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Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices. A distinction is made between undirected graphs, where edges link two vertices symmetrically, directed graphs, where edges link two vertices asymmetrically. Graphs are one of the prime objects of study in discrete mathematics. Refer to the glossary of graph theory for basic definitions in graph theory. Definitions in graph theory vary; the following are some of the more basic ways of defining graphs and related mathematical structures. In one restricted but common sense of the term, a graph is an ordered pair G = comprising: V a set of vertices. To avoid ambiguity, this type of object may be called an undirected simple graph. In the edge, the vertices x and y are called the endpoints of the edge; the edge is said to be incident on x and on y. A vertex may exist in a graph and not belong to an edge. Multiple edges are two or more edges.

In one more general sense of the term allowing multiple edges, a graph is an ordered triple G = comprising: V a set of vertices. To avoid ambiguity, this type of object may be called an undirected multigraph. A loop is an edge. Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex x is the edge or is incident on =, not in. So to allow loops the definitions must be expanded. For undirected simple graphs, E ⊆ should become E ⊆. For undirected multigraphs, ϕ: E → should become ϕ: E →. To avoid ambiguity, these types of objects may be called an undirected simple graph permitting loops and an undirected multigraph permitting loops respectively. V and E are taken to be finite, many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. Moreover, V is assumed to be non-empty, but E is allowed to be the empty set; the order of a graph is its number of vertices. The size of a graph is its number of edges.

The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n/2; the edges of an undirected simple graph permitting loops G induce a symmetric homogeneous relation ~ on the vertices of G, called the adjacency relation of G. Specifically, for each edge, its endpoints x and y are said to be adjacent to one another, denoted x ~ y. A directed graph or digraph is a graph. In one restricted but common sense of the term, a directed graph is an ordered pair G = comprising: V a set of vertices. To avoid ambiguity, this type of object may be called a directed simple graph. In the edge directed from x to y, the vertices x and y are called the endpoints of the edge, x the tail of the edge and y the head of the edge; the edge is called the inverted edge of. The edge is said to be incident on x and on y. A vertex may exist in a graph and not belong to an edge.

A loop is an edge. Multiple edges are two or more edges. In one more general sense of the term allowing multiple edges, a directed graph is an ordered triple G = comprising: V a set of vertices. To avoid ambiguity, this type of object may be called a directed multigraph. Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex x is the edge or is incident on, not in. So to allow loops the definitions must be expanded. For directed simple graphs, E ⊆ should become E ⊆ V2. For directed multigraphs, ϕ: E → should become ϕ: E → V2. To avoid ambiguity, these types of objects may be called a directed simple graph permitting loops and a directed multigraph permitting loops respectively; the edges of a directed simple graph permitting loops G is a homogeneous relation ~ on the vertices of G, called t

Great American Train Show

The Great American Train Show is the name of what was, for two decades, the largest traveling model train show in the United States. The company was incorporated in 1985 and went defunct in 2006. During the 1990s, the company operated as many as 90 train shows every year in 40 different states; the Great American Train Show, or GATS, was founded by David K. Swanson as an outgrowth of a monthly train show in Wheaton, near Chicago, itself an outgrowth of a local model railroad club. Beginning in 1982, Swanson began running train shows in cities outside the Chicago area expanding the operation to the west coast by the mid-1980s and to the east coast by the mid-1990s. GATS was run as a subsidiary of the NIART company, but following incorporation as GATS Limited became itself a parent to a series of subsidiaries which existed at one point or another; these included: Great Midwest Train Show Great Western Train Show Great American Travel Service GA Technology Services Computer Renaissance Computer CentralIn 2001, the company was sold by Swanson to Elmo Geoghegan, a former Bob's Big Boy employee who had worked for GATS as a show manager.

Geoghegan moved the emphasis of the company from train shows towards computer shows, a change which proved financially disastrous. In 2004, the company's name was changed to Great Western & Atlantic Train Show, in April 2006 the company ceased all operations; the core business of GATS consisted of running consumer shows focused on model railroading. These train shows consisted of several dozen vendors selling a variety of model railroad related merchandise, as well as several operating model train layouts. Shows were opened to the public over the course of two days and Sunday. Public attendance could range from around 1,500 to as many as 10,000 attendees over the course of the two-day show; as show promoter, GATS performed facility and show operation personnel contracting, exhibitor registration, floor plan design, on-site management functions, as well as conducting advertising and public relations campaigns to draw attendees. After selling GATS, Swanson went on to purchase Greenberg Shows, an east coast train show company, in competition with GATS.

This operation was folded into the Great Train Expo company, created by Swanson when GATS went defunct. Swanson founded the World's Greatest Hobby on Tour, a model railroad industry trade and consumer show. Huckaby, George. "Interview with Dave Swanson". Trolleyville Times. Retrieved April 13, 2007. Huckaby, George. "Rest in Peace - Great Western & Atlantic Train Show". Trolleyville Times. Retrieved April 13, 2007. "What we know about GATS Limited/Hobbies Unlimited". Great Train Expo. April 24, 2006. Archived from the original on March 14, 2007. Retrieved April 13, 2007. Huckaby, George. "GATS Sold!". Trolleyville Times. Retrieved April 13, 2007. GATS goes out of business, from the Trolleyville Times Archived GATS website, from the Wayback Machine Greenberg Shows, longtime competitor to GATS Great Train Expo, founded by David K. Swanson in 2006 Former GATS official website

Laohwangping Airfield

Laohwangping Airfield is a former World War II United States Army Air Forces airfield, located 1 mile west of Jiuzhou Zhen in the People’s Republic of China. The airfield was the primary home of the 35th Photographic Reconnaissance Squadron, which flew unarmed P-38 Lightning photo-reconnaissance aircraft from the airfield beginning in February 1945 until the end of the war, it was the home of the 23d Fighter Group 76th Fighter Squadron, equipped with P-51 Mustangs in February 1945. Records indicate this was most a temporary facility, with a compressed earth runway with tents and small wooden buildings used for a support facility, it appears to have been closed and dismantled in September 1945 after the war ended, with the land being returned to agricultural use. The available satellite image of this area of China is low resolution and it is difficult to determine if any relics of the airfield exist; this article incorporates public domain material from the Air Force Historical Research Agency website http://www.afhra.af.mil/

WXTA

WXTA is a commercial radio station located in Edinboro, broadcasting to the Erie, area. WXTA airs a country music format branded as "Nash FM 97-9". Signing-on in October 1988 as WMYJ, the station operated with an Adult Contemporary format for a year as "Y-98." WMYJ was purchased by Bob Winters over the summer of 1989. The format switch to country happened at noon on September 11, 1989, the station took the calls WXTA. WXTA was sold to Jim Embrescia's Media One Group in 1996, Regent Communications in 1999, Citadel Broadcasting in 2004, Cumulus Media in 2011. Former air personalities at WXTA include "Uncle Fred" Horton, Bobby Reed, Ron Kline, Mike McKay, Natalie Massing, John Cunningham, "Big John" Jacobs, Ed Beeler, Bobbi Weston, Brian Williams, Rick Shigo, Chris Atkins, Dale Thompson, Chet Price, Bill Shannon, Laura Luke, Jim Mirabello, Adam Reese, John Gallagher, Tom Lavery, Sammy James, Jay Foyst, Herb Palmer, Dan "Shoeless" Sheldon, Ellie McVay, Truckin' Tom, Cindy Wear, Becca Lynn and the syndicated "Big D & Bubba."

Former syndicated programming on WXTA includes "After Midnite with Blair Garner", "Country Gold Saturday Night" and "Neon Nights with Lia". Morning Personality "Uncle Fred" Horton died on March 4, 2008. Current on-air personalities include Jim Griffey, Nicole Dohoda and Chuck Stevens. Stevens is the stations Program Director. On-air features include Sheffield in the Morning, Country Cartunes at 5, Kickin' it with Kix, Retro Country. On February 3, 2014, WXTA along with 9 other Cumulus-owned country music stations made the switch to going under the Nash FM branding. WXTA official website Query the FCC's FM station database for WXTA Radio-Locator information on WXTA Query Nielsen Audio's FM station database for WXTA

Cirsium

Cirsium is a genus of perennial and biennial flowering plants in the Asteraceae, one of several genera known as thistles. They are more known as plume thistles; these differ from other thistle genera in having feathered hairs to their achenes. The other genera have a pappus of simple unbranched hairs, they are native to Eurasia and northern Africa, with about 60 species from North America. Thistles are known for their effusive flower heads purple, rose or pink yellow or white; the radially symmetrical disc flowers are at the end of the branches and are visited by many kinds of insects, featuring a generalised pollination syndrome. They have erect stems and prickly leaves, with a characteristic enlarged base of the flower, spiny; the leaves are alternate, some species can be hairy. Extensions from the leaf base down the stem, called wings, can be lacking, conspicuous, or inconspicuous, they can spread by seed, by rhizomes below the surface. The seed has pappus, which can carry them far by wind. Cirsium thistles are used as food plants by the larvae of some Lepidoptera species - see list of Lepidoptera that feed on Cirsium.

The seeds are attractive to small finches such as American goldfinch. Most species are considered weeds by agricultural interests. Cirsium vulgare is listed as a noxious weed in nine US states; some species in particular are cultivated in gardens and wildflower plantings for their aesthetic value and/or to support pollinators such as butterflies. Some species dubbed weeds by various interest groups can provide these benefits. Cirsium vulgare, for instance, ranked in the top 10 for nectar production in a UK plants survey conducted by the AgriLand project, supported by the UK Insect Pollinators Initiative. Bull thistle was a top producer of nectar sugar in another study in Britain, ranked third with a production per floral unit of. Not only does it provide abundant nectar, it provides seeds and floss for birds, such as the American goldfinch, Spinus tristis, supports the larvae of a Painted Lady butterfly, Vanessa cardui. A great many native North American plants have weed in their common names, despite their beneficial qualities, such as Asclepias tuberosa known as butterflyweed.

Some other common species are: Cirsium palustre, Cirsium oleraceum. Some ecological organizations, such as the Xerces Society, have attempted to raise awareness of the benefits of thistles, to counteract the general agricultural and home garden labeling of thistles as unwanted weeds; the monarch butterfly, Danaus plexippus for instance, was highlighted as relying upon thistles such as Tall thistle, Cirsium altissimum, for its migration, as a important nectar source. Although such organizations focus on the benefits of native thistles, non-native thistles, such as Cirsium vulgare in North America, may provide similar benefits to wildlife; some prairie and wildflower seed production companies supply bulk seed for native North American thistle species, for wildlife habitat restoration, although availability tends to be low. Thistles are valued by bumblebees for their high nectar production. Certain species of Cirsium, like Cirsium monspessulanum, Cirsium pyrenaicum and Cirsium vulgare, have been traditionally used as food in rural areas of southern Europe.

Cirsium oleraceum is cultivated as a food source in India. The word'Cirsium' derives from the Greek word kirsos meaning'swollen vein'. Thistles were used as a remedy against swollen veins; the flower blooms April to August. HybridsCirsium × canalense – canal thistle Cirsium × crassum – thistle Cirsium × erosum – glory thistle Cirsium × iowense – Iowa thistle Cirsium × vancouverense – Vancouver thistle

Oriri

Oriri is a long poem by the birth control pioneer Marie C. Stopes, published by Heinemann as a short book, its subject is love between a "He" and a "She", it is written in semi-dramatic form, with other members of the cast including "Spirits of Air", "Spirits of Earth" and other "Elementals". Stopes writes in the "Argument" that prefaces the poem: "Interwoven in the tale is a crystallisation of most of what matters fundamentally in the sciences of geology and physiology, in the art of love, in religion." The poem brings together Stopes' scientific knowledge with her theory of "erogamic love", the latter of, an important component of the theories that underlay her most influential and controversial publication, the sex manual Married Love. In her biography of Stopes, Ruth Hall suggests that the poem was written when Stopes was in love with Keith Briant, a recent Oxford graduate, her junior