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one to one relation is called an injective function For the terms one to one many to one etc we can at least say that they are household words But the term mapping cardinalities is very confusing even to your fellow computer scientists I ve never heard it before It suggests we re mapping cardinalities onto each other which we re not and it suggests infinite cardinalities but they are finite and typically very small Booh P S It turn out that Database System Concepts like many other textbook in introductory courses on relational database s goes at some length to keep the presentation informal of what is fundamentally a formal mathematical theory The authors purposely avoid all reference to actual mathematical notation or terminology the phrase mapping cardinalities is just a symptom of this general approach While informal the resulting description is still accurate but it is very incomplete as soon as you try to use it for anything but trivial cases you run into undefined areas I like it 2 C s idea by kozmund Tue May 08 2001 at 17 54 14 In this usage of map the intended meaning is to determine the relation between cardinalities To quote the dictionary sitting next to me cardinality is the number of elements in a given mathematical set Combine the two and we have determine the relation between the number of elements in either set This is exactly what we are striving for Since this is a phrase often used when talking about relational databases it is safe to asssume that the intent is to define legal ways for tables to relate to one another One common mistake made by mathematicians and even other computer scientists is in mistaking the terminology of one discipline for another For example most

Original URL path: http://everything2.com/title/Mapping+Cardinalities?author_id=6479 (2016-02-12)

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re mapping cardinalities onto each other which we re not and it suggests infinite cardinalities but they are finite and typically very small Booh P S It turn out that Database System Concepts like many other textbook in introductory courses on relational database s goes at some length to keep the presentation informal of what is fundamentally a formal mathematical theory The authors purposely avoid all reference to actual mathematical notation or terminology the phrase mapping cardinalities is just a symptom of this general approach While informal the resulting description is still accurate but it is very incomplete as soon as you try to use it for anything but trivial cases you run into undefined areas I like it 2 C s cardinality virtues of a good programmer number theory set logic cardinality of subsets one to one ICQ Integrity Constraints Cool Breeze tab completion organic chemistry Georg Cantor set theory dannye trials Eleven or One Ramsey Clark HTTP Request Line caught stealing Vamps and Tramps conjunctive normal form Ground effect relational database model terminology Sign in Login Password remember me Lost password Sign up Need help accounthelp everything2 com Recommended Reading About Everything2 User Picks 2016 New Hampshire Democratic

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for good reasons too A humbling experience The E2 editors were trying to turn it more and more into an objective encyclopedia It s good to have one but for me the specific value of E2 s content is the subjective edge as dwyn explained E2 got better and better even though it s also lost a lot of great content over time It started out as the radically simple and beautiful Everything 1 It then went through spectacular growing pains at one point its policies software and help files were so horribly deficient that I was considering legal action In the course of many months the issues were addressed with busloads of small improvements We now have clear policies great software features and excellent help documents I found the process very interesting but it s hard to retrace due to the no noding about noding embargo see also the update User Bookmarks Sort by name Sort by date March of the Critics 2012 Sign in Login Password remember me Lost password Sign up Need help accounthelp everything2 com Recommended Reading About Everything2 User Picks 2016 New Hampshire Democratic Primary Thomas Atkins transfinite induction September 11 2002 Not All Who

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positions described and evaluated from Radical Apathy another interesting blog Do we need God from Pandeism Hellre kontroversiell än snäll another blog RationalWiki s Pandeism Page Thelemapedia s Pandeism page VisWiki s Pandeism page Keywen s Pandeism page Wikipedia s Pandeism page Wiktionary s Pandeism page Wikiquote s Pandeism page The Parallels of Pandeism by Bernardo Kastrup Ph D Encyclopedia Britannica s page on pandeism Alex Ashman BBC News Metaphysical Isms Useful Notes on Pandeism from TV Tropes LiveJournal Pandeism group User Bookmarks Sort by name Sort by date 3D Turnip A BUSINESS PROPOSITION QUEST 2012 Akashic Records Alternatives to Everything2 An apology a love note and a remembrance applesauce Aristotle and the University of Paris Batcave Beware the leader who bangs the drums of war Beyond Belief A Buddhist Critique of Christianity Bhagavad Gita Brahman and the fractal nature of the universe due to the chaotic nature of string theory Coconut tart Crackling ice cubes Creativity the essence of being human December 21 2012 Determinism and super determinism donut Drafts Eight legged essay electromagnetic hypersensitivity Every single ancestor of yours for five hundred million years has successfully mated foundationalism Gløgg God and the Universe God is not a deceiver Gödel s theorem applied to God Helping your kid brother die Herbs to get in touch with the element of water Hilbert s Nullstellensatz Hip hop quadrant system How Lucifer lost his superuser privileges How many bits are in the human genome How much firewood equals an atomic bomb humanism Husserl and the Orange I got a good giggle out of this immediate inference rules Imprimatur Introduction to Finite Mathematics Japanese Cucumber Salad Jesus Among Other Gods The Absolute Claims Of The Christian Message Leek Potato and Artichoke Soup magik maltheism Mushroom sauce with port November 14 2009 October 20

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Open archived version from archive - Small gaps between primes - Everything2.com

the Erdös method they obtained Δ 2 3 8 or about 0 46650 In 1977 Huxley combined the Erdös method and the Hardy Littlewood Bombieri Davenport method to obtain Δ 0 44254 Then in 1986 Maier used his discovery that certain interval s contain a factor of more primes than average intervals Working in these intervals and combining all of the above methods he proved that Δ 0 2486 which was the best result until now However Dan Goldston and Cem Yildirim have recently written a manuscript which was presented in a lecture at the American Institute of Mathematics which advances the theory of small gaps between primes by a huge amount First of all they show that Δ 0 exactly Moreover they can prove that for infinitely many n the inequality P n 1 P n log p n 8 9 holds Goldston s and Yildirim s approach begins with the methods of Hardy Littlewood and Bombieri Davenport They have discovered an extraordinary way to approximate on average sums over prime K tuples We believe after work of Gallagher using the Hardy Littlewood conjectures for the distribution of prime K tuples that the prime numbers in a short interval N N Γlog N are distributed like a Poisson random variable with parameter Γ Goldston and Yildirim exploit this model in choosing approximations They ultimately use the theory of orthogonal polynomial s to express the optimal approximation in terms of Laguerre polynomial s Hardy and Littlewood could have proven this theorem under the assumption of the Generalized Riemann Hypothesis the Bombieri Vinogradov theorem allows for the unconditional treatment This new approach opens the door for much further work It is clear from the manuscript that the savings of an exponent of 1 9 in the power of log P n is not the best that the method will allow There are at least two possible refinements One is in the examination of lower order terms that arise in his method Can they be used to enhance the argument The other is in the error term Gallagher found in summing the singular series arising from the Hardy Littlewood K tuple conjecture There is reason to believe that this error term can be improved possibly using ideas in recent work of Montgomery and Soundararajan Beyond Pair Correlation This work is astonishing in another regard They have actually proven that for any fixed number r the inequality P n r P n Log P n 8r 8r 1 holds for infinitely many n In 1977 Helmut Maier burst onto the analytic number scene with his tour de force proof that the largest gaps known to hold for two consecutive primes could be proven for each gap of r consecutive gaps for any fixed r Goldston and Yildirim have achieved a similar sort of result for consecutive small gaps at the same time in that they have demolished all previous records for one small gap It is clear is that at least part of

Original URL path: http://everything2.com/title/Small+gaps+between+primes?author_id=1327081 (2016-02-12)

Open archived version from archive - Small gaps between primes (idea) by Gartogg - Everything2.com

with the Hardy Littlewood approach proved unconditionally Δ 1 2 and then using the Erdös method they obtained Δ 2 3 8 or about 0 46650 In 1977 Huxley combined the Erdös method and the Hardy Littlewood Bombieri Davenport method to obtain Δ 0 44254 Then in 1986 Maier used his discovery that certain interval s contain a factor of more primes than average intervals Working in these intervals and combining all of the above methods he proved that Δ 0 2486 which was the best result until now However Dan Goldston and Cem Yildirim have recently written a manuscript which was presented in a lecture at the American Institute of Mathematics which advances the theory of small gaps between primes by a huge amount First of all they show that Δ 0 exactly Moreover they can prove that for infinitely many n the inequality P n 1 P n log p n 8 9 holds Goldston s and Yildirim s approach begins with the methods of Hardy Littlewood and Bombieri Davenport They have discovered an extraordinary way to approximate on average sums over prime K tuples We believe after work of Gallagher using the Hardy Littlewood conjectures for the distribution of prime K tuples that the prime numbers in a short interval N N Γlog N are distributed like a Poisson random variable with parameter Γ Goldston and Yildirim exploit this model in choosing approximations They ultimately use the theory of orthogonal polynomial s to express the optimal approximation in terms of Laguerre polynomial s Hardy and Littlewood could have proven this theorem under the assumption of the Generalized Riemann Hypothesis the Bombieri Vinogradov theorem allows for the unconditional treatment This new approach opens the door for much further work It is clear from the manuscript that the savings of an exponent of 1 9 in the power of log P n is not the best that the method will allow There are at least two possible refinements One is in the examination of lower order terms that arise in his method Can they be used to enhance the argument The other is in the error term Gallagher found in summing the singular series arising from the Hardy Littlewood K tuple conjecture There is reason to believe that this error term can be improved possibly using ideas in recent work of Montgomery and Soundararajan Beyond Pair Correlation This work is astonishing in another regard They have actually proven that for any fixed number r the inequality P n r P n Log P n 8r 8r 1 holds for infinitely many n In 1977 Helmut Maier burst onto the analytic number scene with his tour de force proof that the largest gaps known to hold for two consecutive primes could be proven for each gap of r consecutive gaps for any fixed r Goldston and Yildirim have achieved a similar sort of result for consecutive small gaps at the same time in that they have demolished all previous records

Original URL path: http://everything2.com/user/Gartogg/writeups/Small+gaps+between+primes (2016-02-12)

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s me most recent writeup Is Jerusalem really sacred to Islam Send private message to Gartogg OK my current goal is to attain a Node Fu of 10 15 I ve decided to decline XP for voting It s a simple deal I want to have my XP be a measure of how well I m writing I don t need the XP and I d love to see my Node fu drop to a point that is maintainable and then slowly begin to rise again as my writeups gasp actually improve What s really important dead rat game thing by zhaus Spinning the rat on the tail needs more and more practice but can you do it It is to be done to play the game to fun Having the rat the First Person spin it over it s tail Run away he does then so other friends left look at rat spin fun Then he make fun of other friend s because they re not knowing what fun comes next He take hose and sprays friends so fun They get mad but he talk to them they were silly fun to watch dead rat Then all are funny happy Took a lot to live a lot like you Wow Brainfuck User Bookmarks Sort by name Sort by date Seems I might have stolen the blue part of her rainbow but all I really did was make it bigger a way bigger blue Took a lot to live a lot like you Sign in Login Password remember me Lost password Sign up Need help accounthelp everything2 com Recommended Reading About Everything2 User Picks 2016 New Hampshire Democratic Primary Thomas Atkins transfinite induction September 11 2002 Not All Who Are Lost Wandered Super bowl babies Editor Picks Godot the musical

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