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# Topic: Supremum

###### In the News (Sun 18 Aug 19)

 Supremum - Wikipedia, the free encyclopedia In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S. In particular, note the third example where the supremum of a set of rationals is irrational (which means that the rationals are incomplete). The difference between the supremum of a set and the greatest element of a set may not be immediately obvious. en.wikipedia.org /wiki/Supremum   (1669 words)

 Complete lattice - Wikipedia, the free encyclopedia The supremum is given by the union and the infimum by the intersection of subsets. The supremum of finite sets is given by the least common multiple and the infimum by the greatest common divisor. The supremum is given by the union of open sets and the infimum by the interior of the intersection. en.wikipedia.org /wiki/Complete_lattice   (2082 words)

 Lattice   (Site not responding. Last check: 2007-10-21) More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant - the definition is independent of that choice). The supremum is given by the least common multiple and the infimum by the greatest common divisor. The supremum is given by the subgroup generated by the union of the groups and the infimum is given by the intersection. bopedia.com /en/wikipedia/l/la/lattice_1.html   (1356 words)

 Essential supremum and essential infimum - Wikipedia, the free encyclopedia Then, in the same way as the supremum of f is defined to be the smallest upper bound, the essential supremum is defined as the smallest essential upper bound. The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. Thus, the essential supremum and the essential infimum of these functions are both 2. en.wikipedia.org /wiki/Essential_supremum_and_essential_infimum   (365 words)

 PlanetMath: essential supremum This allows us to generalize the maximum of a function in a useful way. This is version 1 of essential supremum, born on 2002-02-17. why does this reduce to the usual notion of supremum when f is continuous? planetmath.org /encyclopedia/EssentialSupremum.html   (87 words)

 Convergence to the neighborhood of the optimal value function However, crude discretization of the environment and a robust controller, which is part of the environment, exhibits itself as a varying environment. Supremum norm was computed against event values belonging to the last time step. Supremum norm, in turn, is zero for the last point (not shown). www.ai.mit.edu /projects/jmlr/papers/volume3/szita02a/html/node16.html   (301 words)

 Supremum   (Site not responding. Last check: 2007-10-21) This is sometimes called the supremum axiom and expresses the completeness of the real numbers. If the supremum value belongs to the set then we can say there is a largest element in the set. In a lattice every nonempty finite subset has a supremum, and in a complete lattice every subset has a supremum. www.termsdefined.net /su/supremum.html   (467 words)

 Analysis WebNotes: Chapter 02, Class 05 A set need not have a supremum just because it is bounded above, but in order for it to have a supremum, it must, in particular, be bounded above. On the other hand a set may have no greatest element and still have a supremum (for example, this set studied in Class 4). We'll prove that this last set has no supremum in the rationals later on, after we have the definition of the real numbers to work with. www.math.unl.edu /~webnotes/classes/class05/class05.htm   (373 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The element $a$ of $X$ is called the {\bf supremum} or {\bf least upper bound} of $S$ if \begin{enumerate} \item[(i)] $a$ is itself an upper bound of $S$, and \item[(ii)] no element $b$ of $X$ which is less than $a$ can be an upper bound for $S$. On the other hand a set may have no greatest element and still have a supremum (for example, #ref(1-set)this set studied in Class 4). We'll prove that this last set has no supremum in the rationals later on, after we have #ref(Real_Axioms)the definition of the real numbers to work with. www.math.unl.edu /~webnotes/src/classes-1997/class05.wfy   (586 words)

 Renyi Sample Size Page   (Site not responding. Last check: 2007-10-21) The algorithm used for computing sample size for the supremum log-rank is described in detail in the technical report "A sample size formula for the supremum log-rank statistic" by Kevin Hasegawa Eng and Michael R. Kosorok, which can be downloaded by clicking here. As part of this package, the function surv.Rtest for computing the supremum weighted log-rank and its p-value is included. The final optional argument is the error permitted for the supremum p-value (with default 1.0e-8). www.biostat.wisc.edu /~kosorok/renyi.html   (470 words)

 Brownian Motion in Space - Algorithms Firstly there is the process of simulating the brownian motion itself, and secondly there is the process by which each of the simulated occurrences is checked to find the supremum. Finally, for the chosen supremum, a refined value of the trajectory is recalculated using a numerical double integral. Thus the interpolated values for H(s,t) are used to identify the supremum, but then the value of this supremum is refined using a more accurate but time-consuming double integral. www.mcs.vuw.ac.nz /~ray/Brownian/algorithms.shtml   (939 words)

 ON THE SUPREMUM OF A FAMILY OF SINGULAR COMPACTIFICATIONS   (Site not responding. Last check: 2007-10-21) It has been previously shown that the supremum of singular compactifications need not itself be a singular compactification. We provide necessary and sufficient conditions which describe when the supremum of a family of singular compactifications is a singular compactification. We also show that there are compactifications which are not the supremum of a family of singular compactifications. math.la.asu.edu /~rmmc/rmj/Vol28-1/ANDR   (143 words)

 Metamath Proof Explorer - Real and Complex Numbers   (Site not responding. Last check: 2007-10-21) But the supremum is pi, which is not a rational number. As you go diagonally down the list of decimal expansions of real numbers, mismatching the list digit by digit, at each point you will have a new rational number with one more digit (that is different from all numbers in the list up to that point). The supremum of this list of rational numbers is a real number (with an infinite number of digits after the decimal point) that is not on the list. us.metamath.org /mpegif/mmcomplex.html   (2040 words)

 Section (vii) The Overwhelming Linguistic and Conceptual Complexity of the Notions of Sup and Inf As in Extract 6.4 most students appear in a linguistic unease (in parallel with their difficulty with the new concept) with the terminology commonly used for supremum and infimum, that is least upper and greatest lower bound. In her response there is evidence of a conception regarding the supremum of a set according to which anything less than the supremum is necessarily contained in the set. Conclusion: In the above, evidence was given of difficulties embedded in the notion of supremum and infimum of a set, and in particular of the notion of infimum as the greatest lower bound of a set. www.uea.ac.uk /~m011/thesis/chapter6/6vii.htm   (881 words)

 Formal Foundations of Computer Science 1 -- 6.4 Sequences and Series The elements of an infinite sequence either become arbitrarily large or their size is bounded. As above example shows, the infimum and supremum of a sequence need not be among the values of the sequence; in this case the sequence values arbitrarily close approach the bound without ever actually reaching it: values of the given sequence ; please note that for infinite sequences supremum respectively infimum need not be among the values of the sequence; please compare with the actual definitions of "sup" and "inf". www.risc.uni-linz.ac.at /education/courses/ws99/formal/report/index_47.html   (553 words)

 Convergence of Markov chains in the relative supremum norm, Lars Holden Convergence of Markov chains in the relative supremum norm, Lars Holden If the detailed balance condition and a weak continuity condition are satisfied, then the strong Doeblin condition is equivalent to convergence in the relative supremum norm. We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.jap/1014843084   (193 words)

 Section (vi) The Unsettling Character of the Logical Conjunctions in the Definitions of ST and ST and ... When discussing however the existence of the supremum it seems that, even though the two conditions, upper-boundedness and non-emptiness, are logically equivalent, in the mathematician's mind the former carries more weight than the latter. In order to prove that a number is the supremum of a set one has to show that this number is an upper bound of the set and also that it is its least upper bound. Finally the uncertainty generated by one of the students' presentation highlighted the necessity for the novices to acquire a clear and minimally-ambiguous semantic and linguistic command of their writing or presenting style. www.uea.ac.uk /~m011/thesis/chapter6/6vi.htm   (1296 words)

 Definition of Upper Bound and Least Upper Bound (Supremum)   (Site not responding. Last check: 2007-10-21) The least upper bound, called the supremum, of a set S, is defined as a quantity M such that no member of the set exceeds M, but if ε is any positive quantity, however small, there is a member that exceeds M - ε. The least upper bound of a function, f, is defined as a quantity M such that f(x) ≤ M for all x in its domain, but if ε is any positive quantity, however small, there is an x in the domain such that f(x) exceeds M - ε. Let S be the supremum of A. By the definition of supremum, given any small positive number, ε, you can find an element of the set (and thus an element S mcraefamily.com /MathHelp/CalculusLimitUpperBound.htm   (457 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Using logical symbols only, formulate the following two propositions: the subset A of the set of real numbers is bounded; and an upper bound of the subset A of the set of real numbers is m. Give the definition of the supremum of a subset of the field of real numbers. State the supremum principle for the field of real numbers. www.sftw.umac.mo /~fstitl/topics/exam2.html   (517 words)

 Geometric Series and Convergence Theorems Before we present this test, we need to discuss a rather sophisticated idea used with it-the limit supremum. The limit supremum is a powerful idea because the limit supremum of a sequence always exists, which is not true for the ordinary limit. However, Example 4.20 illustrates the fact that, if the limit of a sequence does exist, then it will be the same as the limit supremum. math.fullerton.edu /mathews/c2003/ComplexGeometricSeriesMod.html   (384 words)

 PlanetMath: limit superior , to be the supremum of all the limit points of We can generalize the above definition to the case of a mapping Cross-references: limit, diverges, infimum, converges, equivalent, sequence, inclusion mapping, mapping, supremum, limit point, real numbers planetmath.org /encyclopedia/SupremumLimit.html   (171 words)

 Basic Theorem on Concept Lattices The formula for the supremum is substantiated correspondingly. The statement about the supremum of the concept lattice is proved correspondingly as follows: To be able to use the formula proved above for the supremum of a subset of a concept lattice, we rewrite the concept wwwtcs.inf.tu-dresden.de /~sertkaya/ms_thesis/node24.html   (1420 words)

 On the supremum distribution of integrated stationary Gaussian processes with negative linear drift, Jinwoo Choe, Ness ... In this paper we study the supremum distribution of a class of Gaussian processes having stationary increments and negative drift using key results from Extreme Value Theory. We focus on deriving an asymptotic upper bound to the tail of the supremum distribution of such processes. We discuss the importance of the bound, its applicability to queueing problems, and show numerical examples to illustrate its performance. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aap/1029954270   (210 words)

 Completeness   (Site not responding. Last check: 2007-10-21) But the supremum of the same set does exist in , a finite set, the supremum and infimum always exist. In this case the supremum is the largest element of www.math.pitt.edu /~sparling/23014/23014notes6/node12.html   (87 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-21) Date: 02/02/98 at 21:52:29 From: Doctor Pete Subject: Re: Explain supremum Hi, The concept of supremum, or least upper bound, is as follows: Let S = {a[n]}, the sequence with terms a[0], a[1],... S has a supremum, called sup S, if for every n, a[n] <= sup S (i.e. This is why the supremum is also called the "least upper bound," for a bound is a number which a function, sequence, or set, never exceeds. mathforum.org /library/drmath/view/51467.html   (290 words)

 GLEASON'S THEOREM HAS A CONSTRUCTIVE PROOF Without the law of excluded middle, you can find x so that f(x) is close to the supremum, and so get an almost diagonal matrix for B. That's true in a classical context, where any bounded function has a supremum and an infimum, but it doesn't necessarily hold in a constructive context. Let M be the supremum of f and m the infimum. www.math.fau.edu /Richman/Docs/glhasrev.html   (2118 words)

 MySQL Internals Manual :: 13.1.3 The Infimum and Supremum Records   (Site not responding. Last check: 2007-10-21) , an infimum is lower than the the lowest possible real value (negative infinity) and a supremum is greater than the greatest possible real value (positive infinity). code comments distinguish between "the infimum and supremum records" and the "user records" (all other kinds). considers the infimum and supremum to be part of the header or not. dev.mysql.com /doc/internals/en/innodb-infimum-and-supremum-records.html   (221 words)

 Time: Hitoshi Kitada's Home Page   (Site not responding. Last check: 2007-10-21) A minimal element of U(A) is called a *supremum* of A. Theorem: Let A be a set of real numbers that is bounded above. Since a supremum of A is a maximal element of A and A=L(B), it follows that Since a supremum of A is a maximal element of A and A=L(B), it follows that L(B) has a maximal element, as claimed," you seem to "deduce" that sup(A) belongs to the set A from Theorem 11. www.kitada.com   (7039 words)

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