彼女の論文は、数学界のトップに位置する学術誌である『Annals of Mathematics』『Inventiones Mathematicae』『Journal of the American Mathematical Society』に掲載された。数学者の大半が、生涯一度もこのような偉業を成し遂げることはないとファーブは言う。「彼女はそれを博士論文でやり遂げたんだ」。
① 「ハブ」の存在
現実社会の友人ネットワークについて考えてみると、大多数の人は友人の数は数名だが、「友人の数がずば抜けて多い」人物が何人かはいる。これはウェブでも同様で、全ドキュメント(1999年で10億以上と言われる)の90％以上はリンクされる数は10以下であるが、ごく少数のページは100万近くリンクされている。後者はネットワーク上では「ずば抜けて枝の多い頂点」であり、ハブと呼ばれる。このハブは現実に存在するにもかかわらず、エルデシュのランダムネットワークでやワッツ・ストロガッツのスモールワールド・ネットワークでは生じない。では、ハブが生じるネットワークとはどのようなものなのか？
② ベキ法則
1900年代、イタリアの経済学者ヴィルフレード・パレートは、「収入分布は“ベキ法則”にしたがう」ことを発見。これは「世の中にはごく一握りのきわめて収入の多い人たちがおり、人口の大多数はわずかな収入しかない」ということを表す法則であり、後にパレートの法則とか「80対20の法則」などと呼ばれた(世の中のお金の80％は人口の20％の人という一握りの人たちが持っており、お金の20％はその他大勢の80％が持っている、ということ)。
③ スケールフリー・ネットワーク
ベキ法則は、正規分布(釣鐘型の分布)とは違って、①どこにもピークがなく、なめらかに減少する、②分布のすそ野は正規分布よりも広い、③ごく少数のきわめて大きい事象と無数の小さい事象が共存する状態を表すなどの特徴を持つ。バラバシは、枝の数と頂点の数がベキ法則に分布をスケールフリー・ネットワークと呼んだ。
…以上が、Ramsey-Cass-Koopmans モデルの設定である。資本蓄積、技術、資源の各制約を前提に、消費者は効用が最大になる消費を選ぶのである。主体（消費者）が最適な値を選ぶという意味で、c_tはcontrol variable とも呼ばれる2。また、各期において、主体が所与とする変数をstate variable と呼ぶ。このモデルの場合は、k_t、A_tがstate variable である。
State variable は、主体が環境を変えられる変数と変えられない変数に分けられる。前者をendogenous state variable という場合もあり、このモデルでは資本蓄積を通して変更されるk_tである。後者はexogenous state variable という場合もあり、このモデルの場合はA_tである。
…解をclosed form で得ることは難しい。
従って、そのようなモデルを扱う場合、定常状態周りで解がどのような振る舞いなのか、対数線形近似して求めるという手法が用いられる。1 次の近似を行えば、変数同士の関係を線形で表すことができるので、行列による差分方程式の表現が可能となる。そして、この行列による表現を操作すれば、policy function を求めることができるのである。
一般に、k_tは過去からの蓄積があるため、backward に解かれる。λ_tは横断面条件を満たすために、forward に解かれる。これらを上手く識別し、それぞれの変数についての比較的簡単な解法を与えたのがBlanchard and Kahn (1980)である。
、ごく最近の展開としては、経済主体間の情報の非対称性を強調する
Angeletos and La’o [2009] “Noisy Business Cycles,” NBER Working Paper
14982.
は、リーマンショック以降の、今後のマクロ動学分析の方向性を示しているも
のとして注目されている。ルーカスの誤認識モデルやケインズの美人投票のよう
に、マクロ経済学が 40 年周期で期待形成、とくに市場全体の期待形成に関心が
戻るのは興味深い。
There are also logical issues that will emerge. To be explicit, the proof of RH in this paper is by contradiction and this is not accepted as valid in ZF, it does require choice. I fully expect that the most general version of the Riemann Hypothesis will be an undecidable problem in the Godel sense.
トッド関数は関・ベルヌーイ数の母関数としてあらわれるだけでなく、黒体輻射に関するプランクの公式やリーマン・ロッホ問題の位相幾何公式にもその姿を現す。そしてトッド関数を通常のコホモロジー理論と K 理論とにおけるオイラー類の比と理解し直すことで、アティヤ=シンガー指数定理の公式にも結びついていく。（参考資料：「調和級数から指数定理へ - 日本数学会」）
Frank Plumpton Ramsey (/?ramzi/; 22 February 1903 ? 19 January 1930) was a British philosopher, mathematician and economist who made fundamental contributions to abstract algebra before his death at the age of 26. He was a close friend of Ludwig Wittgenstein and was instrumental(楽器・役立つ) in translating Wittgenstein's Tractatus Logico-Philosophicus (『論理哲学論考』or『論考』1921) into English, as well as persuading Wittgenstein to return to philosophy and Cambridge. Like Wittgenstein, he was a member of the Cambridge Apostles, the intellectual secret society, from 1921.
Life
Ramsey was born on 22 February 1903 in Cambridge where his father Arthur Stanley Ramsey (1867?1954), also a mathematician, was President of Magdalene College. His mother was Mary Agnes Stanley (1875?1927). He was the eldest of two brothers and two sisters, and his brother Michael Ramsey, the only one of the four siblings who was to remain Christian, later became Archbishop of Canterbury. He entered Winchester College in 1915 and later returned to Cambridge to study mathematics at Trinity College. While studying mathematics at Trinity College, Ramsey became a student to John Maynard Keynes, and an active member in the Apostles, a Cambridge discussion group. In 1923, he received his bachelor's degree in mathematics, passing his examinations with the result of first class with distinction. He was named Senior Wrangler.[2] Easy-going, simple and modest, Ramsey had many interests besides his scientific work. Even as a teenager Ramsey exhibited both a profound ability and, as attested by his brother, an extremely diverse range of interests:
He was interested in almost everything. He was immensely widely read in English literature; he was enjoying classics though he was on the verge of plunging into being a mathematical specialist; he was very interested in politics, and well-informed; he had got a political concern and a sort of left-wing caring-for-the-underdog kind of outlook about politics.
??Michael Ramsey, Quoted in Mellor
In 1923, Ramsey was befriended by Geoffrey and Margaret Pyke, then on the point of founding the Malting House School in Cambridge; the Pykes took Ramsey into their family, taking him on holiday and asking him to be the godfather of their young son. Margaret found herself to be the object of his affection, Ramsey recording in his diary:
One afternoon I went out alone with her on Lake Orta and became filled with desire and we came back and lay on two beds side by side she reading, I pretending to, but with an awful conflict in my mind. After about an hour I said (she was wearing her horn spectacles and looking superlatively beautiful in the Burne Jones style) ‘Margaret will you fuck with me?’[3]
Margaret wanted time to consider his proposition and thus began an uncomfortable dance between them, which contributed to Ramsey's depressive moods in early 1924; as a result he travelled to Vienna for psychoanalysis. He, like many of his contemporaries, including his Viennese flatmate and fellow Apostle Lionel Penrose (also in analysis with Siegfried Bernfeld), was intellectually interested in psychoanalysis. Ramsey's analyst was Theodor Reik, a disciple of Freud. As one of the justifications for undertaking the therapy, he asserted in a letter to his mother that unconscious impulses might even affect the work of a mathematician. While in Vienna, he visited Wittgenstein in Puchberg, was befriended by the Wittgenstein family and visited A.S. Neill's experimental school four hours from Vienna at Sonntagsberg. In the summer of 1924, he continued his analysis by joining Reik at Dobbiaco (in South Tyrol), where a fellow analysand was Lewis Namier. Ramsey returned to England in October 1924; with John Maynard Keynes's support he became a fellow of King's College, Cambridge. He joined a Psychoanalysis Group in Cambridge with fellow members Arthur Tansley, Lionel Penrose, Harold Jeffreys, John Rickman and James Strachey, the qualification for membership of which was a completed psychoanalysis.
Ramsey married Lettice Baker in September 1925, the wedding taking place in a Register Office since Ramsey was, as his wife described him, a ‘militant atheist'. After Ramsey's death, Lettice Ramsey opened a photography studio in Cambridge with photographer Helen Muspratt.[4] The marriage produced two daughters. Despite his atheism, Ramsey was quite tolerant towards his brother when the latter decided to become a priest in the Church of England.[5]
In 1926 he became a university lecturer in mathematics and later a Director of Studies in Mathematics at King's College. The Vienna Circle manifesto (1929) lists three of his publications[6][7][8] in a bibliography of closely related authors.
Ramsey and Wittgenstein
When I. A. Richards and C. K. Ogden, both Fellows of Magdalene, first met Ramsey, he expressed his interest in learning German. According to Richards, he mastered the language "in almost hardly over a week",[9] although other sources show he had taken one year of German in school.[10] Ramsey was then able, at the age of 19, to make the first draft of the translation of the German text of Wittgenstein's Tractatus Logico Philosophicus. Ramsey was impressed by Wittgenstein's work and after graduating as Senior Wrangler in the Mathematical Tripos of 1923 he made a journey to Austria to visit Wittgenstein, at that time teaching in a primary school in the small community of Puchberg am Schneeberg. For two weeks Ramsey discussed the difficulties he was facing in understanding the Tractatus. Wittgenstein made some corrections to the English translation in Ramsey's copy and some annotations and changes to the German text that subsequently appeared in the second edition in 1933.
Ramsey and John Maynard Keynes cooperated to try to bring Ludwig Wittgenstein back to Cambridge (he had been a student there before World War I). Once Wittgenstein had returned to Cambridge, Ramsey became his nominal supervisor. Wittgenstein submitted the Tractatus Logico-Philosophicus as his doctoral thesis. G.E. Moore and Bertrand Russell acted as examiners. Later, the three of them arranged financial aid for Wittgenstein to help him continue his research work.
In 1929 Ramsey and Wittgenstein regularly discussed issues in mathematics and philosophy with Piero Sraffa, an Italian economist who had been brought to Cambridge by Keynes after Sraffa had aroused Benito Mussolini's ire by publishing an article critical of the Fascist regime in the Manchester Guardian. The contributions of Ramsey to these conversations were acknowledged by both Sraffa and Wittgenstein in their later work.
Early death
Suffering from chronic liver problems, Ramsey developed jaundice(黄疸になる:developに(病気に)なるって意味があるのか？英辞郎だと問題が生じる) after an abdominal operation(abdominalは腹部の。開腹手術らしい) and died on 19 January 1930 at Guy's Hospital in London at the age of 26.
He is buried in the Parish of the Ascension Burial Ground in Cambridge; his parents are buried in the same plot.[11]
Ramsey's notes and manuscripts were acquired by Nicholas Rescher for the Archives of Scientific Philosophy and the University of Pittsburgh. This collection contains only a few letters but a great many drafts of papers and book chapters, some still unpublished. Other papers, including his diary and letters and memoirs by his widow Lettice Ramsey and his father, are held in the Modern Archives, King's College, Cambridge.
The Decision Analysis Society[12] annually awards the Frank P. Ramsey Medal[13] to recognise substantial contributions to decision theory and its application to important classes of real decision problems.
Work
Mathematical logic
One of the theorems proved by Ramsey in his 1928 paper On a problem of formal logic now bears his name (Ramsey's theorem). While this theorem is the work Ramsey is probably best remembered for, he only proved it in passing, as a minor lemma along the way to his true goal in the paper, solving a special case of the decision problem for first-order logic, namely the decidability of what is now called the Bernays?Schonfinkel?Ramsey class of first-order logic, as well as a characterisation of the spectrum of sentences in this fragment of logic. Alonzo Church would go on to show that the general case of the decision problem for first-order logic is unsolvable (see Church's theorem). A great amount of later work in mathematics was fruitfully developed out of the ostensibly minor lemma, which turned out to be an important early result in combinatorics, supporting the idea that within some sufficiently large systems, however disordered, there must be some order. So fruitful, in fact, was Ramsey's theorem that today there is an entire branch of mathematics, known as Ramsey theory, which is dedicated to studying similar results.
Philosophy
His philosophical works included Universals (1925), Facts and propositions (1927) (which proposed a redundancy theory of truth), Universals of law and of fact (1928), Knowledge (1929), Theories (1929), On Truth (1929), and General propositions and causality (1929). Wittgenstein mentions him in the introduction to his Philosophical Investigations as an influence.
Economics
Keynes and Pigou encouraged Ramsey to work on economics as "From a very early age, about sixteen I think, his precocious mind was intensely interested in economic problems" (Keynes, 1933). Ramsey responded to Keynes's urging by writing three papers in economic theory all of which were of fundamental importance, though it was many years before they received their proper recognition by the community of economists.
Ramsey's three papers, described below in detail, were on subjective probability and utility (1926), optimal allocation (1927) and optimal one-sector economic growth (1928). The economist Paul Samuelson described them in 1970 as "three great legacies ? legacies that were for the most part mere by-products of his major interest in the foundations of mathematics and knowledge."[14]
A mathematical theory of saving
This significant paper was published in The Economic Journal, and involved "a strategically beautiful application of the calculus of variations" (Paul Samuelson)[15] to determine the optimal amount an economy should invest (save) rather than consume so as to maximise future utility, or in Ramsey's words "how much of its income should a nation save?" (Ramsey, 1928).
Keynes described the article as "one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject, the power and elegance of the technical methods employed, and the clear purity of illumination with which the writer's mind is felt by the reader to play about its subject. The article is terribly difficult reading for an economist, but it is not difficult to appreciate how scientific and aesthetic(審美眼・審美眼のある esθ[e]tik) qualities are combined in it together."[16] The Ramsey model is today acknowledged as the starting point for optimal accumulation theory although its importance was not recognised until many years after its first publication.
The main contributions of the model were firstly the initial question Ramsey posed on how much savings should be and secondly the method of analysis, the intertemporal maximisation (optimisation) of collective or individual utility by applying techniques of dynamic optimisation. Tjalling C. Koopmans and David Cass modified the Ramsey model incorporating the dynamic features of population growth at a steady rate and of Harrod-neutral technical progress again at a steady rate, giving birth to a model named the Ramsey?Cass?Koopmans model where the objective now is to maximise household's utility function.
A Contribution to the Theory of Taxation
Author(s): F. P. Ramsey
Source: The Economic Journal, Vol. 37, No. 145 (Mar., 1927), pp. 47-61 https://www.jstor.org/stable/pdf/2222721.pdf
A contribution to the theory of taxation
In this paper Ramsey's contribution to economic theory was the elegant concept of Ramsey pricing. This is applicable in situations where a (regulated) monopolist wants to maximise consumer surplus whilst at the same time ensuring that its costs are adequately covered. This is achieved by setting the price such that the markup over marginal cost is inversely proportional to the price elasticity of demand for that good. Like its predecessor this paper was published in The Economic Journal in 1927. Ramsey poses the question that is to be solved at the beginning of the article: "a given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how much should these rates be adjusted in order that the decrement of utility may be a minimum?" (Ramsey 1927). The problem was suggested to him by the economist Arthur Pigou and the paper was Ramsey's answer to the problem.
Truth and probability
Keynes in his A Treatise on Probability (1921) argued against the subjective approach in epistemic probabilities. For Keynes, subjectivity of probabilities doesn't matter as much, as for him there is an objective relationship between knowledge and probabilities, as knowledge is disembodied and not personal.
Ramsey in his article disagrees with Keynes's approach as for him there is a difference between the notions of probability in physics and in logic. For Ramsey probability is not related to a disembodied body of knowledge but is related to the knowledge that each individual possesses alone. Thus personal beliefs that are formulated by this individual knowledge govern probabilities, leading to the notions of subjective probability and Bayesian probability. Consequently, subjective probabilities can be inferred by observing actions that reflect individuals' personal beliefs. Ramsey argued that the degree of probability that an individual attaches to a particular outcome can be measured by finding what odds the individual would accept when betting on that outcome.
Ramsey suggested a way of deriving a consistent theory of choice under uncertainty that could isolate beliefs from preferences while still maintaining subjective probabilities.
Despite the fact that Ramsey's work on probabilities was of great importance again no one paid any attention to it until the publication of Theory of Games and Economic Behavior of John von Neumann and Oskar Morgenstern in 1944 (1947 2nd ed.)
ある数がほとんど整数（ほとんどせいすう、英: almost integer）であるとは、整数ではないが、整数に非常に近いことを意味する。どれほど近ければ十分であるのか明確な決まりはないが、一見して整数に近いとは分からないのに、近似値を計算すると驚くほど整数に近い数で、小数点以下の部分が「.000…」または「.999…」のように、0か9が数個連続する場合、このように表現される。例えば、「インドの魔術師」の異名をもつシュリニヴァーサ・ラマヌジャンは
65537 は {\displaystyle 2^{2^{4}}+1}2^{{2^{4}}}+1 の形で表され、2018年2月現在知られているうちで最大のフェルマー素数である。カール・フリードリヒ・ガウスは1801年に出版した『整数論の研究』において、p がフェルマー素数ならば正 p 角形は定規とコンパスで作図可能であることを証明した。また、逆に、奇素数 p に対して正 p 角形が作図可能ならば、p はフェルマー素数であることも証明した
