fbpx
Wikipedia

Almost all

In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if is a set, "almost all elements of " means "all elements of but those in a negligible subset of ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null.

In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of " means "a negligible amount of elements of ".

Meanings in different areas of mathematics

Prevalent meaning

Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) but finitely many".[1][2] This use occurs in philosophy as well.[3] Similarly, "almost all" can mean "all (elements of an uncountable set) but countably many".[sec 1]

Examples:

Meaning in measure theory

 
The Cantor function as a function that has zero derivative almost everywhere

When speaking about the reals, sometimes "almost all" can mean "all reals but a null set".[6][7][sec 2] Similarly, if S is some set of reals, "almost all numbers in S" can mean "all numbers in S but those in a null set".[8] The real line can be thought of as a one-dimensional Euclidean space. In the more general case of an n-dimensional space (where n is a positive integer), these definitions can be generalised to "all points but those in a null set"[sec 3] or "all points in S but those in a null set" (this time, S is a set of points in the space).[9] Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory,[10][11][sec 4] or in the closely related sense of "almost surely" in probability theory.[11][sec 5]

Examples:

Meaning in number theory

In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1". That is, if A is a set of positive integers, and if the proportion of positive integers in A below n (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive integers are in A.[16][17][sec 7]

More generally, let S be an infinite set of positive integers, such as the set of even positive numbers or the set of primes, if A is a subset of S, and if the proportion of elements of S below n that are in A (out of all elements of S below n) tends to 1 as n tends to infinity, then it can be said that almost all elements of S are in A.

Examples:

  • The natural density of cofinite sets of positive integers is 1, so each of them contains almost all positive integers.
  • Almost all positive integers are composite.[sec 7][proof 1]
  • Almost all even positive numbers can be expressed as the sum of two primes.[4]: 489 
  • Almost all primes are isolated. Moreover, for every positive integer g, almost all primes have prime gaps of more than g both to their left and their right; that is, there is no other prime between pg and p + g.[18]

Meaning in graph theory

In graph theory, if A is a set of (finite labelled) graphs, it can be said to contain almost all graphs, if the proportion of graphs with n vertices that are in A tends to 1 as n tends to infinity.[19] However, it is sometimes easier to work with probabilities,[20] so the definition is reformulated as follows. The proportion of graphs with n vertices that are in A equals the probability that a random graph with n vertices (chosen with the uniform distribution) is in A, and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them.[21] Therefore, equivalently to the preceding definition, the set A contains almost all graphs if the probability that a coin flip-generated graph with n vertices is in A tends to 1 as n tends to infinity.[20][22] Sometimes, the latter definition is modified so that the graph is chosen randomly in some other way, where not all graphs with n vertices have the same probability,[21] and those modified definitions are not always equivalent to the main one.

The use of the term "almost all" in graph theory is not standard; the term "asymptotically almost surely" is more commonly used for this concept.[20]

Example:

Meaning in topology

In topology[24] and especially dynamical systems theory[25][26][27] (including applications in economics),[28] "almost all" of a topological space's points can mean "all of the space's points but those in a meagre set". Some use a more limited definition, where a subset only contains almost all of the space's points if it contains some open dense set.[26][29][30]

Example:

Meaning in algebra

In abstract algebra and mathematical logic, if U is an ultrafilter on a set X, "almost all elements of X" sometimes means "the elements of some element of U".[31][32][33][34] For any partition of X into two disjoint sets, one of them will necessarily contain almost all elements of X. It is possible to think of the elements of a filter on X as containing almost all elements of X, even if it isn't an ultrafilter.[34]

Proofs

  1. ^ The prime number theorem shows that the number of primes less than or equal to n is asymptotically equal to n/ln(n). Therefore, the proportion of primes is roughly ln(n)/n, which tends to 0 as n tends to infinity, so the proportion of composite numbers less than or equal to n tends to 1 as n tends to infinity.[17]

See also

References

Primary sources

  1. ^ Cahen, Paul-Jean; Chabert, Jean-Luc (3 December 1996). Integer-Valued Polynomials. Mathematical Surveys and Monographs. Vol. 48. American Mathematical Society. p. xix. ISBN 978-0-8218-0388-2. ISSN 0076-5376.
  2. ^ Cahen, Paul-Jean; Chabert, Jean-Luc (7 December 2010) [First published 2000]. "Chapter 4: What's New About Integer-Valued Polynomials on a Subset?". In Hazewinkel, Michiel (ed.). Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications. Vol. 520. Springer. p. 85. doi:10.1007/978-1-4757-3180-4. ISBN 978-1-4419-4835-9.
  3. ^ Gärdenfors, Peter (22 August 2005). The Dynamics of Thought. Synthese Library. Vol. 300. Springer. pp. 190–191. ISBN 978-1-4020-3398-8.
  4. ^ a b Courant, Richard; Robbins, Herbert; Stewart, Ian (18 July 1996). What is Mathematics? An Elementary Approach to Ideas and Methods (2nd ed.). Oxford University Press. ISBN 978-0-19-510519-3.
  5. ^ Movshovitz-hadar, Nitsa; Shriki, Atara (2018-10-08). Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook. World Scientific. p. 38. ISBN 978-981-320-864-3. This can also be expressed in the statement: 'Almost all prime numbers are odd.'
  6. ^ a b Korevaar, Jacob (1 January 1968). Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration. Vol. 1. New York: Academic Press. pp. 359–360. ISBN 978-1-4832-2813-6.
  7. ^ Natanson, Isidor P. (June 1961). Theory of Functions of a Real Variable. Vol. 1. Translated by Boron, Leo F. (revised ed.). New York: Frederick Ungar Publishing. p. 90. ISBN 978-0-8044-7020-9.
  8. ^ Sohrab, Houshang H. (15 November 2014). Basic Real Analysis (2 ed.). Birkhäuser. p. 307. doi:10.1007/978-1-4939-1841-6. ISBN 978-1-4939-1841-6.
  9. ^ Helmberg, Gilbert (December 1969). Introduction to Spectral Theory in Hilbert Space. North-Holland Series in Applied Mathematics and Mechanics. Vol. 6 (1st ed.). Amsterdam: North-Holland Publishing Company. p. 320. ISBN 978-0-7204-2356-3.
  10. ^ Vestrup, Eric M. (18 September 2003). The Theory of Measures and Integration. Wiley Series in Probability and Statistics. United States: Wiley-Interscience. p. 182. ISBN 978-0-471-24977-1.
  11. ^ a b Billingsley, Patrick (1 May 1995). (PDF). Wiley Series in Probability and Statistics (3rd ed.). United States: Wiley-Interscience. p. 60. ISBN 978-0-471-00710-4. Archived from the original (PDF) on 23 May 2018.
  12. ^ Niven, Ivan (1 June 1956). Irrational Numbers. Carus Mathematical Monographs. Vol. 11. Rahway: Mathematical Association of America. pp. 2–5. ISBN 978-0-88385-011-4.
  13. ^ Baker, Alan (1984). A concise introduction to the theory of numbers. Cambridge University Press. p. 53. ISBN 978-0-521-24383-4.
  14. ^ Granville, Andrew; Rudnick, Zeev (7 January 2007). Equidistribution in Number Theory, An Introduction. Nato Science Series II. Vol. 237. Springer. p. 11. ISBN 978-1-4020-5404-4.
  15. ^ Burk, Frank (3 November 1997). Lebesgue Measure and Integration: An Introduction. A Wiley-Interscience Series of Texts, Monographs, and Tracts. United States: Wiley-Interscience. p. 260. ISBN 978-0-471-17978-8.
  16. ^ Hardy, G. H. (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. p. 50.
  17. ^ a b Hardy, G. H.; Wright, E. M. (December 1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. pp. 8–9. ISBN 978-0-19-853310-8.
  18. ^ Prachar, Karl (1957). Primzahlverteilung. Grundlehren der mathematischen Wissenschaften (in German). Vol. 91. Berlin: Springer. p. 164. Cited in Grosswald, Emil (1 January 1984). Topics from the Theory of Numbers (2nd ed.). Boston: Birkhäuser. p. 30. ISBN 978-0-8176-3044-7.
  19. ^ a b Babai, László (25 December 1995). "Automorphism Groups, Isomorphism, Reconstruction". In Graham, Ronald; Grötschel, Martin; Lovász, László (eds.). Handbook of Combinatorics. Vol. 2. Netherlands: North-Holland Publishing Company. p. 1462. ISBN 978-0-444-82351-9.
  20. ^ a b c Spencer, Joel (9 August 2001). The Strange Logic of Random Graphs. Algorithms and Combinatorics. Vol. 22. Springer. pp. 3–4. ISBN 978-3-540-41654-8.
  21. ^ a b Bollobás, Béla (8 October 2001). Random Graphs. Cambridge Studies in Advanced Mathematics. Vol. 73 (2nd ed.). Cambridge University Press. pp. 34–36. ISBN 978-0-521-79722-1.
  22. ^ Grädel, Eric; Kolaitis, Phokion G.; Libkin, Leonid; Marx, Maarten; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (11 June 2007). Finite Model Theory and Its Applications. Texts in Theoretical Computer Science (An EATCS Series). Springer. p. 298. ISBN 978-3-540-00428-8.
  23. ^ Buckley, Fred; Harary, Frank (21 January 1990). Distance in Graphs. Addison-Wesley. p. 109. ISBN 978-0-201-09591-3.
  24. ^ Oxtoby, John C. (1980). Measure and Category. Graduate Texts in Mathematics. Vol. 2 (2nd ed.). United States: Springer. pp. 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).
  25. ^ Baratchart, Laurent (1987). "Recent and New Results in Rational L2 Approximation". In Curtain, Ruth F. (ed.). Modelling, Robustness and Sensitivity Reduction in Control Systems. NATO ASI Series F. Vol. 34. Springer. p. 123. doi:10.1007/978-3-642-87516-8. ISBN 978-3-642-87516-8.
  26. ^ a b Broer, Henk; Takens, Floris (28 October 2010). Dynamical Systems and Chaos. Applied Mathematical Sciences. Vol. 172. Springer. p. 245. doi:10.1007/978-1-4419-6870-8. ISBN 978-1-4419-6870-8.
  27. ^ Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V. (30 April 1997). Dynamics of One-Dimensional Maps. Mathematics and Its Applications. Vol. 407. Springer. p. 33. doi:10.1007/978-94-015-8897-3. ISBN 978-94-015-8897-3.
  28. ^ Yuan, George Xian-Zhi (9 February 1999). KKM Theory and Applications in Nonlinear Analysis. Pure and Applied Mathematics; A Series of Monographs and Textbooks. Marcel Dekker. p. 21. ISBN 978-0-8247-0031-7.
  29. ^ Albertini, Francesca; Sontag, Eduardo D. (1 September 1991). "Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems". In Bonnard, Bernard; Bride, Bernard; Gauthier, Jean-Paul; Kupka, Ivan (eds.). Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory. Vol. 8. Birkhäuser. p. 29. doi:10.1007/978-1-4612-3214-8. ISBN 978-1-4612-3214-8.
  30. ^ De la Fuente, Angel (28 January 2000). Mathematical Models and Methods for Economists. Cambridge University Press. p. 217. ISBN 978-0-521-58529-3.
  31. ^ Komjáth, Péter; Totik, Vilmos (2 May 2006). Problems and Theorems in Classical Set Theory. Problem Books in Mathematics. United States: Springer. p. 75. ISBN 978-0387-30293-5.
  32. ^ Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer (24 September 2007). The Classical Fields: Structural Features of the Real and Rational Numbers. Encyclopedia of Mathematics and Its Applications. Vol. 112. Cambridge University Press. p. 155. ISBN 978-0-521-86516-6.
  33. ^ Schoutens, Hans (2 August 2010). The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics. Vol. 1999. Springer. p. 8. doi:10.1007/978-3-642-13368-8. ISBN 978-3-642-13367-1.
  34. ^ a b Rautenberg, Wolfgang (17 December 2009). A Concise to Mathematical Logic. Universitext (3rd ed.). Springer. pp. 210–212. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1221-3.

Secondary sources

  1. ^ Schwartzman, Steven (1 May 1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Spectrum Series. Mathematical Association of America. p. 22. ISBN 978-0-88385-511-9.
  2. ^ Clapham, Christopher; Nicholson, James (7 June 2009). The Concise Oxford Dictionary of mathematics. Oxford Paperback References (4th ed.). Oxford University Press. p. 38. ISBN 978-0-19-923594-0.
  3. ^ James, Robert C. (31 July 1992). Mathematics Dictionary (5th ed.). Chapman & Hall. p. 269. ISBN 978-0-412-99031-1.
  4. ^ Bityutskov, Vadim I. (30 November 1987). "Almost-everywhere". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Vol. 1. Kluwer Academic Publishers. p. 153. doi:10.1007/978-94-015-1239-8. ISBN 978-94-015-1239-8.
  5. ^ Itô, Kiyosi, ed. (4 June 1993). Encyclopedic Dictionary of Mathematics. Vol. 2 (2nd ed.). Kingsport: MIT Press. p. 1267. ISBN 978-0-262-09026-1.
  6. ^ "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-11.
  7. ^ a b Weisstein, Eric W. "Almost All". MathWorld. See also Weisstein, Eric W. (25 November 1988). CRC Concise Encyclopedia of Mathematics (1st ed.). CRC Press. p. 41. ISBN 978-0-8493-9640-3.
  8. ^ Itô, Kiyosi, ed. (4 June 1993). Encyclopedic Dictionary of Mathematics. Vol. 1 (2nd ed.). Kingsport: MIT Press. p. 67. ISBN 978-0-262-09026-1.

almost, mathematics, term, almost, means, negligible, amount, more, precisely, displaystyle, almost, elements, displaystyle, means, elements, displaystyle, those, negligible, subset, displaystyle, meaning, negligible, depends, mathematical, context, instance, . In mathematics the term almost all means all but a negligible amount More precisely if X displaystyle X is a set almost all elements of X displaystyle X means all elements of X displaystyle X but those in a negligible subset of X displaystyle X The meaning of negligible depends on the mathematical context for instance it can mean finite countable or null In contrast almost no means a negligible amount that is almost no elements of X displaystyle X means a negligible amount of elements of X displaystyle X Contents 1 Meanings in different areas of mathematics 1 1 Prevalent meaning 1 2 Meaning in measure theory 1 3 Meaning in number theory 1 4 Meaning in graph theory 1 5 Meaning in topology 1 6 Meaning in algebra 2 Proofs 3 See also 4 References 4 1 Primary sources 4 2 Secondary sourcesMeanings in different areas of mathematics EditPrevalent meaning Edit Further information Cofinite set Throughout mathematics almost all is sometimes used to mean all elements of an infinite set but finitely many 1 2 This use occurs in philosophy as well 3 Similarly almost all can mean all elements of an uncountable set but countably many sec 1 Examples Almost all positive integers are greater than 1012 4 293 Almost all prime numbers are odd 2 is the only exception 5 Almost all polyhedra are irregular as there are only nine exceptions the five platonic solids and the four Kepler Poinsot polyhedra If P is a nonzero polynomial then P x 0 for almost all x if not all x Meaning in measure theory Edit Further information Almost everywhere The Cantor function as a function that has zero derivative almost everywhere When speaking about the reals sometimes almost all can mean all reals but a null set 6 7 sec 2 Similarly if S is some set of reals almost all numbers in S can mean all numbers in S but those in a null set 8 The real line can be thought of as a one dimensional Euclidean space In the more general case of an n dimensional space where n is a positive integer these definitions can be generalised to all points but those in a null set sec 3 or all points in S but those in a null set this time S is a set of points in the space 9 Even more generally almost all is sometimes used in the sense of almost everywhere in measure theory 10 11 sec 4 or in the closely related sense of almost surely in probability theory 11 sec 5 Examples In a measure space such as the real line countable sets are null The set of rational numbers is countable so almost all real numbers are irrational 12 Georg Cantor s first set theory article proved that the set of algebraic numbers is countable as well so almost all reals are transcendental 13 sec 6 Almost all reals are normal 14 The Cantor set is also null Thus almost all reals are not in it even though it is uncountable 6 The derivative of the Cantor function is 0 for almost all numbers in the unit interval 15 It follows from the previous example because the Cantor function is locally constant and thus has derivative 0 outside the Cantor set Meaning in number theory Edit Further information Asymptotically almost surely In number theory almost all positive integers can mean the positive integers in a set whose natural density is 1 That is if A is a set of positive integers and if the proportion of positive integers in A below n out of all positive integers below n tends to 1 as n tends to infinity then almost all positive integers are in A 16 17 sec 7 More generally let S be an infinite set of positive integers such as the set of even positive numbers or the set of primes if A is a subset of S and if the proportion of elements of S below n that are in A out of all elements of S below n tends to 1 as n tends to infinity then it can be said that almost all elements of S are in A Examples The natural density of cofinite sets of positive integers is 1 so each of them contains almost all positive integers Almost all positive integers are composite sec 7 proof 1 Almost all even positive numbers can be expressed as the sum of two primes 4 489 Almost all primes are isolated Moreover for every positive integer g almost all primes have prime gaps of more than g both to their left and their right that is there is no other prime between p g and p g 18 Meaning in graph theory Edit In graph theory if A is a set of finite labelled graphs it can be said to contain almost all graphs if the proportion of graphs with n vertices that are in A tends to 1 as n tends to infinity 19 However it is sometimes easier to work with probabilities 20 so the definition is reformulated as follows The proportion of graphs with n vertices that are in A equals the probability that a random graph with n vertices chosen with the uniform distribution is in A and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them 21 Therefore equivalently to the preceding definition the set A contains almost all graphs if the probability that a coin flip generated graph with n vertices is in A tends to 1 as n tends to infinity 20 22 Sometimes the latter definition is modified so that the graph is chosen randomly in some other way where not all graphs with n vertices have the same probability 21 and those modified definitions are not always equivalent to the main one The use of the term almost all in graph theory is not standard the term asymptotically almost surely is more commonly used for this concept 20 Example Almost all graphs are asymmetric 19 Almost all graphs have diameter 2 23 Meaning in topology Edit In topology 24 and especially dynamical systems theory 25 26 27 including applications in economics 28 almost all of a topological space s points can mean all of the space s points but those in a meagre set Some use a more limited definition where a subset only contains almost all of the space s points if it contains some open dense set 26 29 30 Example Given an irreducible algebraic variety the properties that hold for almost all points in the variety are exactly the generic properties sec 8 This is due to the fact that in an irreducible algebraic variety equipped with the Zariski topology all nonempty open sets are dense Meaning in algebra Edit In abstract algebra and mathematical logic if U is an ultrafilter on a set X almost all elements of X sometimes means the elements of some element of U 31 32 33 34 For any partition of X into two disjoint sets one of them will necessarily contain almost all elements of X It is possible to think of the elements of a filter on X as containing almost all elements of X even if it isn t an ultrafilter 34 Proofs Edit The prime number theorem shows that the number of primes less than or equal to n is asymptotically equal to n ln n Therefore the proportion of primes is roughly ln n n which tends to 0 as n tends to infinity so the proportion of composite numbers less than or equal to n tends to 1 as n tends to infinity 17 See also EditAlmost Almost everywhere Almost surelyReferences EditPrimary sources Edit Cahen Paul Jean Chabert Jean Luc 3 December 1996 Integer Valued Polynomials Mathematical Surveys and Monographs Vol 48 American Mathematical Society p xix ISBN 978 0 8218 0388 2 ISSN 0076 5376 Cahen Paul Jean Chabert Jean Luc 7 December 2010 First published 2000 Chapter 4 What s New About Integer Valued Polynomials on a Subset In Hazewinkel Michiel ed Non Noetherian Commutative Ring Theory Mathematics and Its Applications Vol 520 Springer p 85 doi 10 1007 978 1 4757 3180 4 ISBN 978 1 4419 4835 9 Gardenfors Peter 22 August 2005 The Dynamics of Thought Synthese Library Vol 300 Springer pp 190 191 ISBN 978 1 4020 3398 8 a b Courant Richard Robbins Herbert Stewart Ian 18 July 1996 What is Mathematics An Elementary Approach to Ideas and Methods 2nd ed Oxford University Press ISBN 978 0 19 510519 3 Movshovitz hadar Nitsa Shriki Atara 2018 10 08 Logic In Wonderland An Introduction To Logic Through Reading Alice s Adventures In Wonderland Teacher s Guidebook World Scientific p 38 ISBN 978 981 320 864 3 This can also be expressed in the statement Almost all prime numbers are odd a b Korevaar Jacob 1 January 1968 Mathematical Methods Linear Algebra Normed Spaces Distributions Integration Vol 1 New York Academic Press pp 359 360 ISBN 978 1 4832 2813 6 Natanson Isidor P June 1961 Theory of Functions of a Real Variable Vol 1 Translated by Boron Leo F revised ed New York Frederick Ungar Publishing p 90 ISBN 978 0 8044 7020 9 Sohrab Houshang H 15 November 2014 Basic Real Analysis 2 ed Birkhauser p 307 doi 10 1007 978 1 4939 1841 6 ISBN 978 1 4939 1841 6 Helmberg Gilbert December 1969 Introduction to Spectral Theory in Hilbert Space North Holland Series in Applied Mathematics and Mechanics Vol 6 1st ed Amsterdam North Holland Publishing Company p 320 ISBN 978 0 7204 2356 3 Vestrup Eric M 18 September 2003 The Theory of Measures and Integration Wiley Series in Probability and Statistics United States Wiley Interscience p 182 ISBN 978 0 471 24977 1 a b Billingsley Patrick 1 May 1995 Probability and Measure PDF Wiley Series in Probability and Statistics 3rd ed United States Wiley Interscience p 60 ISBN 978 0 471 00710 4 Archived from the original PDF on 23 May 2018 Niven Ivan 1 June 1956 Irrational Numbers Carus Mathematical Monographs Vol 11 Rahway Mathematical Association of America pp 2 5 ISBN 978 0 88385 011 4 Baker Alan 1984 A concise introduction to the theory of numbers Cambridge University Press p 53 ISBN 978 0 521 24383 4 Granville Andrew Rudnick Zeev 7 January 2007 Equidistribution in Number Theory An Introduction Nato Science Series II Vol 237 Springer p 11 ISBN 978 1 4020 5404 4 Burk Frank 3 November 1997 Lebesgue Measure and Integration An Introduction A Wiley Interscience Series of Texts Monographs and Tracts United States Wiley Interscience p 260 ISBN 978 0 471 17978 8 Hardy G H 1940 Ramanujan Twelve Lectures on Subjects Suggested by His Life and Work Cambridge University Press p 50 a b Hardy G H Wright E M December 1960 An Introduction to the Theory of Numbers 4th ed Oxford University Press pp 8 9 ISBN 978 0 19 853310 8 Prachar Karl 1957 Primzahlverteilung Grundlehren der mathematischen Wissenschaften in German Vol 91 Berlin Springer p 164 Cited in Grosswald Emil 1 January 1984 Topics from the Theory of Numbers 2nd ed Boston Birkhauser p 30 ISBN 978 0 8176 3044 7 a b Babai Laszlo 25 December 1995 Automorphism Groups Isomorphism Reconstruction In Graham Ronald Grotschel Martin Lovasz Laszlo eds Handbook of Combinatorics Vol 2 Netherlands North Holland Publishing Company p 1462 ISBN 978 0 444 82351 9 a b c Spencer Joel 9 August 2001 The Strange Logic of Random Graphs Algorithms and Combinatorics Vol 22 Springer pp 3 4 ISBN 978 3 540 41654 8 a b Bollobas Bela 8 October 2001 Random Graphs Cambridge Studies in Advanced Mathematics Vol 73 2nd ed Cambridge University Press pp 34 36 ISBN 978 0 521 79722 1 Gradel Eric Kolaitis Phokion G Libkin Leonid Marx Maarten Spencer Joel Vardi Moshe Y Venema Yde Weinstein Scott 11 June 2007 Finite Model Theory and Its Applications Texts in Theoretical Computer Science An EATCS Series Springer p 298 ISBN 978 3 540 00428 8 Buckley Fred Harary Frank 21 January 1990 Distance in Graphs Addison Wesley p 109 ISBN 978 0 201 09591 3 Oxtoby John C 1980 Measure and Category Graduate Texts in Mathematics Vol 2 2nd ed United States Springer pp 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 Grotschel and Lovasz 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 1987 Recent and New Results in Rational L2 Approximation In Curtain Ruth F ed Modelling Robustness and Sensitivity Reduction in Control Systems NATO ASI Series F Vol 34 Springer p 123 doi 10 1007 978 3 642 87516 8 ISBN 978 3 642 87516 8 a b Broer Henk Takens Floris 28 October 2010 Dynamical Systems and Chaos Applied Mathematical Sciences Vol 172 Springer p 245 doi 10 1007 978 1 4419 6870 8 ISBN 978 1 4419 6870 8 Sharkovsky A N Kolyada S F Sivak A G Fedorenko V V 30 April 1997 Dynamics of One Dimensional Maps Mathematics and Its Applications Vol 407 Springer p 33 doi 10 1007 978 94 015 8897 3 ISBN 978 94 015 8897 3 Yuan George Xian Zhi 9 February 1999 KKM Theory and Applications in Nonlinear Analysis Pure and Applied Mathematics A Series of Monographs and Textbooks Marcel Dekker p 21 ISBN 978 0 8247 0031 7 Albertini Francesca Sontag Eduardo D 1 September 1991 Transitivity and Forward Accessibility of Discrete Time Nonlinear Systems In Bonnard Bernard Bride Bernard Gauthier Jean Paul Kupka Ivan eds Analysis of Controlled Dynamical Systems Progress in Systems and Control Theory Vol 8 Birkhauser p 29 doi 10 1007 978 1 4612 3214 8 ISBN 978 1 4612 3214 8 De la Fuente Angel 28 January 2000 Mathematical Models and Methods for Economists Cambridge University Press p 217 ISBN 978 0 521 58529 3 Komjath Peter Totik Vilmos 2 May 2006 Problems and Theorems in Classical Set Theory Problem Books in Mathematics United States Springer p 75 ISBN 978 0387 30293 5 Salzmann Helmut Grundhofer Theo Hahl Hermann Lowen Rainer 24 September 2007 The Classical Fields Structural Features of the Real and Rational Numbers Encyclopedia of Mathematics and Its Applications Vol 112 Cambridge University Press p 155 ISBN 978 0 521 86516 6 Schoutens Hans 2 August 2010 The Use of Ultraproducts in Commutative Algebra Lecture Notes in Mathematics Vol 1999 Springer p 8 doi 10 1007 978 3 642 13368 8 ISBN 978 3 642 13367 1 a b Rautenberg Wolfgang 17 December 2009 A Concise to Mathematical Logic Universitext 3rd ed Springer pp 210 212 doi 10 1007 978 1 4419 1221 3 ISBN 978 1 4419 1221 3 Secondary sources Edit Schwartzman Steven 1 May 1994 The Words of Mathematics An Etymological Dictionary of Mathematical Terms Used in English Spectrum Series Mathematical Association of America p 22 ISBN 978 0 88385 511 9 Clapham Christopher Nicholson James 7 June 2009 The Concise Oxford Dictionary of mathematics Oxford Paperback References 4th ed Oxford University Press p 38 ISBN 978 0 19 923594 0 James Robert C 31 July 1992 Mathematics Dictionary 5th ed Chapman amp Hall p 269 ISBN 978 0 412 99031 1 Bityutskov Vadim I 30 November 1987 Almost everywhere In Hazewinkel Michiel ed Encyclopaedia of Mathematics Vol 1 Kluwer Academic Publishers p 153 doi 10 1007 978 94 015 1239 8 ISBN 978 94 015 1239 8 Ito Kiyosi ed 4 June 1993 Encyclopedic Dictionary of Mathematics Vol 2 2nd ed Kingsport MIT Press p 1267 ISBN 978 0 262 09026 1 Almost All Real Numbers are Transcendental ProofWiki proofwiki org Retrieved 2019 11 11 a b Weisstein Eric W Almost All MathWorld See also Weisstein Eric W 25 November 1988 CRC Concise Encyclopedia of Mathematics 1st ed CRC Press p 41 ISBN 978 0 8493 9640 3 Ito Kiyosi ed 4 June 1993 Encyclopedic Dictionary of Mathematics Vol 1 2nd ed Kingsport MIT Press p 67 ISBN 978 0 262 09026 1 Retrieved from https en wikipedia org w index php title Almost all amp oldid 1129166685, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.