corps . The general public can make proposals at

11060-541: The United States, despite the usage of "the Americas " as the totality of North and South America. "The States" is an established colloquial shortening of the name, used particularly from abroad; "stateside" is the corresponding adjective or adverb. The first inhabitants of North America migrated from Siberia across the Bering land bridge about 12,000 years ago; the Clovis culture , which appeared around 11,000 BC,

11200-624: The United States, except Iran , North Korea , and Bhutan . Though Taiwan does not have formal diplomatic relations with the U.S., it maintains close unofficial relations. The United States regularly supplies Taiwan with military equipment to deter potential Chinese aggression. Its geopolitical attention also turned to the Indo-Pacific when the United States joined the Quadrilateral Security Dialogue with Australia, India, and Japan. The United States has

11340-509: The United States. The American prison system is largely heterogenous, with thousands of relatively independent systems operating across federal, state, local, and tribal levels. In 2023, "these systems [held] almost 2 million people in 1,566 state prisons, 98 federal prisons, 3,116 local jails, 1,323 juvenile correctional facilities, 181 immigration detention facilities, and 80 Indian country jails, as well as in military prisons , civil commitment centers, state psychiatric hospitals, and prisons in

11480-572: The broadcasting masts for radio and television stations, civil divisions, regional and historic names, individual buildings, roads, and triangulation station names. The databases were initially available on paper (2 to 3 spiral-bound volumes per state), on microfiche , and on magnetic tape encoded (unless otherwise requested) in EBCDIC with 248-byte fixed-length records in 4960-byte blocks . The feature classes for association with each name included (for examples) "locale" (a "place at which there

11620-457: The contiguous United States are in the State of California, about 84 miles (135 km) apart. At an elevation of 20,310 feet (6,190.5 m), Alaska's Denali is the highest peak in the country and continent. Active volcanoes are common throughout Alaska's Alexander and Aleutian Islands , and Hawaii consists of volcanic islands. The supervolcano underlying Yellowstone National Park in

11760-423: The coordinates were taken to be those of a primary civic feature such as the city hall or town hall , main public library , main highway intersection, main post office, or central business district regardless of changes over time; these coordinates are called the "primary point". Secondary coordinates were only an aid to locating which topographic map(s) the feature extended across, and were "simply anywhere on

11900-556: The country's GDP. The U.S. has 42% of the world's nuclear weapons —the second-largest share after Russia. The United States has the third-largest combined armed forces in the world, behind the Chinese People's Liberation Army and Indian Armed Forces . The military operates about 800 bases and facilities abroad, and maintains deployments greater than 100 active duty personnel in 25 foreign countries. State defense forces (SDFs) are military units that operate under

12040-469: The country's territory would expand with the admission of new states , rather than the expansion of existing states. The U.S. Constitution was drafted at the 1787 Constitutional Convention to overcome the limitations of the Articles. It went into effect in 1789, creating a federal republic governed by three separate branches that together ensured a system of checks and balances . George Washington

12180-713: The development of self-governance, and the First Great Awakening , a series of Christian revivals , fueled colonial interest in religious liberty . For a century, the American colonists had been providing their own troops and materiel in conflicts with indigenous peoples allied with Britain's colonial rivals , especially France, and the Americans had begun to develop a sense of self-defense and self-reliance separate from Britain. The French and Indian War (1754–1763) took on new significance for all North American colonists after Parliament under William Pitt

12320-476: The domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate ( Upsampling ) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic content of the original signal be preserved without creating aliased harmonic content of

12460-422: The entire United States and that were abridged versions of the data in the other 57: one for the 50,000 most well known populated places and features, and one for most of the populated places. The files were compiled from all of the names to be found on USGS topographic maps, plus data from various state map sources. In phase 1, elevations were recorded in feet only, with no conversion to metric, and only if there

12600-580: The environment for labor unions to begin to flourish . This period eventually ended with the advent of the Progressive Era , which was characterized by significant reforms. Pro-American elements in Hawaii overthrew the Hawaiian monarchy ; the islands were annexed in 1898. That same year, Puerto Rico , the Philippines , and Guam were ceded to the U.S. by Spain after the latter's defeat in

12740-418: The feature and on the topographic map with which it is associated". River sources were determined by the shortest drain, subject to the proximities of other features that were clearly related to the river by their names. The USGS Topographic Map Names database (TMNDB hereafter) was also 57 computer files containing the names of maps: 56 for 1:24000 scale USGS maps as with the NGNDB, the 57th being (rather than

12880-544: The feature or place name are cited. Variant names, alternatives to official federal names for a feature, are also recorded. Each feature receives a permanent, unique feature record identifier, sometimes called the GNIS identifier. The database never removes an entry, "except in cases of obvious duplication." The GNIS was originally designed for four major purposes: to eliminate duplication of effort at various other levels of government that were already compiling geographic data, to provide standardized datasets of geographic data for

13020-541: The federal government often applied policies of Indian removal or assimilation . The Trail of Tears (1830–1850) was a U.S. government policy that forcibly removed and displaced most Native Americans living east of the Mississippi River to lands far to the west. These and earlier organized displacements prompted a long series of American Indian Wars west of the Mississippi. The Republic of Texas

13160-606: The first nuclear weapons and used them against the Japanese cities of Hiroshima and Nagasaki in August 1945, ending the war. The United States was one of the " Four Policemen " who met to plan the post-war world , alongside the United Kingdom , Soviet Union , and China . The U.S. emerged relatively unscathed from the war, with even greater economic power and international political influence . After World War II,

13300-424: The function at intermediate points, such as x = 2.5. {\displaystyle x=2.5.} We describe some methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and smoothness of the resulting interpolant function. The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method

13440-406: The function which we want to interpolate by g , and suppose that x lies between x a and x b and that g is twice continuously differentiable. Then the linear interpolation error is In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including polynomial interpolation and spline interpolation (described below),

13580-435: The government and others, to index all of the names found on official U.S. government federal and state maps, and to ensure uniform geographic names for the federal government. Phase 1 lasted from 1978 to 1981, with a precursor pilot project run over the states of Kansas and Colorado in 1976, and produced 5 databases. It excluded several classes of feature because they were better documented in non-USGS maps, including airports,

13720-584: The help of their teachers, a professor of linguistics, and a man who had been conducting a years-long project to collect Native American placenames in the area, in changing the names of several places that they had spotted in class one day and challenged for being racist, including renaming "Negrohead Creek" to an Athabascan name Lochenyatth Creek and "Negrohead Mountain" to Tl'oo Khanishyah Mountain, both of which translate to "grassy tussocks" in Lower Tanana and Gwichʼin respectively. Likewise, in researching

13860-437: The integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar). A key feature of mimetic interpolation is that vector calculus identities are satisfied, including Stokes' theorem and the divergence theorem . As a result, mimetic interpolation conserves line, area and volume integrals. Conservation of line integrals might be desirable when interpolating

14000-463: The integration of Army National Guard units and personnel into the U.S. Army and (since 1947) the U.S. Air Force. There are about 18,000 U.S. police agencies from local to national level in the United States. Law in the United States is mainly enforced by local police departments and sheriff departments in their municipal or county jurisdictions. The state police departments have authority in their respective state , and federal agencies such as

14140-412: The interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to least squares approximation. Approximation theory studies how to find the best approximation to

14280-487: The intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by In this case we get f (2.5) = 0.5972. Like polynomial interpolation, spline interpolation incurs

14420-422: The location of the feature's mouth, or of the approximate center of the area of the feature. Such approximate centers were "eye-balled" estimates by the people performing the digitization, subject to the constraint that centers of areal features were not placed within other features that are inside them. alluvial fans and river deltas counted as mouths for this purpose. For cities and other large populated places,

14560-482: The military draft (leading to the end of conscription in 1973) and wide opposition to U.S. intervention in Vietnam (with the U.S. totally withdrawing in 1975). A societal shift in the roles of women was significantly responsible for the large increase in female paid labor participation during the 1970s, and by 1985 the majority of American women aged 16 and older were employed. The late 1980s and early 1990s saw

14700-612: The most attractive to the population are the most vulnerable. The U.S. is one of 17 megadiverse countries containing large numbers of endemic species : about 17,000 species of vascular plants occur in the contiguous United States and Alaska, and over 1,800 species of flowering plants are found in Hawaii, few of which occur on the mainland. The United States is home to 428 mammal species, 784 birds, 311 reptiles, 295 amphibians , and around 91,000 insect species. There are 63 national parks , and hundreds of other federally managed parks, forests, and wilderness areas , managed by

14840-712: The most such names being Arizona. One of the two standard reference works for placenames in Arizona is Byrd Howell Granger's 1983 book Arizona's Names: X Marks the Place , which contains many additional names with racial slurs not in the GNIS database. Despite "Nigger" having been removed from federal government use by Stewart Udall , its replacement "Negro" still remained in GNIS names in 2015, as did " Pickaninny ", " Uncle Tom ", and " Jim Crow " and 33 places named "Niggerhead". There were 828 names containing "squaw", including 11 variations on "Squaw Tit" and "Squaw Teat", contrasting with

14980-662: The most violent, the California genocide of thousands of Native inhabitants, lasted into the early 1870s, just as additional western territories and states were created. During the colonial period, slavery had been legal in the American colonies , though the practice began to be significantly questioned during the American Revolution. States in the North enacted abolition laws , though support for slavery strengthened in Southern states , as inventions such as

15120-660: The north and Mexico to the south, with the states of Alaska to the northwest and the archipelagic Hawaii in the Pacific Ocean . The United States also asserts sovereignty over five major island territories and various uninhabited islands . The country has the world's third-largest land area , largest exclusive economic zone , and third-largest population , exceeding 334 million. Its three largest metropolitan areas are New York , Los Angeles , and Chicago , and its three most populous states are California , Texas , and Florida . Paleo-Indians migrated across

15260-442: The number of data points is infinite or if the function to be interpolated has compact support. Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to Hermite interpolation problems. When each data point is itself a function, it can be useful to see the interpolation problem as a partial advection problem between each data point. This idea leads to

15400-414: The original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process. This table gives some values of an unknown function f ( x ) {\displaystyle f(x)} . Interpolation provides a means of estimating

15540-472: The original signal above the original Nyquist limit of the signal (that is, above fs/2 of the original signal sample rate). An early and fairly elementary discussion on this subject can be found in Rabiner and Crochiere's book Multirate Digital Signal Processing . The term extrapolation is used to find data points outside the range of known data points. In curve fitting problems, the constraint that

15680-600: The point where a dam is thought to be". The National Geographic Names database (NGNDB hereafter) was originally 57 computer files, one for each state and territory of the United States (except Alaska which got two) plus one for the District of Columbia. The second Alaska file was an earlier database, the Dictionary of Alaska Place Names that had been compiled by the USGS in 1967. A further two files were later added, covering

15820-408: The policy of redlining later adopted by the federal Home Owners' Loan Corporation . An explosion of technological advancement accompanied by the exploitation of cheap immigrant labor led to rapid economic expansion during the late 19th and early 20th centuries , allowing the United States to outpace the economies of England, France, and Germany combined. This fostered the amassing of power by

15960-574: The primary colonial grievances was a denial of their rights as Englishmen , particularly the right to representation in the British government that taxed them . To demonstrate their dissatisfaction and resolve, the First Continental Congress met in 1774 and passed the Continental Association , a colonial boycott of British goods that proved effective. The British attempt to then disarm the colonists resulted in

16100-450: The problems of linear interpolation. However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computational complexity ) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see Runge's phenomenon ). Polynomial interpolation can estimate local maxima and minima that are outside

16240-458: The range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at x ≈ 1.566, f ( x ) ≈ 1.003 and a local minimum at x ≈ 4.708, f ( x ) ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive may have an interpolant with negative values, and whose inverse therefore contains false vertical asymptotes . More generally,

16380-515: The rights of African Americans . National infrastructure, including transcontinental telegraph and railroads , spurred growth in the American frontier . From 1865 through 1917 an unprecedented stream of immigrants arrived in the United States, including 24.4 million from Europe. Most came through the port of New York City , and New York City and other large cities on the East Coast became home to large Jewish , Irish , and Italian populations, while many Germans and Central Europeans moved to

16520-614: The rocky Great Basin and Chihuahua , Sonoran , and Mojave deserts. In the northwest corner of Arizona , carved by the Colorado River over millions of years, is the Grand Canyon , a steep-sided canyon and popular tourist destination known for its overwhelming visual size and intricate, colorful landscape. The Sierra Nevada and Cascade mountain ranges run close to the Pacific coast . The lowest and highest points in

16660-409: The same rules as for the NGNDB. So too did names with diacritic characters. Phase 2 was broader in scope than phase 1, extending the scope to a much larger set of data sources. It ran from the end of phase 1 and had managed to completely process data from 42 states by 2003, with 4 still underway and the remaining 4 (Alaska, Kentucky, Michigan, and New York) awaiting the initial systematic compilation of

16800-453: The seven points: Substituting x = 2.5, we find that f (2.5) = ~0.59678. Generally, if we have n data points, there is exactly one polynomial of degree at most n −1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power n . Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of

16940-499: The shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to Chebyshev polynomials . Linear interpolation uses a linear function for each of intervals [ x k , x k+1 ]. Spline interpolation uses low-degree polynomials in each of

17080-573: The six service branches, which are made up of the U.S. Army , Marine Corps , Navy , Air Force , and Space Force . The Coast Guard is administered by the Department of Homeland Security in peacetime and can be transferred to the Department of the Navy in wartime. The United States spent $ 916 billion on its military in 2023, which is by far the largest amount of any country , making up 37% of global military spending and accounting for 3.4% of

17220-545: The sole authority of a state government. SDFs are authorized by state and federal law but are under the command of the state's governor . They are distinct from the state's National Guard units in that they cannot become federalized entities. A state's National Guard personnel, however, may be federalized under the National Defense Act Amendments of 1933 , which created the Guard and provides for

17360-651: The solution of differential equations . TFC achieves this by constructing a constrained functional (a function of a free function), that inherently satisfies given constraints regardless of the expression of the free function. This simplifies solving various types of equations and significantly improves the efficiency and accuracy of methods like Physics-Informed Neural Networks (PINNs). TFC offers advantages over traditional methods like Lagrange multipliers and spectral methods by directly addressing constraints analytically and avoiding iterative procedures, although it cannot currently handle inequality constraints. Interpolation

17500-451: The sources to use. Many more feature classes were included, including abandoned Native American settlements, ghost towns , railway stations on railway lines that no longer existed, housing developments , shopping centers , and highway rest areas . The actual compilation was outsourced by the U.S. government, state by state, to private entities such as university researchers. The Antarctica Geographic Names database (AGNDB hereafter)

17640-543: The states are also represented by local elected governments , which are administrative divisions of the states. States are subdivided into counties or county equivalents , and further divided into municipalities . The District of Columbia is a federal district containing the U.S. capital, Washington, D.C. . The federal district is an administrative division of the federal government. Federally recognized tribes govern 326 Indian reservations . The United States has an established structure of foreign relations, and it has

17780-510: The structure and responsibilities of the national federal government and its relationship with the individual states. The U.S. Constitution is the world's oldest national constitution still in effect (from March 4, 1789). Composed of three branches, all headquartered in Washington, D.C., the federal government is the national government of the United States. It is regulated by a strong system of checks and balances . The three-branch system

17920-466: The use of "Nipple" in names with non- Native American allusions such as "Susies Nipple". United States The United States of America ( USA ), commonly known as the United States ( U.S. ) or America , is a country primarily located in North America . It is a federal union of 50 states and a federal capital district, Washington, D.C. The 48 contiguous states border Canada to

18060-446: The values of a function for a limited number of values of the independent variable . It is often required to interpolate ; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from

18200-582: The world's fourth-longest river system , runs predominantly north–south through the heart of the country. The flat and fertile prairie of the Great Plains stretches to the west, interrupted by a highland region in the southeast. The Rocky Mountains , west of the Great Plains, extend north to south across the country, peaking at over 14,000 feet (4,300 m) in Colorado . Farther west are

18340-583: The world's second-largest diplomatic corps as of 2024 . It is a permanent member of the United Nations Security Council , and home to the United Nations headquarters . The United States is a member of the G7 , G20 , and OECD intergovernmental organizations. Almost all countries have embassies and many have consulates (official representatives) in the country. Likewise, nearly all countries host formal diplomatic missions with

18480-444: The world's oldest surviving federation, and its presidential system of national government has been adopted, in whole or in part, by many newly independent states worldwide following their decolonization . It is a liberal representative democracy "in which majority rule is tempered by minority rights protected by law ". The Constitution of the United States serves as the country's supreme legal document , also establishing

18620-553: The world's tornadoes occur in the country , mainly in Tornado Alley . Overall, the United States receives more high-impact extreme weather incidents than any other country. Extreme weather became more frequent in the U.S. in the 21st century, with three times the number of reported heat waves as in the 1960s. In the American Southwest , droughts became more persistent and more severe. The regions considered as

18760-539: The world, with almost 2 million people incarcerated . An analysis of the World Health Organization Mortality Database from 2010 showed U.S. homicide rates "were 7 times higher than in other high-income countries, driven by a gun homicide rate that was 25 times higher". Interpolation In engineering and science , one often has a number of data points, obtained by sampling or experimentation , which represent

18900-575: Was annexed in 1845, and the 1846 Oregon Treaty led to U.S. control of the present-day American Northwest . Victory in the Mexican–American War resulted in the 1848 Mexican Cession of California, Nevada, Utah, and much of present-day Colorado and the American Southwest . The California gold rush of 1848–1849 spurred a huge migration of white settlers to the Pacific coast, leading to even more confrontations with Native populations. One of

19040-755: Was added in the 1990s and comprised records for BGN-approved names in Antarctica and various off-lying islands such as the South Orkney Islands , the South Shetland Islands , the Balleny Islands , Heard Island , South Georgia , and the South Sandwich Islands . It only contained records for natural features, not for scientific outposts. The media on which one could obtain the databases were extended in

19180-759: Was adopted on July 4, 1776. The political values of the American Revolution included liberty , inalienable individual rights ; and the sovereignty of the people ; supporting republicanism and rejecting monarchy , aristocracy , and all hereditary political power; civic virtue ; and vilification of political corruption . The Founding Fathers of the United States , who included Washington, Jefferson, John Adams , Benjamin Franklin , Alexander Hamilton , John Jay , James Madison , Thomas Paine , and many others, were inspired by Greco-Roman , Renaissance , and Enlightenment philosophies and ideas. The Articles of Confederation were ratified in 1781 and established

19320-446: Was an actual elevation recorded for the map feature. They were of either the lowest or highest point of the feature, as appropriate. Interpolated elevations, calculated by interpolation between contour lines , were added in phase 2. Names were the official name, except where the name contained diacritic characters that the computer file encodings of the time could not handle (which were in phase 1 marked with an asterisk for update in

19460-583: Was elected the country's first president under the Constitution, and the Bill of Rights was adopted in 1791 to allay skeptics' concerns about the power of the more centralized government. His resignation as commander-in-chief after the Revolutionary War and his later refusal to run for a third term as the country's first president established a precedent for the supremacy of civil authority in

19600-567: Was undocumented, and it was (in the words of a 1986 report from the Engineer Topographic Laboratories of the United States Army Corps of Engineers ) "an unreasonable determination", with the likes of Cayuga Lake being labelled a "tank". The USACE report assumed that "tank" meant "reservoir", and observed that often the coordinates of "tanks" were outside of their boundaries and were "possibly at

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