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State of West Virginia
Flag Seal
Nickname(s): Mountain State
Motto(s): Montani semper liberi (Mountaineers Are Always Free)
Before Statehood Known as
The Virginia
Official language(s) none (de facto English)
Capital Charleston
Largest city Charleston (Capital)
Largest metro area Charleston metro area
Area Ranked 41st
- Total 24,230 sq mi
(62,755 km²)
- Width 130 miles (210 km)
- Length 240 miles (385 km)
- % water 0.6
- Latitude 37° 12′ N to 40° 39′ N
- Longitude 77° 43′ W to 82° 39′ W
Population Ranked 37th in the U.S.
- Total 1,855,364 (2011 est)[1]
- Density 77.1/sq mi (29.8/km2)
Ranked 29th in the U.S.
- Median income $38,029 (48th)
- Highest point Spruce Knob[2][3]
4,862 ft (1482 m)
- Mean 1,500 ft (460 m)
- Lowest point Potomac River
at Virginia border[2][3]
240 ft (73 m)
Admission to Union June 20, 1863 (35)
Governor Earl Ray Tomblin (D)
U.S. Senators Jay Rockefeller (D)
Joe Manchin (D)
Time zone Eastern: UTC-5/-4
Abbreviations WV US-WV
Web site wv.gov
West Virginia is a state in the Appalachia/Upland South region of the United States. West Virginia broke away from Virginia during the American Civil War and was admitted to the Union as a separate state on June 20, 1863. It is the only state formed as a direct result of the Civil War, and the only state to form by seceding from a preexisting state.
West Virginia is one of the Border States. The state's Northern Panhandle, and North-Central region, is tied to Pittsburgh, Pennsylvania. Also, those in the Eastern Panhandle are connected with the Washington, D.C. suburbs in Maryland and Virginia, and southern West Virginians often consider themselves Southerners. Finally, the towns and farms along the mid-Ohio River have an appearance and culture somewhat resembling the Midwest.[4]
1.1 Climate
2.1 European exploration and settlement
2.2 Trans-Allegheny Virginia, 1776-1861
2.3 Separation from Virginia
3 Law and government
3.1 State and local
3.2 Politics
4.1 Resources
4.2 Agriculture and industry
4.3 Transportation
7 Culture and points of interest
The state is noted for its great natural beauty, its historically significant logging and coal mining industries, and its labor history. It is also well known as a tourist destination for those people interested in outdoor activities such as skiing, whitewater rafting, rock climbing, fishing, hiking, and hunting.
West Virginia is bordered by Pennsylvania to the north; by Ohio to the north and west; by Kentucky to the west; by Maryland to the north and east; and by Virginia to the east and south. The Ohio and Potomac rivers form parts of the boundaries.
West Virginia is the only state in the nation located entirely within the Appalachian Mountain range, and in which all areas are mountainous; for this reason it is nicknamed "The Mountain State." About 75 percent of the state is within the Cumberland Plateau and Allegheny Plateau regions. Though the relief is not high, the plateau region is extremely rugged in most areas.
Shaded relief map of the Cumberland Plateau and Ridge-and-valley Appalachians.
On the eastern state line with Virginia, high peaks in the Monongahela National Forest region give rise to an island of colder climate and ecosystems similar to those of northern New England and eastern Canada. The highest point in the state is atop Spruce Knob, which at 4,863 feet (1,482 m)[2] is covered in a boreal forest of dense spruce trees at altitudes above 4,000 feet (1,220 m). Spruce Knob lies within the Monongahela National Forest and is a part of the Spruce Knob-Seneca Rocks National Recreation Area.[5] A total of six wilderness areas can also be found within the forest. Outside the forest to the south, the New River Gorge is a 1,000-foot (304 m) deep canyon carved by the New River. The National Park Service manages that part of the gorge and river designated as the New River Gorge National River, one of only 15 rivers in the U.S. with this level of protection.
The native vegetation for most of the state was originally mixed hardwood forest of oak, chestnut, maple, beech, and white pine, with willow and American sycamore along the state's waterways. Many of the areas are rich in biodiversity and scenic beauty. Ecologically, most of West Virginia falls into the Appalachian mixed mesophytic forests ecoregion.
The underlying rock strata are sandstones, shales, bituminous coal beds, and limestones laid down in a near shore environment from sediments derived from mountains to the east, in a shallow inland sea on the west. Some beds illustrate a coastal swamp environment, some river delta, some shallow water. Sea level rose and fell many times during the Mississippian and Pennsylvanian eras, giving a variety of rock strata. The Appalachian Mountains are some of the oldest on earth, having formed over 300 million years ago.
The summit of Spruce Knob is often covered in clouds.
The climate of West Virginia borders on a humid subtropical climate (Koppen climate classification Cfa) in the lower elevations of the extreme southwestern part of the state (including Huntington) and parts of the Eastern Panhandle east of the Appalachians with hot, humid summers and milder winters. The rest of the state has a humid continental climate (Koppen climate classification Dfa, or Dfb at the higher elevations) with warm to hot, humid summers and cold winters, increasing in severity with elevation. However, the weather is subject in all parts of the state to change. The hardiness zones range from zone 5b in the central Appalachian mountains to zone 7a in the warmest parts of the lowest elevations. In the Eastern Panhandle and the Ohio River Valley temperatures are warm enough to see and grow subtropical plants, such as Southern magnolia (Magnolia grandiflora), Crepe Myrtle, Albizia julibrissin, American Sweetgum, and even the occasional needle palm and sabal minor. These plants don't thrive as well in other parts of the state.
Average January temperatures range from around 28°F (-2°C) near the Cheat River to 43°F (5°C) along sections of the border with Kentucky. July averages range from 67°F (19°C) along the North Branch Potomac River to 76°F (24°C) in the western part of the state. It is cooler in the mountains than in the lower sections of the state.
Annual precipitation ranges from less than 32 inches (810 mm) in the lower eastern section to more than 56 inches (1,400 mm) in higher parts of the Allegheny Front. Slightly more than half the rainfall occurs from April to September. Dense fogs are common in many valleys of the Kanawha section, especially the Tygart Valley. Snow usually lasts only a few days in the lower sections but may persist for weeks in the higher mountain areas. An average of 34 inches (86 cm) of snow falls annually in Charleston, although during the winter of 1995-1996, more than three times that amount fell as several cities in the state established new records for snowfall.
Many ancient human-made earthen mounds from various mound builder cultures survive, especially in the areas of Moundsville, South Charleston, and Romney. Although little is known about these civilizations, artifacts found there are evidence of a complex, stratified culture with metal tools and implements.
By 1600, organized tribes such as the Delaware and Shawnee had moved into the area. The Iroquois Confederacy of Cayuga, Mohawk, Oneida, Onondaga, and Seneca, and eventually the Tuscarora began to move into the area from present-day New York. This Six Nation Confederacy dominated the area, but allowed other tribes to remain in the area.
The Ohio Valley area of what was then western Virginia began to attract European colonists during the 1600s. As a result, the Native Americans were forced west, while many succumbed to disease and warfare. The Six Nations tribes formed alliances with both the British and French, who were fighting each other for dominance of the "New World."[6]
European exploration and settlement
Thomas Lee, the first manager of the Ohio Company of Virginia.
In 1671, General Abram Wood, at the direction of Royal Governor William Berkeley of the Virginia Colony, sent a party that discovered Kanawha Falls. In 1716, Governor Alexander Spotswood and about thirty horsemen made an excursion into what is now Pendleton County. John Van Metre, an Indian trader, entered the northern region in 1725. The same year, German settlers from Pennsylvania founded New Mecklenburg, the present Shepherdstown, on the Potomac River.
King Charles II of England, in 1661, granted to a company of gentlemen the land between the Potomac and Rappahannock rivers, known as the Northern Neck. The grant finally came into the possession of Thomas Fairfax, 6th Lord Fairfax of Cameron, and in 1746, a stone was erected at the source of the North Branch Potomac River to mark the western limit of the grant. A considerable part of this land was surveyed by George Washington between 1748 and 1751. His diary indicates that many squatters, mostly German, were living along the South Branch Potomac River. Christopher Gist, a surveyor for the first Ohio Company, explored the country along the Ohio River north of the mouth of the Kanawha River, between 1751 and 1752, with the aim of establishing a fourteenth colony named Vandalia. Many settlers crossed the mountains after 1750, encountering Native American resistance. Apparently, few Native Americans lived within the present limits of the state, but the region was a common hunting ground, crossed also by many war trails. During the French and Indian War, the scattered settlements were mostly destroyed.
Native American attacks continued until after the American Revolutionary War. During the war, the settlers in Western Virginia were generally active Whigs and many served in the Continental Army.
Trans-Allegheny Virginia, 1776-1861
Social conditions in western Virginia were entirely unlike those in the eastern portion of the state. A movement to create a state beyond the Alleghenies was revived and a petition for the establishment of "Westsylvania" was presented to Congress, with the argument that the mountains made an almost impassable barrier on the east. The rugged nature of the country made slavery unprofitable, and time only increased the social, political, economic, and cultural differences.
The convention that met in 1829, to form a new constitution for Virginia, against the protest of the counties beyond the mountains, required a property qualification for suffrage and gave the slave-holding counties the benefit of three-fifths of their slave population in apportioning the state's representation in the U.S. House of Representatives. As a result, every county beyond the Alleghenies except one voted to reject the constitution, which nevertheless passed because of eastern support. Though the Virginia Constitution of 1850 provided for white male suffrage, the distribution of representation among the counties continued to give control to the section east of the Blue Ridge Mountains.
Separation from Virginia
John S. Carlile, a leader during the First Wheeling Convention
West Virginia is the only state in the Union to secede from another state, Virginia, in U.S. history.[7] On April 17, 1861, fifteen of the forty-six delegates from the area located in the present state of West Virginia voted to secede from the United States.[8] Almost immediately after the vote to proceed with secession prevailed in the Virginia General Assembly, a mass meeting at the town of Clarksburg recommended that each county in northwestern Virginia send delegates to a convention to meet in Wheeling on May 13, 1861. When this First Wheeling Convention met, 425 delegates from 25 counties were present, with more than one-third of the delegates from the northern panhandle area.[9] On May 23 1861, secession from the U.S. was ratified by a large majority in the state as a whole, with the western counties voting approximately 34,677 against and 19,121 for.[10]
The Second Wheeling Convention met as agreed on June 11, and declared that, since the Secession Convention had been called without the consent of the people, all its acts were void, and that all who adhered to it had vacated their offices. The Wheeling Conventions, and the delegates themselves, were never actually elected by public ballot to act on behalf of western Virginia.[11] An act for the reorganization of the government was then passed on June 19. The Wheeling Convention members met on June 20, and organized a state government, electing two U.S. senators who were recognized at Washington, D.C. At that point, therefore, there were two state governments in Virginia, one pledging allegiance to the United States and one to the Confederacy.
The Wheeling Convention, which had taken a recess until August 6, then reassembled on August 20, and called for a popular vote on the formation of a new state and for a convention to frame a constitution if the vote should be favorable. At the election on October 24, 1861, 18,489 votes were cast for the new state and only 781 against.[12] The convention began on November 26, 1861, and finished its work on February 18, 1862, and the instrument was ratified (18,162 for and 514 against) on April 11, 1862.
Harpers Ferry (as it appears today) changed hands a dozen times during the American Civil War.
On May 13, the state legislature of the reorganized government approved the formation of the new state. An application for admission to the Union was made to Congress, and on December 31, 1862, an enabling act was approved by President Abraham Lincoln admitting West Virginia, on the condition that a provision for the gradual abolition of slavery be inserted in the Constitution. The Convention was reconvened on February 12, 1863, and the demand was met. The revised constitution was adopted on March 26 1863, and on April 20 1863, President Lincoln issued a proclamation admitting the state at the end of sixty days (June 20 1863).
First Confederate Memorial, Romney.
The question of the constitutionality of the formation of the new state was brought before the Supreme Court of the United States: The Virginia General Assembly repealed the act of secession and, in 1866, brought suit against West Virginia asking the court to declare the counties a part of Virginia. The U.S. Congress, however, passed a joint resolution recognizing the transfer on March 10, 1866. The Supreme Court, in 1870, decided in favor of West Virginia.[13]
During the American Civil War, West Virginia suffered comparatively little. George B. McClellan's forces gained possession of the greater part of the territory in the summer of 1861, and Union control was never seriously threatened, in spite of the attempt by Robert E. Lee in the same year. In 1863, General John D. Imboden, with 5,000 Confederates, overran a considerable portion of the state. Bands of guerrillas burned and plundered in some sections, and were not entirely suppressed until after the war ended.
The area which became West Virginia furnished about an equal amount of soldiers to the Federal and Confederate governments, approximately 22-25,000 each.[14] The Wheeling government disenfranchised returning Confederate soldiers in 1865.[15] In addition, Confederates' property could be confiscated, and in 1866, a constitutional amendment disfranchised all who had given aid and comfort to the Confederacy. Shifting political control in the state led to the abrogation of the 1866 amendment, and eventually, an entirely new state constitution was adopted on August 22, 1872.
Law and government
The West Virginia State Capitol.
West Virginia's capital and seat of government is the city of Charleston, located in the southwest area of the state. Charleston, founded in 1794, was built on a settlement near the mouth of the Elk River dating from 1788. Once West Virginia was finally proclaimed a state by President Lincoln, a new state capital had to be named. The capital shifted between Charleston and Wheeling for several years until Charleston was finally chosen by popular vote in 1877. The first capitol building was built in 1885. The current structure was dedicated in 1932, after the first burned down.[16]
The governor of West Virgina is elected every four years, on the same day as the U.S. Presidential election, and is sworn in during the following January. Governors of West Virginia can serve two consecutive terms, but must sit out a term before serving a third term in office. Other elected officials in the executive branch include the Agriculture Commissioner, Auditor, Attorney General, Secretary of State, and Treasurer.[17]
The West Virginia Legislature is bicameral, consisting of the House of Delegates and the Senate. It is a citizen's legislature, meaning that legislative office is not a full-time occupation, but rather a part-time position. Consequently, the legislators often hold a full-time job in their community of residence. Typically, the legislature is in session for 60 days between January and early April, closing at midnight on the last day of the session. During the remainder of the year, monthly interim sessions are held in preparation for the regular session. Legislators also gather periodically for "special" sessions when called by the governor.
West Virginia's highest court is the Supreme Court of Appeals. The Supreme Court of Appeals of West Virginia is the busiest appellate court of its type in the United States. West Virginia is one of 11 states with a single appellate court. The state constitution allows for the creation of an intermediate court of appeals, but the Legislature has never created one. The Supreme Court is made up of five justices, elected in partisan elections to 12-year terms. For the purpose of courts of general jurisdiction, the state is divided into 31 judicial circuits. Each circuit is made up of one or more counties. Circuit judges are elected in partisan elections to serve eight-year terms.
West Virginia is one of twelve states that does not have a death penalty.
West Virginia is an alcoholic beverage control state. It retains a monopoly on wholesaling of distilled spirits and does not operate retail outlets.
West Virginia counties administer and collect property taxes. Municipalities may levy license and gross receipts taxes on businesses located within the city limits and a hotel occupancy tax on lodging places in the city. Although the Department of Tax and Revenue plays a major role in the administration of this tax, less than one-half of one percent of the property tax collected goes to state government. The primary beneficiaries of the property tax are county boards of education. The WV Department of Tax and Revenue supervises and otherwise assists counties and municipalities in their work of assessment and tax rate determination.
West Virginia has a history of supporting Democratic candidates in state and local elections. It also has a very strong tradition of union membership. While the state continued its Democratic tradition by supporting Bill Clinton by large margins in 1992 and 1996, a majority of West Virginia voters supported George W. Bush in 2000 and 2004.
The most consistent support for Democrats is found in the coal region of southern West Virginia (especially McDowell, Mingo, Logan, Wyoming, and Boone Counties). Republicans dominate east of the Allegheny Mountains, especially in the state's Eastern Panhandle and Potomac Highlands. The Northern Panhandle and North-Central West Virginia regions usually split right down the middle in terms of being Republican or Democrat.
West Virginia State Symbols
Motto: Montani semper liberi (Latin, "Mountaineers are Always Free")
Slogan: Open for Business
Bird: Northern Cardinal, (Cardinalis cardinalis)
Animal: Black Bear, (Ursus americanus)
Fish: Brook Trout, (Salvelinus fontinalis)
Insect: European Honey Bee, (Apis mellifera)
Flower: Rhododendron, (Rhododendron maximum)
Tree: Sugar Maple, (Acer saccharum)
Song: "The West Virginia Hills"
"This Is My West Virginia"
"West Virginia, My Home Sweet Home"
Gemstone: Silicified Mississippian Fossil Coral(Lithostrotionella)
Colors: Old Gold and Blue
Butterfly: Monarch Butterfly, (Danaus plexippus)
Fruit: Golden Delicious Apple, (Malus domestica)
The economy of West Virginia is now one of the most fragile of any state in the U.S. According to U.S. Census Bureau data, West Virginia is the third lowest in per capita income,[18] ahead of only Arkansas and Mississippi. It also ranks last in median household income.[19] The proportion of West Virginia's adult population with a bachelor's degree is the lowest in the U.S., at 15.3 percent.[20]
Salt mining had been underway since the eighteenth century, though it had largely played out by the time of the American Civil War, when the red salt of Kanawha County was a valued commodity of first Confederate, and later Union forces. Later, more sophisticated mining methods would restore West Virginia's role as a major producer of salt.
In the second half of the nineteenth century, the discovery of major deposits on bituminous coal led to the development of extensive transport networks and the expansion of industry throughout the U.S. The residents (both Native Americans and early European settlers) had known of the underlying coal and used it for heating and fuel. These very small "personal" mines were the only practical development, pre-railroads. After the Civil War, the new railroads were a practical means to transport large quantities of coal to expanding U.S. and export markets. As the anthracite mines of northwestern New Jersey and Pennsylvania were playing out, investors and industrialists focused on new interest in West Virginia's coal. Geologists such as Dr. David T. Ansted surveyed potential coal fields and invested in land and early mining projects.
The completion of the Chesapeake and Ohio Railway (C&O) across the state to the new city of Huntington on the Ohio River, in 1872, opened access to the New River Coal Field. The C&O built a huge coal pier at Newport News, Virginia at the Virginia port of Hampton Roads. In 1881, the owners of the newly re-named Norfolk and Western Railroad (N & W) built the line to capture the coal production. Roanoke, Virginia, emerged as a trans-shipment point for West Virginia's coal on its way to the ports. In 1889, in the southern part of the state, along the Norfolk and Western rail lines, the coal center of Bluefield, West Virginia, was founded. The "capital" of the enormous Pocahontas coalfield, Bluefield would be the largest city in the southern part of the state for decades.
Bituminous coal seam in southwestern West Virginia
After Reconstruction, the new 35th state benefited from development of its mineral resources more than any other single economic activity. In northern West Virgina, and elsewhere, the older Baltimore and Ohio Railroad (B&O) and other lines expanded to take advantage of coal opportunities as well. The B&O developed coal piers in Baltimore and at several ports on the Great Lakes. Other significant carriers of WV coal were the Western Maryland Railway (WM), Southern Railway (SOU), and the Louisville and Nashville Railroad (L&N).
By 1900, only the coalfields in the most rugged terrain of southern West Virginia were untapped. Within this area west of the New River Coalfield in Raleigh and Wyoming counties lay the Winding Gulf Coalfield, later promoted as the "Billion Dollar Coalfield." A short-line railroad was chartered to stretch between the C&O, along the Kanawha River, and the N&W at Matoaka, a distance of about 80 miles. With financial support from major industrialists, this railroad was eventually built across the state to the Hampton Roads port. After merging with the N & W, the two railroads created the modern Norfok Southern system.
As coal mining and related work became a major employment activities in the state, there was considerable labor strife as working conditions, safety issues, and economic concerns arose. Even in the twenty-first century, mining safety and ecological concerns are challenging the state whose coal continued to power electrical generating plants in many other states.
Coal is not the only valuable mineral found in West Virginia, as the state was the site of the 1928 discovery of the 34.48 carat (6.896 g) Jones Diamond. West Virginia also has some oil reserves but currently only has a few small to medium sized oil and natural gas fields being tapped.
Agriculture and industry
The Veterans Memorial Bridge, which carries U.S. Route 22.
Farming is very limited in West Virginia because of the state's mountainous terrain. The chemical industry has its roots in the earliest salt mining carried out by Native Americans in the 1700s, using the salt brines of the Little Kanawha River. After the Civil War, when the salt mines started to run out, the manufacture of simple salt compounds like bromine and potassium salts began. Eventually, more complex manufacturing developed. Abundant natural resources, including carbon from coal, and hydropower from mountainous rivers fueled the expansion of the chemical industry all along the Kanawha Valley. The start of World War I led to increased demand for chemical compounds, and with the closing off of Germany's chemical industry, the allies and the rest of the world turned to West Virginia. DuPont, Carbide, and Monsanto, among others, expanded throughout the twentieth century. The chemical industry has expanded to include petrochemicals and synthetic materials. The industry has provided thousands of jobs and extensive infrastructure to West Virginia, however, concerns about exposure and epidemiologic evidence of cancer and other diseases resulting from air- and water-borne pollution, and more recently, potential terrorist threats have led to considerable regulation of the industry.[21]
The New River Gorge Bridge.
Highways form the backbone of transportation systems in West Virginia, with over 37,300 miles of public roads in the state.[22] Airports, railroads, and rivers complete the commercial transportation modes for West Virginia. Commercial air travel is facilitated by airports in Charleston, Huntington, Beckley, Bluefield, Lewisburg, Clarksburg, Martinsburg, Morgantown, Wheeling, and Parkersburg. West Virginia University in Morgantown boasts a PRT (personal rapid transit) system, the state's only single rail public transit system. Developed by Boeing, the WVU School of Engineering and the U.S. Department of Transportation, it was a model for low-capacity light transport designed for smaller cities.
Rail lines in the state were once more prevalent, but many have been discontinued because of increased automobile traffic. Many old tracks have been converted to rail trails for recreational use, and the state is still served by a few commercial lines for hauling coal and by Amtrak. In 2006, Norfolk Southern, along with the West Virginia and U.S. Government approved a plan to modify many of the rail tunnels in West Virginia, especially in the southern half of the state, to allow for double stacked cars. This is expected to also help bring economic growth to the southern half of the state.
Because of the mountainous nature of the entire state, West Virginia has several notable tunnels and bridges. The most famous of these is the New River Gorge Bridge, which was at a time the longest steel single-arch bridge in the world with a 3,031-foot (924 m) span. The bridge is also pictured on the West Virginia state quarter. The Veterans Memorial Bridge (Weirton-Steubenville Bridge) was at its time of construction one of only three cable-stayed steel girder trusses in the United States. It connects Steubenville, Ohio, with Weirton, West Virginia, along U.S. Route 22.
Education is overseen at the state level by the West Virginia Department of Education, which controls the State Board of Education, the WV Schools for the Deaf and Blind, Office of Education Performance Audits, and the Regional Education Service Agencies.
West Virginia University in Morgantown, Marshall University in Huntington, and West Virgina State University in Institute are among the public colleges and universities in the state.
West Virginia population density map.
As of 2005, West Virginia had an estimated population of 1,816,856, an increase of 4,308, or 0.2 percent, from the prior year and an increase of 8,506, or 0.5 percent, since the year 2000. This includes a natural decrease since the last census of 3,296 people (that is 108,292 births minus 111,588 deaths) and an increase from net migration of 14,209 people into the state. Immigration from outside the United States resulted in a net increase of 3,691 people, and migration within the country produced a net increase of 10,518 people.
Only 1.1 percent of the state's residents were foreign-born, placing West Virginia last among the 50 states in that statistic. It has the lowest percentage of residents that speak a language other than English in the home (2.7 percent).
The five largest ancestry groups in West Virginia are: American (23.2 percent), German (17.2 percent), Irish (13.5 percent) (Most actually Scots-Irish), English (12 percent), Italian (4.8 percent).
Culture and points of interest
West Virgina has a culture of strong self-reliance developed through its early, and continued, isolation. Largely unaffected by circumstances taking place outside the mountain settlements, its rural regions retain a unique culture. Musical instruments, ballads, and handicrafts from generations past remain. The state has taken advantage of its unique cultural heritage and traditions, promoting them through tourism.
One of West Virginia's greatest natural resources is its pristine mountain environment. There are over 35 state parks. Numerous festivals—ethnic, heritage, craft, and food, among others—take place annually in these parks and throughout the state.[23]
↑ Annual Estimates of the Resident Population for the United States, Regions, States, and Puerto Rico: April 1, 2010 to July 1, 2011 (CSV). 2011 Population Estimates. United States Census Bureau, Population Division (December 2011). Retrieved December 21, 2011.
↑ 2.0 2.1 2.2 Elevations and Distances in the United States. United States Geological Survey (2001). Retrieved October 24, 2011.
↑ 3.0 3.1 Elevation adjusted to North American Vertical Datum of 1988.
↑ West Virginia Division of Culture and History, An Introduction to West Virginia's Ethnic Communities. Retrieved December 25, 2007.
↑ Monongahela National Forest, Welcome to the Monongahela National Forest. Retrieved December 25, 2007.
↑ West Virginia Division of Culture and History, Native American Clashes with European Settlers. Retrieved December 25, 2007.
↑ West Virginia Division of Culture and History, A State of Convenience, the Creation of West Virginia, Reorganized Government of Virginia Approves Separation. Retrieved December 25, 2007.
↑ Charles Ambler, The History of West Virginia (1933), p. 309.
↑ J. McGregor, The Disruption of Virginia, p. 193.
↑ Richard O. Curry, A House Divided, p. 147.
↑ Ambler, p. 318.
↑ Curry, p. 86.
↑ Find Law, Virginia v. West Virginia, 78 U.S. 39 (1870). Retrieved June 16, 2008.
↑ West Virginia Division of Culture and History, Researching your Civil War Ancestor. Retrieved December 25, 2007.
↑ Charles Ambler, "Disfranchisement in West Virginia," Yale Review, 1905, p. 41
↑ Charleston Convention & Visitors Bureau, Your Doorway to West Virginia. Retrieved December 25, 2007.
↑ State of West Virginia, Executive Branch. Retrieved December 25, 2007.
↑ U.S. Census Bureau, State Rankings—Personal Income Per Capita In Constant (2000) Dollars 2004. Retrieved December 25, 2007.
↑ U.S. Census Bureau,State Rankings—Median Household Income, 2003. Retrieved December 25, 2007.
↑ U.S. Census Bureau, State Rankings—Persons 25 Years Old And Over With A Bachelor's Degree Or More, 2004. Retrieved December 25, 2007.
↑ Nathan Cantrell, West Virginia's chemical Industry, West Virginia Historical Society. Retrieved December 25, 2007.
↑ West Virginia Department of Transportation, Roadways. Retrieved December 25, 2007.
↑ Encyclopædia Britannica, West Virginia, Encyclopædia Britannica Online. Retrieved December 25, 2007.
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Milnes, Gerald. 1999. Play of a Fiddle Traditional Music, Dance, and Folklore in West Virginia. Lexington, Ky: University Press of Kentucky. ISBN 0813120802.
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Rice, Otis K. 1985. West Virginia a History. Lexington, Ky: University of Kentucky. ISBN 081311540X.
Seltzer, Curtis. 1985. Fire in the Hole Miners and Managers in the American Coal Industry. Lexington, Ky: University Press of Kentucky. ISBN 0813115264.
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Williams, John Alexander, and Topper Sherwood. 1993. West Virginia a History for Beginners. Charleston, [W. Va.]: Appalachian Editions. ISBN 0962748633.
Williams, John Alexander. 1976. West Virginia a Bicentennial History. New York: Norton. ISBN 0393055906.
Williams, John Alexander. 1976. West Virginia and the Captains of Industry. Morgantown: West Virginia University Library. ISBN 0870122371.
Williams, John Alexander. 2002. Appalachia a History. Chapel Hill: University of North Carolina Press. ISBN 0807860522.
This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.
All links retrieved August 8, 2013.
West Virginia Legislature. Constitution of West Virginia.
United States Geological Survey. Science In Your Backyard: West Virginia. R
West Virginia Wild and Wonderful. http://www.wvtourism.com/default.aspx West Virginia Division of Tourism].
Political divisions of the United States
States Alabama | Alaska | Arizona | Arkansas | California | Colorado | Connecticut | Delaware | Florida | Georgia | Hawaii | Idaho | Illinois | Indiana | Iowa | Kansas | Kentucky | Louisiana | Maine | Maryland | Massachusetts | Michigan | Minnesota | Mississippi | Missouri | Montana | Nebraska | Nevada | New Hampshire | New Jersey | New Mexico | New York | North Carolina | North Dakota | Ohio | Oklahoma | Oregon | Pennsylvania | Rhode Island | South Carolina | South Dakota | Tennessee | Texas | Utah | Vermont | Virginia | Washington | West Virginia | Wisconsin | Wyoming
Federal district District of Columbia
Insular areas American Samoa | Baker Island | Guam | Howland Island | Jarvis Island | Johnston Atoll | Kingman Reef | Midway Atoll | Navassa Island | Northern Mariana Islands | Palmyra Atoll | Puerto Rico | Virgin Islands | Wake Island
West Virginia history
History of "West Virginia"
Retrieved from http://www.newworldencyclopedia.org/p/index.php?title=West_Virginia&oldid=972237


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And even for there is no proper or a stable update for versions above 11.2, it is pointless to remain for a specific about Cydia download iOS 11.3 in a short time. If there will be an update from CoolStar for 11.2.2 and respective sessions as announced, we can hopefully remain for them to bang 11.3 as well before long. Consequently, stay tuned for we do not know how they have been planned for.
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It's that time of year again, folks! A little more than two months until Dragon*Con! 71 Days. This of course means time to get serious!
1 – A Tentative Costume List – This is the one that absolutely changes the most! Things have to be made, costume pieces are scattered about and it's hard to find all of the pieces, group photoshoots, you don't fit a costume anymore…All kinds of factors can change the list in a heartbeat! The way I format my costuming list has every piece and accessory needed for a costume, whether or not it is finished, and then finally whether or not it is packed. I think this format is the most handy and ensures that you don't leave something behind!
2 – A Tentative Schedule – That's right, I like to stalk all of the fan tracks and try to figure out WHAT to do – what I want to do, what I have to do, etc. Like this year for example, I'm supposed to be on a couple of panels so I have to find that out ASAP and plan accordingly. Also there are set photo op times with celebs so you have to plan around those, too. This is another one that can change abruptly because let's face it – sometimes you lose track of time at D*Con because of all of the fun you're having, so you just forget to go to something..
* A side note about my schedule, I WILL schedule times to go eat something each day because frankly it's dumb to skip meals – even at Dragon.
3 – Room Deposits – If you are the room holder for your group, this is CRUCIAL!!! I keep a running tally of what each person has paid me for their share of the room. I let them pay in increments as they can, with a deadline of August 1st. This way people have time to pay and they risk losing their spot if they don't cough up the dough. And also that way no one gets overcharged because I have an accurate running total with formulas in the spreadsheet calculating what their remaining total is. While this one doesn't change that often, I feel like it is one of the most important spreadsheets.
Now one may think that with ALL of these spreadsheets, how can anyone focus on the fun of D*Con? It's actually not that difficult. The costume list gets chucked once everything is packed and ready to go! That puts me at 2. Everyone should have paid their share of the room by the time I get there so that cuts out another one. That only leaves me with the schedule which I usually just print out and go from there. And I stray from that schedule quite a bit. I use categories like "MUST GO", "Want To Go", and "Maybe Check Out". It's easy. And color coded. It fits my OCD quite well!
Another fantastic idea to help stay organized with groups and what not is Google+ on your smartphone! You can chat with certain people and figure out where events have moved to, what time, etc. Things get hectic and they change all of the time at the con.
I hope some of these tips and tricks help some of you out and help make your Pre-Dragon*Con experience a little LESS stressful! Remember, we're all there to have FIVE AWESOME DAYS OF FUN! See you all there!


text:
Monday | May 8th, 2017
Substance Lacking In Lobbying Executive Order
President Donald Trump's latest Executive Order (EO) was titled, "Promoting Free Speech and Religious Liberty" but whether it does either remains to be seen. Observers said the EO doesn't change existing regulations much, if at all, since the Internal Revenue Service (IRS) so rarely enforces the Johnson Amendment, which prohibits charities from directly endorsing political candidates and electioneering .
The American Civil Liberties Union (ACLU) initially promised to file suit against the administration regarding the executive order but determined, after a review, that the order does not meaningfully alter the ability of religious institutions or individuals to intervene in the political process, according to Executive Director Anthony Romero. The EO signing "was an elaborate photo-op with no discernible policy outcome," he said. "The order portends but does not yet do harm to the provision of reproductive health services," he added.
"The directive to federal agencies to explore religious-based exceptions to healthcare does cue up a potential future battle, but as of now, the status quo has not been changed," Romero said. "What President Trump did was merely provide a faux sop to religious conservatives and kick the can down the road on religious exemptions on reproductive health care services," he said in a statement.
The EO states that it will be the policy of the executive branch to "vigorously enforce federal law's robust protections of religious freedom," and will honor and enforce those protections. The Treasury Secretary will ensure that the department does not take adverse action against any organization for "speaking about moral or political issues from a religious perspective, where speech of similar character has not ordinarily been treated as participation or intervention in a political campaign on behalf or opposition to a candidate for office."
How that differs from existing policy or law, if at all, is unclear. Charities and churches are allowed to comment publicly on issues of the day so long as political activity is not a substantial portion of their work and they do not directly endorse a candidate.
In another section of the EO, the Treasury Secretary, Labor Secretary and Health and Human Services Secretary are directed "to consider issuing amended regulations, consistent with applicable law to address conscience-based objections to the preventive-care mandate."
This EO is consistent with other executive orders that administration has issued, according to John Cooney, a partner in the regulatory group of Washington, D.C., law firm Venable LLP. "All administrations use executive orders when they think it would serve the purpose a president has in mind," said Cooney, who ran the executive order process in the Reagan White House.
"This is just a policy preference but that doesn't mean that it doesn't have an impact on the calculus of an agency," Cooney said. This EO instructed several secretaries to consider issuing revised regulations on the Johnson Amendment and the preventive care mandate.
The president is instructing appointees, as heads of federal agencies, within the range of discretion they possess under federal statutes. "All administrations have done this when they want to announce their policies publicly," Cooney said. It's the most formal format you can use and it's covered extensively by the press and ultimately published in the Federal Register, he said. The same instructions could have been passed along quietly from the president or staff to department secretaries. "That kind of communication is more typical," he said.
Since Reagan, most presidents have averaged anywhere from 30 to 50 executive orders per year during their tenure, according to the American Presidents Project. President Trump so far has issued 24.
Administrations have used executive orders to express policy preferences of a president and want obtain broad, public dissemination. Sometimes statutes delegate authority directly to the president as is most often seen in the case of terms and conditions under which federal contracts can be procured. President Trump revoked orders by the Obama administration, such as providing greater overtime payments. That's one where the president can change underlying law but it depends upon Congress giving the authority. More recently, President Trump issued an executive order about designations of protected national monuments and management of federal lands.
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Refactor use of this in React functional component? I have this code in and example app and need to use it in a functional component:
 ref={(c) => { this._carousel = c; }}

How can I assign the ref in an functional component?

A: You can use the React.useRef hook.
Example:
import React from "react";

function MyAwesomeFxnComponent() {
  const ref = React.useRef(null);
  return (
      <Carousel ref={ref} />
  );
}

A: With functional components you can use refs like this:
// Import react useRef hooks
import React, { useRef } from 'react';

export function CarouselWrapper {
  // create your reference
  const carouselRef = useRef(null);

  return (
   // Pass ref to your carousel component
   <Carousel ref={carouselRef} />
  )
};

Here you can read docs https://reactjs.org/docs/hooks-reference.html#useref

redpajama_set_name: RedPajamaStackExchange


The different flowers have their unique meanings in the sense of love. The best way of confessing your everlasting love with your partner is to send beautiful flowers. The flowers also work to rebuild the broken relationships. You can make happy your loved ones with the awesome arrangement of different colors flowers.
This can be the best gift for your special someone. You can get more varieties of gifts from many online portals for every occasion like birthday gifts, anniversary gifts, beautiful gifts for raksha bandhan, and many more like these. You can make everyone happy with special gifts by surprising them with the help of online delivery of gifts.
Here are some popular flowers to confess your deep love to your loving ones.
A rose is best to express romantic feelings with your loving partner. The red rose symbolizes the love and passion. The red rose bouquet can express your deep love feelings in a unique way to your partner. A rose is also to admire the natural beauty of the person.
The pink rose is used to express the friendly nature and cheering up the friend. A pink rose can be used at the beginning of the love story of couples. The pink rose is also the symbol of gratefulness and admiration in the relationships. You can surprise your beloved with the red and pink heart shaped roses bouquet to express your strong feelings.
The carnations flower is also a beautiful way to express some amazing love feelings with your partner. A carnation represents an attractive love and dedication in love with your beloved. The beautiful bouquet of lovely carnations can bring your love to another level. The carnations flowers are also the best choice for the first wedding anniversary. The arrangement of red and white carnations together represents respect and pure love with good luck charm to your better half.
The gerberas daisy represents the simplicity and innocence of true love. A love life also looks better when we make it simple. These gerberas are best to explain the reality of the better understandings in love relationship. The elegant gerberas daisy is also to symbolize the happiness and cheerfulness of the relationship. The gerbera also comes in different colors.
The orange colors are for the shining of the love life. The combination of pink and white gerberas give the message of innocence in love and commendation of having better love in life. The red gerbera is for the dedication in love within the beautiful relationship. The gerbera flowers always turn towards the sun which makes it best for the symbol of the sun shining the love life. It is best for the new beginning of the relationship.
The lily flower is basically very soft and elegant in nature. This lily flower symbolizes the soft corner of heart and gracefulness of the person. It can be dedicated to the person to express the royal feel in the relationship. It is the best choice to give to the newly married couple for the beginning of the relationship. The lilies also important in different cultures. The white lily is the symbol of purity. The pink color lilies are used to represent the motivation in different challenges of life.
The blue iris is the awesome flower to express the deep emotions and sentiments in the relationship. The blue iris flower is also able to attract the love with its unique charming effect. The blue iris flower bouquet is best for sharing the emotional attachments with your loving partner.
You can get all the flowers from online portals for every occasion. Also, you can get the designer and delicious cakes for every occasion like the special designer cakes for birthday, cakes for anniversary, cakes for wedding and many more like these. The combo of flowers and cakes will add more spark in your relationship. So, order your flowers now, and make your relationship stronger by sending beautiful flowers and gifts on every occasion.


Like Merchant Ships: Groceries for "The Masses"
Dawn (Frugal for Life) has picked up the discussion about low incomes and healthy food. Since I feed my family on a tight budget, this is a topic near and dear to my heart. I thought the Grist article had a few good points, but its discussion about low income nutrition is as far from reality as the $90 Farmer's Dinners it describes.
Yes, people with motivation can find healthy food on any budget. All kinds of creative people online have demonstrated that fact. However, many people on a poverty level income lack adequate resources for healthy shopping. (And I am not including the families who have chosen to live below poverty level as a form of self-sufficiency or downsizing here.) Let's face it, if your family is hungry, a loaf of white bread is not so objectionable. If you get most of your information from television, you don't stand a chance in the grocery. Reading skills may be poor or nonexistent; picking up a copy of Superfoods at the library is not an option. If you don't have a computer or you work while the library is open, you can't just click away at the thousands of whole foods recipes online. If you lack transportation, the grocery with good produce may be reachable only once a month. Even if you have a yard to grow your own food, you need tools and energy to make a garden happen.
People in poverty can overcome these obstacles. I'm simply trying to point out that it's not as easy as the organic-fed, well-educated liberals would have you believe. At least not for everyone, and certainly not for an economic class of Americans whose healthy survival skills have been bred out by generations of "helpful" social programs. Let them eat cake, right?
I am a huge advocate for healthy and organic foods, but I totally agree with this post. There is plenty of research that shows that inner city groceries have higher prices and less selection than suburban groceries, making it even more difficult for many low income families to feed themselves, whether healthy foods or not.
Even if people knew what to eat and better ways to shop, it is still extremely difficult to eat a mostly organic diet on a small income.
La Leche League offers some nutritional advice at their meetings, and they concentrate on the basics of shopping. They advocate shopping mainly the outside aisles of the store (that's where the meat, dairy and veggies usually are). They also advocate trying to eat foods in as close to their natural state as possible. I think those are basic, doable ideas for improving family nutrition regardless of income.
I think our agricultural extension service offers some classes in community centers within low income neighborhoods, too. I'm not sure my LLL group ever discussed healthy shopping in the year I attended, but since all the other moms were of similar backgrounds, why bother?
here, here. Well said. I'm a little tired of the well fed organic conservatives and liberals myself. I'm a progressive conservative.
I wanted to add that La Leche League has a cookbook, called Whole Foods For The Whole Family that is very good for eating a healthier diet.
It's been one of my go-to recipe books for over 15 years, and it's still useful.
Very interesting post, Meredith. As someone who is at the poverty level ON PAPER, this is also a subject that is dear to me.
I think one of the hardest things to overcome for many people is habit. Many people eat what they grew up eating, and they cook they way they've seen it cooked. Even people who aren't poor are this way. My dad, who is an RN, still uses tons of butter, lard, and salt in everything he cooks. He loves white bread, white pasta, and lots of potatoes. He knows what he should be eating, but thinks his diet tastes "so much better". To learn to think about food, and cook it, differently (if nothing else!)is "too hard".
I see the same thing with my farmer relatives. They grow alot of their own food and raise their own beef. But the way they prepare it for eating? Well, if the green beans are black and floating in bacon grease, and there's gravy on just about everything, it doesn't really matter if it is organic. I once commented about it (nicely). They are convinced that they ARE eating healthy, it's homegrown after all.
All that to say, that even if you do buy super-healthy foods, organic and all that, it is just as important to know what to do with it once you've got it. That involves more than availability. It involves a new set of skills. When you're in survival mode, that may be the last thing on your mind. Even when you're not, its sometimes daunting.
"If you get most of your information from television, you don't stand a chance in the grocery."
I just read today about a correlation that kids who watch more televison eat more junk food.


text:
Q: Can we make communication between Jnlp and Browser? I am invoking applet through JNLP from my application (which is hosted on tomcat server) and it is working fine. Now from applet I need to send the response back to the application or browser.
In my search, I found that we can do this using web socket. But not sure from where to start.
Can you please let me know any idea on this.

A: I think you need some kind of push notifications. One way of doing this as you already mentioned is with websockets and there are bunch of tutorials on the web as well as a question here on stackoverflow How to implement push to client using Java EE 7 WebSockets?
The second way is with server polling (frequently sending requests and checking for new events/states/subscribers etc.)


text:
Bicycling while female in Afghanistan
Nicole Capozziello
Young cyclists in Bamiyan province in Afghanistan. Still image by Jenny Nichols from the film Afghan Cycles, directed by Sarah Menzies, which is making its U.S. premiere at SIFF.
"I have a feeling of joy when I ride a bicycle and I don't want to get off of it. I want the ride to last even longer," says Frozan, a young woman on Afghanistan's National Cycling Team, at the beginning of the new documentary Afghan Cycles.
These moments of joy, as women cycle through the streets and countryside of their home country, are the pulse of this inspiring film, which will have its U.S. premiere this weekend at SIFF.
The screening will be a culmination of years of work for local filmmaker Sarah Menzies, who took her first trip to Afghanistan in 2013. "Originally, I thought it would be a short film that would do well in the adventure film world," Menzies says. "I quickly realized we needed to go much deeper with this story to contextualize what these women are up against."
This is Menzies' first feature film. Her 2013 short film, Catch It, took her to Norway to chronicle the story of French surfer Léa Brassy, while her most recent film, A Steelhead Quest, had her wading through North American rivers with fisherwoman Terry Meyers.
During the five years of making Afghan Cycles, Menzies spent time on the ground hearing the stories and everyday experiences of people across Afghanistan, capturing a different perspective of a country that has rarely avoided headlines over the last couple of decades. Afghanistan's political instability has contributed to a culture where, in some areas, it's a radical act for a woman to merely be outside the house. "The condition for girls in Afghanistan is full of fear," says one woman in the film.
Unsurprisingly, bicycling while female, even with a hard scarf under your helmet, presents its own risks. Beyond the regular fear of accidents, women who want to cycle also have to get their family's permission and, once they do, they face constant threats, including motorists hurling everything from disapproving glances to rocks.
Despite the beating sun and undercurrent of anxiety about security, Afghan Cycles shows the National Cycling Team members fiercely cycling on, with eyes fixed on the road ahead of them, representing themselves and their nation. For Frozan and her teammates, a bicycle not only represents transportation and freedom but hope for more equality for women and girls in the future. "I don't have the freedom to ride my bicycle anytime I want," says Frozan. "But I want that in the future."
Zahra leads the pack on a 2014 training ride with the Women's National Cycling Team of Afghanistan. (Image by Jenny Nichols/Afghan Cycles)
In the years since Menzies began making the film, she's seen attitudes and politics change in the country. "There is no women's National Cycling Team at the moment," says Menzies. "But from what I understand, there are more women riding bikes right now than when the team started. Club teams, inspired by all the press that the National Team had gotten, are still popping up." Though its presence angered many in Afghanistan, the National Cycling Team has also inspired people around the world, leading to a Nobel Peace Prize nomination in 2016.
The filmmaker hopes to keep drumming up inspiration with her film, which had its world premiere at Hot Docs Canadian International Documentary Film Festival just last month.
In addition to continuing to screen Afghan Cycles at festivals across the country, Menzies hopes to launch an impact campaign, partnering with cycling groups as well as Afghani women organizations.
"Even though this film is about very specific women in Afghanistan, their story is universal," Menzies says. "A bicycle can give people independence and mobility to get around safely and efficiently. And we want to harness that, allowing it to be a conversation as we talk about women's rights and equality."
If you go: Afghan Cycles plays at SIFF Cinema Uptown in Seattle, 6 p.m. May 20 and 4 p.m. May 21 and in Shoreline at 1 p.m. May 27. Director Sarah Menzies and Web-designer Kaitlin Bailey are scheduled to attend all three screenings.
ArtSEA: Seattle starts 2023 with a scream
Plus, more ways to kick off the new year, from artful calendars to gallery shows to a bottomless pit.
/ January 5
One of the largest and most expensive projects in the city's history brings Northwest Indigenous works to the fore.
Margo Vansynghel
/ January 25
A Seattle photographer's firsthand account of the Jan. 6 chaos
Nate Gowdy discusses the Capitol attack and his new photo book 'Insurrection.'
Nicole Capozziello is a freelance writer on arts, food and social justice.


Elizabeth is a nationally known contributor to major Catholic magazines. You may access her previous articles from this page.
Reprint Notice | Articles may be reprinted with the permission of Elizabeth Ficocelli. All articles must be reprinted in entirety including publisher credit. All articles on this site are copyrighted and the sole property of the author.


text:
\section{Introduction}

Our basic task is to estimate estimate the size of a largest bin in a special case of the balls and bins model. This models simply means that the balls are randomly thrown into bins. The process of their placement is of a various study -- its randomness, independence and other properties lead to various bin sizes.
The most simple model is to use fully random functions or some kind of their approximation to place the balls. There is a plenty of results, i.e. estimates of bin sizes, for various placement processes.

When the balls are thrown independently at random to the bins the expected size of the largest bin is $\Theta\left(\frac{\log m}{\log \log m}\right)$.

One of the first results were shown by Carter and Wegman \cite{cw} and this model was used to design universal and perfect hashing. They showed that the expected size of a bin is a constant when the placement is done by the functions which we will refer to as simple linear functions. These functions are two-wise independent and thus achieve $O(\sqrt{m})$ expected size of a largest bin.

It is also possible to use functions with higher degrees of independence and obtain better bounds. There are lower bounds for on the speed of such functions, size needed to represent and the size of the largest bin and independence they achieve given by Siegel \cite{siegel}.

The need to improve the size of the largest bins lead to two-choice paradigm. Out of two bins, hence we use two functions, the balls is placed into the smaller one. In this model the size of the largest bin is $O(\log \log m)$ where $m$ is the number of balls and bins shown by Azar et al \cite{azar} and improved by V\"{o}cking \cite{vocking}.

Nowadays more complicated family of functions are studied in \cite{wieder}. The functions no longer rely on high degree of independence but are designed so that they achieve small largest bins even with high probability.

Our model exhibits the use of simple linear functions and the balls are chosen from an interval in $\mathbb{Z}_p$. Such model has a constant size of largest bins.

\section{Notation and definitions}
\label{sec:notation}
We refer to the set $\{0, \dots, k - 1\}$ as to $[k]$.
In the whole text we assume that $p$ is a fixed prime.
The set of chosen balls is denoted by $S \subset [p]$.
The number of bins is the same as the number of balls and is denoted by $m$, i.e. $|S| = m$.

For each pair $(a, b) \in [p]^2$ we define the function $h'_{a, b}$ as $h'_{a, b}(x) = (ax + b) \bmod p$ and the function $h_{a, b}$ as $h_{a, b}(x) = h'_{a, b}(x) \bmod m$.

The multiset of simple linear functions mapping $[p]$ to the range $[m]$ is denoted by $\hlin$ and is defined as $\hlin = \{h_{a, b} \mid a, b \in [p] \}$.
For a function $h \in \hlin$ we define the size of $i$-th bin as $\bin{h}{S}{i} = |S \cap h^{-1}(i)|$ and the maximal size of the bin as $\lbin{h}{S} = \max_{i \in [m]} \bin{h}{S}{i}$.

In the following text we fix the probability space to be formed by a uniform choice of $h \in \hlin$.
The notation $\vbin{S}{i}$ and $\vlbin{S}$ then refers to the random variables formed by the mentioned random uniform choice.

For an element $x$ we define the value $l(x, a, b) = \left\lfloor\frac{ax + b}{p}\right\rfloor$; that is how many ``leaps'' are created by applying the function $h_{a, b}$ on the element $x$ in the field $\mathbb{Z}_p$.

\section{Collision probability for three elements}

We first study the probability of collision of three arbitrary elements.
By collision of the elements we understand the event when all of the elements are mapped to the same element in $[m]$ by the randomly chosen linear function.

We fix three different elements $x, y, z \in [p]$ and we count the number of pairs $(a, b) \in [p]^2$ such that $|h_{a, b}(\{x, y, z\})| = 1$.

We start by simplifying to the case when $x = 0, y = 1$ and the third element $z = d$ for a suitable $d \in [p]$ such that $d > 1$ depending on the choice of $x, y, z$.

\begin{lemma}[Transformation lemma]
\label{lemma:transformation}
Let $x, y, z \in [p]$ be arbitrary different elements. Moreover assume that $i_x, i_y, i_z \in [m]$. Then there exist an element $d \in [p]$ such that
\[
\prob{h(x) = i_x, h(y) = i_y,  h(z) = i_z} = \prob{h(0) = i_x, h(1) = i_y, h(d) = i_z}.
\]
\end{lemma}
\begin{proof}
The idea of the proof is simple. We show that there is a one-to-one map between simple linear functions mapping $x, y, z$ to $i_x, i_y, i_z$ and simple linear functions transforming $0, 1, d$ to the same elements.

In the first part of the proof we observe that combining simple linear functions with a linear function in $\mathbb{Z}_p$ does not change the the probability space.
There is a single linear function transforming $0, 1$ to $x, y$ in $\mathcal{Z}_p$ which we refer to as $h'_{\alpha,\beta}$.
Finally we choose $d$ so that $h'_{\alpha, \beta}(d) = z$ and the proof is finished.

We show that the elements $x, y, z$ can be transformed to the elements $0, 1, d$ so that the probability of the mappings from the statement of the lemma remains the same.

Choose $\alpha, \beta \in [p]$ so that $\alpha \neq 0$.
Observe that the mapping $(\gamma, \delta) \mapsto (\alpha \gamma, \beta \gamma + \delta)$ is a one-to-one map on $[p]^2$.
If there is another pair $(\epsilon, \phi)$ such that $(\alpha \epsilon, \beta \epsilon + \phi) = (\alpha \gamma, \beta \gamma + \delta)$, then $\gamma = \epsilon$ and $\delta = \phi$. Thus the mapping is injective.
Also for arbitrary $(r, s) \in [p]^2$ the element $(\alpha^{-1}r, s - \beta\alpha^{-1}r)$ is mapped to $(\alpha \alpha^{-1}r, \beta r + s - \beta\alpha^{-1}r) = (r, s)$.

The compound function $h'_{a, b} \circ h'_{\alpha, \beta}$ is exactly equal to the function $h'_{\alpha a, \beta a + b}$; this also follows from the fact that the set of all linear functions in $\mathbb{Z}_p$ forms a group with the operation of compounding functions.

Let $H' = \{h'_{a, b} \mid (a, b) \in [p]^2\}$.
From the previous we can conclude that the combination of a function $h' \in H'$ with a fixed function $h'_{\alpha, \beta}$ is a one-to-one map in the space of functions $H'$.
Also observe that the composition of a function $h_{a, b} \in \hlin$ with $h'_{\alpha, \beta}$ can not change the probability (count of the functions) of mapping arbitrary three elements to a their prescribed images.

There is also a single function $(\alpha, \beta) \in [p] ^ 2$, i.e. a single function $h'_{\alpha, \beta}$, transforming the elements $0$ and $1$ to $x$ and $y$ in the field $[p]$ without taking modulo $m$. It is the function $\beta = x$ and $\alpha = y - x$.
To prove the lemma we choose $d \in [p]$ such that $h'_{\alpha, \beta}(d) = z$, i.e. $d = \alpha ^ {-1}(z - \beta)$.
\end{proof}

Lemma \ref{lemma:transformation} shows that the probability properties, e.g. collision, mapping to the prescribed elements, for the elements $x, y, z$ are the same as for the elements $\{0, 1, d\}$ where $d$ comes from the previous lemma.

Next we estimate the collision probability for the elements $0, 1, d$.

\begin{lemma}[Probability of collision of three elements]
\label{lemma:probability-3-elements}
Let $d \in [p]$ be arbitrary element.
\[
\prob{|h(\{0, 1, d\})| = 1 } = \frac{1 + \max\left(1, \frac{p}{dm}\right)\left(1 + \frac{d}{m}\right)}{p}.
\]
\end{lemma}
\begin{proof}
We count the number of functions $h \in \hlin$ such that $h(\{0, 1, d\}) = \{y\}$ for some $y \in [m]$.
For each $x \in [p]$ it holds that $l(x, a, b) \in [x]$ and $h(x) = (ax + b - l(x, a, b)p) \bmod m$.

Whenever the elements $0, 1$ and $d$ are mapped to the same element $y$ it must hold that $h(0) = h(1)$ and $h(0) = h(d)$. Hence
\begin{align*}
	b \bmod p \bmod m & = (a + b) \bmod p \bmod m \\
	b \bmod p \bmod m & = (da + b) \bmod p \bmod m. \\
\end{align*}
From which we obtain the following sequence of equations
\begin{align*}
	m & \mid a + b - l(1, a, b)p - b \\
	m & \mid da + b - l(d, a, b)p - b, \\
\end{align*}
\begin{align*}
	m & \mid a - l(1, a, b)p \\
	m & \mid da - l(d, a, b)p, \\
\end{align*}
\begin{align*}
	m & \mid da - dl(1, a, b)p \\
	m & \mid (dl(1, a, b) - l(d, a, b))p. \\
\end{align*}
Since $p$ is a prime we conclude the fact that $m \mid dl(1, a, b) - l(d, a, b)$.
We estimate the collision probabilities from the two statements following from the previous formulas:
\begin{align}
	m & \mid a - l(1, a, b)p \label{3-prob-1-statement} \\
	m & \mid (dl(1, a, b) - l(d, a, b))p. \label{3-prob-2-statement}
\end{align}

The statement (\ref{3-prob-2-statement}) roughly means that out of $d$ possible values for $l(d, a, b)$ only the $1 / m$ fraction may generate the collision of the three elements. Notice that for a fixed $l \in [d]$ it holds that $\{a \in [p] \mid l(d, a, b) = l\}$ equals is a subinterval of $[p]$.
From (\ref{3-prob-1-statement}) we can observe that only the $1 / m$ fraction from the possible values of $a$ lying in the appropriate intervals allowed by valid values of $l(d, a, b)$ are causing collisions.

For the rest of the proof fix the value of $b$.
First, we show that the values of $a$ such that $l(d, a, b) = l \in [d]$ form disjoint intervals in $[p]$ each of size at most $\lceil p/d \rceil$.
Then we count the number of values $a$ in an interval causing collisions -- using (\ref{3-prob-1-statement}).
And finally we count the number of the valid intervals.

Let $l(d, a, b) = l$, then it holds that $l \leq \frac{da + b}{p} < l + 1$. Immediately we get that $a \in \left[\frac{pl - b}{d}, \frac{p(l + 1) - b}{d}\right) \cap \mathcal{Z}$. The total number of values of $a$, i.e. integers, in each valid interval is at most $\lceil p / d \rceil$. The ceiling must be applied. For example assume an interval of length of 1.5 starting at point 0.8 -- it contains two integer points 1 and 2. This happens whenever $\frac{pl - b}{d}$ is an integer.

Now fix the value $l \in [d]$ such that $l(d, a, b) = l$.
In order to estimate the number of values of $a$ causing the collisions we split into two cases according to the value of $l(1, a, b)$.

\subparagraph{The first case, $l(1, a, b) = 0$.}
From the two previous statements we conclude that
\begin{align*}
	m & \mid a \\
	m & \mid l. \\
\end{align*}

\subparagraph{The second case, $l(1, a, b) = 1$.}
As in the first case it must hold that
\begin{align*}
	m & \mid a - p \\
	m & \mid d - l. \\
\end{align*}

In both cases, there are at most $\lceil d/m \rceil$ values of $l$ satisfying the second condition.
Also for each satisfying value of $l$ there are at most $\lceil \lceil\frac{p}{d}\rceil / m \rceil$ values of $a$ causing the collision.

In both cases and for each $b$ it holds that the probability of collision of the three elements is bounded by
\[
\frac{\left(1 + \frac{1 + \frac{p}{d}}{m}\right)\left(1 + \frac{d}{m}\right)}{p} =
\begin{cases}
	\frac{1}{m^2} + \frac{1}{dm} + O\left(p^{-1}\right) & \mbox{if } p/dm \geq 1 \\
	\frac{1}{m^2} + \frac{d}{pm} + O\left(p^{-1}\right) & \mbox{otherwise, i.e. } \frac{p}{m} \leq d < p.
\end{cases}
\]
\end{proof}

The worst possible case is for $d = 2$ and the probability is roughly $1/2m$.
When $d \geq p/m$, the formula is a great overestimate as shown in Figure~1.

\begin{figure}[h]
	\label{fig:probability-3}
	\centering
	\includegraphics[width=8cm]{coll-3-21787-512-scaled}
	\caption{The function of probability of collision of the elements $0, 1, d$ with respect to $d$. Notice that the probability is decreasing in the part when $d \leq p / m$ and is almost symmetric. In this figure $m = 512$ and $p = 21787$.}
\end{figure}

\begin{corollary}
\label{co:d-elements}
Let $d < p / m$. Then $\prob{|h([d])| = 1} \leq \frac{1}{(d - 1) m} + 1/m^2 + O(p^{-1})$.
\end{corollary}
\begin{proof}
When all the elements from $[d]$ collide, then the elements $\{0, 1, d\}$ must collide as well. The probability of the collision of $\{0, 1, d\}$ is hence a valid upper bound on the probability of the collision of the whole interval. The statement is then a direct application of Lemma~\ref{lemma:probability-3-elements}.
\end{proof}

For completeness we just show a simple fact that our probability estimate is tight when we have a stronger assumption, namely we assume $p > 3m^2$.

\begin{lemma}
\label{lm:0-d-prob-lower-bound}
If $d \leq m$ and $p > 3m^2$, then $\prob{|h(\{0, 1, \ldots, d - 1\})| = 1} = \Omega\left(\frac{1}{dm}\right).$
\end{lemma}
\begin{proof}
For a fixed $b$, if $a < (p - b)/d$ and $m \mid a$, then the elements $\{0, 1, \ldots, d - 1\}$ collide.
For each $b$ there are at least $\lfloor (p - b)/dm \rfloor$ such values of $a$.

We conclude that the number of pairs $(a, b)$ making the elements collide is at least
\[
\sum_{b \in [p]} \frac{p - b}{dm} - 1 = \frac{(p + 1)p}{2dm} - p \geq \frac{p ^ 2 - 2pdm}{2dm} \geq \frac{p^2}{6dm}.
\]

Thus the resulting probability is at least $\frac{1}{6dm}$.
\end{proof}

\section{The expected size of most loaded bins}

First we study the role of the parameter $b$ in the hash function $h_{a, b}$.

The following lemma states that the effect of $b$ on $\vlbin{S}$ is not asymptotic since it more or less only shifts the largest bin.
\begin{lemma}
\label{lm:b-zero}
Assume that  $a, b \in [p]$ and $S \subseteq [p]$. Then \[ \frac{1}{2} \lbin{h_{a, b}}{S} \leq \lbin{h_{a, 0}}{S} \leq 2\lbin{h_{a, b}}{S} . \]
\end{lemma}
\begin{proof}
Let $L \subseteq S$ be elements of bin $y$, i.e. $h_{a, b}(L) = y$.
For each $x \in L$ we have that
\begin{align*}
h_{a, 0}(x)
	& = ax \bmod p \bmod m \\
	& = (ax + b - b) \bmod p \bmod m \\
	& = \begin{cases}
((ax + b) \bmod p - b \bmod p)\bmod m & \mbox{ if } (ax + b) \bmod p \geq b \\
(p + (ax + b) \bmod p - b \bmod p) \bmod m & \mbox{ otherwise.} \\
\end{cases}
\end{align*}

Notice that the two possible new bins are either $(y - b) \bmod m$ or $(p + y - b) \bmod m$.
The lemma now follows from the following two observation.
First each original bin is either shifted and keeps its size or is split into two possibly uneven shifted bins -- hence $\frac{1}{2} \lbin{h_{a, b}}{S} \leq \lbin{h_{a, 0}}{S}$.
And notice that each new bin can only contain elements from at most two different original bins and thus $\lbin{h_{a, 0}}{S} \leq 2\lbin{h_{a, b}}{S}$.
\end{proof}

For completeness let us mention that the change of the sign of $a$ has almost no effect on $\vlbin{S}$.
\begin{lemma}
\label{lemma:sign-a}
Assume that  $a \in [p]$ and $S \subseteq [p]$ such that $0 \not\in S$. Then \[ \lbin{h_{a, 0}}{S} = \lbin{h_{p - a, 0}}{S} . \]
\end{lemma}
\begin{proof}
Similarly as in the proof of the previous lemma. Let $L \subseteq S$ be elements of bin $y$, i.e. $h_{a, 0}(L) = y$.
Let $x \in L$, then $h_{p - a, 0}(x) = (p - a)x \bmod p \bmod m = (p - ((ax) \bmod p))\bmod m = (p - y) \bmod m$.
Observe that $(p - a)x \bmod p = p - ((ax) \bmod p)$ holds only when $x \neq 0$.
The bin $y$ is thus moved to the bin $(p - y) \bmod m$ and the lemma holds.
\end{proof}

Obviously allowing zero makes only a negligible change.
\begin{corollary}
Let $S \subseteq [p]$, then
\[
\lbin{h_{a, 0}}{S} -1 \leq \lbin{h_{p - a, 0}}{S} \leq \lbin{h_{a, 0}}{S} + 1.
\]
\end{corollary}

For the choice of $S = [m]$ we show that the expected size of a most loaded bin is within $O(1)$.
This can be compactly formulated as follows.
\begin{theorem}
\label{thm:interval-constant}
Assume that $p \geq m^2$, then
\[
\expect{\vlbin{[m]}} = O(1).
\]
\end{theorem}
\begin{proof}
By Lemma~\ref{lm:b-zero} we may assume that the chosen function has $b = 0$ without asymptotically increasing the expected size of the largest bin. In the proof of the claims we thus assume that the chosen linear function is exactly the function $h_{a, 0}$. Moreover we assume that $a \neq 0$. Notice that this assumption adds exactly $m/p$ to the computed expected value which is $O(1)$.

Observe that each bin is formed by a single arithmetic progression. Notice that since $p$ is a prime it holds that $(-p) \bmod m$ is co-prime with $m$.
The reason can be stated as follows.
Let $x_1 < x_2$ be two elements in a single bin, then for $d = x_2 - x_1$ it holds that $m \divides ad \bmod p$ or $m \divides p - (ad \bmod p)$.

All the solutions of the equation $ax \bmod p \bmod m = 0$ where $x \in [m]$ form a finite arithmetic progression.
For the proof of the previous statement notice that since $p$ is a prime it holds that $(-p) \bmod m$ is co-prime with $m$.

In addition a difference $d$ and a given length $l$, $l \geq 3$, there is a canonical value $x \in [m]$ such that if there is a bin of size at least $l$, then there is another bin formed by an arithmetic progression of length at least $l$ with the same difference $d$ having $x$ as the minimal element. If $ad \bmod p < p/2$, we choose $x = \operatorname{argmin}_{x \in [m - ld]} ax \bmod p$. Otherwise we put $x = \operatorname{argmax}_{x \in [m - ld]} ax \bmod p$.

After establishing the previous facts we simply compute the expected value of $\vlbin{[m]}$ using the following idea. Now we allow $b$ to have arbitrary value.

Assume that $\vlbin{[m]} > l \geq 3$, then there is an arithmetic progression chosen from $[m]$ of size at least $l/2$ collapsing into a single bin, here we use Lemma~\ref{lm:b-zero}. Since for a fixed difference and length we have its canonical position there are at most $m/l$ possible arithmetic progressions from which we choose from. By Corollary~\ref{co:d-elements} we upper bound the probability of the collapse of the arithmetic progression as
\[
\frac{m}{l} \left(\frac{1}{(l/2 - 1)m} + 1/m^2 + O(p^{-1})\right) \leq O(l^{-2}).
\]

Hence for $l \geq 3$ we have
\[
\prob{\vlbin{[m]} \geq l} = O(l^{-2}).
\]

Then we simply conclude that
\[
\expect{\vlbin{[m]}} \leq O(1) + \sum_{l = 1}^m O\left(\frac{1}{l^2}\right) = O(1).
\]

\end{proof}

We can conclude the main result, i.e. each set transformable to $[m]$ in $\mathbb{Z}_p$ has constant sized largest bins.
\begin{corollary}
Let $S \subseteq [p]$, $a, b \in [p]$.
If $\forall x \in [m] \colon (ax + b) \bmod p \in S$, then $\expect{\vlbin{S}} = O(1)$.
\end{corollary}
\begin{proof}
Direct corollary of Theorem~\ref{thm:interval-constant} since by Lemma~\ref{lemma:transformation} (extended to all the elements of $S$) the probabilistic properties of $S$ do not change under the transformation $x \mapsto (ax + b) \bmod p$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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