The general logical form of the definition is: A subset S of a space X is a totally bounded set if and only if, given any size E, there exist a natural number n and a family A1, A2, ..., An of subsets of X, such that S is contained in the union of the family (in other words, the family is a finite cover of S), and such that each set Ai in the family is of size E (or ...

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Totally bounded space, Totally bounded space - Definitions, Totally bounded space - Examples and nonexamples, Totally bounded space - Relationships with compactness and completeness, Totally bounded space - Use of the axiom of choice

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The experiments whose results are under certain conditions summarized by the Tsirelson bound or by the CHSH inequality concern measurements obtained by a pair of observers, A and B, who each can detect one signal at a time in one of two distinct own channels or outcomes: for instance A detecting and counting a signal either as (A↑) or (A↓), and B detecting and counting a signal either as (B «), or (B »). Signals are to be considered and counted only if A and B detect them trial-by-trial together; i.e. ...

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Tsirelson's bound, Tsirelson's bound - Derivation following Tsirelson's elementary proof, Tsirelson's bound - The role of Landau's identity in deriving Tsirelson's inequality, Tsirelson's bound - Application to EPR experiments, Tsirelson's bound - Tsirelson's bound as bound for objective local theories, Tsirelson's bound - Comparison with the CHSH inequality

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There is a nice relationship between total boundedness and compactness: A space is compact if and only if it is both totally bounded and Cauchy complete. This can be seen as a generalisation of the Heine-Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundedness with total boundedness (and also replace closedness with completeness). There is a complementary relationship between total boundedness and the process of Cauchy completion: A space is totally bounded if and only if its Cauchy completion is totally bounded. (This corresponds to the ...

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Totally bounded space, Totally bounded space - Definitions, Totally bounded space - Examples and nonexamples, Totally bounded space - Relationships with compactness and completeness, Totally bounded space - Use of the axiom of choice

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Bound Brook New Jersey - Local government. The Mayor of Bound Brook is Frank J. Ryan. Members of the Borough Council are Council President Debbie Cozza, Ed Gabrielski, Carey Pilato, Jeff Thompson and Javier Vasquez. Bound Brook New Jersey - Federal state and county representation. Bound Brook is in the Seventh Congressional District a ...

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Bound Brook New Jersey, Bound Brook New Jersey - History, Bound Brook New Jersey - Geography, Bound Brook New Jersey - Demographics, Bound Brook New Jersey - Government, Bound Brook New Jersey - Local government, Bound Brook New Jersey - Federal state and county representation, Bound Brook New Jersey - Famous residents

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The town was first settled in 1681. A wooden bridge over the Raritan River was erected as early as 1761 and named Queen's Bridge in 1767. Later it became a covered bridge. During the American Revolutionary War the bridge was used repeatedly by both sides including during the Battle of Bound Brook in 1777. In 1875 the wooden bridge was replaced by a steel pipe truss bridge, which was replaced by a steel g ...

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Bound Brook New Jersey, Bound Brook New Jersey - History, Bound Brook New Jersey - Geography, Bound Brook New Jersey - Demographics, Bound Brook New Jersey - Government, Bound Brook New Jersey - Local government, Bound Brook New Jersey - Federal state and county representation, Bound Brook New Jersey - Famous residents

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Bound Brook is located at 40°33'55" North, 74°32'22" West (40.565203, -74.539513)GR1. According to the United States Census Bureau, the borough has a total area of 4.4 km² (1.7 mi²). 4.4 km² (1.7 mi²) of it is land and none of the area is covered with water. ...

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Bound Brook New Jersey, Bound Brook New Jersey - History, Bound Brook New Jersey - Geography, Bound Brook New Jersey - Demographics, Bound Brook New Jersey - Government, Bound Brook New Jersey - Local government, Bound Brook New Jersey - Federal state and county representation, Bound Brook New Jersey - Famous residents

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Given four operators (F, G, U, and V) together with a product operation (∙) defined for any pair of these four operators, and given that the following four pairs of operators commute: F ∙ U = U ∙ F, F ∙ V = V ∙ F, G ∙ U = U ∙ G, and G ∙ V = V ∙ G, then it follows that: F ∙ U + F ∙ V + U ∙ G − V ∙ G = 1/√2 F ∙ F + 1/√2 G ∙ G + 1/√2 U ∙ U + 1/ ...

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Tsirelson's bound, Tsirelson's bound - Derivation following Tsirelson's elementary proof, Tsirelson's bound - The role of Landau's identity in deriving Tsirelson's inequality, Tsirelson's bound - Application to EPR experiments, Tsirelson's bound - Tsirelson's bound as bound for objective local theories, Tsirelson's bound - Comparison with the CHSH inequality

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Before stating a precise definition of free variable and bound variable (or dummy variable), we present some examples that perhaps make these two concepts clearer than the definition would (unfortunately the term dummy variable is used by many statisticians to mean an indicator variable or some variant thereof; the name is really not apt for that purpose, but magnificently conveys the intuition behind the definition of t ...

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Free variables and bound variables, Free variables and bound variables - Examples, Free variables and bound variables - Variable-binding operators, Free variables and bound variables - Formal explanation

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As of the censusGR2 of 2000, there are 10,155 people, 3,615 households, and 2,461 families residing in the borough. The population density is 2,292.9/km² (5,953.7/mi²). There are 3,802 housing units at an average density of 858.5/km² (2,229.0/mi²). The racial makeup of the borough is 82.57% White, 2.52% African American, 0.31% Native American, 2.88% Asian, 0.07% Pacific Islander, 8.67% from other races, and 2.99% from two or more races. 34.87 ...

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Bound Brook New Jersey, Bound Brook New Jersey - History, Bound Brook New Jersey - Geography, Bound Brook New Jersey - Demographics, Bound Brook New Jersey - Government, Bound Brook New Jersey - Local government, Bound Brook New Jersey - Federal state and county representation, Bound Brook New Jersey - Famous residents

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Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science but in all cases they are purely syntactic properties of expressions and variables in them. For this section we can summarize syntax by identifying expressions with trees whose leaf nodes are variables, function constants or predicate constants and whose nodes are logical operators. Variable-binding operators are logical operators that occur in almost every formal language. Indeed languages which do not have them are either extremely inexpressi ...

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Free variables and bound variables, Free variables and bound variables - Examples, Free variables and bound variables - Variable-binding operators, Free variables and bound variables - Formal explanation

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The most basic of improper integrals are integrals such as: As stated above, this need not be defined as an improper integral, since it can be construed as a Lebesgue integral instead. Nonetheless, for purposes of actually computing this integral, it is more convenient to treat it as an improper integral, i.e., to evaluate it when the upper bound of integration is finite and then take the limit as that bound approaches ∞. The antiderivative of the function being inte ...

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Improper integral, Improper integral - Infinite bounds of integration, Improper integral - Vertical asymptotes at bounds of integration, Improper integral - Cauchy principal values

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The Time-Bound Programme (TBP) approach constitutes one of the means put in place by the International Programme on the Elimination of Child Labour (IPEC) to assist countries in fulfilling their obligations under the convention. TBPs are designed as a comprehensive framework that governments can use to chart a course of action with well-defined targets. They comprise a set of integrated and coordinated policies and interventions with clear goals, specific targets and a defined time frame, aimed at preventing and eliminating a c ...

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Time-Bound Programmes for the Eradication of the Worst forms of Child Labour, Time-Bound Programmes for the Eradication of the Worst forms of Child Labour - Time-bound measures, Time-Bound Programmes for the Eradication of the Worst forms of Child Labour - Time-bound programmes

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When certain proteins are synthesized by a ribosome, it can become "membrane-bound", associated with the membrane of the nucleus and the rough endoplasmic reticulum (in eukaryotes only) for the time of synthesis. They insert the freshly produced polypeptide chains directly into the ER, from where they are transported to their destinations. Bound ribosomes usually produce proteins that are used within the cell membrane ...

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Ribosome, Ribosome - Overview, Ribosome - Free ribosomes, Ribosome - Membrane bound ribosomes, Ribosome - Atomic structure, Ribosome - External link

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The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. If an ordered set S has the property that every nonempty subset of S having an upper bound also has a least upper bound, then S is said to have the least-upper-bound property. As noted above, the set R of all real numbers has the least-upper-bound property. Similarly, the set Z of integers has the least-upper-bound property; if S is a nonempty subset o ...

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Supremum, Supremum - Supremum of a set of real numbers, Supremum - Approximation property, Supremum - Additive property, Supremum - Comparison property, Supremum - Suprema within partially ordered sets, Supremum - Comparison with other order theoretical notions, Supremum - Greatest elements, Supremum - Maximal elements, Supremum - Minimal upper bounds, Supremum - Least-upper-bound property

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Consider This integral involves a function with a vertical asymptote at x = 0. One can evaluate this integral by evaluating from b to 1, and then take the limit as b approaches 0. One should note that the antiderivative of the above function is (3)(x1/3); which can be evaluated by direct substitution to give the value 3 × (1 − ...

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Improper integral, Improper integral - Infinite bounds of integration, Improper integral - Vertical asymptotes at bounds of integration, Improper integral - Cauchy principal values

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Function names assume a min-heap: For pairing heaps the insert, decreaseKey and merge operations are conjectured to be O(1) amortized complexity but this has not yet been proven. ...

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Heap, Heap - Variants, Heap - Comparison of theoretic bounds for variants, Heap - Heap applications

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Since the empty set has no members, when it is considered as a subset of any ordered set, then any member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set. When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely "negative infinity", denoted which is defined to be les ...

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Empty set, Empty set - Notation, Empty set - Properties, Empty set - Common problems, Empty set - Axiomatic set theory, Empty set - Does it exist or is it necessary?, Empty set - Operations on the empty set, Empty set - Bounds, Empty set - The empty set and zero, Empty set - Category theory

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Suppose H is a Hilbert space, with inner product <.,.>. Consider a continuous linear operator A : H → H (this is the same as a bounded operator). Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator A* : H → H with the following property: This operator A* is the adjoint of A. ...

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Hermitian adjoint, Hermitian adjoint - Definition for bounded operators, Hermitian adjoint - Properties, Hermitian adjoint - Hermitian operators, Hermitian adjoint - Adjoints of unbounded operators, Hermitian adjoint - Other adjoints

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For a Hilbert space H, the continuous linear operators A : H → H are of particular interest. Such a continuous operator is bounded in the sense that it maps bounded sets to bounded sets. This allows to define its norm as The sum and the composition of two continuous linear operators is again continuous and linear. For y in H, the map that sends x to <y, Ax> is linear and continuous, and according to the Riesz representation theorem can theref ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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In his seminal The Art of Computer Programming, Don Knuth discussed a number of lower bounds for the number of comparisons required to locate the kth smallest entry of an unorganized list of n items (using only comparisons). There's a trivial lower bound of n − 1 for the minimum or maximum entry. To see this, consider a tournament where each game represents one comparison. Since every player except the winner of the tournament must lose a game before we know the winner, ...

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Selection algorithm, Selection algorithm - Selection with sorting algorithm, Selection algorithm - Linear minimum/maximum algorithms, Selection algorithm - Nonlinear general section algorithm, Selection algorithm - Partition based general selection algorithm, Selection algorithm - Linear general selection algorithm, Selection algorithm - Selection as incremental sorting, Selection algorithm - Using data structures to select in sublinear time, Selection algorithm - Selecting k smallest or largest elements, Selection algorithm - Lower bounds, Selection algorithm - Language support

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An identity involving four operators (F, G, U, and V) and a product operation (·) has been pointed out by L. J. Landau [2]: Given that the following four pairs of operators commute: F · U = U · F, F · V = V · F, G · U = U · G, and G · V = V · G, and given the normalization constraints F · F = G · G, and U · U = V · V, then Landau's identity holds: See also:

Tsirelson's bound, Tsirelson's bound - Derivation following Tsirelson's elementary proof, Tsirelson's bound - The role of Landau's identity in deriving Tsirelson's inequality, Tsirelson's bound - Application to EPR experiments, Tsirelson's bound - Tsirelson's bound as bound for objective local theories, Tsirelson's bound - Comparison with the CHSH inequality

Read more here: » Tsirelson's bound: Encyclopedia II - Tsirelson's bound - The role of Landau's identity in deriving Tsirelson's inequality