理解できないから、またまた検索したら、下記ヒットした。ご存知と思うが、自分のメモとして貼る 後述”It is a hard (and often open) problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers.”とか
https://en.wikipedia.org/wiki/Lottery Lottery (抜粋) Contents [hide] 1 Classical history 2 Medieval history 3 Ticket gallery 4 Early modern history 4.1 France, 1539?1789 4.2 England, 1566?1826 4.3 Early United States 1612?1900 4.4 German-speaking countries 4.5 Spain, 1763 5 Modern history by country 5.1 Australia 5.2 Canada 5.3 Mexico 5.4 Spain 5.5 Thailand 5.6 United Kingdom 5.7 United States 6 Mathematical analysis 6.1 Probability of winning 7 Scams and frauds 8 Payment of prizes 9 See also 10 References 11 Further reading 12 External links (引用終り)
https://en.wikipedia.org/wiki/Lottery (抜粋) Early modern history France, 1539?1789 King Francis I of France discovered the lotteries during his campaigns in Italy and decided to organize such a lottery in his kingdom to help the state finances. The first French lottery, the Loterie Royale, was held in 1539 and was authorized with the edict of Chateaurenard. This attempt was a fiasco, since the tickets were very costly and the social classes which could afford them opposed the project. During the two following centuries lotteries in France were forbidden or, in some cases, tolerated.
Mathematical analysis The purchase of lottery tickets cannot be accounted for by decision models based on expected value maximization. The reason is that lottery tickets cost more than the expected gain, as shown by lottery mathematics, so someone maximizing expected value should not buy lottery tickets. Yet, lottery purchases can be explained by decision models based on expected utility maximization, as the curvature of the utility function can be adjusted to capture risk-seeking behavior. More general models based on utility functions defined on things other than the lottery outcomes can also account for lottery purchase. In addition to the lottery prizes, the ticket may enable some purchasers to experience a thrill and to indulge in a fantasy of becoming wealthy. If the entertainment value (or other non-monetary value) obtained by playing is high enough for a given individual, then the purchase of a lottery ticket could represent a gain in overall utility. In such a case, the disutility of a monetary loss could be outweighed by the combined expected utility of monetary and non-monetary gain, thus making the purchase a rational decision for that individual. (引用終り)
https://en.wikipedia.org/wiki/Lottery_mathematics Lottery mathematics (抜粋) Contents [hide] 1 Choosing 6 from 49 2 Odds of getting other possibilities in choosing 6 from 49 3 Pick8-32 odds and calculations 4 Powerballs And Bonus Balls 5 Minimum number of tickets for a match 6 References 7 External links
Minimum number of tickets for a match It is a hard (and often open) problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers. In the 5-from-90 lotto, the minimum number of tickets that can guarantee a ticket with at least 2 matches is 100.[3]
References 1. Zabrocki, Mike (2003-03-01). "Calculating the Probabilities of Winning Lotto 6/49,Version 3" (PDF). Retrieved 2016-08-14. http://garsia.math.yorku.ca/~zabrocki/math5020f03/lot649/lot649v3.pdf
http://garsia.math.yorku.ca/~zabrocki/math5020f03/ Math 5020 Fundamentals of Mathematics for Teachers Professor Mike Zabrocki
(March 1, 2004) I revised the draft of the explanation of Lottery 6/49 to produce version 3. At this point I don't have much momentum on this project, but please offer your comments on the forum (I got none this last week except for one negative one). See my remarks on the forum. I will bring this up in class tonight.
https://en.wikipedia.org/wiki/Fierz_identity Fierz identity (抜粋) In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz.
There is a version of the Fierz identities for Dirac spinors and there is another version for Weyl spinors. And there are versions for other dimensions besides 3+1 dimensions.
Spinor bilinears can be thought of as elements of a Clifford Algebra. Then the Fierz identity is the concrete realization of the relation to the exterior algebra. The identities for a generic scalar written as the contraction of two Dirac bilinears of the same type can be written with coefficients according to the following table. (引用終り)
これ、4番目(PDFが直接落ちるのでURL取れず。下記で代用) a mathematical model for the lottery - RACO www.raco.cat/index.php/Questiio/article/download/26594/26428 このページを訳す MS Nikulin 著 - ?1992 - ?関連記事 the prize winners, being those holding the tickets with those corres- ponding numbers.” From “Educated Guessing” Samuel Kotz (1983, Marcel Dekker):. “A lottery is a game of chance with low stakes and potentially high winnings, which account for the widespread appeal of this type of gambling. In its simplests form, a player bets on a number and wins if the state also selects that number. While we usually view a lottery as a game, many applications exist in the real world. For example,.
これ、5番目(PDFではないが、Lottery Software が詳しそうだ) http://saliu.com/strategy.html Basics of a Lotto Strategy Based on: Sums (Sum-Totals); Odd Even; Low High Numbers By Ion Saliu, Founder of Lottery Mathematics (抜粋) I. Introduction to Lottery Strategies, Filtering, Number Grouping II. Pick-3 Lottery Software for Low or High, Odd or Even Digit Grouping III. Lotto Software for Low / High, Odd / Even Numbers, Plus Lotto Skipping IV. True Lottery Filters, Filtering to Create the Best Lotto, Lottery Strategies, Systems V. Essential Resources in Lotto, Lottery Strategy, Systems, Software
Resources, links to the best in lotto software, lottery strategies.
つづく 0420現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/07(木) 22:23:23.82ID:jVxSMgKf>>419 つづき 5. Resources in Lottery Software, Systems, Strategies, Lotto Wheeling Introduction to Lottery Mathematics: Probabilities, Appearance, Repeat, Affinity or Number Affiliation, Wheels, Systems, Strategies. The Starting Strategy Page: Lottery Software, Strategy, Systems. Presenting software to create free winning lotto, lottery strategies, systems based on mathematics. Get your lotto systems or wheels, the best lottery, lotto software, combinations, winning numbers. Lotto, Lottery Software, Excel Spreadsheets: Programming, Strategies. Read a genuine analysis of Excel spreadsheets applied to lottery and lotto developing of software, systems, and strategies. Combining Excel analysis with powerful lottery and lotto software programmed by this author, Parpaluck. MDIEditor Lotto WE: Lottery Software Manual, Book, ebook, Help. ~ Also applicable to LotWon lottery, lotto software; plus Powerball, Mega Millions, Euromillions. Visual Tutorial, Book, Manual: Lottery Software, Lotto Apps, Programs. Basic Manual for Lotto Software, Lottery Software. Sum-Totals for Lottery, Lotto Games ? Pick 3 4 Lotteries, Lotto 5, 6, Powerball, Mega Millions, Euromillions. Lotto Software for Groups of Numbers: Odd, Even, Low, High, Sums, Frequencies, User's Groups. Lottery Software Sum-Totals, Sums: Lotto, Powerball, Mega Millions, Euromillions. Lottery Skip Systems: Lotto, Powerball, Mega Millions, Euromillions. Lotto, Lottery Strategy in Reverse: Not-to-Win Leads to Not-to-Lose or WIN. Lottery Utility Software: Pick-3, 4 Lottery, Lotto-5, 6, Powerball, Mega Millions, Euromillions. Theory, Analysis of Deltas in Lotto, Lottery Software, Strategy, Systems. The Best Strategy for Lottery, Gambling, Sports Betting, Horse Racing, Blackjack, Roulette. Lotto Software for Groups of Numbers. "The Start Is the Hardest Part":Play a Lotto Strategy, Lottery Strategies Download lottery software, lotto software つづく 0421現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/07(木) 22:27:03.74ID:jVxSMgKf sage 0422現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/07(木) 22:28:11.68ID:jVxSMgKf>>420 つづき (上記の原文にはURLのリンクが貼ってあるよ)
あと、数学公式としては ”余録3: http://mathworld.wolfram.com/HypergeometricDistribution.html Hypergeometric Distribution -- from Wolfram MathWorld (抜粋) It therefore also describes the probability of obtaining exactly i correct balls in a pick-N lottery from a reservoir of r balls” で尽きているような気がしてきたね 0429現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/08(金) 07:31:01.17ID:pI4TAlAF>>428 補足
>当然、内容は数分間の流し読みですよ(^^
キーワード下記 PDF Lottery mathematics Odds of getting other possibilities summation 78 件 (0.55 秒)
<前振りで数学的な構造> (>>284-285より) http://mathforum.org/kb/message.jspa?messageID=5432910 Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 (抜粋) 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.
** 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) (引用終り)
趣旨を日本語にすると ruler functionとか、改良トマエ関数で、 f(x) = 1/q if x = p/q ↓ f^r = 1/q^r となって
1)指数r=2なら:nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. 2)指数r > 2なら:differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. 3)指数1/q^rより早く減衰する関数1/w(q) :differentiable on a set whose complement has Hausdorff dimension zero. (前振り終り)
で、「証明すべきこと」は、1/q^rで、Hausdorff dimension 1 - 2/rで、rが大きくなると、どんどんHausdorff dimensionが1に近づく。つまり、differentiableな範囲が大きくなる 指数1/q^rより早く減衰する関数1/w(q)では、”a set whose complement has Hausdorff dimension zero”ですよ
しかし、指数1/q^rより早く減衰する関数1/w(q)でも、微分不可の部分が残って、Hausdorff dimension zeroにもかかわらず、 ”Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.”(>>285より) だと。つまり、証明すべきは、ここで、”指数1/q^rより早く減衰する関数1/w(q)でも、微分不可の部分が残って、Hausdorff dimension zeroにもかかわらず、「fail to be differentiable is large」なのだ”ということなのだ
重複を厭わず、下記追加引用 (下記より) ”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.”
ここで、”Let g be continuous and discontinuous on sets of points that are each dense in the reals.” とあるでしょ。この”each dense in the reals”を覚えておいてね。あとで使う(^^
(>>285より) http://mathforum.org/kb/message.jspa?messageID=5432910 Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 (抜粋) ** 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.)
There are 22 items below. I found 4 of them on the internet, I provide the complete text for 9 of them, and I give some idea of what the remaining 9 items involve.
1.で、(>>443)英文では”each dense in the reals” ”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.”
という例の場合は、R − ∪[p∈Q] { p } には開区間が全く存在しない。 だから、そのイメージの仕方は間違っている。 0465現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 22:18:51.30ID:OrUOLzdR 哀れな素人さんのために Philosophy本だが、検索ヒットしたので貼る(^^ http://publish.uwo.ca/~jbell/The%20Continuous.pdf The Continuous Infinitesimal Mathematics Philosophy JL Bell 著 - ?2005 Preface This book has a double purpose. First, to trace the historical development of the concepts of the continuous and the infinitesimal; and second, to describe the ways in which these two concepts are treated in contemporary mathematics. So the first part of the book is largely philosophical, while the second is almost exclusively mathematical. In writing the book I have found it necessary to thread my way through a wealth of sources, both philosophical and mathematical; and it is inevitable that a number of topics have not received the attention they deserve. Still, the thread itself, if tangled in places, has been luminous. “Only connect ... Live in fragments no longer,” says E. M. Forster, and that is what I have tried to do here.