几乎所有
数学中,几乎所有(英语:Almost all)表示“除了极少数可忽略的以外,其他都是”。更准确的说法,若是集合,“集合中几乎所有的元素”表示“集合中,不考虑在某个可忽略子集内元素的其他元素。”“可忽略”的具体意思则依上下文而定,可能是有限集合、可数集或零测集。
相反的,几乎没有(almost no)表示“只有极少数可忽略的是”,“集合中几乎没有的元素”表示“集合中,只有在某个可忽略子集内的元素”。
不同数学领域中的意思
[编辑]普遍的意思
[编辑]数学里的“几乎所有”有时会指“无限集合中的元素,只有有限多个不符合,其余都符合”的情形[1][2]。此用法也会用在哲学上[3]。“几乎所有”也可以指“不可数集中的元素,只有可数数量的不符合,其余都符合”的情形[sec 1]。
例如:
- 几乎所有正实数都超过1012[4]:293。
- 几乎所有质数都是奇数(只有2例外)[5]
- 几乎所有多面体都是非正多面体(只有九个例外,五个柏拉图立体和四个星形正多面体)。
- 若P是非零多项式,则P(x) ≠ 0对几乎所有的x都成立。
量测理论中的意思
[编辑]![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/49/CantorEscalier.svg/250px-CantorEscalier.svg.png)
在探讨实数时,有时“几乎所有”是指“除了在某个零测集以外的所有实数。”[6][7][sec 2]。同様地,若S是某个实数集合,则“几乎所有在集合S里的数字”是指“除了在某个零测集以外,集合S的所有实数。”[8]数线可以视为是一维的欧几里得空间。在更广义的n维空间(n为正整数),其定义则推广为“除了在某个零测集以外,空间里的所有点。”[sec 3]或是“除了在某个零测集以外,集合S里的所有点。” (此时,S是空间中点的集合)[9]。更广义的说法,“几乎所有”在测度理论中有时是指几乎处处[10][11][sec 4],或是概率论中的几乎必然[11][sec 5]。
例子:
- 在测度空间(例如实数)里,可数集是零测集。有理数的集合可数,因此几乎所有的实数都是无理数[12]。
- 康托尔的第一篇集合论论文证明了代数数的集合也是可数的,因此几乎所有的实数都是超越数[13][sec 6]。
- 几乎所有实数都是正规数[14]。
- 康托尔集也是零测集,虽然康托尔集不可数,但几乎所有实数都不在内[6]。
- 康托尔函数的导数在单位区间内几乎所有数字下均为0[15]。这是以上范例的结果,因为康托尔函数是局部常数函数,在康托尔集以下,其导数为0。
数论中的意思
[编辑]数论中的“几乎所有正整数”可以指“自然密度为1集合里的正整数”。也就是说,若A是一个正整数的集合,当n趋近无限大时,小于n,在集合A里的正整数数量,除以小于n的正整数数量,比值趋近于1,则几乎所有整数都是在集合A内[16][17][sec 7]。
若再进一步推广,令S是正整数的无穷集合,例如正的偶数集合或是质数集合,若A是S的子集合,且当n趋近无限大时,若集合A里小于n的元素数量,除以集合S里小于n的元素数量,比值趋近于1,则可以说几乎所有集合S里的元素都在集合A里。
例子:
- 正整数的余有限集其自然密度为1,因此每一个余有限集都包括几乎所有的正整数。
- 几乎所有正整数都是合数[sec 7][proof 1]
- 几乎所有正的偶数都可以表示为二个质数的和[4]:489。
- 几乎所有质数都不是孪生素数。进一步说,针对每一个正整数g,几乎所有质数的间隙都大于g,几乎所有质数和其较大质数以及较小质数的间隔都都大于g,也就是说,在p − g和p + g之间没有其他的质数[18]。
拓扑学中的意思
[编辑]在topology[19],特别是动力系统理论中[20][21][22](包括经济学的应用)[23] ,拓扑空间内几乎所有的点可以指“除了在某个贫乏集以外,所有此空间内的点。”有些则用更限定的定义,子集包括空间内几乎所有的点,若这个子集包括某个开集的稠密集[21][24][25]。
例子:
代数中的意思
[编辑]在抽象代数和数理逻辑中,若U是集合X的超滤子,“集合X内几乎所有元素”有时是指“U的部分元素内的元素”[26][27][28][29]。针对任何将X分为二个不交集的集合划分,其中一个不交集包括X里几乎所有的元素。[29]
证明
[编辑]相关条目
[编辑]参考资料
[编辑]一次文献
[编辑]- ^ Cahen, Paul-Jean; Chabert, Jean-Luc. Integer-Valued Polynomials. Mathematical Surveys and Monographs 48. American Mathematical Society. 3 December 1996: xix. ISBN 978-0-8218-0388-2. ISSN 0076-5376.
- ^ Cahen, Paul-Jean; Chabert, Jean-Luc. Chapter 4: What's New About Integer-Valued Polynomials on a Subset?. Hazewinkel, Michiel (编). Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications 520. Springer. 7 December 2010: 85 [First published 2000]. ISBN 978-1-4419-4835-9. doi:10.1007/978-1-4757-3180-4.
- ^ Gärdenfors, Peter. The Dynamics of Thought. Synthese Library 300. Springer. 22 August 2005: 190–191. ISBN 978-1-4020-3398-8.
- ^ 4.0 4.1 Courant, Richard; Robbins, Herbert; Stewart, Ian. What is Mathematics? An Elementary Approach to Ideas and Methods 2nd. Oxford University Press. 18 July 1996. ISBN 978-0-19-510519-3.
- ^ Movshovitz-hadar, Nitsa; Shriki, Atara. Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook. World Scientific. 2018-10-08: 38. ISBN 978-981-320-864-3 (英语).
This can also be expressed in the statement: 'Almost all prime numbers are odd.'
- ^ 6.0 6.1 Korevaar, Jacob. Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration 1. New York: Academic Press. 1 January 1968: 359–360. ISBN 978-1-4832-2813-6.
- ^ Natanson, Isidor P. Theory of Functions of a Real Variable 1. 由Boron, Leo F.翻译 revised. New York: Frederick Ungar Publishing. June 1961: 90. ISBN 978-0-8044-7020-9.
- ^ Sohrab, Houshang H. Basic Real Analysis 2. Birkhäuser. 15 November 2014: 307. ISBN 978-1-4939-1841-6. doi:10.1007/978-1-4939-1841-6.
- ^ Helmberg, Gilbert. Introduction to Spectral Theory in Hilbert Space. North-Holland Series in Applied Mathematics and Mechanics 6 1st. Amsterdam: North-Holland Publishing Company. December 1969: 320. ISBN 978-0-7204-2356-3.
- ^ Vestrup, Eric M. The Theory of Measures and Integration. Wiley Series in Probability and Statistics. United States: Wiley-Interscience. 18 September 2003: 182. ISBN 978-0-471-24977-1.
- ^ 11.0 11.1 Billingsley, Patrick. Probability and Measure (PDF). Wiley Series in Probability and Statistics 3rd. United States: Wiley-Interscience. 1 May 1995: 60. ISBN 978-0-471-00710-4. (原始内容 (PDF)存档于23 May 2018).
- ^ Niven, Ivan. Irrational Numbers. Carus Mathematical Monographs 11. Rahway: Mathematical Association of America. 1 June 1956: 2–5. ISBN 978-0-88385-011-4.
- ^ Baker, Alan. A concise introduction to the theory of numbers. Cambridge University Press. 1984: 53. ISBN 978-0-521-24383-4.
- ^ Granville, Andrew; Rudnick, Zeev. Equidistribution in Number Theory, An Introduction. Nato Science Series II 237. Springer. 7 January 2007: 11. ISBN 978-1-4020-5404-4.
- ^ Burk, Frank. Lebesgue Measure and Integration: An Introduction. A Wiley-Interscience Series of Texts, Monographs, and Tracts. United States: Wiley-Interscience. 3 November 1997: 260. ISBN 978-0-471-17978-8.
- ^ Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. 1940: 50.
- ^ 17.0 17.1 Hardy, G. H.; Wright, E. M. An Introduction to the Theory of Numbers 4th. Oxford University Press. December 1960: 8–9. ISBN 978-0-19-853310-8.
- ^ Prachar, Karl. Primzahlverteilung. Grundlehren der mathematischen Wissenschaften 91. Berlin: Springer. 1957: 164 (德语). Cited in Grosswald, Emil. Topics from the Theory of Numbers 2nd. Boston: Birkhäuser. 1 January 1984: 30. ISBN 978-0-8176-3044-7.
- ^ Oxtoby, John C. Measure and Category. Graduate Texts in Mathematics 2 2nd. United States: Springer. 1980: 59,68. ISBN 978-0-387-90508-2. While Oxtoby does not explicitly define the term there, Babai has borrowed it from Measure and Category in his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham, Grötschel and Lovász's Handbook of Combinatorics (vol. 2), and Broer and Takens note in their book Dynamical Systems and Chaos that Measure and Category compares this meaning of "almost all" to the measure theoretic one in the real line (though Oxtoby's book discusses meagre sets in general topological spaces as well).
- ^ Baratchart, Laurent. Recent and New Results in Rational L2 Approximation. Curtain, Ruth F. (编). Modelling, Robustness and Sensitivity Reduction in Control Systems. NATO ASI Series F 34. Springer. 1987: 123. ISBN 978-3-642-87516-8. doi:10.1007/978-3-642-87516-8.
- ^ 21.0 21.1 Broer, Henk; Takens, Floris. Dynamical Systems and Chaos. Applied Mathematical Sciences 172. Springer. 28 October 2010: 245. ISBN 978-1-4419-6870-8. doi:10.1007/978-1-4419-6870-8.
- ^ Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V. Dynamics of One-Dimensional Maps. Mathematics and Its Applications 407. Springer. 30 April 1997: 33. ISBN 978-94-015-8897-3. doi:10.1007/978-94-015-8897-3.
- ^ Yuan, George Xian-Zhi. KKM Theory and Applications in Nonlinear Analysis. Pure and Applied Mathematics; A Series of Monographs and Textbooks. Marcel Dekker. 9 February 1999: 21. ISBN 978-0-8247-0031-7.
- ^ Albertini, Francesca; Sontag, Eduardo D. Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems. Bonnard, Bernard; Bride, Bernard; Gauthier, Jean-Paul; Kupka, Ivan (编). Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory 8. Birkhäuser. 1 September 1991: 29. ISBN 978-1-4612-3214-8. doi:10.1007/978-1-4612-3214-8.
- ^ De la Fuente, Angel. Mathematical Models and Methods for Economists. Cambridge University Press. 28 January 2000: 217. ISBN 978-0-521-58529-3.
- ^ Komjáth, Péter; Totik, Vilmos. Problems and Theorems in Classical Set Theory. Problem Books in Mathematics. United States: Springer. 2 May 2006: 75. ISBN 978-0387-30293-5.
- ^ Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer. The Classical Fields: Structural Features of the Real and Rational Numbers. Encyclopedia of Mathematics and Its Applications 112. Cambridge University Press. 24 September 2007: 155. ISBN 978-0-521-86516-6.
- ^ Schoutens, Hans. The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics 1999. Springer. 2 August 2010: 8. ISBN 978-3-642-13367-1. doi:10.1007/978-3-642-13368-8.
- ^ 29.0 29.1 Rautenberg, Wolfgang. A Concise to Mathematical Logic. Universitext 3rd. Springer. 17 December 2009: 210–212. ISBN 978-1-4419-1221-3. doi:10.1007/978-1-4419-1221-3.
二次文献
[编辑]- ^ Schwartzman, Steven. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English
. Spectrum Series. Mathematical Association of America. 1 May 1994: 22. ISBN 978-0-88385-511-9.
- ^ Clapham, Christopher; Nicholson, James. The Concise Oxford Dictionary of mathematics. Oxford Paperback References 4th. Oxford University Press. 7 June 2009: 38. ISBN 978-0-19-923594-0.
- ^ James, Robert C. Mathematics Dictionary 5th. Chapman & Hall. 31 July 1992: 269. ISBN 978-0-412-99031-1.
- ^ Bityutskov, Vadim I. Almost-everywhere. Hazewinkel, Michiel (编). Encyclopaedia of Mathematics 1. Kluwer Academic Publishers. 30 November 1987: 153. ISBN 978-94-015-1239-8. doi:10.1007/978-94-015-1239-8.
- ^ Itô, Kiyosi (编). Encyclopedic Dictionary of Mathematics 2 2nd. Kingsport: MIT Press. 4 June 1993: 1267. ISBN 978-0-262-09026-1.
- ^ Almost All Real Numbers are Transcendental - ProofWiki. proofwiki.org. [2019-11-11].
- ^ 7.0 7.1 埃里克·韦斯坦因. Almost All. MathWorld. See also Weisstein, Eric W. CRC Concise Encyclopedia of Mathematics 1st. CRC Press. 25 November 1988: 41. ISBN 978-0-8493-9640-3.
- ^ Itô, Kiyosi (编). Encyclopedic Dictionary of Mathematics 1 2nd. Kingsport: MIT Press. 4 June 1993: 67. ISBN 978-0-262-09026-1.