In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann, it is based on the earlier Banach–Tarski paradox, in turn based on the Hausdorff paradox. Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would have the same area as the original; this would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators; the group of area-preserving transformations contains such subgroups, this opens the possibility of performing paradoxical decompositions using them.
The following is an informal description of the method found by von Neumann. Assume that we have a free group H of area-preserving linear transformations generated by two transformations, σ and τ, which are not far from the identity element. Being a free group means that all its elements can be expressed uniquely in the form σ u 1 τ v 1 σ u 2 τ v 2 ⋯ σ u n τ v n for some n, where the u s and v s are all non-zero integers, except the first u and the last v. We can divide this group into two parts: those that start on the left with σ to some non-zero power and those that start with τ to some power. If we operate on any point in Euclidean 2-space by the various elements of H we get what is called the orbit of that point. All the points in the plane can thus be classed into orbits, of which there are an infinite number with the cardinality of the continuum. Using the axiom of choice, we can choose one point from each orbit and call the set of these points M. We exclude the origin, a fixed point in H.
If we operate on M by all the elements of H, we generate each point of the plane once. If we operate on M by all the elements of A or of B, we get two disjoint sets whose union is all points but the origin. Now we take some figure such as the unit disk. We choose another figure inside it, such as a smaller square, centred at the origin. We can cover the big figure with several copies of the small figure, albeit with some points covered by two or more copies. We can assign each point of the big figure to one of the copies of the small figure. Let us call the sets corresponding to each copy C 1, C 2, …, C m. We shall now make a one-to-one mapping of each point in the big figure to a point in its interior, using only area-preserving transformations. We take the points belonging to C 1 and translate them so that the centre of the C 1 square is at the origin. We take those points in it which are in the set A defined above and operate on them by the area-preserving operation σ τ; this puts them into set B.
We take the points belonging to B and operate on them with σ2. They will now still be in B. We proceed in this manner, using σ3τ on the A points from C2 and σ4 on its B points, so on. In this way, we have mapped all points from the big figure in a one-to-one manner to B type points not too far from the centre, within the big figure. We can make a second mapping to A type points. At this point we can apply the method of the Cantor-Bernstein-Schroeder theorem; this theorem tells us that if we have an injection from set D to set E, an injection from E to D there is a one-to-one correspondence between D and E. In other words, having a mapping from the big figure to a subset of the A points in it, we can make a mapping from the big figure to all the A points in it. We can make a mapping from the big figure to all the B points in it. So looking at this the other way round, we can separate the figure into its A and B points, map each of these back into the whole figure! This sketch glosses over some things, such as.
It turns out that more sets are necessary to work around this. The paradox for the square can be strengthened as follows: Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps; this has consequences
Bppunalikevirus is a genus of viruses in the order Caudovirales, in the family Podoviridae. Bacteria serve as natural hosts, contains only two species; this genus of temperate phages is named after its first sequenced member, Bordetella phage BPP-1, which serves as its type species. Group: dsDNA Viruses in Bppunalikevirus are non-enveloped, with icosahedral and Head-tail geometries, T=7 symmetry; the diameter is around 67 nm. Genomes are linear, around 42kb in length, encodes about 50 proteins; the Bordetella phages of this genus contains an RNA-directed DNA polymerase which plays a role in tropism. Viral replication is cytoplasmic. Entry into the host cell is achieved by adsorption into the host cell. Dna templated. Bacteria serve as the natural host. Transmission routes are passive diffusion. Viralzone: Bppunalikevirus ICTV
Diderich de Thurah was a military cadet, a naval officer in the Royal Danish-Norwegian navy and fabrikmester, artist and publisher. He studied with, worked under, Knud Benstrup, he proved an unworthy successor as Fabrikmester. In life he achieved notability in translating Lutheran texts from English and German, into Danish. Born in Aarhus on 1 May 1704, Diderich de Thurah moved to Ribe when his father, Laurids Thura, became bishop of Ribe in 1713, he and his younger brother, Lauritz de Thurah, met King Frederik IV when the monarch was visiting Ribe and chose the two boys for military service. In 1719, he went to Copenhagen as a military cadet, a landkadet in Danish, to receive an education at the Military Cadet Academy, his older brother, became a priest and poet. In 1723, Diderich de Thurah received an immediate commission into the Royal Danish navy after presenting a fine carved model in amber of the ship-of-the-line Anne Sophie to the king, he chose to specialise in ship construction, in April 1724, he was sent on an extended study tour to England together with another Danish officer, Lieutenant Bragenæs.
There was some discussion over his salary, still that of a cadet, he could not make his travel expenses stretch. He received his lieutenant's salary from 1 January 1725, but continued to plead that lack of funds led to him missing out on several learning opportunities. By 1726, the Danish admiralty was complaining about the lack of written reports with which it could gauge the two lieutenants' industry and the scientific advances he had been sent to learn, they were ordered to justify themselves. In late 1726, a book of plans for a 72-gun ship-of-the-line was produced, the two lieutenants sought a further year to profit from their studies. In November 1727, they were recalled to Copenhagen. In 1728, Bragenæs and Thurah worked up designs for a 56-gun ship based on English principles – such a ship had to have good sailing qualities and sufficient storage for provisions for longer voyages. In October 1729, together with the experienced shipbuilders Benstrup and Judichær, they critically examined a ship designed by senior lieutenant Krag while that ship was still on the stocks.
Benstrup was not happy with Thurah's professional abilities, relationships soured. Thurah was responsible in 1731 for the purchase of timber for shipbuilding, which necessitated a visit to Holland. Promoted in 1732 and 1734 to the rank of senior lieutenant, he became head of naval shipbuilding, fabrikmester in 1734. Further promoted to captain and commander, Thurah continued to have problems with his finances, requiring a gift from royal funds to settle with his creditors, he was reprimanded for exceeding his authority in the shipyard by removing Turesen from the supervision of the building of the ships Jylland and Dannebrog, ordered to curb his animosity and attend more professionally to his duties. Although he retained the title of fabrikmester until 1758, his designs were not acceptable for ships of the fleet, the Frenchman Laurent Barbé was appointed as chief ship designer. From 1758, he was given the post of head of customs. A charge of abuse of authority was laid against Thurah in 1764 by a ship owner in Aalborg, when the latter's ship De To Sønner was arrested, a charge vigorously contested by Thurah.
Few ships are credited to Deiderich de Thurah by the Royal Danish Naval Museum: Dannebroge ship-of-the-line Fyen ship-of-the-line Langeland brig Møen brig Windhunden yacht or kongebaad Unnamed boat He was an artist, known for carving amber and ivory. A carving in amber of the Judichaer designed ship-of-the-line Dronning Anne Sophie was presented to the monarch in 1723 and was on display in the king's art collection. In 1729 and 1730, additional to his shipyard duties, he was art tutor to the 7-year-old crown prince. On leaving the navy, until his death in 1788, Thurah was busy for many years with literary work and translations in relation to religious art and tracts. Amongst the pieces he translated from English were works by James Blair, Thomas Newton, William Sherlock and Lady Pakington. In this career, he proved more successful than in his earlier life as a naval officer. Thurah died in Copenhagen on 1 March 1788 and was buried in St. Peter's Church, the German-speaking Lutheran church; some input from Danish Wikipedia articles da:Diderich de Thurah and da:Fabrikmester and da:Knud Nielsen Benstrup Bjerg H C: Diderich de Thurah in Dansk Biografisk Leksikon, 3 edition.
Gyldendal 1979-84. Accessed 8 September 2019 Kornerup B: Albert Thura in Dansk Biografisk Leksikon, 3 ed. Gyldendal 1979-84. Accessed 18 September 2019 T. A. Topsøe-Jensen og Emil Marquard “Officerer i den dansk-norske Søetat 1660-1814 og den danske Søetat 1814-1932“ Royal Danish Naval Museum website for Database > Avancerede > Set Konstructør to "Thurah" > Søg Royal Danish Naval Museum - List of Danish Warships Royal Danish Naval Museum - Skibregister for individual ships record cards where they exist
NHS South East Coast was a strategic health authority of the National Health Service in England. It operated in the South East region, along with NHS South Central, providing coterminosity with the local government office region. NHS South East Coast was established on 1 July 2006 as one of 10 Strategic Health Authorities in England. NHS South East Coast contained 26 NHS organisations, including 6 Foundation Trusts; these comprise: eight Primary Care Trusts. Abolished April 2013 Brighton and Hove City PCT East Sussex Downs and Weald PCT Eastern and Coastal Kent PCT Hastings and Rother PCT Medway PCT Surrey PCT West Kent PCT West Sussex PCT Ashford and St Peter's Hospitals NHS Trust Brighton and Sussex University Hospitals NHS Trust Dartford and Gravesham NHS Trust East Kent Hospitals University NHS Foundation Trust East Sussex Hospitals NHS Trust Frimley Park NHS Foundation Trust Maidstone and Tunbridge Wells NHS Trust The Medway NHS Foundation Trust Royal Surrey County Hospital NHS Trust Royal West Sussex NHS Trust Surrey and Sussex Healthcare NHS Trust The Queen Victoria Hospital NHS Foundation Trust Worthing and Southlands Hospitals NHS Trust Official website
The MACC Fund - Midwest Athletes Against Childhood Cancer, Inc.- was founded on December 10, 1976, during the retirement of Jon McGlocklin from the Milwaukee Bucks. Jon's friend, Longtime Milwaukee Bucks Broadcaster Eddie Doucette, his wife, had a son, diagnosed with cancer at the age of 2. Brett's diagnosis coincided with the timing of Jon's retirement, it was decided to use Jon's departure from professional basketball to announce the start of the MACC Fund. During a half-time on the floor of the Milwaukee Arena, the MACC Fund came into being. Jon McGlocklin was named the organization's President, an office he holds today, Eddie was named Vice- President. Living in California since 1978, Eddie is the now the MACC Fund's Honorary Vice-President. Brett Doucette, the MACC Fund's first success story, is married and living in Arizona, his dad Eddie is working as a consultant with the NBA and Jon does color commentary for Bucks televised games with Honorary MACC Fund Athletic Board member, Jim Paschke.
Local athletes and community-minded business people comprised the Fund's original Board of Directors. All three groups are still involved. In addition, tens of thousands of people from throughout Wisconsin and the Midwest have embraced the MACC Fund and its mission to offer a special gift of "Hope Through Research to Children with Cancer and Related Blood Disorders." Fund-raising is based on the concept of "a good time for a good cause." Special events led the way in the beginning and still comprise about 75% of the Fund's annual income. Brewers home runs and doubles, Packers sacks and interceptions and Bucks 3-point baskets and opening tips raised funds from generous sponsors and cooperative sports teams. Cause-related marketing programs with donations for each purchase followed suit along with a telethon called MACC*ATHON. Women for MACC took up the fight in 1982 with its own impressive fund-raising events. A yearly telethon/auction of signed and game-used sports memorabilia to benefit the fund on WTMJ-TV has been a staple of the holiday season since the late 1980s, along with the "MACC Star", a yearly Christmas ornament available for sale designed by a child undergoing cancer treatment.
The MACC Fund's largest annual event is the Trek 100 Ride for Hope, which has raised over $8 million since 1990. Since 2009, the Trek 100 has been held in early June and begins and ends at the home of Trek Bicycle Corporation in Waterloo, Wisconsin. Riders raise at least $2 following routes that range from 19 to 100 miles. Packers quarterback Aaron Rodgers raised $50k for the fund after winning Celebrity Jeopardy. Traditional donations, memorials and foundation support and estate planning gifts fill in the other 25% of the Fund's annual income. Total income in 2007 was $3.8 million. The MACC Fund has a full-time staff of 5; the 10-year average administrative and indirect fund-raising expense is about 25%. The MACC Fund has contributed $33 million for pediatric cancer research thanks to the generosity and commitment of people who have taken the time to care for a child with cancer; this research impacts children throughout the country since the successful protocols become the standard for patient care everywhere.
Long term commitments take the total to $39 million. The primary beneficiary of MACC Fund support is the MACC Fund Center for Cancer and Blood Disorders, a collaborative effort of the Medical College of Wisconsin and Children's Hospital of Wisconsin. Research is conducted in the 6-story MACC Fund Research Center of the Medical College of Wisconsin; the MACC Fund supports research at the University of Wisconsin's Paul P. Carbone Comprehensive Cancer Center where research will be conducted in the MACC Fund Pediatric Cancer Research Wing upon its completion in the fall of 2009. Translational research is supported at the MACC Fund Center for Cancer and Blood Disorders at Children's Hospital of Wisconsin; the MACC Fund committed nearly 1.5 million for the Midwest Children's Cancer Center's annual budget in fiscal 2008. This is about 30% of the Center's annual budget. In addition, support to the UW Cancer Center totaled $600,000 in 2008; the MACC Fund Scientific Review Board was formed in 1980 to review the work of the Midwest Children's Cancer Center and to review grants submitted from outside this Center.
It is made up of leading pediatric oncology practitioners from around the country. The Board does a biennial site review of the Cancer Center; the current Review Board is headed by Dr. Kevin Shannon of the University of California, San Francisco Comprehensive Cancer Center. Past MACC Fund Scientific Review Board Chairs have come from prestigious medical schools such as Yale, Harvard, UCLA and The University of Southern California; the research that the MACC Fund supports has had a significant impact. The overall cure rate for all types of childhood cancer has risen from 20% to 80% in the past 30 years. Wilms tumor of the kidney is now 90% curable. Acute lymphocytic leukemia is about 85% curable, and the Midwest Children's Cancer Center has performed more successful non-related bone marrow transplants than any other Center in the country. Official website
Oxford Friend is an Oxfordshire-based LGBTQ+ charity providing emotional support and counselling to the lesbian, gay and transgender community. The original volunteers were part of the local branch of the Campaign for Homosexual Equality. Listening services are provided by telephone, email and instant messaging; the charity provides training on LGBTQ+ issues for educational institutions and other organisations in and around Oxfordshire. Oxford Friend raises money via a grant from Oxford City Council and external training packages; the group was established in 1980 and registered as a charity in 1988. In 2006 the group was awarded the Queen's Award for Voluntary Service. Facebook page Twitter page Website Brighton Gay and Lesbian Switchboard Gay & Lesbian Switchboard of New York London Friend Switchboard