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wiki_7600_chunk_0 | Mixed models | In matrix notation a linear mixed model can be represented as y = X β + Z u + ϵ {\displaystyle {\boldsymbol {y}}=X{\boldsymbol {\beta }}+Z{\boldsymbol {u}}+{\boldsymbol {\epsilon }}} where y {\displaystyle {\boldsymbol {y}}} is a known vector of observations, with mean E ( y ) = X β {\displaystyle E({\boldsymbol {y}})=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7601_chunk_0 | Multinomial distribution | In matrix notation, E ( X ) = n p , {\displaystyle \operatorname {E} (\mathbf {X} )=n\mathbf {p} ,\,} and Var ( X ) = n { diag ( p ) − p p T } , {\displaystyle \operatorname {Var} (\mathbf {X} )=n\lbrace \operatorname {diag} (\mathbf {p} )-\mathbf {p} \mathbf {p} ^{\rm {T}}\rbrace ,\,} with pT = the row vector tr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7602_chunk_0 | Multivariate Pólya distribution | In matrix notation, E ( X ) = n p , {\displaystyle \operatorname {E} (\mathbf {X} )=n\mathbf {p} ,\,} and var ( X ) = n { diag ( p ) − p p T } ( n + α 0 1 + α 0 ) , {\displaystyle \operatorname {var} (\mathbf {X} )=n\lbrace \operatorname {diag} (\mathbf {p} )-\mathbf {p} \mathbf {p} ^{\rm {T}}\rbrace \left({\frac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7603_chunk_0 | Sylvester's determinant identity | In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851.Given an n-by-n matrix A {\displaystyle A} , let det ( A ) {\displaystyle \det(A)} denote its determinant. Choos... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7604_chunk_0 | Frobenius covariant | In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices Ai associated with the eigenvalues and eigenvectors of A.: pp.403, 437–8 They are named after the mathematician Ferdinand Frobenius. Each covariant is a projection on the eigenspace associated with ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7605_chunk_0 | Perron–Frobenius theorem | In matrix theory, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real. The corresponding eigenvector can be chosen to have strictly positive components, an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7606_chunk_0 | Rule of Sarrus | In matrix theory, the Rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French mathematician Pierre Frédéric Sarrus.Consider a 3 × 3 {\displaystyle 3\times 3} matrix M = , {\displaystyle M={\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7607_chunk_0 | Rule of Sarrus | {\displaystyle {\begin{aligned}\det(M)&=\det {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}\\&=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.\end{aligned}}} A similar scheme based on diagonals works for 2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7608_chunk_0 | Single cell analysis | In matrix-assisted laser desorption and ionization (MALDI), the sample is incorporated in a chemical matrix that is capable of absorbing energy from a laser. Similar to SIMS, ionization happens in vacuum. Laser irradiation ablates the matrix material from the surface and results in charged gas phase matrix particles, t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7609_chunk_0 | Sylvester matroid | In matroid theory, a Sylvester matroid is a matroid in which every pair of elements belongs to a three-element circuit (a triangle) of the matroid. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7610_chunk_0 | Binary matroid | In matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2). That is, up to isomorphism, they are the matroids whose elements are the columns of a (0,1)-matrix and whose sets of elements are independent if and only if the corresponding columns are linearly independent in GF(2). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7611_chunk_0 | Gammoid | In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph. The concept of a gammoid was introduced and shown to be a matroid by Hazel Perfect (1968), based on considerations related to Menger's theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7612_chunk_0 | Matroid girth | In matroid theory, a mathematical discipline, the girth of a matroid is the size of its smallest circuit or dependent set. The cogirth of a matroid is the girth of its dual matroid. Matroid girth generalizes the notion of the shortest cycle in a graph, the edge connectivity of a graph, Hall sets in bipartite graphs, ev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7613_chunk_0 | Closure (mathematics) | In matroid theory, the closure of X is the largest superset of X that has the same rank as X. The transitive closure of a set. The algebraic closure of a field. The integral closure of an integral domain in a field that contains it. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7614_chunk_0 | Closure (mathematics) | The radical of an ideal in a commutative ring. In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset. In formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language. In g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7615_chunk_0 | Gate teleportation | The main components needed for teleportation include a sender, the information (a qubit), a traditional channel, a quantum channel, and a receiver. An interesting fact is that the sender does not need to know the exact contents of the information that is being sent. The measurement postulate of quantum mechanics—when a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7616_chunk_0 | Gate teleportation | For actual teleportation, it is required that an entangled quantum state or Bell state be created for the qubit to be transferred. Entanglement imposes statistical correlations between otherwise distinct physical systems by creating or placing two or more separate particles into a single, shared quantum state. This int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7617_chunk_0 | Gate teleportation | Any changes that one particle of the entanglement undergoes, the other particle will also undergo that change, causing the entangled particles to act as one quantum state. These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7618_chunk_0 | Gate teleportation | Thus, an observation resulting from a measurement choice made at one point in spacetime seems to instantaneously affect outcomes in another region, even though light hasn't yet had time to travel the distance; a conclusion seemingly at odds with special relativity. This is known as the EPR paradox. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7619_chunk_0 | Gate teleportation | However such correlations can never be used to transmit any information faster than the speed of light, a statement encapsulated in the no-communication theorem. Thus, teleportation as a whole can never be superluminal, as a qubit cannot be reconstructed until the accompanying classical information arrives. The sender ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7620_chunk_0 | Gate teleportation | Of this changed state, the particles in Alice's possession are then sent to an analyzer that will measure the change of the entangled state. The "change" measurement will allow the receiver to recreate the original information that the sender had resulting in the information being teleported or carried between two peop... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7621_chunk_0 | Gate teleportation | The quantum channel is the communication mechanism that is used for all quantum information transmission and is the channel used for teleportation (relationship of quantum channel to traditional communication channel is akin to the qubit being the quantum analog of the classical bit). However, in addition to the quantu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7622_chunk_0 | Gate teleportation | Because of this need for the traditional channel, the speed of teleportation can be no faster than the speed of light (hence the no-communication theorem is not violated). The main advantage with this is that Bell states can be shared using photons from lasers making teleportation achievable through open space having n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7623_chunk_0 | Gate teleportation | For example, qubits can be encoded in the degrees of freedom of electrons surrounding the atomic nucleus or in the degrees of freedom of the nucleus itself. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them.As of 2015, the quantum ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7624_chunk_0 | Glycine receptor | In mature adults, glycine is a inhibitory neurotransmitter found in the spinal cord and regions of the brain. As it binds to a glycine receptor, a conformational change is induced, and the channel created by the receptor opens. As the channel opens, chloride ions are able to flow into the cell which results in hyperpol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7625_chunk_0 | Glycine receptor | This is called the "shunting" effect and can be explained by Ohm's Law. As the receptor is activated, the membrane conductance is increased and the membrane resistance is decreased. According to Ohm's Law, as resistance decreases, so does voltage. A decreased postsynaptic voltage results in a decreased release of neuro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7626_chunk_0 | Dollo's law of irreversibility | In maximum parsimony, Dollo parsimony refers to a model whereby a character is gained only one time and can never be regained if it is lost. For example, the evolution and repeated loss of teeth in vertebrates could be well-modeled under Dollo parsimony, whereby teeth made from hydroxyapatite evolved only once at the o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7627_chunk_0 | Odiogo | In may 2005 Swedish Radio first published podcast for the listerners to download. The first podcasts offered were Ekots lördagsintervju i P1, Spanarna i P1 and Salva i P3 but shortly after the offer was expanded to include several other titles. In February 2006, following London radio station LBC's successful launch of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7628_chunk_0 | Odiogo | The second series of the podcast was distributed through audible.co.uk and was the first major podcast to charge consumers to download the show (at a rate of 95 pence per half-hour episode). The first series of The Ricky Gervais Show podcast had been freely distributed by the Positive Internet Company and marketed thro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7629_chunk_0 | Odiogo | In July 2009, the company VoloMedia is awarded the "Podcast patent" by the USPTO in patent number 7,568,213. Dave Winer, the co-inventor of podcasting (with Adam Curry), points out that his invention predated this patent by two years.On February 2, 2006, Virginia Tech (Virginia Polytechnic Institute and State Universit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7630_chunk_0 | Mean-field approximation | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7631_chunk_0 | Ditone | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7632_chunk_0 | Prokhorov's theorem | 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 "Prokhor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7633_chunk_0 | Monotone class theorem | 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 inductio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7634_chunk_0 | Carathéodory's extension theorem | 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. Consequent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7635_chunk_0 | Bounded convergence theorem | 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 add... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7636_chunk_0 | Ham-sandwich theorem | 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 conti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7637_chunk_0 | Ham-sandwich theorem | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7638_chunk_0 | Kakutani's theorem (measure theory) | 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-measu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7639_chunk_0 | Continuity set | 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)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7640_chunk_0 | Lebesgue measure | 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 d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7641_chunk_0 | Push-forward measure | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7642_chunk_0 | Radonifying function | 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 Rad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7643_chunk_0 | Egorov's Theorem | 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 p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7644_chunk_0 | Positive and negative sets | {\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\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7645_chunk_0 | Filtration (algebra) | 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 {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7646_chunk_0 | Filtration (algebra) | 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 equipp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7647_chunk_0 | Filtration (algebra) | {\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 {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7648_chunk_0 | Extended real number | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7649_chunk_0 | Pointwise convergence | 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 everyw... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7650_chunk_0 | Pointwise convergence | 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 )... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7651_chunk_0 | Projection (measure theory) | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7652_chunk_0 | Projection (measure theory) | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7653_chunk_0 | Projection (measure theory) | 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 pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7654_chunk_0 | Tangent measure | 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 theor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7655_chunk_0 | John von Neumann | 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 so... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7656_chunk_0 | John von Neumann | "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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7657_chunk_0 | John von Neumann | 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 measurabl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7658_chunk_0 | John von Neumann | 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 lectur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7659_chunk_0 | Euler measure | In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7660_chunk_0 | Poisson random process | 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 \textstyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7661_chunk_0 | Poisson random process | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7662_chunk_0 | Poisson random process | }}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 \texts... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7663_chunk_0 | Carathéodory extension theorem | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7664_chunk_0 | Radon integral | 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 wit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7665_chunk_0 | Likelihood ratio | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7666_chunk_0 | Arsis and thesis | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7667_chunk_0 | Instrument calibration | 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7668_chunk_0 | Total differential | 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 c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7669_chunk_0 | Total differential | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7670_chunk_0 | Random-fuzzy variable | 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 det... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7671_chunk_0 | Random-fuzzy variable | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7672_chunk_0 | Random-fuzzy variable | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7673_chunk_0 | Random-fuzzy variable | 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 possi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7674_chunk_0 | Shifting standards model | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7675_chunk_0 | Carbonate-associated sulfate | 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 powderi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7676_chunk_0 | Carbonate-associated sulfate | 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, thes... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7677_chunk_0 | Carbonate-associated sulfate | 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. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7678_chunk_0 | Cow's Grass | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7679_chunk_0 | Cow's Grass | 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 ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7680_chunk_0 | Paintball marker | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7681_chunk_0 | Notch (engineering) | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7682_chunk_0 | Notch (engineering) | 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 configur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7683_chunk_0 | Bonded seal | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7684_chunk_0 | Compliant mechanism | 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 joint... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7685_chunk_0 | Compression seal fitting | 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7686_chunk_0 | Crosshead guide | 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 cy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7687_chunk_0 | Fillet (mechanics) | In mechanical engineering, a fillet is a rounding of an interior or exterior corner of a part designed in CAD. An interior or exterior corner, with an angle or type of bevel, is called a "chamfer". Fillet geometry, when on an interior corner is a line of concave function, whereas a fillet on an exterior corner is a lin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7688_chunk_0 | Fillet (mechanics) | Depending on a geometric modelling kernel different CAD software products may provide different fillet functionality. Usually fillets can be quickly designed onto parts using 3D solid modeling engineering by picking edges of interest and invoking the function. Smooth edges connecting two simple flat features are genera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7689_chunk_0 | Jaw coupling | In mechanical engineering, a jaw coupling is a type of general purpose power transmission coupling that also can be used in motion control (servo) applications. It is designed to transmit torque (by connecting two shafts) while damping system vibrations and accommodating misalignment, which protects other components fr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7690_chunk_0 | Key (engineering) | In mechanical engineering, a key is a machine element used to connect a rotating machine element to a shaft. The key prevents relative rotation between the two parts and may enable torque transmission. For a key to function, the shaft and rotating machine element must have a keyway and a keyseat, which is a slot and po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7691_chunk_0 | Kinematic chain | In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained motion that is the mathematical model for a mechanical system. As the word chain suggests, the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7692_chunk_0 | Kinematic chain | These joints are generally modeled as holonomic constraints. A kinematic diagram is a schematic of the mechanical system that shows the kinematic chain. The modern use of kinematic chains includes compliance that arises from flexure joints in precision mechanisms, link compliance in compliant mechanisms and micro-elect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7693_chunk_0 | Parallel force system | In mechanical engineering, a parallel force system is a situation in which two forces of equal magnitude act in the same direction within the same plane, with the counter force in the middle. An example of this is a see saw. The children are applying the two forces at the ends, and the fulcrum in the middle gives the c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7694_chunk_0 | Rolling-element bearings | In mechanical engineering, a rolling-element bearing, also known as a rolling bearing, is a bearing which carries a load by placing rolling elements (such as balls or rollers) between two concentric, grooved rings called races. The relative motion of the races causes the rolling elements to roll with very little rollin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7695_chunk_0 | Rolling-element bearings | Rolling-element bearings have the advantage of a good trade-off between cost, size, weight, carrying capacity, durability, accuracy, friction, and so on. Other bearing designs are often better on one specific attribute, but worse in most other attributes, although fluid bearings can sometimes simultaneously outperform ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7696_chunk_0 | Eccentric (mechanism) | In mechanical engineering, an eccentric is a circular disk (eccentric sheave) solidly fixed to a rotating axle with its centre offset from that of the axle (hence the word "eccentric", out of the center).It is used most often in steam engines, and used to convert rotary motion into linear reciprocating motion to drive ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7697_chunk_0 | Overconstrained mechanism | In mechanical engineering, an overconstrained mechanism is a linkage that has more degrees of freedom than is predicted by the mobility formula. The mobility formula evaluates the degree of freedom of a system of rigid bodies that results when constraints are imposed in the form of joints between the links. If the link... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7698_chunk_0 | Overconstrained mechanism | If the links in the system move planes parallel to a fixed plane, or in concentric spheres about a fixed point, then the mobility formula is M = 3 ( N − 1 − j ) + ∑ i = 1 j f i . {\displaystyle M=3(N-1-j)+\sum _{i=1}^{j}f_{i}.} If a system of links and joints has mobility M = 0 or less, yet still moves, then it is call... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7699_chunk_0 | Backlash (engineering) | In mechanical engineering, backlash, sometimes called lash, play, or slop, is a clearance or lost motion in a mechanism caused by gaps between the parts. It can be defined as "the maximum distance or angle through which any part of a mechanical system may be moved in one direction without applying appreciable force or ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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