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With a successful run of 12 years and over 2,700 episodes, the Jack and Stench Show is among the longest-running monetized podcasts. In March 2007, the Cambridge CARET Centre also helped to give birth to the first as-live podcast channel for women politicians in the UK and globally called Women's Parliamentary Radio. A former BBC correspondent and political editor in the East, Boni Sones OBE, worked with three other broadcast journalists—Jackie Ashley, Deborah McGurran, and Linda Fairbrother—to create an online radio station where women MPs of all parties could be interviewed impartially.
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https://en.wikipedia.org/wiki/Odiogo
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The MP3 files could be streamed or downloaded. Their resulting 550 interviews over 15 years can now be found in one of four audio archives nationally at the British Library, the London School of Economics, The History of Parliament Trust and the Churchill Archives University of Cambridge. Sones has also written four books about these podcast interviews and archives, which are in all the major libraries in the UK.
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https://en.wikipedia.org/wiki/Odiogo
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The Adam Carolla Show started as a regular weekday podcast in March 2009; by March 2011, 59.6 million episodes had been downloaded in total, claiming a record; however, as previously mentioned, Gervais's podcast had already received five times Carolla's downloads by the time the record was supposedly set. The BBC noted in 2011 that more people (eight million in the UK or about 16% of the population, with half listening at least once a week—a similar proportion to the USA) had downloaded podcasts than had used Twitter.Besides the aforementioned Adam Carolla Show, 2009 saw a huge influx of many other popular new comedy podcasts, including the massively successful talk-style podcasts with a comedic bent such as WTF with Marc Maron, The Joe Rogan Experience, How Do Podcast, and the David Feldman Show. 2009 also saw the launch of the surrealist comedy show Comedy Bang!
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https://en.wikipedia.org/wiki/Odiogo
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Bang! (which was known as Comedy Death-Ray Radio at the time), which was later turned into a TV show with the same name.
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https://en.wikipedia.org/wiki/Odiogo
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With a run of eight years (as of October 2013), the various podcasts provided by Wrestling Observer/Figure Four Online, including Figure Four Daily and the Bryan and Vinny Show with host Bryan Alvarez, and Wrestling Observer Radio with hosts Alvarez and Dave Meltzer, have produced over 6,000 monetized podcasts at a subscription rate of $10.99 per month. Their subscription podcast model launched in June 2005. Alvarez and Meltzer were co-hosts in the late 1990s at Eyada.com, the first Internet-exclusive live streaming radio station, broadcasting out of New York City.In 2014, This American Life launched the first season of their Serial podcast.
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https://en.wikipedia.org/wiki/Odiogo
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The podcast was a surprise success, achieving 68 million downloads by the end of Season 1 and becoming the first podcast to win a Peabody Award. The program was referred to as a "phenomenon" by media outlets and popularized true crime podcasts. True crime programs such as My Favorite Murder, Crimetown, and Casefile were produced after the release of Serial and each of these titles became successful in their own right.
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https://en.wikipedia.org/wiki/Odiogo
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From 2012 to 2013, surveys showed that the number of podcast listeners had dropped for the first time since 2008. However, after Serial debuted, audience numbers rose by 3%. Podcasting reached a new stage of growth in 2017 when The New York Times debuted The Daily news podcast. The Daily is designed to match the fast pace of modern news, and the show features original reporting and recordings of the newspaper's top stories. As of May 2019, it has the highest unique monthly US audience of any podcast.
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https://en.wikipedia.org/wiki/Odiogo
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In mean field theory, the mean field appearing in the single-site problem is a time-independent scalar or vector quantity. However, this isn't always the case: in a variant of mean field theory called dynamical mean field theory (DMFT), the mean field becomes a time-dependent quantity. For instance, DMFT can be applied to the Hubbard model to study the metal–Mott-insulator transition.
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https://en.wikipedia.org/wiki/Mean-field_approximation
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In meantone systems, there are two different semitones. This results because of the break in the circle of fifths that occurs in the tuning system: diatonic semitones derive from a chain of five fifths that does not cross the break, and chromatic semitones come from one that does. The chromatic semitone is usually smaller than the diatonic. In the common quarter-comma meantone, tuned as a cycle of tempered fifths from E♭ to G♯, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively. Extended meantone temperaments with more than 12 notes still retain the same two semitone sizes, but there is more flexibility for the musician about whether to use an augmented unison or minor second. 31-tone equal temperament is the most flexible of these, which makes an unbroken circle of 31 fifths, allowing the choice of semitone to be made for any pitch.
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https://en.wikipedia.org/wiki/Just_chromatic_semitone
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In meantone temperaments, the major tone and minor tone are replaced by a "mean tone" which is somewhere in between the two. Two of these tones make a ditone or major third. This major third is exactly the just (5:4) major third in quarter-comma meantone. This is the source of the name: the note exactly halfway between the bounding tones of the major third is called the "mean tone".
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https://en.wikipedia.org/wiki/Ditone
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In measure and probability theory in mathematics, a convex measure is a probability measure that — loosely put — does not assign more mass to any intermediate set "between" two measurable sets A and B than it does to A or B individually. There are multiple ways in which the comparison between the probabilities of A and B and the intermediate set can be made, leading to multiple definitions of convexity, such as log-concavity, harmonic convexity, and so on. The mathematician Christer Borell was a pioneer of the detailed study of convex measures on locally convex spaces in the 1970s.
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https://en.wikipedia.org/wiki/Convex_measure
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In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of almost surely in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero.
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https://en.wikipedia.org/wiki/Almost_everywhere
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When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term almost everywhere is abbreviated a.e.
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https://en.wikipedia.org/wiki/Almost_everywhere
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; in older literature p.p. is used, to stand for the equivalent French language phrase presque partout.A set with full measure is one whose complement is of measure zero. In probability theory, the terms almost surely, almost certain and almost always refer to events with probability 1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a probability space. Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for almost all elements (though the term almost all can also have other meanings).
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https://en.wikipedia.org/wiki/Almost_everywhere
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In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements.
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https://en.wikipedia.org/wiki/Prokhorov's_theorem
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In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets G {\displaystyle G} is precisely the smallest 𝜎-algebra containing G . {\displaystyle G.} It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.
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https://en.wikipedia.org/wiki/Monotone_class_theorem
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In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure. The theorem is also sometimes known as the Carathéodory–Fréchet extension theorem, the Carathéodory–Hopf extension theorem, the Hopf extension theorem and the Hahn–Kolmogorov extension theorem.
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https://en.wikipedia.org/wiki/Carathéodory's_extension_theorem
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In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables.
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https://en.wikipedia.org/wiki/Bounded_convergence_theorem
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In measure theory, Stone & Tukey (1942) proved two more general forms of the ham sandwich theorem. Both versions concern the bisection of n subsets X1, X2, …, Xn of a common set X, where X has a Carathéodory outer measure and each Xi has finite outer measure. Their first general formulation is as follows: for any continuous real function f: S n × X → R {\displaystyle f\colon S^{n}\times X\to \mathbb {R} } , there is a point p of the n-sphere Sn and a real number s0 such that the surface f(p,x) = s0 divides X into f(p,x) < s0 and f(p,x) > s0 of equal measure and simultaneously bisects the outer measure of X1, X2, …, Xn. The proof is again a reduction to the Borsuk-Ulam theorem.
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https://en.wikipedia.org/wiki/Ham-sandwich_theorem
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This theorem generalizes the standard ham sandwich theorem by letting f(s,x) = s1x1 + … + snxn. Their second formulation is as follows: for any n + 1 measurable functions f0, f1, …, fn over X that are linearly independent over any subset of X of positive measure, there is a linear combination f = a0f0 + a1f1 + … + anfn such that the surface f(x) = 0, dividing X into f(x) < 0 and f(x) > 0, simultaneously bisects the outer measure of X1, X2, …, Xn. This theorem generalizes the standard ham sandwich theorem by letting f0(x) = 1 and letting fi(x), for i > 0, be the i-th coordinate of x.
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https://en.wikipedia.org/wiki/Ham-sandwich_theorem
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In measure theory, a branch of mathematics that studies generalized notions of volumes, an s-finite measure is a special type of measure. An s-finite measure is more general than a finite measure, but allows one to generalize certain proofs for finite measures. The s-finite measures should not be confused with the σ-finite (sigma-finite) measures.
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https://en.wikipedia.org/wiki/S-finite_measure
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In measure theory, a branch of mathematics, Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable product measures. It gives an "if and only if" characterisation of when two such measures are equivalent, and hence it is extremely useful when trying to establish change-of-measure formulae for measures on function spaces. The result is due to the Japanese mathematician Shizuo Kakutani. Kakutani's theorem can be used, for example, to determine whether a translate of a Gaussian measure μ {\displaystyle \mu } is equivalent to μ {\displaystyle \mu } (only when the translation vector lies in the Cameron–Martin space of μ {\displaystyle \mu } ), or whether a dilation of μ {\displaystyle \mu } is equivalent to μ {\displaystyle \mu } (only when the absolute value of the dilation factor is 1, which is part of the Feldman–Hájek theorem).
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https://en.wikipedia.org/wiki/Kakutani's_theorem_(measure_theory)
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In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that μ ( ∂ B ) = 0 , {\displaystyle \mu (\partial B)=0\,,} where ∂ B {\displaystyle \partial B} is the (topological) boundary of B. For signed measures, one asks that | μ | ( ∂ B ) = 0 . {\displaystyle |\mu |(\partial B)=0\,.} The class of all continuity sets for given measure μ forms a ring.Similarly, for a random variable X, a set B is called continuity set if Pr = 0. {\displaystyle \Pr=0.}
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https://en.wikipedia.org/wiki/Continuity_set
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In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than more general measures and show a variety of different properties depending on the sets they are defined on.
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https://en.wikipedia.org/wiki/Finite_measure
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In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration.
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https://en.wikipedia.org/wiki/Lebesgue_measure
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Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.
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https://en.wikipedia.org/wiki/Lebesgue_measure
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In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure.A property that is true of the elements of a conull set is said to be true almost everywhere. == References ==
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https://en.wikipedia.org/wiki/Conull_set
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In measure theory, a generic property is one that holds almost everywhere. The dual concept is a null set, that is, a set of measure zero.
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https://en.wikipedia.org/wiki/Generic_property
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In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.
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https://en.wikipedia.org/wiki/Push-forward_measure
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In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the second space. It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure.
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https://en.wikipedia.org/wiki/Radonifying_function
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In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian physicist and geometer, who published independent proofs respectively in 1910 and 1911. Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's theorem for integrable functions.
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https://en.wikipedia.org/wiki/Egorov's_Theorem
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In measure theory, given a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} and a signed measure μ {\displaystyle \mu } on it, a set A ∈ Σ {\displaystyle A\in \Sigma } is called a positive set for μ {\displaystyle \mu } if every Σ {\displaystyle \Sigma } -measurable subset of A {\displaystyle A} has nonnegative measure; that is, for every E ⊆ A {\displaystyle E\subseteq A} that satisfies E ∈ Σ , {\displaystyle E\in \Sigma ,} μ ( E ) ≥ 0 {\displaystyle \mu (E)\geq 0} holds. Similarly, a set A ∈ Σ {\displaystyle A\in \Sigma } is called a negative set for μ {\displaystyle \mu } if for every subset E ⊆ A {\displaystyle E\subseteq A} satisfying E ∈ Σ , {\displaystyle E\in \Sigma ,} μ ( E ) ≤ 0 {\displaystyle \mu (E)\leq 0} holds. Intuitively, a measurable set A {\displaystyle A} is positive (resp. negative) for μ {\displaystyle \mu } if μ {\displaystyle \mu } is nonnegative (resp.
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https://en.wikipedia.org/wiki/Positive_and_negative_sets
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nonpositive) everywhere on A . {\displaystyle A.} Of course, if μ {\displaystyle \mu } is a nonnegative measure, every element of Σ {\displaystyle \Sigma } is a positive set for μ .
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https://en.wikipedia.org/wiki/Positive_and_negative_sets
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{\displaystyle \mu .} In the light of Radon–Nikodym theorem, if ν {\displaystyle \nu } is a σ-finite positive measure such that | μ | ≪ ν , {\displaystyle |\mu |\ll \nu ,} a set A {\displaystyle A} is a positive set for μ {\displaystyle \mu } if and only if the Radon–Nikodym derivative d μ / d ν {\displaystyle d\mu /d\nu } is nonnegative ν {\displaystyle \nu } -almost everywhere on A . {\displaystyle A.} Similarly, a negative set is a set where d μ / d ν ≤ 0 {\displaystyle d\mu /d\nu \leq 0} ν {\displaystyle \nu } -almost everywhere.
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https://en.wikipedia.org/wiki/Positive_and_negative_sets
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In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is an increasing sequence of σ {\displaystyle \sigma } -algebras on a measurable space. That is, given a measurable space ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} , a filtration is a sequence of σ {\displaystyle \sigma } -algebras { F t } t ≥ 0 {\displaystyle \{{\mathcal {F}}_{t}\}_{t\geq 0}} with F t ⊆ F {\displaystyle {\mathcal {F}}_{t}\subseteq {\mathcal {F}}} where each t {\displaystyle t} is a non-negative real number and t 1 ≤ t 2 ⟹ F t 1 ⊆ F t 2 . {\displaystyle t_{1}\leq t_{2}\implies {\mathcal {F}}_{t_{1}}\subseteq {\mathcal {F}}_{t_{2}}.} The exact range of the "times" t {\displaystyle t} will usually depend on context: the set of values for t {\displaystyle t} might be discrete or continuous, bounded or unbounded.
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https://en.wikipedia.org/wiki/Filtration_(algebra)
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For example, t ∈ { 0 , 1 , … , N } , N 0 , or {\mbox{ or }}[0,+\infty ).} Similarly, a filtered probability space (also known as a stochastic basis) ( Ω , F , { F t } t ≥ 0 , P ) {\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)} , is a probability space equipped with the filtration { F t } t ≥ 0 {\displaystyle \left\{{\mathcal {F}}_{t}\right\}_{t\geq 0}} of its σ {\displaystyle \sigma } -algebra F {\displaystyle {\mathcal {F}}} . A filtered probability space is said to satisfy the usual conditions if it is complete (i.e., F 0 {\displaystyle {\mathcal {F}}_{0}} contains all P {\displaystyle \mathbb {P} } -null sets) and right-continuous (i.e. F t = F t + := ⋂ s > t F s {\displaystyle {\mathcal {F}}_{t}={\mathcal {F}}_{t+}:=\bigcap _{s>t}{\mathcal {F}}_{s}} for all times t {\displaystyle t} ).It is also useful (in the case of an unbounded index set) to define F ∞ {\displaystyle {\mathcal {F}}_{\infty }} as the σ {\displaystyle \sigma } -algebra generated by the infinite union of the F t {\displaystyle {\mathcal {F}}_{t}} 's, which is contained in F {\displaystyle {\mathcal {F}}}: F ∞ = σ ( ⋃ t ≥ 0 F t ) ⊆ F .
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https://en.wikipedia.org/wiki/Filtration_(algebra)
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{\displaystyle {\mathcal {F}}_{\infty }=\sigma \left(\bigcup _{t\geq 0}{\mathcal {F}}_{t}\right)\subseteq {\mathcal {F}}.} A σ-algebra defines the set of events that can be measured, which in a probability context is equivalent to events that can be discriminated, or "questions that can be answered at time t {\displaystyle t} ". Therefore, a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information. A typical example is in mathematical finance, where a filtration represents the information available up to and including each time t {\displaystyle t} , and is more and more precise (the set of measurable events is staying the same or increasing) as more information from the evolution of the stock price becomes available.
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https://en.wikipedia.org/wiki/Filtration_(algebra)
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In measure theory, it is often useful to allow sets that have infinite measure and integrals whose value may be infinite. Such measures arise naturally out of calculus. For example, in assigning a measure to R {\displaystyle \mathbb {R} } that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering improper integrals, such as ∫ 1 ∞ d x x {\displaystyle \int _{1}^{\infty }{\frac {dx}{x}}} the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as f n ( x ) = { 2 n ( 1 − n x ) , if 0 ≤ x ≤ 1 n 0 , if 1 n < x ≤ 1 {\displaystyle f_{n}(x)={\begin{cases}2n(1-nx),&{\mbox{if }}0\leq x\leq {\frac {1}{n}}\\0,&{\mbox{if }}{\frac {1}{n}}
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https://en.wikipedia.org/wiki/Extended_real_number
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In measure theory, length is most often generalized to general sets of R n {\displaystyle \mathbb {R} ^{n}} via the Lebesgue measure. In the one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely, the length of an open interval is first defined as ℓ ( { x ∈ R ∣ a < x < b } ) = b − a . {\displaystyle \ell (\{x\in \mathbb {R} \mid a
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https://en.wikipedia.org/wiki/Length
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In measure theory, one talks about almost everywhere convergence of a sequence of measurable functions defined on a measurable space. That means pointwise convergence almost everywhere, that is, on a subset of the domain whose complement has measure zero. Egorov's theorem states that pointwise convergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set. Almost everywhere pointwise convergence on the space of functions on a measure space does not define the structure of a topology on the space of measurable functions on a measure space (although it is a convergence structure).
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https://en.wikipedia.org/wiki/Pointwise_convergence
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For in a topological space, when every subsequence of a sequence has itself a subsequence with the same subsequential limit, the sequence itself must converge to that limit. But consider the sequence of so-called "galloping rectangles" functions, which are defined using the floor function: let N = floor ( log 2 n ) {\displaystyle N=\operatorname {floor} \left(\log _{2}n\right)} and k = n {\displaystyle k=n} mod 2 N , {\displaystyle 2^{N},} and let Then any subsequence of the sequence ( f n ) n {\displaystyle \left(f_{n}\right)_{n}} has a sub-subsequence which itself converges almost everywhere to zero, for example, the subsequence of functions which do not vanish at x = 0. {\displaystyle x=0.} But at no point does the original sequence converge pointwise to zero. Hence, unlike convergence in measure and L p {\displaystyle L^{p}} convergence, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.
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https://en.wikipedia.org/wiki/Pointwise_convergence
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In measure theory, projection maps often appear when working with product (Cartesian) spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be measurable. Sometimes for some reasons product spaces are equipped with 𝜎-algebra different than the product 𝜎-algebra. In these cases the projections need not be measurable at all. The projected set of a measurable set is called analytic set and need not be a measurable set.
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https://en.wikipedia.org/wiki/Projection_(measure_theory)
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However, in some cases, either relatively to the product 𝜎-algebra or relatively to some other 𝜎-algebra, projected set of measurable set is indeed measurable. Henri Lebesgue himself, one of the founders of measure theory, was mistaken about that fact.
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https://en.wikipedia.org/wiki/Projection_(measure_theory)
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In a paper from 1905 he wrote that the projection of Borel set in the plane onto the real line is again a Borel set. The mathematician Mikhail Yakovlevich Suslin found that error about ten years later, and his following research has led to descriptive set theory. The fundamental mistake of Lebesgue was to think that projection commutes with decreasing intersection, while there are simple counterexamples to that.
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https://en.wikipedia.org/wiki/Projection_(measure_theory)
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In measure theory, tangent measures are used to study the local behavior of Radon measures, in much the same way as tangent spaces are used to study the local behavior of differentiable manifolds. Tangent measures (introduced by David Preiss in his study of rectifiable sets) are a useful tool in geometric measure theory. For example, they are used in proving Marstrand's theorem and Preiss' theorem.
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https://en.wikipedia.org/wiki/Tangent_measure
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In measure theory, the "problem of measure" for an n-dimensional Euclidean space Rn may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of Rn?" The work of Felix Hausdorff and Stefan Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution (because of the Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character" - the existence of a measure could be determined by looking at the properties of the transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, and is not solvable for higher dimensions.
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https://en.wikipedia.org/wiki/John_von_Neumann
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"Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space." Around 1942 he told Dorothy Maharam how to prove that every complete σ-finite measure space has a multiplicative lifting, however he did not publish this proof and she later came up with a new one.In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions.
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https://en.wikipedia.org/wiki/John_von_Neumann
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A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions". He proved this in the positive, and in later papers with Stone discussed various generalizations and algebraic aspects of this problem. He also proved by new methods the existence of disintegrations for various general types of measures.
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https://en.wikipedia.org/wiki/John_von_Neumann
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Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for compact groups. He had to create entirely new techniques to apply this to locally compact groups. He also gave a new, ingenious proof for the Radon–Nikodym theorem. His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.
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https://en.wikipedia.org/wiki/John_von_Neumann
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In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function.
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https://en.wikipedia.org/wiki/Euler_measure
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In measure theory, the Poisson point process can be further generalized to what is sometimes known as the general Poisson point process or general Poisson process by using a Radon measure Λ {\displaystyle \textstyle \Lambda } , which is a locally finite measure. In general, this Radon measure Λ {\displaystyle \textstyle \Lambda } can be atomic, which means multiple points of the Poisson point process can exist in the same location of the underlying space. In this situation, the number of points at x {\displaystyle \textstyle x} is a Poisson random variable with mean Λ ( x ) {\displaystyle \textstyle \Lambda ({x})} . But sometimes the converse is assumed, so the Radon measure Λ {\displaystyle \textstyle \Lambda } is diffuse or non-atomic.A point process N {\displaystyle \textstyle {N}} is a general Poisson point process with intensity Λ {\displaystyle \textstyle \Lambda } if it has the two following properties: the number of points in a bounded Borel set B {\displaystyle \textstyle B} is a Poisson random variable with mean Λ ( B ) {\displaystyle \textstyle \Lambda (B)} .
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https://en.wikipedia.org/wiki/Poisson_random_process
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In other words, denote the total number of points located in B {\displaystyle \textstyle B} by N ( B ) {\displaystyle \textstyle {N}(B)} , then the probability of random variable N ( B ) {\displaystyle \textstyle {N}(B)} being equal to n {\displaystyle \textstyle n} is given by: Pr { N ( B ) = n } = ( Λ ( B ) ) n n ! e − Λ ( B ) {\displaystyle \Pr\{N(B)=n\}={\frac {(\Lambda (B))^{n}}{n!
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https://en.wikipedia.org/wiki/Poisson_random_process
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}}e^{-\Lambda (B)}} the number of points in n {\displaystyle \textstyle n} disjoint Borel sets forms n {\displaystyle \textstyle n} independent random variables.The Radon measure Λ {\displaystyle \textstyle \Lambda } maintains its previous interpretation of being the expected number of points of N {\displaystyle \textstyle {N}} located in the bounded region B {\displaystyle \textstyle B} , namely Λ ( B ) = E . {\displaystyle \Lambda (B)=\operatorname {E} .} Furthermore, if Λ {\displaystyle \textstyle \Lambda } is absolutely continuous such that it has a density (which is the Radon–Nikodym density or derivative) with respect to the Lebesgue measure, then for all Borel sets B {\displaystyle \textstyle B} it can be written as: Λ ( B ) = ∫ B λ ( x ) d x , {\displaystyle \Lambda (B)=\int _{B}\lambda (x)\,\mathrm {d} x,} where the density λ ( x ) {\displaystyle \textstyle \lambda (x)} is known, among other terms, as the intensity function.
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https://en.wikipedia.org/wiki/Poisson_random_process
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In measure theory, the following implications hold between measures: So every probability measure is a sub-probability measure, but the converse is not true. Also every sub-probability measure is a finite measure and a σ-finite measure, but the converse is again not true.
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https://en.wikipedia.org/wiki/Sub-probability_measure
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In measure theory, we are not interested in semi-rings and rings themselves, but rather in σ-algebras generated by them. The idea is that it is possible to build a pre-measure on a semi-ring S {\displaystyle S} (for example Stieltjes measures), which can then be extended to a pre-measure on R ( S ) , {\displaystyle R(S),} which can finally be extended to a measure on a σ-algebra through Caratheodory's extension theorem. As σ-algebras generated by semi-rings and rings are the same, the difference does not really matter (in the measure theory context at least). Actually, Carathéodory's extension theorem can be slightly generalized by replacing ring by semi-field.The definition of semi-ring may seem a bit convoluted, but the following example shows why it is useful (moreover it allows us to give an explicit representation of the smallest ring containing some semi-ring).
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https://en.wikipedia.org/wiki/Carathéodory_extension_theorem
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In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes both Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. Lebesgue–Stieltjes integrals, named for Henri Leon Lebesgue and Thomas Joannes Stieltjes, are also known as Lebesgue–Radon integrals or just Radon integrals, after Johann Radon, to whom much of the theory is due. They find common application in probability and stochastic processes, and in certain branches of analysis including potential theory.
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https://en.wikipedia.org/wiki/Radon_integral
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In measure-theoretic probability theory, the density function is defined as the Radon–Nikodym derivative of the probability distribution relative to a common dominating measure. The likelihood function is this density interpreted as a function of the parameter, rather than the random variable. Thus, we can construct a likelihood function for any distribution, whether discrete, continuous, a mixture, or otherwise. (Likelihoods are comparable, e.g. for parameter estimation, only if they are Radon–Nikodym derivatives with respect to the same dominating measure.) The above discussion of the likelihood for discrete random variables uses the counting measure, under which the probability density at any outcome equals the probability of that outcome.
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https://en.wikipedia.org/wiki/Likelihood_ratio
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In measured data, congested traffic most often occurs in the vicinity of highway bottlenecks, e.g., on-ramps, off-ramps, or roadwork. A transition from free flow to congested traffic is known as traffic breakdown. In Kerner’s three-phase traffic theory traffic breakdown is explained by a phase transition from free flow to synchronized flow (called as F →S phase transition). This explanation is supported by available measurements, because in measured traffic data after a traffic breakdown at a bottleneck the downstream front of the congested traffic is fixed at the bottleneck. Therefore, the resulting congested traffic after a traffic breakdown satisfies the definition of the "synchronized flow" phase.
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https://en.wikipedia.org/wiki/Three_phase_traffic_theory
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In measured music, the terms arsis and thesis "are used respectively for unstressed and stressed beats or other equidistant subdivisions of the bar". Thus in music the terms are used in the opposite sense of poetry, with the arsis being the upbeat, or unstressed note preceding the downbeat. A fugue per arsin et thesin these days generally refers to one where one of the entries comes in with displaced accents (the formerly strong beats becoming weak and vice versa). An example is the bass line at bar 37 of no. 17 of Bach's Das Wohltemperierte Clavier. In the past, however, a fugue per arsin et thesin could also mean one where the theme was inverted.
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https://en.wikipedia.org/wiki/Arsis_and_thesis
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In measurement of the I/O performance of five filesystems with five storage configurations—single SSD, RAID 0, RAID 1, RAID 10, and RAID 5 it was shown that F2FS on RAID 0 and RAID 5 with eight SSDs outperforms EXT4 by 5 times and 50 times, respectively. The measurements also suggest that the RAID controller can be a significant bottleneck in building a RAID system with high speed SSDs.
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https://en.wikipedia.org/wiki/Standard_RAID_levels
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In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of known accuracy, a device generating the quantity to be measured such as a voltage, a sound tone, or a physical artifact, such as a meter ruler. The outcome of the comparison can result in one of the following: no significant error being noted on the device under test a significant error being noted but no adjustment made an adjustment made to correct the error to an acceptable levelStrictly speaking, the term "calibration" means just the act of comparison and does not include any subsequent adjustment. The calibration standard is normally traceable to a national or international standard held by a metrology body.
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https://en.wikipedia.org/wiki/Instrument_calibration
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In measurement, the Coggeshall slide rule, also called a carpenter's slide rule, was a slide rule designed by Henry Coggeshall in 1677 to help in measuring the dimensions, surface area, and volume of timber. With his original design and later improvements, Coggeshall's slide rule brought the tool its first practical use outside of mathematical study. It would remain popular for the next few centuries.The Coggeshall rule consisted of two rulers, each a foot (30 cm) long, which were put together in various ways. The most common and convenient arrangement was to have one of the rulers slide within a groove made along the middle of the other, like an ordinary linear slide rule, as shown in the figure below.
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https://en.wikipedia.org/wiki/Coggeshall_slide_rule
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Another form had one ruler sliding alongside the other, and a third form had a common two-foot folding ruler with a groove along one side in which a thin sliding piece was inserted that carried Coggeshall's lines. Coggeshall first described this apparatus in a paper he released in London titled, "Timber-measure by a line of more ease, dispatch and exactness, then any other way now in use, by a double scale: after the countrey-measure, by the length and quarter of the circumference in round timber, and by the length and side of the square in squared timber, and square equal in flat timber: as also stone-measure and gauging of vessels by the same near and exact way, likewise a diagonal scale of 100 parts in a quarter of an inch, very easie both to make and use.
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https://en.wikipedia.org/wiki/Coggeshall_slide_rule
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"After improving the design, he republished his work under the title "A Treatise of Measuring by a Two-foot Rule, which slides to a Foot" (1682). He released a highly modified version in 1722 titled "The Art of Practical Measuring easily performed by a Two-foot Rule which slides to a Foot." By 1767, seven revised editions had been released.
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https://en.wikipedia.org/wiki/Coggeshall_slide_rule
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In measurement, the term "linear foot" (sometimes incorrectly referred to as "lineal foot") refers to the number of feet in a length of material (such as lumber or fabric) without regard to the width; it is used to distinguish from surface area in square foot.
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https://en.wikipedia.org/wiki/Linear_feet
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In measurement, the total differential is used in estimating the error Δ f {\displaystyle \Delta f} of a function f {\displaystyle f} based on the errors Δ x , Δ y , … {\displaystyle \Delta x,\Delta y,\ldots } of the parameters x , y , … {\displaystyle x,y,\ldots } . Assuming that the interval is short enough for the change to be approximately linear: and that all variables are independent, then for all variables, This is because the derivative f x {\displaystyle f_{x}} with respect to the particular parameter x {\displaystyle x} gives the sensitivity of the function f {\displaystyle f} to a change in x {\displaystyle x} , in particular the error Δ x {\displaystyle \Delta x} . As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign.
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https://en.wikipedia.org/wiki/Total_differential
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From this principle the error rules of summation, multiplication etc. are derived, e.g.: That is to say, in multiplication, the total relative error is the sum of the relative errors of the parameters. To illustrate how this depends on the function considered, consider the case where the function is f ( a , b ) = a ln b {\displaystyle f(a,b)=a\ln b} instead. Then, it can be computed that the error estimate is with an extra 'ln b' factor not found in the case of a simple product. This additional factor tends to make the error smaller, as ln b is not as large as a bare b.
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https://en.wikipedia.org/wiki/Total_differential
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In measurements, the measurement obtained can suffer from two types of uncertainties. The first is the random uncertainty which is due to the noise in the process and the measurement. The second contribution is due to the systematic uncertainty which may be present in the measuring instrument. Systematic errors, if detected, can be easily compensated as they are usually constant throughout the measurement process as long as the measuring instrument and the measurement process are not changed.
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https://en.wikipedia.org/wiki/Random-fuzzy_variable
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But it can not be accurately known while using the instrument if there is a systematic error and if there is, how much? Hence, systematic uncertainty could be considered as a contribution of a fuzzy nature. This systematic error can be approximately modeled based on our past data about the measuring instrument and the process.
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https://en.wikipedia.org/wiki/Random-fuzzy_variable
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Statistical methods can be used to calculate the total uncertainty from both systematic and random contributions in a measurement. But, the computational complexity is very high and hence, are not desirable. L.A.Zadeh introduced the concepts of fuzzy variables and fuzzy sets.
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https://en.wikipedia.org/wiki/Random-fuzzy_variable
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Fuzzy variables are based on the theory of possibility and hence are possibility distributions. This makes them suitable to handle any type of uncertainty, i.e., both systematic and random contributions to the total uncertainty.Random-fuzzy variable (RFV) is a type 2 fuzzy variable, defined using the mathematical possibility theory, used to represent the entire information associated to a measurement result. It has an internal possibility distribution and an external possibility distribution called membership functions. The internal distribution is the uncertainty contributions due to the systematic uncertainty and the bounds of the RFV are because of the random contributions. The external distribution gives the uncertainty bounds from all contributions.
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https://en.wikipedia.org/wiki/Random-fuzzy_variable
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In measures of length, the quarter (qr.) was ¼ of a yard, formerly an important measure in the cloth trade. 3 qr. was a Flemish ell, 4 quarters were a yard, 5 qr.
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https://en.wikipedia.org/wiki/Quarter_(unit)
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was an (English) ell, and 6 qr. was an aune or French ell. Each quarter was made up of 4 nails. Its metric equivalent was formerly reckoned as about 0.228596 m, but the International Yard and Pound Agreement set it as 0.2286 exactly in 1959.
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https://en.wikipedia.org/wiki/Quarter_(unit)
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In measuring political social capital, it is common to take the sum of society's membership of its groups. Groups with higher membership (such as political parties) contribute more to the amount of capital than groups with lower membership, although many groups with low membership (such as communities) still add up to be significant. While it may seem that this is limited by population, this need not be the case as people join multiple groups.
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https://en.wikipedia.org/wiki/Social_capital
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In a study done by Yankee City, a community of 17,000 people was found to have over 22,000 different groups. How a group relates to the rest of society also affects social capital, but in a different manner. Strong internal ties can in some cases weaken the group's perceived capital in the eyes of the general public, as in cases where the group is geared towards crime, distrust, intolerance, violence or hatred towards others. The Ku Klux Klan is an example of this kind of organizations.
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https://en.wikipedia.org/wiki/Social_capital
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In measuring stereotype accuracy, researchers often assume that assessments are stable across time and situation. However, research based on the shifting standards model shows just the opposite, that stereotypes are unstable and depend largely on how the participant chooses a reference point for making their assessment.
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https://en.wikipedia.org/wiki/Shifting_standards_model
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In measuring the abundance and isotopic composition of CAS, it is important to know exactly what is being measured: CAS within particular shell fragments, corals, microbialites, cements, or otherwise. The first step is therefore to separate out the desired component for measurement. This could mean drilling and powdering a rock (if the CAS measurement of the whole rock is desired) or sorting sediments by visual identification of particular microfossils or mineral phases, using fine tweezers and drills under a microscope.
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https://en.wikipedia.org/wiki/Carbonate-associated_sulfate
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The fragments, sediments, or powders should be cleaned (likely by sonication) and exposed only to deionized and filtered water, so that no contaminant sulfur species are introduced, and the original CAS is not further reduced, oxidized, or otherwise altered. Next, the clean samples must be measured. In one method, these samples are "digested" in an acid, likely HCl, which will liberate CAS from inclusions or the mineral lattice by dissolving the calcite mineral.
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https://en.wikipedia.org/wiki/Carbonate-associated_sulfate
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The resulting sulfate ions are precipitated (often by mixture with barium chloride to produce barium sulfate), and the solid sulfate precipitate is filtered, dried, and transferred to an elemental analysis pipeline, which may involve the combustion of the sample and the mass balance of its various combustion products (which should include CO2 and SO2). Knowledge of the ratio of sulfur to oxygen and other components in the elemental analysis pipeline allows one to calculate the amount of sulfate introduced to the pipeline by the sample. This, along with the precise measurement of the original sample's mass and volume, yields a sulfate concentration for the original sample.
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https://en.wikipedia.org/wiki/Carbonate-associated_sulfate
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The "combustion" and reaction to SO2 can also bypassed by instead passing the acid-dissolved sample through an ion chromatography column, wherein different ions' polarity determines the strength of their interactions with polymers in the column, such that they are retained in the column for different amounts of time. The concentration of CAS may also be measured by spectroscopic methods. This could mean using the characteristic X-ray-induced fluorescence of sulfur, oxygen, carbon, and other elements in the sample to determine the abundance and ratios of each component, or the energy spectrum of an electron beam transmitted through the sample. It is also important to calibrate your measurement using standards of a known sulfate concentration, so that the strength/intensity of the signal associated with each sample can be mapped to a particular abundance.
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https://en.wikipedia.org/wiki/Carbonate-associated_sulfate
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In measuring unsaturation in fatty acids, the traditional method is the iodine number. Iodine adds stoichiometrically to double bonds, so their amount is reported in grams of iodine spent per 100 grams of oil. The standard unit is a dimensionless stoichiometry ratio of moles double bonds to moles fatty acid. A similar quantity, bromine number, is used in gasoline analysis.
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https://en.wikipedia.org/wiki/Cow's_Grass
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In pulp and paper industry, a similar kappa number is used to measure how much bleaching a pulp requires. Potassium permanganate is added to react with the unsaturated compounds (lignin and uronic acids) in the pulp and back-titrated. Originally with chlorine bleaching the required quantity of chlorine could be then calculated, although modern methods use multiple stages. Since the oxidizable compounds are not exclusively lignin and the partially pulped lignin does not have a single stoichiometry, the relation between the kappa number and the precise amount of lignin is inexact.
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https://en.wikipedia.org/wiki/Cow's_Grass
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In meat processing, biopreservation has been extensively studied in fermented meat products and ready to eat meat products. The use of native or artificially-introduced microbial population to improve animal health and productivity, and/or to reduce pathogenic organisms, has been termed a probiotic or competitive enhancement approach. Competitive enhancement strategies that have been developed include competitive exclusion, addition of a microbial supplement (probiotic) that improves gastrointestinal health, and adding a limiting, non-host digestible nutrient (prebiotic) that provides an existing (or introduced) commensal microbial population a competitive advantage in the gastrointestinal tract. Each of these approaches utilizes the activities of the native microbial ecosystem against pathogens by capitalizing on the natural microbial competition. Generally speaking, competitive enhancement strategies offer a natural 'green' method to reduce pathogens in the gut of food animals.
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https://en.wikipedia.org/wiki/Biopreservation
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In mechanical and poppet-based electropneumatic markers, the valve is usually designed to accommodate a specific operating pressure. Low pressure valves provide quieter operation and increased gas efficiency when tuned properly. However, excessively low pressure can decrease gas efficiency as dramatically as excessively high pressure.
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https://en.wikipedia.org/wiki/Paintball_marker
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Additionally, the valve must be set to release enough air to fire the paintball. If the valve is not tuned properly, insufficient air to fire the paintball may reach the bolt. This phenomenon, known as "shoot-down", causes fired paintballs to gradually lose range, and can also occur at high rates of fire. Some markers have integral or external chambers, called low-pressure chambers, which hold a large volume of gas behind the valve to prevent shoot-down. Tuning can also prevent air blowing up the feed tube upon firing, which disrupts the feeding of paintballs into the marker.
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https://en.wikipedia.org/wiki/Paintball_marker
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In mechanical design, the prototypical "male" component is a threaded bolt, but an alignment post, a mounting boss, or a sheet metal tab connector can also be considered as male. Correspondingly, a threaded nut, an alignment hole, a mounting recess, or sheet metal slot connector is considered to be female. While some mechanical designs are "one-off" custom setups not intended to be repeated, there is an entire fastener industry devoted to manufacturing mass-produced or semi-custom components. To avoid unnecessary confusion, conventional definitions of fastener gender have been defined and agreed upon.
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https://en.wikipedia.org/wiki/Gender_of_connectors_and_fasteners
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In mechanical engineering and materials science, a notch refers to a V-shaped, U-shaped, or semi-circular defect deliberately introduced into a planar material. In structural components, a notch causes a stress concentration which can result in the initiation and growth of fatigue cracks. Notches are used in materials characterization to determine fracture mechanics related properties such as fracture toughness and rates of fatigue crack growth. Notches are commonly used in material impact tests where a morphological crack of a controlled origin is necessary to achieve standardized characterization of fracture resistance of the material.
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https://en.wikipedia.org/wiki/Notch_(engineering)
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The most common is the Charpy impact test, which uses a pendulum hammer (striker) to strike a horizontal notched specimen. The height of its subsequent swing-through is used to determine the energy absorbed during fracture. The Izod impact strength test uses a circular notched vertical specimen in a cantilever configuration. Charpy testing is conducting with U- or V-notches whereby the striker contacts the specimen directly behind the notch, whereas the now largely obsolete Izod method involves a semi-circular notch facing the striker. Notched specimens are used in other characterization protocols, such as tensile and fatigue tests.
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https://en.wikipedia.org/wiki/Notch_(engineering)
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In mechanical engineering, Yoshimura buckling is a triangular mesh buckling pattern found in thin-walled cylinders under compression along the axis of the cylinder, producing a corrugated shape resembling the Schwarz lantern. The same pattern can be seen on the sleeves of Mona Lisa.This buckling pattern is named after Yoshimaru Yoshimura (吉村慶丸), the Japanese researcher who provided an explanation for its development in a paper first published in Japan in 1951, and later republished in the United States in 1955. Unknown to Yoshimura, the same phenomenon had previously been studied by Theodore von Kármán and Qian Xuesen in 1941.The crease pattern for folding the Schwarz lantern from a flat piece of paper, a tessellation of the plane by isosceles triangles, has also been called the Yoshimura pattern based on the same work by Yoshimura.
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https://en.wikipedia.org/wiki/Yoshimura_buckling
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The Yoshimura creasing pattern is related to both the Kresling and Hexagonal folds, and can be framed as a special case of the Miura fold. Unlike the Miura fold which is rigidly deformable, both the Yoshimura and Kresling patterns require panel deformation to be folded to a compact state. == References ==
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https://en.wikipedia.org/wiki/Yoshimura_buckling
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In mechanical engineering, a bonded seal is a type of washer used to provide a seal around a screw or bolt. Originally made by Dowty Group, they are also known as Dowty seals or Dowty washers. Now widely manufactured, they are available in a range of standard sizes and materials A bonded seal consists of an outer annular ring of a hard material, typically steel, and an inner annular ring of an elastomeric material that acts as a gasket. It is the compression of the elastomeric part between the faces of the parts on either side of the bonded seal that provides the sealing action.
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https://en.wikipedia.org/wiki/Bonded_seal
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The elastomeric material, typically nitrile rubber, is bonded by heat and pressure to the outer ring, which holds it in place. This structure increases resistance to bursting, increasing the pressure rating of the seal. Because the bonded seal itself acts to retain the gasket material, there is no need for the parts to be sealed to be shaped to retain the gasket.
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https://en.wikipedia.org/wiki/Bonded_seal
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This results in simplified machining and greater ease of use as compared to some other seals, such as O-rings. Some designs come with an additional flap of rubber on the internal diameter to locate the bonded seal at the centre of the hole; these are called self-centring bonded washers. == References ==
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https://en.wikipedia.org/wiki/Bonded_seal
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In mechanical engineering, a cam follower, also known as a track follower, is a specialized type of roller or needle bearing designed to follow cam lobe profiles. Cam followers come in a vast array of different configurations, however the most defining characteristic is how the cam follower mounts to its mating part; stud style cam followers use a stud while the yoke style has a hole through the middle.
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https://en.wikipedia.org/wiki/Cam_follower
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In mechanical engineering, a compliant mechanism is a flexible mechanism that achieves force and motion transmission through elastic body deformation. It gains some or all of its motion from the relative flexibility of its members rather than from rigid-body joints alone. These may be monolithic (single-piece) or jointless structures. Some common devices that use compliant mechanisms are backpack latches and paper clips. One of the oldest examples of using compliant structures is the bow and arrow.
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https://en.wikipedia.org/wiki/Compliant_mechanism
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In mechanical engineering, a compression seal fitting, also known as a sealing gland, is intended to seal some type of element (probe, wire, conductor, pipe, tube, fiber optic cable, etc.) when the element must pass through a pressure or environmental boundary. A compression seal fitting may serve several purposes: It restrains the element from moving as a result of a pressure difference. It prohibits the leakage of gas or liquid media along the element. In some cases, it electrically isolates the element from the mounting device.A compression seal fitting, unlike an epoxy seal or gasket, uses mechanical components and an axial force to compress a soft sealant inside a body which then creates a seal. An epoxy seal differs in that it is composed of some type of compound which is poured into a mold in an attempt to create a seal.
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https://en.wikipedia.org/wiki/Compression_seal_fitting
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In mechanical engineering, a crosshead is a mechanical joint used as part of the slider-crank linkages of long reciprocating engines (either internal combustion or steam) and reciprocating compressors to eliminate sideways force on the piston. Also, the crosshead enables the connecting rod to freely move outside the cylinder. Because of the very small bore-to-stroke ratio on such engines, the connecting rod would hit the cylinder walls and block the engine from rotating if the piston was attached directly to the connecting rod like on trunk engines. Therefore, the longitudinal dimension of the crosshead must be matched to the stroke of the engine.
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https://en.wikipedia.org/wiki/Crosshead_guide
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In mechanical engineering, a diaphragm seal is a flexible membrane that seals and isolates an enclosure. The flexible nature of this seal allows pressure effects to cross the barrier but not the material being contained. Common uses for diaphragm seals are to protect pressure sensors from the fluid whose pressure is being measured.
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https://en.wikipedia.org/wiki/Diaphragm_seal
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In mechanical engineering, a docking sleeve or mounting boss is a tube or enclosure used to couple two mechanical components together, or for chilling, or to retain two components together; this permits two equally sized appendages to be connected via insertion and fixing within the construction. Docking sleeves may be physically solid or flexible, their implementation varying widely according to the required application of the device. The most common application is the plastic appendage that receives a screw in order to attach two parts. == References ==
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https://en.wikipedia.org/wiki/Docking_sleeve
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In mechanical engineering, a face seal is a seal in which the sealing surfaces are normal to the axis of the seal. Face seals are typically used in static applications and are used to prevent leakage in the radial direction with respect to the axis of the seal. Face seals are often located in a groove or cavity on a flange.
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https://en.wikipedia.org/wiki/Face_seal
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