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Ulam spiral

The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers. Ulam and Gardner emphasized the striking appearance in the spiral of prominent diagonal and vertical lines containing large numbers of primes. Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, certain such polynomials, such as Euler's prime-generating polynomial x2 − x + 41, are believed to produce a high density of prime numbers; the Ulam spiral is connected with major unsolved problems in number theory such as Landau's problems. In particular, no quadratic polynomial has been proved to generate infinitely many primes, much less to have a high asymptotic density of them, although there is a well-supported conjecture as to what that asymptotic density should be.

In 1932, more than thirty years prior to Ulam's discovery, the herpetologist Laurence Klauber constructed a triangular, non-spiral array containing vertical and diagonal lines exhibiting a similar concentration of prime numbers. Like Ulam, Klauber noted the connection with prime-generating polynomials, such as Euler's; the Ulam spiral is constructed by writing the positive integers in a spiral arrangement on a square lattice: and marking the prime numbers: In the figure, primes appear to concentrate along certain diagonal lines. In the 200×200 Ulam spiral shown above, diagonal lines are visible, confirming that the pattern continues. Horizontal and vertical lines with a high density of primes, while less prominent, are evident. Most the number spiral is started with the number 1 at the center, but it is possible to start with any number, the same concentration of primes along diagonal and vertical lines is observed. Starting with 41 at the center gives a diagonal containing an unbroken string of 40 primes, the longest example of its kind.

According to Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of "a long and boring paper" at a scientific meeting. These hand calculations amounted to "a few hundred points". Shortly afterwards, with collaborators Myron Stein and Mark Wells, used MANIAC II at Los Alamos Scientific Laboratory to extend the calculation to about 100,000 points; the group computed the density of primes among numbers up to 10,000,000 along some of the prime-rich lines as well as along some of the prime-poor lines. Images of the spiral up to 65,000 points were displayed on "a scope attached to the machine" and photographed; the Ulam spiral was described in Martin Gardner's March 1964 Mathematical Games column in Scientific American and featured on the front cover of that issue. Some of the photographs of Stein and Wells were reproduced in the column. In an addendum to the Scientific American column, Gardner mentioned the earlier paper of Klauber. Klauber describes his construction as follows, "The integers are arranged in triangular order with 1 at the apex, the second line containing numbers 2 to 4, the third 5 to 9, so forth.

When the primes have been indicated, it is found that there are concentrations in certain vertical and diagonal lines, amongst these the so-called Euler sequences with high concentrations of primes are discovered." Diagonal and vertical lines in the number spiral correspond to polynomials of the form f = 4 n 2 + b n + c where b and c are integer constants. When b is the lines are diagonal, either all numbers are odd, or all are depending on the value of c, it is therefore no surprise that all primes other than 2 lie in alternate diagonals of the Ulam spiral. Some polynomials, such as 4 n 2 + 8 n + 3, while producing only odd values, factorize over the integers and are therefore never prime except when one of the factors equals 1; such examples correspond to diagonals that are nearly so. To gain insight into why some of the remaining odd diagonals may have a higher concentration of primes than others, consider 4 n 2 + 6 n + 1 and 4 n 2 + 6 n + 5. Compute remainders upon division by 3 as n takes successive values 0, 1, 2, ….

For the first of these polynomials, the sequence of remainders is 1, 2, 2, 1, 2, 2, …, while for the second, it is 2, 0, 0, 2, 0, 0, …. This implies that in the sequence of values taken by the second polynomial, two out of every three are divisible by 3, hence not prime, while in the sequence of values taken by the first polynomial, none are divisible by 3, thus it seems plausible that the first polynomial will produce values with a higher density of primes than will the second. At the least, this observation gives little reason to believe that the corresponding diagonals will be dense with primes. One should, of course

Juan López (archbishop)

Juan López, O. P. was a Roman Catholic prelate who served as Archbishop of Bishop of Cebu. Juan López was born in Spain. On February 10, 1663, he was ordained a priest of the Order of Friars Preachers. On April 23, 1663, Pope Alexander VII appointed him Bishop of Cebu where he succeeded Juan Velez, Bishop Elect of Cebu, who died before his consecration. On January 4, 1665, he was consecrated bishop by Marcos Ramírez de Prado y Ovando, Bishop of Michoacán. On November 14, 1672, Pope Clement X appointed him Archbishop of Manila where he served until his death on February 12, 1674. Cheney, David M. "Archdiocese of Cebu". Retrieved June 18, 2018. Chow, Gabriel. "Metropolitan Archdiocese of Cebu". Retrieved June 18, 2018. Cheney, David M. "Archdiocese of Manila". Retrieved March 25, 2018. Chow, Gabriel. "Metropolitan Archdiocese of Manila". Retrieved March 25, 2018

Bosnia and Herzegovina football league system

The Bosnia and Herzegovina football league system is a series of connected leagues for football clubs in Bosnia and Herzegovina. The system is hierarchical, with relegation between leagues at different levels; the top division is organized by Football Association of Bosnia and Herzegovina, the second and third levels by entity associations, lower levels by cantonal or regional associations. The Premier League of Bosnia and Herzegovina is at the top of the structure, it has 16 teams. The League champion is the champion of Bosnia and Herzegovina and has the right to play in the UEFA Champions League qualifying rounds; the bottom two teams of the table are relegated to the second level of competition. The second tier is divided into two leagues – the First League of the Federation of Bosnia and Herzegovina and the First League of Republika Srpska – containing 16 and 12 clubs. Relegated teams from the Premier League are demoted to the second-level competition in the next season. Geographical location is the criterion for decided decision in which one of these two leagues the teams will play.

The winner of each league is promoted to the Premier League, bottom teams are relegated. The number of relegated teams depends on how many teams enter from the Premier League and the Third Divisions. Third-level football is more dispersed; each league is associated with a different geographical area. In Federation of Bosnia and Herzegovina, there are four second divisions, in Republika Srpska two second divisions. Same principle of promotion and relegation is used, so the league winners are promoted to the appropriate second-level league, clubs at the bottom of the table are relegated to lower levels. There are cantonal football associations in nine of the ten cantons which make up the Federation of Bosnia and Herzegovina, they each organize clubs depending on the number of clubs they control. In Republika Srpska, which has no cantons, there are instead four football regions, each of which has a league; the first leagues in the Federation cantons and the regional leagues in Republika Srpska constitute the fourth level of club football.

Further down, there are many different types of organization. Some cantons have second leagues, there are municipality and inter-municipality leagues for clubs not associated with any cantonal or regional league. Bosnia and Herzegovina Football Cup starts in spring with qualification matches; those matches are played between lower league clubs on region level. Best teams qualify to entity cups. In the first round of national cup, 12 Premier League teams are joined by 12 teams from Federation and 8 teams from Republika Srpska. First round is played over one leg while remaining rounds are played over two legs

Long Way to Heaven

Long Way to Heaven is the fifth studio album by the Canadian heavy metal band Helix. This album was their third for Capitol Records, there were bigger expectations from the band after the success of the previous Walkin' the Razor's Edge; the first single was "cowritten by Paul Hackman and Bob Halligan, Jr.. The song received heavy airplay in the U. S. gaining "double breaker" status, in Canada was added to heavy video play on MuchMusic. Q107 in Toronto had the song riding at #1 for several weeks on their "Top Ten at Ten", their first tour to kick off the album was in Sweden where they became the first Canadian rock band to tour that country extensively. For this they achieved their first #1 album in that country. Returning to North America, they toured with Keel, as well as headlining dates, they played odd one-off dates with Meat Loaf and Heart. While headlining in Newfoundland, a local band called; this band featured Rainer and Cindy Wiechmann, who joined Helix 19 years in 2004, on lead guitar and backup vocals respectively.

The second single released from the album was "The Kids Are all Shakin'", a song inspired by a fan letter from Poland. The album went platinum in Canada and they headlined their first Canadian tour with the Headpins as the supporting act; the Long Way to Heaven album was released for the first time on CD in September 1999. The Kids Are All Shakin' - 3:48 Deep Cuts the Knife - 4:01 Ride the Rocket - 3:24 Long Way to Heaven - 3:34 House On Fire - 4:15 Christine - 3:34 Without You - 3:40 School of Hard Knocks - 4:06 Don't Touch the Merchandise - 2:47 Bangin' Off-A-The Bricks - 3:15 There was a remix made of "The Kids Are All Shakin'", used in the music video; the audio track was not available for purchase until the Helix Deep Cuts compilation of 1999. Brian Vollmer - vocals Paul Hackman - guitar and vocals Brent "The Doctor" Doerner - guitar and vocals Daryl Gray - bass and vocals Greg "Fritz" Hinz - drums Sleaze Roxx

Darius Van Arman

Darius Van Arman is an American businessman, co-founder, co-owner of Secretly Group, an independent label group, based in Indiana. Van Arman's father was a math professor. In 1995 he studied mathematics at the University of Virginia. Van Arman founded the Jagjaguwar record label in 1996 in Virginia. In 1999, Van Arman dropped out of school, moved to Bloomington and became partners with Chris Swanson, together began the Secretly Group. Van Arman serves on the boards of A2IM, SoundExchange and Merlin; the Secretly Group includes the Secretly Canadian and Jagjaguwar labels, Dead Oceans and The Numero Group. Together these labels have represented the following artists: The War on Drugs, Major Lazer, Syl Johnson, Bon Iver and Dinosaur Jr

James Milne (railway manager)

Sir James Milne, K. C. V. O. C. S. I. was an Irish railway manager in Great Britain. He was General Manager of the Great Western Railway from 1929 to 1947, deputy chairman of the Railway Executive Committee from 1938 to 1947. Milne was born in Dublin, Ireland, in 1883, he attended Campbell College in Belfast and moved to Great Britain to study Engineering at the Victoria University of Manchester, graduating in 1904. Milne joined the GWR 1904 as a pupil engineer in the locomotive department, he moved to Paddington and gained operational and traffic experience. In 1912 Milne married Nora Rebekah Morse, daughter of Levi Lapper Morse. Milne joined the Ministry of Transport when it was set up in 1919 as Director of Statistics until 1921, he served on the Geddes Committee on National Expenditure and the India Retrenchment Committee, chaired by Lord Inchcape. Milne was appointed Companion of the Order of the Star of India in 1923. Milne returned to the GWR as assistant general manager in 1922, replaced Pole as General Manager in 1929.

He continued Pole's work on the GWR's advertising and corporate image, introducing the Gill Sans typeface in advertising and the GWR monogram on advertising and rolling stock. He was knighted in 1932, appointed Knight Commander of the Royal Victorian Order in 1936. During his tenure he helped set up Railway Air Services, a joint venture between the major British railway companies and Imperial Airways; the GWR investigated electrification but thought it not suitable or economic for its network. From 1938 Milne continued as General Manager but was deputy chairman of the Railway Executive Committee, a government body responsible for running British railways during the Second World War. Milne was a member of the Road Transport Advisory Committee; the REC's work continued after the end of the war through to nationalisation in 1948. In 1940 Milne was elected as a GWR director but could not take up the role as the REC was a government body. On 29 July 1944 Paddington station had to be closed because of large crowds trying to leave London for the August Bank holiday and to escape flying bombs.

The GWR had locomotives and coaches available, but were not allowed to run extra trains because of wartime restrictions. Milne had to threaten to involve the Prime Minister, Winston Churchill, before the Ministry of War Transport relented and allowed the extra trains to run. Milne opposed state ownership of the railways, but was still offered the chairmanship of the Railway Executive of the British Transport Commission, being formed to manage the proposed nationalised British Railways. Milne declined the offer and retired from the GWR at the end of 1947. In 1948 an ex-GWR locomotive, Castle class No. 7001 Denbigh Castle, was renamed as No. 7001 Sir James Milne. Milne died in 1958. Dickinson, Geoff. "Levi Lapper Morse J. P." My Primitive Methodist Ancestors. Retrieved 22 December 2014. "No. 32830". The London Gazette. 1 June 1923. P. 3945. "No. 33785". The London Gazette. 29 December 1931. P. 2. "No. 15241". The Edinburgh Gazette. 7 January 1936. P. 19. Maggs, Colin. A History of the Great Western Railway. Stroud, Gloucestershire, UK: Amberley Publishing.

ISBN 978-1-4456-1277-5. OCLC 855536026. Nock, O. S.. The Great Western Railway in the 20th Century. London: Ian Allan. ISBN 9780711002272. OCLC 251662074. Semmens, Peter. A History of the Great Western Railway: 3. Wartime and the Final Years 1939-48. @Steam past. London: Guild Publishing. ISBN 978-0-043-85106-7. OCLC 786175335. University of Manchester; the Victoria University of Manchester: Register of Graduates Up to July 1st, 1908. Manchester: University Press. "Road Transport in an Emergency". The Glasgow Herald. 6 October 1938. Retrieved 22 December 2014. "Milne, Sir James". Who's Who. 1920–2015. A & C Black, an imprint of Bloomsbury Publishing plc. Retrieved 21 January 2015. Photograph of James Milne