cited paper published in 1969 Clauser et al, 1969 . They derived the CHSHinequality , which ... that there exist underlying local hidden variables. The inequality must be obeyed under local realism but can be infringed under certain conditions in quantum mechanics. Statement of the inequality The usual form of the CHSHinequality is 1 &minus 2 S 2 ... 2 can be found. If it is numerically greater than 2 it has infringed the CHSHinequality ... of the CHSHinequality The original 1969 derivation will not be given here since it is not easy to follow ... leq 2 pm E a prime, b prime E a prime, b , math which includes the CHSHinequality. Derivation from Clauser and Horne s 1974 inequality In their 1974 paper Clauser, 1974 , Clauser and Horne show that the CHSH ... Aspect Aspect s second experiment in 1982 have used the CHSHinequality, estimating the terms using ...In physics , the CHSH Bell test is an application of Bell s theorem , intended to distinguish between ... local hidden variable theories . CHSH stands for John Clauser John C lauser , Michael Horne Michael ... inequality could be applied in real experiments with real imperfect detectors, though it was later proved Bell, 1971 that the inequality itself was equally valid. The occurrence of zero outcomes ... 2 math , which is greater than 2, and CHSH violations are therefore predicted by the theory of quantum ... a constant distribution and is unaffected by the choice of detector setting. A typical CHSH experiment ... these being the ones for which the QM formula gives the greatest violation of the inequality. For each ... all local hidden variable theories. The CHSH paper lists many preconditions or reasonable and or presumable ... of interest is given in Clauser and Horne s 1974 paper, in which they start from the CH74 inequality. It would appear from both these later derivations that the only assumptions really needed for the inequality ... rho lambda d lambda qquad 6 math Then, applying the triangle inequality to both sides, using ... more details
lowercase wikibooks Guide to Unix Explanations Choice of Shell Choosing a shell chsh an abbreviation of change shell is a command computing command on Unix like operating systems that is used to change a login Unix shell shell . Users can either supply the path computing pathname of the shell that they wish to change to on the command line, or supply no arguments, in which case tt chsh tt allows the user to change the shell interactively. ref name LTBS cite book title Learning the Bash Shell Unix shell programming author Cameron Newham and Bill Rosenblatt pages 272 date 2005 publisher O Reilly id ISBN 0596009658 ref tt chsh tt is a setuid program that modifies the tt etc passwd tt file, and only allows ordinary users to modify their own login shells. The superuser can modify the shells of other ... chsh tt . ref name LTBS ref name PGUMOXU cite book title A Practical Guide To Unix For Mac Os X Users ... 2002 publisher O Reilly id ISBN 0596003439 ref On most systems, when tt chsh tt is invoked without .... On Mac OS X , if invoked without the tt s tt option, tt chsh tt displays a text file in the default ... tt chsh s bin bash tt greatly simplifies the task of changing shells. Depending on the system, tt chsh tt may or may not prompt the user for a password before changing the shell, or entering interactive mode. On some systems, use of tt chsh tt by non root users is disabled entirely by the sysadmin. ref name U On many Linux distribution s, the tt chsh tt command is a Pluggable Authentication ... tt module can be used to deny tt chsh tt access to individual users, by specifying a file of the usernames ... O Reilly id ISBN 1565926641 &mdash some examples of invoking tt chsh tt with the tt s tt and tt l tt options See also Comparison of command shells External links man 1 chsh change your login shell apple man page chsh Linux stub Category Unix user management and support related utilities Category Standard Unix programs de Chsh it Chsh pl Chsh ... more details
Wiktionary inequalityInequality may refer to In mathematics Inequality mathematics Inequalities book Inequalities book 1934 , a mathematics book by G. H. Hardy , J. E. Littlewood , and George Polya G. Polya In healthcare Health disparities Healthcare inequality In economics Economic inequality Income inequality metrics International inequality Income inequality in the United States Wealth inequality in the United States In the social sciences Participation inequality Social inequality Social equality Social stratification disambig da Ulighed es Desigualdad eo Neegala o fr In galit sh Nejednakost razvrstavanje ... more details
Wirtinger s inequality is either of two inequality mathematics inequalities named after Wilhelm Wirtinger Wirtinger s inequality for functions Wirtinger inequality 2 forms mathdab ... more details
The following pages deal with inequalities due to Mikhail Gromov mathematician Mikhail Gromov Bishop&ndash Gromov inequality Gromov s inequality for complex projective space Gromov s systolic inequality for essential manifolds L vy&ndash Gromov inequality disambig ... more details
In mathematics , Berger inequality may refer to Berger s inequality for Einstein manifolds the Berger&ndash Kazdan comparison theorem . mathdab ... more details
Information inequality may mean in statistics, the Cram r Rao bound , an inequality for the variance of an estimator based on the information in a sample in information theory, inequalities in information theory describes various inequalities specific to that context. in sociology, Information Inequality and Social Barriers also in sociology, information inequity disambig ... more details
In mathematics , the Kantorovich inequality is a particular case of the Cauchy Schwarz inequality , which is itself a generalization of the triangle inequality . The triangle inequality states that the length of two sides of any triangle, added together, will be equal to or greater than the length of the third side. In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of linear programming . See vector space , inner product , and normed vector space for other examples of how the basic ideas inherent in the triangle inequality line segment and distance can be generalized into a broader context. More formally, the Kantorovich inequality can be expressed this way Let math p i geq 0, quad 0 a leq x i leq b text for i 1, dots ,n. math Let math A n 1,2, dots ,n . math Then math begin align & qquad left sum i 1 n p ix i right left sum i 1 n frac p i x i right & leq frac a b 2 4ab left sum i 1 n p i right 2 frac a b 2 4ab cdot min left left sum i in X p i sum j in Y p j right 2 , , X cup Y A n , X cap Y varnothing right . end align math The Kantorovich inequality is used in convergence analysis it bounds the convergence rate of Cauchy s steepest descent . Equivalents of the Kantorovich inequality have arisen in a number of different fields. For instance, the Bunyakovsky inequality , the Wielandt inequality , and the Cauchy&ndash Schwarz inequality are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of the H lder inequality . The Kantorovich inequality is named after Soviet economist, mathematician, and Nobel Prize winner Leonid Kantorovich , a pioneer in the field of linear programming . References MathWorld urlname KantorovichInequality title Kantorovich Inequality PlanetMath urlname KantorovichInequality title Cauchy Schwarz inequality http carbon.cudenver.edu ... inequality External links http www groups.dcs.st and.ac.uk history Mathematicians Kantorovich.html ... more details
In mathematics, Peetre s inequality, named after Jaak Peetre , says that for any real number t and any Vector space vector s x and y in R sup n sup , the following inequality holds math left frac 1 x 2 1 y 2 right t le 2 t 1 x y 2 t . math References planetmath id 4681 title Peetre s inequality mathanalysis stub Category Linear algebra Category Inequalities km ... more details
In probability and statistics , a correlation inequality is one of a number of inequalities satisfied by the correlation function s of a model. Such inequalities are of particular use in statistical mechanics and in percolation theory . ref cite book mr 0421547 last Ginibre first J. chapter Correlation inequalities in statistical mechanics. title Mathematical aspects of statistical mechanics pages 27&ndash 45 year 1972 publisher Amer. Math. Soc. location Providence, R. I. ref Examples include Bell s inequality FKG inequality Griffiths inequality , and its generalisation, the Ginibre inequality References Reflist External links springer id c c110420 first P.C. last Fishburn Category Probabilistic inequalities Category Statistical inequalities Category Statistical mechanics Category Inequalities statistics stub ... more details
Bonnesen s inequality is an inequality mathematics inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve . It is a strengthening of the classical isoperimetry isoperimetric inequality . More precisely, consider a planar simple closed curve of length math L math bounding a domain of area math F math . Let math r math and math R math denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality math L 2 4 pi F geq pi 2 R r 2. , math The term math pi 2 R r 2 math in the right hand side is known as the em isoperimetric defect em . Loewner s torus inequality with isosystolic defect is a Systolic geometry systolic analogue of Bonnesen s inequality. References Bonnesen, T. Sur une am lioration de l in galit isop rimetrique du cercle et la d monstration d une in galit de Minkowski, C. R. Acad. Sci. Paris 172 1921 , 1087 1089. Yu. D. Burago and V. A. Zalgaller, Geometric inequalities . Translated from the Russian by A. B. Sosinski . Springer Verlag, Berlin, 1988. ISBN 3 540 13615 0. Category Elementary geometry Category Geometric inequalities ... more details
seealso Social equality Horizontal inequality is the inequality economical, social or other that does not follow from a difference in an inherent quality such as intelligence, attractiveness or skills for people or profitability for corporations. In sociology, this is particularly applicable to forced inequality between different subcultures living in the same society. In economics, horizontal inequality is seen when people of similar origin, intelligence, etc. still do not have equal success and have different status, income and wealth. Traditional economic theory predicts that horizontal inequality should not exist in a free market. However, horizontal inequality is observed in real and simulated free market systems. The Pareto optimal economy is one traditional approach to the problem. Even in simulated systems, inequality of perfectly identical actors arises, to give the rich and poor . ref Eric Beinhocker. The Origin of Wealth Evolution, Complexity, and the Radical Remaking of Economics. Harvard Business School Press, 2006. ref References reflist Category Sociology Category Socioeconomics Category Income distribution Category Economic problems econ problem stub ... more details
Orphan date October 2011 In mathematics , a componentwise inequality mathematics inequality is an expression of the form math x,y in real n x preceq y iff x i leq y i forall i 1, ldots,n math ref http www.ee.ucla.edu ee236a lectures lineqs.pdf ref ref http www.stanford.edu class ee364a lectures sets.pdf ref ref http www.ece.ucsb.edu roy classnotes 271a ECE271a lecture3 small.pdf ref The Euclidean vector vector s do not have to be real number real , they can be from any space in which the inequality relation is defined. See also Pointwise Pointwise relations References Reflist Category Inequalities algebra stub ... more details
Mathanalysis stub In functional analysis , a mathematical discipline, the Szeg inequality or P lya&ndash Szeg inequality , named after George P lya and G bor Szeg , states that if math 1 leq p infty math and math u mathbb R n rightarrow mathbb R text in W 1,p mathbb R n , math then math int mathbb R n nabla u p , d mathcal H n leq int mathbb R n nabla u p , d mathcal H n. math See also G bor Szeg H lder s inequality Category Sobolev spaces it Disuguaglianza di Polya Szego ... more details
In mathematics , Steffensen s inequality , named after Johan Frederik Steffensen , is an integral inequality mathematics inequality in real analysis. It states that if a , b R is a non negative, monotonic function monotonically decreasing , integrable function and g a , b 0, 1 is another integrable function, then math int b k b f x , dx leq int a b f x g x , dx leq int a a k f x , dx, math where math k int a b g x , dx. math External links MathWorld title Steffensen s Inequality urlname SteffensensInequality Category Inequalities Category Real analysis mathanalysis stub km fi Steffensenin ep yht l ... more details
In the mathematical theory of Kleinian group s, J rgensen s inequality is an inequality involving the traces of elements of a Kleinian group , proved by harvs txt first Troels last J rgensen authorlink Troels J rgensen year 1976 . The inequality states that if A and B generate a non elementary discrete subgroup of SL sub 2 sub C , then math text Tr A 2 4 text Tr ABA 1 B 1 2 ge 1. , math References Citation last1 J rgensen first1 Troels title On discrete groups of M bius transformations jstor 2373814 mr 0427627 year 1976 journal American Journal of Mathematics issn 0002 9327 volume 98 issue 3 pages 739 749 DEFAULTSORT Jorgensen s Inequality Category Kleinian groups ... more details
Housing inequality refers to the differences in the quality of house housing that exist within a given ... on the negative aspects of inequality . ref Sen 2004 p. 61 ref The term may apply regionally across ... of varying racial or social backgrounds. ref Pryce 2009 p. 145 ref Housing inequality is directly related to concepts of Racism in the United States racial inequality , Social inequality , Income inequality in the United States income inequality , and Wealth inequality in the United States wealth inequality ... market forces , Housing discrimination , and Housing Segregation . Housing inequality is also often ... of poverty. ref Yinger 2001p. 360 ref Residential inequality is especially relevant to discussions ... . ref Sen 1999 p. 87 ref Relation to economic inequality Housing inequality is a type of Economic inequality . This is due to the fact that disparities in housing explain variations in the conversion ... Yinger 2001 ref explains urban residential inequality as a result of natural housing market forces ... economic factors as no single cause can explain housing inequality. In the United States, Thomas Shapiro ... inequality is an inequality of neighborhood amenities. Neighborhood amenities include factors ... effects of housing inequality by restricting access to Personal finance household wealth . ref Krivo and Kaufman 2004 ref The effects of housing inequality are necessarily related to economic inequality as they greatly affect the freedoms available to an individual. Proposed remedies There have been a number of plans proposed to remedy the negative effects of housing inequality. Such plans ... States Scattered site housing International housing inequality While the focus of housing inequality ... to metropolitan areas. International housing inequality is largely characterized by urban disparities ..., housing inequality is increasingly caused by rural to urban migration, increasing urban poverty and inequality, insecure tenure, and globalization. ref UN HABITAT ref All of these factors contribute ... more details
Inequality Reexamined is a book by Amartya Sen first published by Harvard University Press . In it Sen evaluates the different perspectives of the general notion of inequality, focusing mainly on his well known capability approach . The author argues that inequality is a central notion to every social theory that has stood on time. For only if this basic feature is satisfied can a social theory which advocates a set of social arrangements be plausible. Taken the inequality ingredient for granted, the crucial question becomes inequality of what? Sen answers this basic question by advocating his preferred notion of equality which is based on the capability for functionings. Functionings and the role of freedom Confusing date January 2008 Functionings are of two kinds elementary ones such as being in good health, nourished, sheltered and the more complex, social ones such as having self respect, taking part in the life of the community etc. Achievement of an individual is the set of these realized functionings. Whereas capability refers to the real options that someone has in order to pursue his subjective functionings who prefers most. Nevertheless inequalities related to class, gender, communities seem to hinder the extent of human freedom and thus decrease our capability to function. That is why a good society ought to mitigate such discrimination, promoting people s freedom which, according to Sen, is the most valuable element of a satisfactory life. Editions cite book author Amartya Sen title Inequality Reexamined publisher Harvard University Press year 1992 isbn 0 674 45255 0 First Hardcover cite book author Amartya Sen title Inequality Reexamined publisher Harvard University Press year 1995 isbn 0 674 45256 9 Paperback Reprint External links Online version http www.oxfordscholarship.com oso public content economicsfinance 0198289286 toc.html Inequality Reexamined , Oxford Scholarship Online Category Philosophy books Category Sociology books sociology book stub ... more details
This disambiguation page had piped links removed by a bot, per WP MOSDAB . If some links do not seem to belong on this page, they may have originally been piped, so please check before removing links. Thanks In mathematics, Bernstein inequality may refer to Bernstein s inequality mathematical analysis Bernstein inequalities probability theory disambig Category Mathematical disambiguation ... more details
In probability theory , Etemadi s inequality is a so called maximal inequality , an inequality mathematics inequality that gives a bound on the probability that the partial sum s of a Finite set finite collection of independent random variables exceed some specified bound. The result is due to Nasrollah Etemadi . Statement of the inequality Let X sub 1 sub , ..., X sub n sub be independent real valued random variables defined on some common probability space , and let 0. Let S sub k sub denote the partial sum math S k X 1 cdots X k . , math Then math mathbb P left max 1 leq k leq n S k geq 3 alpha right leq 3 max 1 leq k leq n mathbb P left S k geq alpha right . math Remark Suppose that the random variables X sub k sub have common expected value zero. Apply Chebyshev s inequality to the right hand side of Etemadi s inequality and replace by 3. The result is Kolmogorov s inequality with an extra factor of 27 on the right hand side math mathbb P left max 1 leq k leq n S k geq alpha right leq frac 27 alpha 2 mathrm Var S n . math References cite book last Billingsley first Patrick title Probability and Measure publisher John Wiley & Sons, Inc. location New York year 1995 isbn 0 471 00710 2 Theorem 22.5 cite journal last Etemadi first Nasrollah title On some classical results in probability theory journal Sankhy Ser. A volume 47 year 1985 pages 215&ndash 221 mr 0844022 jstor 25050536 issue 2 Category Probabilistic inequalities Category Statistical inequalities ... more details
Carleman s inequality is an inequality mathematics inequality in mathematics , named after Torsten Carleman , who proved it in 1923 ref T. Carleman, Sur les fonctions quasi analytiques , Conf rences faites au cinqui me congres des math maticiens Scandinaves, Helsinki 1923 , 181 196. ref and used it to prove the Denjoy&ndash Carleman theorem on quasi analytic classes. ref cite journal mr 2040885 last1 Duncan first1 John last2 McGregor first2 Colin M. title Carleman s inequality journal Amer. Math. Monthly volume 110 year 2003 issue 5 pages 424&ndash 431 ref ref cite journal mr 1820809 last1 Pe ari first1 Josip last2 Stolarsky first2 Kenneth B. title Carleman s inequality history and new generalizations journal Aequationes Math. volume 61 year 2001 issue 1&ndash 2 pages 49&ndash 62 ref Statement Let a sub 1 sub , a sub 2 sub , a sub 3 sub , ... be a sequence of non negative real number s, then math sum n 1 infty left a 1 a 2 cdots a n right 1 n le e sum n 1 infty a n. math The constant e mathematical constant e in the inequality is optimal, that is, the inequality does not always hold if e is replaced by a smaller number. The inequality is strict it holds with < instead of &le if all the elements in the sequence are positive. Integral version Carleman s inequality has an integral version, which states that math int 0 infty exp left frac 1 x int 0 x ln f t dt right dx leq e int 0 infty f x dx math for any f 0. Carleson s inequality A generalisation, due to Lennart Carleson , states the following ref cite journal first L. last Carleson title A proof of an inequality of Carleman journal Proc. Amer. Math. Soc. volume 5 year 1954 pages 932&ndash 933 ref for any convex function g with g 0 0, and for any 1 p &infin , math int 0 infty x p e g x x dx leq e p 1 int 0 infty x p e g x dx. , math Carleman s inequality follows from the case p 0. Proof One can prove Carleman s inequality by starting with Hardy s inequality ... more details
Social inequality refers to a situation in which individual groups in a society do not have equal social status , social class , and social circle . Areas of social inequality include voting rights , freedom ... Housing inequality housing , travel ing, transport ation, vacation ing and other social goods ... http www.un.org en documents udhr index.shtml a1 ref Social inequality reflects the belief that deviance ... of society. Social inequality involves the belief that some individuals are socially defined .... Causes The reasons for social inequality can vary, but are often broad and far reaching ... can be seen around the globe in the history of all countries. Social inequality is different from economic inequality , though the two are linked. Economic inequality refers to disparities in the distribution of economic assets and income. While economic inequality is caused by the unequal accumulation of wealth , social inequality exists because the lack of wealth in certain areas prohibits ... access to these social goods depends on wealth. Social inequality is linked to racial inequality , gender inequality , and wealth inequality . The way people behave socially, through racism and other ... of inequality on the playing field for blacks and whites . One example he presents reports how a black ... family to acquire wealth, producing social inequality. ref Shapiro, Thomas M. The Hidden ... other factors? Forms of Social Inequality Following are the major types or forms of social inequality. Gender Inequality Main Gender inequality One of the major forms of social inequality is in the form of gender. The emphasis on gender inequality is borne out of the deepening division in the role ... releases WCMS 157562 lang en index.htm ref Racial Inequality Main Racism Racial Inequality is the belief ... and assumptions. ref cite web title Racial Inequality In The Year 2010 url http mylesadamsmlk2010.weebly.com ... their views on these races. This results in racial inequality. Caste Inequality Caste system is another ... more details
For the similarly named inequality involving series Chebyshev s sum inequality In probability theory , Chebyshev s inequality also spelled as Tchebysheff s inequality guarantees that in any sample statistics ... than k standard deviations away from the mean. The inequality has great utility because it can ... Chebyshev s inequality may also refer to the Markov s inequality , especially in the context of analysis ... nowrap k 1 the right hand side is greater than one, so the inequality becomes vacuous, as the probability ... to completely arbitrary distributions unknown except for mean and variance , the inequality generally ..., by Chebyshev s inequality. But if we additionally know that the distribution is normal, we can ..., the bounds provided by Chebyshev s inequality cannot, in general remaining sound for variables of arbitrary ... Equality holds exactly for any distribution that is a linear transformation of this one. Strict inequality ... Chebyshev inequality A one tailed variant with k 0, is ref Grimmett and Stirzaker, problem 7.11.9 ... math Pr X mu geq k sigma leq frac 1 1 k 2 . math The one sided version of the Chebyshev inequality is called Cantelli s inequality, and is due to Francesco Paolo Cantelli . An application distance ... with the expected value, saying the variance exists is equivalent to saying the variance is finite. But this inequality is trivially true if the variance is infinite. Proof using Chebyshev s inequality Setting k 1 in the statement for the one sided inequality gives math Pr X mu geq sigma ... 1 2 . math Thus the median is within one standard deviation of the mean. Proof using Jensen s inequality This proof uses Jensen s inequality twice. We have math begin align left mu m right left mathrm ... E left sqrt X mu 2 right & leq sqrt mathrm E X mu 2 sigma. end align math The first inequality comes from the convex version of Jensen s inequality applied to the absolute value function, which is convex ... a mapsto mathrm E left X a right . , math The third inequality comes from the concave version of Jensen ... more details
In geometry , Pedoe s inequality , named after Daniel Pedoe , states that if a , b , and c are the lengths of the sides of a triangle with area &fnof , and A , B , and C are the lengths of the sides of a triangle with area F , then math A 2 b 2 c 2 a 2 B 2 a 2 c 2 b 2 C 2 a 2 b 2 c 2 geq 16Ff, , math with equality if and only if the two triangles are similarity geometry similar . The expression on the left is not only symmetric under any of the six permutations of the set A , a , B , b , C , c of pairs, but also&mdash perhaps not so obviously&mdash remains the same if a is interchanged with A and b with B and c with C . In other words, it is a symmetric function of the pair of triangles. Pedoe s inequality is a generalization of Weitzenb ck s inequality and of the Hadwiger Finsler inequality . References A Two Triangle Inequality , Daniel Pedoe , The American Mathematical Monthly , volume 70, number 9, page 1012, November, 1963. An Inequality for Two Triangles , D. Pedoe, Proceedings of the Cambridge Philosophical Society , volume 38, part 4, page 397, 1943. External links http www.ele math.com files mia 07 2 full mia 07 32.pdf Pedoe s inequality Category Geometric inequalities Category Triangle geometry ar bs Pedoeova nejednakost de Ungleichung von Pedoe ko it Disuguaglianza di Pedoe nl Ongelijkheid van Pedoe km ru fi Pedoen ep yht l zh yue zh ... more details
In probability theory , Hoeffding s inequality mathematics inequality provides an upper bound on the probability for the sum of random variables to deviate from its expected value . Hoeffding s inequality was proved by Wassily Hoeffding . Let math X 1, dots, X n math be independent random variables . Assume that the math X i math are almost sure ly bounded that is, assume for math 1 leq i leq n math that math Pr X i in a i, b i 1. math Then, for the empirical mean of these variables math overline X X 1 cdots X n n math we have the inequalities Hoeffding 1963, Theorem 2 ref Wassily Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 301 13&ndash 30, March 1963. http links.jstor.org sici?sici 0162 1459 28196303 2958 3A301 3C13 3APIFSOB 3E2.0.CO 3B2 D JSTOR ref math Pr overline X mathrm E overline X geq t leq exp left frac 2t 2n 2 sum i 1 n b i a i 2 right , math math Pr overline X mathrm E overline X geq t leq 2 exp left frac 2t 2n 2 sum i 1 n b i a i 2 right , math which are valid for positive values of t . Here math mathrm E overline X math is the expected value of math overline X math . These inequalities are special cases of the more general Azuma Hoeffding inequality and the even more general Bernstein inequalities in probability theory Bernstein inequality in probability theory , proved by Sergei Bernstein in 1923. They are also special cases of McDiarmid s inequality . Note that the inequalities also hold when the math X i math have been obtained using sampling without replacement in this case the random variables are not independent anymore. A proof of this statement can be found in Hoeffding s paper. For slightly better bounds in the case of sampling without replacement, see for instance the paper by Serfling ref R. J. Serfling, Probability Inequalities for the Sum in Sampling without Replacement ... s inequality Chebyshev s inequality Markov s inequality Chernoff bounds Hoeffding s lemma References ... more details
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