High level people は自分達で勝手に立てたスレ28へどうぞ!sage進行推奨(^^; また、スレ43は、私が立てたスレではないので、私は行きません。そこでは、私はスレ主では無くなりますからね。このスレに不満な人は、そちらへ。 http://rio2016.2ch.net/test/read.cgi/math/1506152332/ 旧スレが512KBオーバー(間近)で、新スレ立てる (スレ主の趣味で上記以外にも脱線しています。ネタにスレ主も理解できていないページのURLも貼ります。関連のアーカイブの役も期待して。)
ディニ微分が、いつごろ論文か判然としないが、 Ulisse Dini (14 November 1845 ? 28 October 1918) と、Books by U. Dini 1907?1915 などとあるので 100年以上前は確実だろう
https://en.wikipedia.org/wiki/Ulisse_Dini Ulisse Dini (抜粋) Ulisse Dini (14 November 1845 ? 28 October 1918) was an Italian mathematician and politician, born in Pisa. He is known for his contribution to real analysis, partly collected in his book "Fondamenti per la teorica delle funzioni di variabili reali".[1]
Life and academic career Dini attended the Scuola Normale Superiore in order to become a teacher. One of his professors was Enrico Betti. In 1865, a scholarship enabled him to visit Paris, where he studied under Charles Hermite as well as Joseph Bertrand, and published several papers. In 1866, he was appointed to the University of Pisa, where he taught algebra and geodesy. In 1871, he succeeded Betti as professor for analysis and geometry. From 1888 until 1890, Dini was rettore[2] of the Pisa University, and of the Scuola Normale Superiore from 1908 until his death in 1918.
He was also active as a politician: in 1871 he was voted into the Pisa city council, and in 1880, he became a member of the Italian parliament.
Honors He has been elected honorary member of the London Mathematical Society.[3]
Work Research activity
Thus, by the year 1877, or seven years from the time he began, he published the treatise, since famous, entitled Foundations for the Theory of Functions of Real Variables (Fondamenti per la teoria delle funzioni di variabili reali).
Much of what Dini here sets forth concerning such topics as continuous and discontinuous functions, the derivative and the conditions for its existence, series, definite integrals, the properties of the incremental ratio, etc., was entirely original with himself and has since come to be regarded everywhere as basal in the real variable theory.
??Walter Burton Ford, (Ford 1920, p. 174).
Books by U. Dini Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale (Pisa, T. Nistri, 1880) Lezioni di analisi infinitesimale. vol. 1 (Pisa, T. Nistri, 1907?1915) Lezioni di analisi infinitesimale.vol. 2 part 1 (Pisa, T. Nistri, 1907?1915) Lezioni di analisi infinitesimale.vol. 2 part 2 (Pisa, T. Nistri, 1907?1915) Fondamenti per la teorica delle funzioni di variabili reali (Pisa, T. Nistri, 1878) (引用終り) 0412現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/22(金) 21:51:38.24ID:UIwpFvOX>>401 >どっちもどっち >ID:KNjgsEZnはただの基地外
ディニ微分については、おっさんの紹介した”Fundamentals of Real Analysis 著者: Sterling K. Berberian”(P220)にもあって下記 (>>390-391) (抜粋) ”5.3.7. Definition. Let g: [a ,b] → R, a < b, and let c ∈ [a ,b] . Write B = [a, b] - {c} and define f: B →R ̄ by the formula
f(x) = g(x) - g(c)/(x - c).
Of course the values of f are in R, but we are being consistent with the foregoing notations; some of the numbers we are about to associate with f may be infinite.
If c ∈ [a, b) then c is approachable from the right by x ∈ B and we define
Of course the values of f are in R, but we are being consistent with the foregoing notations; some of the numbers we are about to associate with f may be infinite.
If c ∈ [a, b) then c is approachable from the right by x ∈ B and we define
These four numbers are called the Dini derivates of g at c; more precisely (for example), (D +g)(c) is the lower right-hand derivate of g at c.” (引用終り)
http://www.artsci.kyushu-u.ac.jp/~ssaito/jpn/ 斎藤新悟のウェブサイトへようこそ! 九州大学基幹教育院准教授。 1981年大阪府生まれ。 東京大学理学部数学科卒業。 University College London, Department of Mathematics博士課程修了。 PhD in Mathematics, University of London. 九州大学学術研究員を経て,2013年4月から現職。
This is a webpage on mathematics and related topics maintained by YAMAGAMI Shigeru under the support of Department of Mathematical Sciences, Ibaraki University.
まず、”ディニ微分”というキーワードが見つかったので、「1.従来の数学の範囲の定理の再証明」の線を調べつつ それ(1項関連)が見つからなければ、その過程で「2.(あるいは)素人の勘違い」ってことがはっきりするだろう 0482132人目の素数さん2017/12/23(土) 21:43:11.49ID:JRmFnvAf>>481 新規の定理って言ってないじゃん まったく話が成り立たねー 0483現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/23(土) 21:50:11.45ID:lrnu6EUA Ulisse Dini (14 November 1845 〜 28 October 1918) was an Italian mathematician and politician, born in Pisa. https://en.wikipedia.org/wiki/Ulisse_Dini
さて、定理1.7 (422 に書いた定理)のそもそもの目的は、変形トマエ函数(Ruler Function)関連で、 「系1.8 有理数の点で不連続、 無理数の点で微分可能となるf : R → R は存在しない」を導くことであった
変形トマエ函数(Ruler Function)関連については、過去スレで取り上げているが、いま一度整理すると (長いが、あとのために抜粋する) http://mathforum.org/kb/message.jspa?messageID=5432910 (>>35より) Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 (抜粋) (注:下記で、f^rなどとして、rの指数による類別をしている) The ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and f(x) = 1/q if x = p/q where p and q are relatively prime integers with q > 0.
It is well-known that f is continuous at each irrational point and discontinuous at each rational point.
** For each r > 2, f^r is differentiable on a set that has c many points in every interval.
The results above can be further refined.
** For each 0 < r < 2, f^r satisfies no pointwise Lipschitz condition. Heuer [15]
** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above.
Let w:Z+ --> Z+ be an increasing function that eventually majorizes every power function. Define f_w(x) = 0 for x irrational, f_w(0) = 1, and f_w(p/q) = 1/w(q) where p and q are relatively prime integers.
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)
Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.
THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, a pointwise Holder condition, or even any specified pointwise modulus of continuity condition on a co-meager set.
(Each co-meager set has c points in every interval.)
--------------------------------------------------------------- [4] Bohus Jurek, "Sur la derivabilite des fonctions a variation bornee", Casopis Pro Pestovani Matematiky a Fysiky 65 (1935), 8-27. [Zbl 13.00704; JFM 61.1115.01]
It appears that Jurek proves some general results concerning the zero Hausdorff h-measure of sets of non-differentiability for bounded variation functions such that the sum of the h-values of the countably many jump discontinuities is finite (special case: h(t) = t^r for a fixed 0 < r < 1). General "h-versions" of the ruler function seem to appear as examples, and V. Jarnik's more precise results about the Hausdorff dimension of Liouville-like Diophantine approximation results are used.
--------------------------------------------------------------- [15] Gerald Arthur Heuer, "Functions continuous at the irrationals and discontinuous at the rationals", American Mathematical Monthly 72 #4 (April 1965), 370-373. [MR 31 #3550; Zbl 131.29201]
Let f(x) = 0 if x is irrational, f(p/q) = |1/q| if p and q are relatively prime integers, and f(0) = 1.
We say that a function g is Lipschitzian at x if there exists a neighborhood U of x and a number M > 0 such that |g(x) - g(y)| <= M*|x - y| for all y in U.
THEOREM 2: The function f^r is: (A) discontinuous at the rationals for every r > 0; (B) continuous but not Lipschitzian at the Liouville numbers, for every r > 0; (C) differentiable at every irrational algebraic number of degree <= r-1, if r > 3.
THEOREM 3: The function f^r is differentiable at every algebraic irrational number if r > 2 (and, by Theorem 1, at none if r <= 2).
THEOREM 4: The function f^2 is Lipschitzian but not differentiable at the points of the set {(1/2)*[m - sqrt(d)]: m is an integer and there exists an integer n such that d = m^2 - 4n is positive but not a perfect square} . [This set is dense in the reals.]
THEOREM 5: If g is a function discontinuous at the rationals and continuous at the irrationals, then there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
(p. 373) "We omit the proof, because it is rather lengthy, and one would hope to generalize the theorem by replacing the rationals by an arbitrary dense set, and possibly to show that the set of points at which g fails to be Lipschitzian is a residual set."
NOTE: Sengupta/Lahiri had essentially obtained this result in 1957 (the points of discontinuity have to form an F_sigma set, however). See my remark in [13] above. This result is also proved in Gerald Arthur Heuer, "A property of functions discontinuous on a dense set", American Mathematical Monthly 73 #4 (April 1966), 378-379 [MR 34 #2791]. Heuer proves that for each 0 < s <= 1 and for each f:R --> R such that {x: f is continuous at x} is dense in R and {x: f is not continuous at x} is dense in R, the set of points where f does not satisfy a pointwise Holder condition of order s is the complement of a first category set (i.e. a co-meager set). By choosing s < 1, we obtain a stronger version of Sengupta/Lahiri's result. By intersecting the co-meager sets for s = 1/2, 1/3, 1/4, ..., we get a co-meager set G such that, for each x in G, f does not satisfy a pointwise Holder condition at x for any positive Holder exponent. (Heuer does not explicitly state this last result.) A metric space version of Heuer's result for an arbitrary given pointwise modulus of continuity condition is essentially given in: Edward Maurice Beesley, Anthony Perry Morse, and Donald Chesley Pfaff, "Lipschitzian points", American Mathematical Monthly 79 #6 (June/July 1972), 603-608 [MR 46 #304; Zbl 239.26004]. See also the last theorem in Norton [17] below.
THEOREM 4: The function f^2 is Lipschitzian but not differentiable at the points of the set {(1/2)*[m - sqrt(d)]: m is an integer and there exists an integer n such that d = m^2 - 4n is positive but not a perfect square} . [This set is dense in the reals.]
THEOREM 5: If g is a function discontinuous at the rationals and continuous at the irrationals, then there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
かな?
特に、THEOREM 5 変形トマエ函数(Ruler Function)のような、有理数で不連続、無理数で連続なる函数では、 ”there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.” だと
1)THEOREM 5: If g is a function discontinuous at the rationals and continuous at the irrationals, then there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
2)「系1.8 有理数の点で不連続、 無理数の点で微分可能となるf : R → R は存在しない」
この二つの比較で、2)の”無理数の点で微分可能”なら、1)THEOREM 5の”continuous at the irrationals”は、満たされる ”there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.”から、有理点以外で必ず”at each point of which g fails to satisfy a Lipschitz condition”なる(無理)点が存在する その(無理)点は、微分不能