On the morning of Stanley's return, however, her strength left her.

On 18 April they received a letter from Emin, who had heard about the expedition a year earlier, and had come down the lake in March after hearing rumors of Stanley's arrival.

On January 13, 2009, Morgan Stanley and Citigroup announced the merger of Smith Barney with Morgan Stanley's Global Wealth Management Group, with Morgan Stanley paying $ 2. 7 billion cash upfront to Citigroup for a 51 % stake in the joint venture.

On June 1, 2009, Morgan Stanley and Citigroup Inc. announced they closed early on the launch of their joint venture that combines Morgan Stanley's wealth management unit ( including many former Dean Witter assets ) with Citi's Smith Barney brokerage division.

On October 19, 2009, Invesco bought Van Kampen Investments and Morgan Stanley's Retail Unit for $ 1. 5 billion.

On June 29, 2009, Papelbon gained his 132nd save, in a 4 – 0 shut out against the Baltimore Orioles, tying Bob Stanley's record for most saves by a Red Sox pitcher.

On 8 April he became the first to prove the quadratic reciprocity law.

On the basis of reciprocity, if one side is allowed to cross-examine his legal opponent at a hearing, the other party must also be given the same opportunity.

" On the other hand, this latter pronouncement was made when Kummer was still excited about the success of his work on reciprocity and when his work on Fermat's Last Theorem was running out of steam, so it may perhaps be taken with some skepticism.

On the other hand, retaliatory aspects i. e. the aspects of trying to get back and cause harm, are known as negative reciprocity.

* O literární vzájemnosti mezi kmeny a nářečími slavskĭmi ( On the literary reciprocity between Slav tribes and vernaculars )

" Lisandro Mendez ’ s ‘ Coyote and Deer ’: On narrative structures, reciprocity, and interactions .” American Indian Quarterly.

He was awarded the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper " On a conjecture by Littlewood and idempotent measures ", and lends his name to the Cohen-Hewitt factorization theorem.

On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics.

On the other hand, Fermat's last theorem has always been known by that name, even before it was proven ; it was never known as " Fermat's conjecture ".

Gödel's first incompleteness theorem first appeared as " Theorem VI " in Gödel's 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems I.

On curved surfaces, the formula for C ( r ) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand – Diquet – Puiseux theorem as

On the other hand, if a equals zero, the theorem holds trivially.

On the other hand, if, then by the deduction theorem, thus the deductive closure of is an element such that,, and.

On the other hand, PFA has the disadvantages that it only works for relatively prime factors ( e. g. it is useless for power-of-two sizes ) and that it requires a more complicated re-indexing of the data based on the Chinese remainder theorem ( CRT ).

In an 1850 paper, On the Solution of the General Equation of the Fifth Degree, Kronecker solved the quintic equation by applying group theory ( though his solution was not in terms of radicals, since this was already proven impossible by Abel – Ruffini theorem ).

1996a, " On the proof of Frege's theorem " in A. Morton and S. P. Stich, eds., Paul Benacerraf and his Critics.

On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields

On the other hand, Birch's theorem shows that if d is any odd natural number, then there is a number N ( d ) such that any form of degree d in more than N ( d ) variables represents 0: the Hasse principle holds trivially.

" On the gap in the theorem of Heegner ", Journal of Number Theory, 1: 16 – 27.

On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not locally symmetric: one of them, known as Berger's first criterion, is a consequence of the Ambrose – Singer theorem, that the curvature generates the holonomy algebra ; the other, known as Berger's second criterion, comes from the requirement that the connection should not be locally symmetric.

The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme ( German: On the theory of associative number systems ) and later rediscovered by Emmy Noether.

On the other hand, Pascal's theorem follows from the above associativity formula, and thus from the associativity of the group operation of elliptic curves by way of continuity.

His works, published in Proceedings of the London Mathematical Society, were " Density integration " ( 53, 1951 ); " On the measure of sum sets ( I ) The theorems of Brunn, Minkowski, and Lusternik, ( with A. M. McBeath )" ( 3, 1953 ); " Linear functions with domain a real countably infinite dimensional space " ( 5, 1955 ); " Linear and bilinear functions with domain contained in a real countably infinite dimensional space " ( 6, 1956 ); " The use of convergence factors in Ward integration " ( 10, 1960 ); " The equivalence of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals " ( 10, 1960 ); " N-variation and N-variational integrals of set functions " ( 11, 1961 ); " Definitions of Riemann type of the variational integrals " ( 11, 1961 ); " Difference-sets and the Banach – Steinhaus theorem " ( 13, 1963 ); " Generalized integrals of vector-valued functions ( 19 1969 )

Debreu subsequently improved Scarf ’ s argument and published it in his 1963 paper: `` On a theorem of Scarf ”.

* Johansen, Nils Voje ; and Ravndal, Finn On the discovery of Birkhoff's theorem version of September 6, 2005.

On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of the Radon – Nikodym theorem.

On any symplectic manifold, one can, at least locally, choose coordinates so as make the symplectic structure constant, by Darboux's theorem ; and, using the associated Poisson bivector, one may consider the above formula.

* 1931 On a generalization of the second theorem of Bourbaki, Proc.

In 1930, in a paper entitled ' On a Problem in Formal Logic ,' Frank P. Ramsey proved a very general theorem ( now known as Ramsey's theorem ) of which this theorem is a simple case.

On a bitterly cold day in January, 1895, accompanied only by Neal Brown as his deputy, Tilghman left the township of Guthrie and headed for Rock Fort and Dunn's ranch.

On a military mission for his native Virginia the youthful George Washington touched off the French and Indian War, then guarded his colony's frontier as head of its militia.

On the glass partition between me and the driver were three signs: one asked for help for the blind, another help for orphans, and the third for relief for the war refugees.

One beat poet composes a poem, `` Lines On A Tijuana John '', which contains a few happy hints for survival.

On the one hand, he does not work for a large agency, but is almost always self-employed.

On this basis, certain extreme kinds of networks will be discussed for illustrative purposes.

On December 21, the day that the Irish House of Commons petitioned for removal of Sir Constantine Phipps, their Tory Lord Chancellor, Molesworth reportedly made this remark on the defense of Phipps by Convocation: `` They that have turned the world upside down, are come hither also ''.

On one visit he stopped at the office of the American, where he was known surreptitiously as `` the Great White Chief '', and for the first time met his managing editor, fat Moses Koenigsberg.

On returning to New York he had a job for several weeks ; ;

On the other hand, the consensus of opinion is that, used with caution and in conjunction with other types of evidence, the native sources still provide a valid rough outline for the English settlement of southern Britain.

On New Year's Eve, Alfred Harcourt drove him up the Hudson to Bill Brown's Training Camp, a well-known establishment for the speedy if temporary rehabilitation of drunkards who could no longer help themselves.

On net balance, in spite of Controller Gerosa's opposition to the new Charter as an invasion of his office, the Controller will have the opportunity for greater usefulness to good government than he has now.

`` On trial in Jakarta for having flown for the Indonesian anti-Communist insurgents, U.S. pilot Alan Lawrence Pope boldly told the court that in supporting the freedom fighters, he was actually defending the sovereignty and independence of Indonesia.

On the other hand, it is no interference with sovereignty to point out defects where they exist, such as that a plan calls for factories without power to run them, or for institutions without trained personnel to staff them.

On September 16, Sam Rayburn will have served as Speaker twice as long as any predecessor and I am proud to join with others in marking this date, and in expressing my esteem for that notable American, Sam Rayburn.

On the basis that all citizens of the state are entitled to benefit equally in the development of its resources, plans for the provision of essential services ( such as water ) will be based on need regardless of arbitrary political boundaries, within the framework of the state plan.

On the other hand, the new authority of $3,825 million proposed for missile procurement ( excluding research and construction ) in 1961 is $581 million higher than for 1960.

On this basis, our already substantial budget for research and development has been further increased in recent years in order to finance the continuing engineering and design work essential to Leesona's future growth in sales and earnings.