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Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, braid groups; this can be regarded as a part of geometric topology. It may be used to refer to the study of topological spaces of dimension 1, though this is more considered part of continuum theory. A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology; the solution by Stephen Smale, in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were intertwined in low dimensions, Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from only weakly linked areas of mathematics. Vaughan Jones' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics.

In 2002, Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard S. Hamilton's Ricci flow, an idea belonging to the field of geometric analysis. Overall, this progress has led to better integration of the field into the rest of mathematics. A surface is a topological manifold; the most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3—for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections; the classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: the sphere. The surfaces in the first two families are orientable, it is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface.

The sphere and the torus have Euler characteristics 2 and 0 and in general the Euler characteristic of the connected sum of g tori is 2 − 2g. The surfaces in the third family are nonorientable; the Euler characteristic of the real projective plane is 1, in general the Euler characteristic of the connected sum of k of them is 2 − k. In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism; each point in TX may be regarded as an isomorphism class of'marked' Riemann surfaces where a'marking' is an isotopy class of homeomorphisms from X to X. The Teichmüller space is the universal covering orbifold of the moduli space. Teichmüller space has a wealth of natural metrics; the underlying topological space of Teichmüller space was studied by Fricke, the Teichmüller metric on it was introduced by Oswald Teichmüller. In mathematics, the uniformization theorem says that every connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.

In particular it admits a Riemannian metric of constant curvature. This classifies Riemannian surfaces as elliptic and hyperbolic according to their universal cover; the uniformization theorem is a generalization of the Riemann mapping theorem from proper connected open subsets of the plane to arbitrary connected Riemann surfaces. A topological space X is a 3-manifold if every point in X has a neighbourhood, homeomorphic to Euclidean 3-space; the topological, piecewise-linear, smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, so there is a prevalence of specialized techniques that do not generalize to dimensions greater than three; this special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, partial differential equations.

3-manifold theory is considered a part of low-dimensional topology or geometric topology. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself.

John Bailhache

John Bailhache was the 10th mayor of Columbus, Ohio. The Columbus City Council appointed him to complete the remaining two-year term of John Brooks after he resigned his office on April 21, 1835, his successor after 1836 was Warren Jenkins. Egger, Charles, ed.. Columbus Mayors. Columbus, Ohio: Columbus Citizen-Journal. Bailhache, John. Brief Sketch of the Life and Editorial Career of John Bailhache of Alton, Illinois. Ohio Historical Society: unpublished manuscript. Bailhache, John. John Bailhache's Diary, 1850: Alton, Illinois to Sacramento, California. A. Johnson; the Biographical Annals of Ohio: A Handbook of the Government and Institutions of the State of Ohio. State of Ohio. 1905. Pp. 255, 720. Fisk, William L.. "John Bailhache: A British Editor in Early Ohio". The Ohio Historical Quarterly. 67: 141–147. An Illustrated History of Southern California. Lewis Publishing. P. 315. ISBN 9785879878806. Klein, Lisa M.. "The Spiritual Journey of Buttles and Bailhache". Be It Remembered: The Story Of Trinity Episcopal Church On Capital Square, Ohio.

Orange Frazer Press Inc. p. 18. ISBN 9781882203260. Thompson, Charles Manfred; the Illinois Whigs Before 1846. University of Illinois. Pp. 157, 159. Van Vugt, William E.. British Buckeyes: The English and Welsh in Ohio, 1700-1900. Kent State University Press. Pp. 212–213. ISBN 9780873388436. William T. Norton, ed.. Centennial History of Madison County and Its People, 1812 to 1912. Unigraphic. P. 118. John Bailhache at Find a Grave John Bailhache at Political Graveyard "Bailhache-Brayman Family Papers, 1796-1922". Chronicling Illinois. Retrieved 2 August 2014. "Illinois Genealogy Trails". Retrieved 4 April 2016

Vladimir Sakhnov

Vladimir Nikolaevich Sakhnov is a former Soviet/Russian cross-country skier who raced from 1983 to 1989. Sakhnov trained at Armed Forces sports society in Alma-Ata, he earned a silver medal in the 4 x 10 km relay at the 1988 Winter Olympics in Calgary. Sakhnov won a silver medal in the 4 × 10 km relay at the 1987 FIS Nordic World Ski Championships, his best individual finish at the Nordic skiing World Championships was a sixth place in the 50 km event at those same championships. Sakhnov's best career finish was second in two World Cup events. All results are sourced from the International Ski Federation. 1 medal – 1 medal – 2 podiums 1 victory 5 podiums Wladimir Sachnov at the International Ski Federation Biography Vladimir Sakhnov at Olympics at Sports-Reference.com Vladimir Sakhnov at the International Olympic Committee

Views on the Arab–Israeli conflict

The Arab–Israeli conflict is the result of numerous factors. Reasons cited for the conflict therefore vary from participant to participant and observer to observer. A powerful example of this divide can be Israelis. In a March, 2005 poll 63% of the Israelis blamed the failure of the Oslo Peace Process on Palestinian violence, but only 5% of the Palestinians agreed. 54% of Palestinians put the blame on continuing Israeli settlement activity, but only 20% of the Israelis agreed. It is therefore difficult to develop a single, objective reason for the conflict, so this article will present some of the arguments made by each side, in turn. There is not a single "Israeli view"; when Israel met Arab leaders who spoke the language of peace to their own people and were willing to take concrete steps for peace, such as President Sadat of Egypt and King Hussein of Jordan, Israel made sacrifices for the sake of peace and reached peace agreements with them. Peacemaking requires concessions and confidence-building measures on both sides.

Just as Israel is willing to address the rights and interests of other parties in the conflict, Israelis insist that their rights and interests must be addressed as well. In 2000, at Camp David, the Palestinians were offered a nominally independent state. Led by Yasser Arafat, the Palestinians rejected this offer; when U. S. President Bill Clinton and the Israelis asked the Palestinians to offer a counter-proposal, Arafat declined and returned to the West Bank. Further negotiations did take place, but they were terminated. In his book The Missing Peace: The Inside Story of the Fight for Middle East Peace, Dennis Ross, the American ambassador and facilitator, writes that the idea the Palestinian state would be a "Bantustan" was a myth, provides maps showing an offer that included contiguous territory; the Palestine Liberation Organization has stated that it is prepared to recognize the state of Israel on the basis of the removal of settlements and retreat from Palestinian territory back to the 1967 borders.

Israel maintains that Hamas refuses to recognize Israel, on grounds that Hamas's'peace offerings' are a ploy. Many if not most Israelis believe that the conflict is a result of Arab attempts to destroy Israel, that only Israeli military power stands between them and annihilation, they characterize the 1948 Arab–Israeli War, the Six-Day War and the Yom Kippur War as attempts to destroy Israel. As evidence of this intent, pro-Israeli literature places a heavy emphasis on statements made by Arab leaders during and preceding the wars; the following quotes are mainstays of these arguments: "If Israel embarks on an aggression against Syria or Egypt.... The battle will be a general one and our basic objective will be to destroy Israel." On May 30, 1967, Nasser proclaimed: "The armies of Egypt, Jordan and Lebanon are poised on the borders of Israel... to face the challenge, while standing behind us are the armies of Iraq, Kuwait and the whole Arab nation. This act will astound the world. Today they will know that the Arabs are arranged for battle, the critical hour has arrived.

We have reached the stage of serious action and not declarations." After Iraq joined the Arab military alliance on June 4, its president Abdur Rahman Aref announced: "The existence of Israel is an error which must be rectified. This is our opportunity to wipe out the ignominy, with us since 1948. Our goal is clear – to wipe Israel off the map." SC 242, the Land for peace formula, was adopted on November 22, 1967 in the aftermath of the Six-Day War and the Khartoum Resolution. It called for withdrawal from occupied territories and for "termination of all claims or states of belligerency" and mutual "acknowledgment of the sovereignty, territorial integrity and political independence" by Israel and the other states in the area, recognized the right of "every state in the area" to live "free from threats or acts of force" within "secure and recognized boundaries". After the Six-Day War, Israel offered to return the Golan Heights to Syria and the Sinai Peninsula to Egypt in exchange for peace treaties and various concessions, but Syria and Egypt refused the offer and this offer of land for peace was soon withdrawn.

Anwar Sadat, the Egyptian President at the time, proposed negotiations towards peace with Israel in the early 1970s but Israel refused the offer, claiming that it held unreasonable preconditions. Israel signed the Camp David Accords with Egypt and subsequently withdrew from all Egyptian territory it occupied. Many, including the original framers of the resolution, have noted that the English-language version of SC 242 did not state all territories occupied during the conflict, recognizing that some territorial adjustments were and rejected previous drafts with the word all; the French language translation of the text did include the definite article. Israel considers it has complied with this sense of the resolution when it returned the Sinai to Egypt in 1982. Israel says that it has demonstrated flexibility and understanding by bringing about the initiation of the peace process, agreeing to painful concessions, implementing them; as opposed to this, many Israelis consider that the predominant Palestinian views of the peace process do not recognize Israel's right to exist, believe that the only real long-term Arab goal is the complete destruction of the Jewish state.

Many Jews and supporters of Israel, most Palestinians and supporters of Palestine, take the view that the existence

Group IV pyridoxal-dependent decarboxylases

In molecular biology, group IV pyridoxal-dependent decarboxylases are a family of enzymes comprising ornithine decarboxylase EC 4.1.1.17, lysine decarboxylase EC 4.1.1.18, arginine decarboxylase EC 4.1.1.19 and diaminopimelate decarboxylaseEC 4.1.1.20. It is known as the Orn/Lys/Arg decarboxylase class-II family. Pyridoxal-5'-phosphate-dependent amino acid decarboxylases can be divided into four groups based on amino acid sequence. Group IV comprises eukaryotic ornithine and lysine decarboxylase and the prokaryotic biosynthetic type of arginine decarboxylase and diaminopimelate decarboxylase. Members of this family while most evolutionary related, do not share extensive regions of sequence similarities; the proteins contain a conserved lysine residue, known, in mouse ODC to be the site of attachment of the pyridoxal-phosphate group. The proteins contain a stretch of three consecutive glycine residues and has been proposed to be part of a substrate-binding region. Group I pyridoxal-dependent decarboxylases Group II pyridoxal-dependent decarboxylases Group III pyridoxal-dependent decarboxylases

Eight Plus

Eight Plus is an album by bassist Ron Carter's Nonet recorded in 1990 and released on the Japanese Victor label. The AllMusic review by Rick Anderson observed "On this album he heads a standard quintet that features two significant innovations: Carter himself plays a half-size piccolo bass and the quintet is further augmented by three cellists. Is the experiment a success? For the most part, yes.... The cellos are a nice touch, their chordal accompaniments give Carter's compositions an unusual and pleasing texture". On All About Jazz, Mark F. Turner stated "Mr. Carter adds a twist by featuring his skills on the piccolo bass. Combine a quartet of cellos with a horn-less jazz quintet, the compositions stretch far beyond the typical jazz environment... For a Nonet recording of a different breed, Eight Plus is worth a listen" while C. Michael Bailey said "The music sounds like hip chamber music; the cellos are not so much plush in their effect as they are rhythmic. They do provide a dense fullness to the music...

This disc is an acquired taste, but like fine Scotch whiskey, it's a taste, more than worth developing". All compositions by Ron Carter except where noted "Eight" – 7:45 "A Blues for Bradley" – 7:45 "Little Waltz" – 9:30 "O. K." – 5:13 "A Song for You" – 8:04 "First Trip" – 5:07 "El Rompe Cabeza" – 9:03 "Just a Closer Walk with Thee" – 1:57 Ron Carter - piccolo bass Stephen Scottpiano Carol Buck, Kermit Moore, Chase Morrison, Rachael Steuermann – cello Leon Maleson – bass Lewis Nashdrums Steve Kroonpercussion