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trion is a basic unit of the neural network model of cortical organization called trion model. this unit represents a highly structured and interconnected aggregate of about a hundred of neurons with the overall diameter of about 0. 7 mm. each trion has three levels of firing activity, and thus a cluster of trions can produce a complex firing pattern which changes rapidly ( millisecond scale ) according to probabilistic ( monte carlo ) rules. = = references = =
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Trion (neural networks)
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u = h u, u † ( t ) = u − 1 ( t ), u ( 0 ) = 1. { \ displaystyle i \ hbar { \ dot { u } } = hu \,, \ quad u ^ { \ dagger } ( t ) = u ^ { - 1 } ( t ) \,, \ quad u ( 0 ) = 1 \,. } alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the hamiltonian, just as the corresponding poisson brackets in classical theory must vanish in order to generate the correct hamilton equations. the formal solution of the field equation is : a k λ ( t ) = a k λ ( 0 ) e − i ω k t + i e 2 π ω k v 0 t d t ′ e k λ ⋅ x ( t ′ ) e i ω k ( t ′ − t ) { \ displaystyle a _ { \ mathbf { k } \ lambda } ( t ) = a _ { \ mathbf { k } \ lambda } ( 0 ) e ^ { - i \ omega _ { k } t } + ie { \ sqrt { \ frac { 2 \ pi } { \ hbar \ omega _ { k } v } } } \ int _ { 0 } ^ { t } dt'\, e _ { \ mathbf { k } \ lambda } \ cdot \ mathbf { \ dot { x } } ( t') e ^ { i \ omega _ { k } \ left ( t'- t \ right ) } } and therefore the equation for akλ may be written : x ¨ + ω 0 2 x = e m e 0 ( t ) + e m e r r ( t ) { \ displaystyle \ mathbf { \ ddot { x } } + \ omega _ { 0 } ^ { 2 } \ mathbf { x } = { \ frac { e } { m } } \ mathbf { e } _ { 0 } ( t ) + { \ frac { e } { m } } \ mathbf { e } _ { rr } ( t ) } where e 0 ( t ) = i k λ 2 π ω k v [ a k λ ( 0 ) e − i ω k t − a k λ † ( 0 ) e i ω k t ] e k λ { \ displaystyle \ mathbf { e } _ { 0
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Zero-point energy
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2nd rev. and enlarged ed. ). berlin heidelberg paris [ etc. ] : springer. isbn 978 - 3 - 540 - 67723 - 9. berggren, karl - fredrik ; °aberg, sven, eds. ( 2001 ). quantum chaos y2k : proceedings of nobel symposium 116, backaskog castle, sweden, june 13 - 17, 2000. stockholm, sweden : physica scripta, the royal swedish academy of sciences. isbn 978 - 981 - 02 - 4711 - 9. reichl, linda e. ( 2004 ). the transition to chaos : conservative classical systems and quantum manifestations. institute for nonlinear science ( 2. [ new ] ed. ). new york heidelberg : springer. isbn 978 - 0 - 387 - 98788 - 0. = = external links = = quantum chaos by martin gutzwiller ( 1992 and 2008, scientific american ) quantum chaos martin gutzwiller scholarpedia 2 ( 12 ) : 3146. doi : 10. 4249 / scholarpedia. 3146 category : quantum chaos scholarpedia what is... quantum chaos by ze'ev rudnick ( january 2008, notices of the american mathematical society ) brian hayes, " the spectrum of riemannium " ; american scientist volume 91, number 4, july – august, 2003 pp. 296 – 300. discusses relation to the riemann zeta function. eigenfunctions in chaotic quantum systems by arnd backer. chaosbook. org
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Quantum chaos
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physical conditions which are within the site, are not weather conditions and which an experienced contractor would have judged at the contract date to have such a small chance of occurring that it would have been unreasonable for her / him to have allowed for them. ice conditions of contract sixth edition clause 12 ( 1 ) if during the execution of the works the contractor shall encounter physical conditions ( other than weather conditions or conditions due to weather conditions ) or artificial obstructions which conditions or obstructions could not in her / his opinion reasonably have been foreseen by an experienced contractor the contractor shall as early as practicable give written notice thereof to the engineer. = = z clauses = = employers often use additional conditions of contract ( " z clauses " ) to amend or delete contract provisions relating to certain obligations, and the efficiency and reform group of the cabinet office in the uk ( formerly the ogc ) has published generic public sector z clauses for use with nec contracts. a standard z clause relating to fair payment for sub - contractors ( often labelled " z5 " ) was recommended for public sector use in 2011, and additional public sector z clauses were later published to reflect the contract termination provisions and other requirements of the public contracts regulations 2015. excessive use of z clauses has been criticised as " onerous " and " poorly drafted " ; nec guidance states that " additional conditions should be used only when absolutely necessary to accommodate special needs ". = = guidance notes and further information = = guidance notes and flow charts are published by the ice, which are supplemented by the frequently asked questions sections of the nec website. prospective users of the nec3 contract are encouraged to study the faq's in order to avoid unintended contract provisions. the often unintended option c scenario where a contractor is paid monies in excess of the target cost plus maximum share provisions is specifically not addressed in the guidance notes or frequently asked questions. other common misinterpretations are minutes of meetings as communications, deleted work and paying for correcting defects. = = footnotes = = = = external links = = nec engineering and construction contract website free nec3 & nec4 guidance notes, flow charts & publications ne consult
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New Engineering Contract
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products of hilbert spaces and the algebras acting on them. von neumann, j. ( 1940 ), " on rings of operators iii ", annals of mathematics, second series, 41 ( 1 ) : 94 – 161, doi : 10. 2307 / 1968823, jstor 1968823. this shows the existence of factors of type iii. von neumann, j. ( 1943 ), " on some algebraical properties of operator rings ", annals of mathematics, second series, 44 ( 4 ) : 709 – 715, doi : 10. 2307 / 1969106, jstor 1969106. this shows that some apparently topological properties in von neumann algebras can be defined purely algebraically. von neumann, j. ( 1949 ), " on rings of operators. reduction theory ", annals of mathematics, second series, 50 ( 2 ) : 401 – 485, doi : 10. 2307 / 1969463, jstor 1969463. this discusses how to write a von neumann algebra as a sum or integral of factors. von neumann, john ( 1961 ), taub, a. h. ( ed. ), collected works, volume iii : rings of operators, ny : pergamon press. reprints von neumann's papers on von neumann algebras. wassermann, a. j. ( 1991 ), operators on hilbert space
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Finite von Neumann algebra
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it has become possible to analyze large datasets with greater accuracy and efficiency. however, method also has its own limitations, such as the lack of training data, imbalanced data, and overfitting. = = future directions = = combining various experimental and computational techniques can provide comprehensive insights into complex structures. integrating data from x - ray crystallography, nmr spectroscopy, and computational modeling enhances accuracy and reliability. continued progress in computational simulations, including quantum chemistry and molecular dynamics, will allow researchers to study larger and more complex systems, aiding in predicting and understanding novel structures. open - access databases and collaborative efforts enable researchers worldwide to share structural data, accelerating scientific progress and fostering innovation. structural chemistry can contribute to the design of eco - friendly materials and catalysts, promoting sustainable practices in the chemical industry. structural chemistry can contribute to the design of eco - friendly materials and catalysts, promoting sustainable practices in the chemical industry. recent development of metal - free nanostructured catalysts is one of the advancements in the field of structural chemistry that has the potential to drive organic transformations in a sustainable manner. = = see also = = chemical structure = = references = =
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Structural chemistry
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k ), methionine ( met, m ), phenylalanine ( phe, f ), proline ( pro, p ), serine ( ser, s ), threonine ( thr, t ), tryptophan ( trp, w ), tyrosine ( tyr, y ), and valine ( val, v ). = = differences from the standard code = = = = see also = = list of all genetic codes : translation tables 1 to 16, and 21 to 31. the genetic codes database. = = references = = this article incorporates text from the united states national library of medicine, which is in the public domain.
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Karyorelict nuclear code
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′ ∗ = n a n ⟨ n ′ | h | n ⟩ = ⟨ n ′ | h | ψ ⟩ { \ displaystyle { \ frac { \ partial \ langle h \ rangle } { \ partial a _ { n'} ^ { * } } } = \ sum _ { n } a _ { n } \ langle n'| h | n \ rangle = \ langle n'| h | \ psi \ rangle } by applying the schrodinger equation and using the orthonormality of the basis states, this further reduces to ∂ ⟨ h ⟩ ∂ a n ′ ∗ = i ∂ a n ′ ∂ t { \ displaystyle { \ frac { \ partial \ langle h \ rangle } { \ partial a _ { n'} ^ { * } } } = i \ hbar { \ frac { \ partial a _ { n'} } { \ partial t } } } similarly, one can show that ∂ ⟨ h ⟩ ∂ a n = − i ∂ a n ∗ ∂ t { \ displaystyle { \ frac { \ partial \ langle h \ rangle } { \ partial a _ { n } } } = - i \ hbar { \ frac { \ partial a _ { n } ^ { * } } { \ partial t } } } if we define " conjugate momentum " variables π n { \ displaystyle \ pi _ { n } } by π n ( t ) = i a n ∗ ( t ) { \ displaystyle \ pi _ { n } ( t ) = i \ hbar a _ { n } ^ { * } ( t ) } then the above equations become ∂ ⟨ h ⟩ ∂ π n = ∂ a n ∂ t, ∂ ⟨ h ⟩ ∂ a n = − ∂ π n ∂ t { \ displaystyle { \ frac { \ partial \ langle h \ rangle } { \ partial \ pi _ { n } } } = { \ frac { \ partial a _ { n } } { \ partial t } }, \ quad { \ frac { \ partial \ langle h \ rangle } { \ partial a _ { n } } } = - { \ frac { \ partial \ pi _ { n } } { \ partial t } } } which is precisely the form of hamilton's equations, with the a n { \ displaystyle a _ { n } } s as
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Hamiltonian (quantum mechanics)
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computing : progress and prospects. washington, dc : the national academies press. doi : 10. 17226 / 25196. isbn 978 - 0 - 309 - 47970 - 7. oclc 1091904777. s2cid 125635007. mermin, n. david ( 2007 ). quantum computer science : an introduction. doi : 10. 1017 / cbo9780511813870. isbn 978 - 0 - 511 - 34258 - 5. oclc 422727925. nielsen, michael ; chuang, isaac ( 2010 ). quantum computation and quantum information ( 10th anniversary ed. ). doi : 10. 1017 / cbo9780511976667. isbn 978 - 0 - 511 - 99277 - 3. oclc 700706156. s2cid 59717455. shor, peter w. ( 1994 ). algorithms for quantum computation : discrete logarithms and factoring. symposium on foundations of computer science. santa fe, new mexico : ieee. pp. 124 – 134. doi : 10. 1109 / sfcs. 1994. 365700. isbn 978 - 0 - 8186 - 6580 - 6. = = further reading = = = = external links = = media related to quantum computer at wikimedia commons learning materials related to quantum computing at wikiversity stanford encyclopedia of philosophy : " quantum computing " by amit hagar and michael e. cuffaro. " quantum computation, theory of ", encyclopedia of mathematics, ems press, 2001 [ 1994 ] introduction to quantum computing for business by koen groenland lectures quantum computing for the determined – 22 video lectures by michael nielsen video lectures by david deutsch lomonaco, sam. four lectures on quantum computing given at oxford university in july 2006
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Quantum process
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of sequencing theory has proceeded more according to the philosophy of applied mathematics. in particular, it has been problem - focused and makes expedient use of approximations, simulations, etc. = = = early uses derived from elementary probability theory = = = the earliest result may be found directly from elementary probability theory. suppose we model the above process taking l { \ displaystyle l } and g { \ displaystyle g } as the fragment length and target length, respectively. the probability of " covering " any given location on the target with one particular fragment is then l / g { \ displaystyle l / g }. ( this presumes l g { \ displaystyle l \ ll g }, which is valid often, but not for all real - world cases. ) the probability of a single fragment not covering a given location on the target is therefore 1 − l / g { \ displaystyle 1 - l / g }, and [ 1 − l / g ] n { \ displaystyle \ left [ 1 - l / g \ right ] ^ { n } } for n { \ displaystyle n } fragments. the probability of covering a given location on the target with at least one fragment is therefore p = 1 − [ 1 − l g ] n. { \ displaystyle p = 1 - \ left [ 1 - { \ frac { l } { g } } \ right ] ^ { n }. } this equation was first used to characterize plasmid libraries, but it may appear in a modified form. for most projects n 1 { \ displaystyle n \ gg 1 }, so that, to a good degree of approximation [ 1 − l g ] n exp ( − n l / g ), { \ displaystyle \ left [ 1 - { \ frac { l } { g } } \ right ] ^ { n } \ sim \ exp ( - nl / g ), } where r = n l / g { \ displaystyle r = nl / g } is called the redundancy. note the significance of redundancy as representing the average number of times a position is covered with fragments. note also that in considering the covering process over all positions in the target, this probability is identical to the expected value of the random variable c { \ displaystyle c }, the fraction of the target coverage. the final result, e ⟨ c ⟩ = 1 − e − r, { \ displaystyle e \ langle c \ rangle =
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DNA sequencing theory
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in mathematics, a jonsson – tarski algebra or cantor algebra is an algebraic structure encoding a bijection from an infinite set x onto the product x×x. they were introduced by bjarni jonsson and alfred tarski ( 1961, theorem 5 ). smirnov ( 1971 ), named them after georg cantor because of cantor's pairing function and cantor's theorem that an infinite set x has the same number of elements as x×x. the term cantor algebra is also occasionally used to mean the boolean algebra of all clopen subsets of the cantor set, or the boolean algebra of borel subsets of the reals modulo meager sets ( sometimes called the cohen algebra ). the group of order - preserving automorphisms of the free jonsson – tarski algebra on one generator is the thompson group f. = = definition = = a jonsson – tarski algebra of type 2 is a set a with a product w from a×a to a and two'projection'maps p1 and p2 from a to a, satisfying p1 ( w ( a1, a2 ) ) = a1, p2 ( w ( a1, a2 ) ) = a2, and w ( p1 ( a ), p2 ( a ) ) = a. the definition for type > 2 is similar but with n projection operators. = = example = = if w is any bijection from a×a to a then it can be extended to a unique jonsson – tarski algebra by letting pi ( a ) be the projection of w−1 ( a ) onto the ith factor. = = references = = jonsson, bjarni ; tarski, alfred ( 1961 ), " on two properties of free algebras ", math. scand., 9 : 95 – 101, mr 0126399, zbl 0111. 02002 smirnov, d. m. ( 1971 ), " cantor algebras with one generator. i. ", algebra and logic, 10 : 40 – 49, doi : 10. 1007 / bf02217801, mr 0296006, zbl 0223. 08006
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Jónsson–Tarski algebra
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iron stress repressed rna ( isrr ) is a cis - encoded antisense rna which regulates the expression of the photosynthetic protein isia. isia expression is activated by the ferric uptake regulator protein ( fur ) under iron stress conditions. isia enhances photosynthesis by forming a ring around photosystem i which acts as an additional antenna complex. isrr is abundant when there is a sufficient iron concentration. isrr is encoded for within the opposite stand of isia gene and contains a conserved stem loop secondary structure. under sufficient iron conditions isrr binds to its complementary region which corresponds to the central third of the isia mrna. the resulting duplex rna is then targeted for degradation. this allows the antisense rna to act as a reversible switch that responds to changes in environmental conditions to modulate the expression of the isia protein. isrr was originally identified within cyanobacteria but may be conserved throughout a number of photosynthetic species from multiple kingdoms. at present, isrr is the only non coding rna identified that has a regulatory role on photosynthetic proteins. = = references = = = = external links = = rfam entry for antisense rna which regulates isia expression
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Iron stress repressed RNA
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growth ". journal of crystal growth. 630 : 127605. bibcode : 2024jcrgr. 63027605a. doi : 10. 1016 / j. jcrysgro. 2024. 127605. s2cid 267414096. https : / / doi. org / 10. 1016 / j. jcrysgro. 2024. 127605 = = external links = = " scheil solidification simulation of 332. 1 foundry aluminium alloy, thermocalc youtube channel ". youtube. archived from the original on 2021 - 12 - 19. retrieved 12 january 2020. " computational phase diagram database of the national institute for materials science of japan, nims ". retrieved 22 september 2021. " matcalc open databases ". retrieved 13 january 2020. " performing scheil - gulliver analysis in matcalc " ( pdf ). retrieved 26 september 2021. " special issue " calphad tools for the metallurgy of solidification " ". retrieved 13 january 2020. " solidification simulation by scheil model and lever rule ". 3 march 2021. retrieved 26 september 2021.
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Scheil equation
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f _ { 1 } \ cdot f _ { 2 }, g \ } = f _ { 1 } \ { f _ { 2 }, g \ } + f _ { 2 } \ { f _ { 1 }, g \ } } jacobi identity : { { h, f }, g } + { { f, g }, h } + { { g, h }, f } ≡ 0 { \ displaystyle \ { \ { h, f \ }, g \ } + \ { \ { f, g \ }, h \ } + \ { \ { g, h \ }, f \ } \ equiv 0 } non - degeneracy : if the point x on m is not critical for f then a smooth function g exists such that { f, g } ( x ) = 0 { \ displaystyle \ { f, g \ } ( x ) \ neq 0 }. given a function f d d t f = ∂ ∂ t f + { f, h }, { \ displaystyle { \ frac { \ mathrm { d } } { \ mathrm { d } t } } f = { \ frac { \ partial } { \ partial t } } f + \ left \ { f, { \ mathcal { h } } \ right \ }, } if there is a probability distribution ρ, then ( since the phase space velocity ( p i, q i ) { \ displaystyle ( { \ dot { p } } _ { i }, { \ dot { q } } _ { i } ) } has zero divergence and probability is conserved ) its convective derivative can be shown to be zero and so ∂ ∂ t ρ = − { ρ, h } { \ displaystyle { \ frac { \ partial } { \ partial t } } \ rho = - \ left \ { \ rho, { \ mathcal { h } } \ right \ } } this is called liouville's theorem. every smooth function g over the symplectic manifold generates a one - parameter family of symplectomorphisms and if { g, h } = 0, then g is conserved and the symplectomorphisms are symmetry transformations. a hamiltonian may have multiple conserved quantities gi. if the symplectic manifold has dimension 2n and there are n functionally independent conserved quantities gi which are in involution ( i. e.
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Hamiltonian mechanics
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, zhenzhen dong ( adesis ), nicole camasso ( jacs ). = = = 2020 meeting and speakers = = = the second annual empowering women in organic chemistry conference was held virtually on thursday, august 13, 2020, and friday, august 14, 2020. 2020 career panel featured shanina sanders johnson ( spelman college ), davita watkins ( univ of mississippi ), niki patel ( merck ), stacy fosu ( abbvie ), beth lorsbach ( corteva agriscience ), sibrina collins ( marburger stem center, lawrence technological university ), shana cyr ( bristol myers squibb ), sherri pietranico - cole ( novartis ), gabby nepomuceno ( california department of toxic substances control ). = = = 2021 meeting and speakers = = = the third annual empowering women in organic chemistry conference was held virtually on thursday, june 24, and friday, june 25, 2021. 2021 career panel featured kay brummond ( univ of pittsburgh ), martha a. sarpong ( glaxosmithkline ), emma radoux ( royal society of chemistry ), callie bryan ( janssen ), kimberly steward ( cargill ) and daisy rosas vargas ( ithaca college ). = = = 2022 meeting and speakers = = = the fourth annual empowering women in organic chemistry conference was a hybrid meeting on thursday, june 23, and friday, june 24, 2022. = = references = =
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Empowering Women in Organic Chemistry
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dq + { \ frac { \ partial l } { \ partial \ zeta } } d \ zeta + pd { \ dot { q } } + { \ frac { \ partial l } { \ partial { \ dot { \ zeta } } } } d { \ dot { \ zeta } } + { \ frac { \ partial l } { \ partial t } } dt } and to replace pd ( dq / dt ) by ( dq / dt ) dp, recall the product rule for differentials, and substitute p d q = d ( q p ) − q d p { \ displaystyle pd { \ dot { q } } = d ( { \ dot { q } } p ) - { \ dot { q } } dp } to obtain the differential of a new function in terms of the new set of variables : d ( l − p q ) = p d q + ∂ l ∂ ζ d ζ − q d p + ∂ l ∂ ζ d ζ + ∂ l ∂ t d t. { \ displaystyle d ( l - p { \ dot { q } } ) = { \ dot { p } } dq + { \ frac { \ partial l } { \ partial \ zeta } } d \ zeta - { \ dot { q } } dp + { \ frac { \ partial l } { \ partial { \ dot { \ zeta } } } } d { \ dot { \ zeta } } + { \ frac { \ partial l } { \ partial t } } dt \,. } introducing the routhian r ( q, ζ, p, ζ, t ) = p q ( p ) − l { \ displaystyle r ( q, \ zeta, p, { \ dot { \ zeta } }, t ) = p { \ dot { q } } ( p ) - l } where again the velocity dq / dt is a function of the momentum p, we have d r = − p d q − ∂ l ∂ ζ d ζ + q d p − ∂ l ∂ ζ d ζ − ∂ l ∂ t d t, { \ displaystyle dr = - { \ dot { p } } dq - { \ frac { \ partial l } { \ partial \ zeta } } d \ zeta + { \ dot { q } } dp - { \ frac { \ partial l } { \ partial { \ dot { \ zeta }
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Routhian mechanics
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within signal processing, in many cases only one image with noise is available, and averaging is then realized in a local neighborhood. results are acceptable if the noise is smaller in size than the smallest objects of interest in the image, but blurring of edges is a serious disadvantage. in the case of smoothing within a single image, one has to assume that there are no changes in the gray levels of the underlying image data. this assumption is clearly violated at locations of image edges, and edge blurring is a direct consequence of violating the assumption. = = description = = averaging is a special case of discrete convolution. for a 3 by 3 neighborhood, the convolution mask m is : m = 1 9 [ 1 1 1 1 1 1 1 1 1 ] { \ displaystyle m = { \ frac { 1 } { 9 } } { \ begin { bmatrix } 1 & 1 & 1 \ \ 1 & 1 & 1 \ \ 1 & 1 & 1 \ \ \ end { bmatrix } } } the significance of the central pixel may be increased, as it approximates the properties of noise with a gaussian probability distribution : m = 1 10 [ 1 1 1 1 2 1 1 1 1 ] { \ displaystyle m = { \ frac { 1 } { 10 } } { \ begin { bmatrix } 1 & 1 & 1 \ \ 1 & 2 & 1 \ \ 1 & 1 & 1 \ \ \ end { bmatrix } } } m = 1 16 [ 1 2 1 2 4 2 1 2 1 ] { \ displaystyle m = { \ frac { 1 } { 16 } } { \ begin { bmatrix } 1 & 2 & 1 \ \ 2 & 4 & 2 \ \ 1 & 2 & 1 \ \ \ end { bmatrix } } } a suitable page for beginners about matrices is at : https : / / web. archive. org / web / 20060819141930 / http : / / www. gamedev. net / reference / programming / features / imageproc / page2. asp the whole article starts on page : https : / / web. archive. org / web / 20061019072001 / http : / / www. gamedev. net / reference / programming / features / imageproc / = = references = =
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Generalized signal averaging
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in statistics and in empirical sciences, a data generating process is a process in the real world that " generates " the data one is interested in. this process encompasses the underlying mechanisms, factors, and randomness that contribute to the production of observed data. usually, scholars do not know the real data generating model and instead rely on assumptions, approximations, or inferred models to analyze and interpret the observed data effectively. however, it is assumed that those real models have observable consequences. those consequences are the distributions of the data in the population. those distributors or models can be represented via mathematical functions. there are many functions of data distribution. for example, normal distribution, bernoulli distribution, poisson distribution, etc. = = references = = https : / / stats. stackexchange. com / questions / 443320 / what - does - a - data - generating - process - dgp - actually - mean
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Data generating process
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weights of a and b in the superposition process. it will never be different from both a and b [ i. e., either a or b ]. the intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states. anton zeilinger, referring to the prototypical example of the double - slit experiment, has elaborated regarding the creation and destruction of quantum superposition : " [ t ] he superposition of amplitudes... is only valid if there is no way to know, even in principle, which path the particle took. it is important to realize that this does not imply that an observer actually takes note of what happens. it is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘ ‘ out there. ’ ’ the absence of any such information is the essential criterion for quantum interference to appear. = = theory = = = = = general formalism = = = any quantum state can be expanded as a sum or superposition of the eigenstates of an hermitian operator, like the hamiltonian, because the eigenstates form a complete basis : | α ⟩ = n c n | n ⟩, { \ displaystyle | \ alpha \ rangle = \ sum _ { n } c _ { n } | n \ rangle, } where | n ⟩ { \ displaystyle | n \ rangle } are the energy eigenstates of the hamiltonian. for continuous variables like position eigenstates, | x ⟩ { \ displaystyle | x \ rangle } : | α ⟩ = d x ′ | x ′ ⟩ ⟨ x ′ | α ⟩, { \ displaystyle | \ alpha \ rangle = \ int dx'| x'\ rangle \ langle x'| \ alpha \ rangle, } where α ( x ) = ⟨ x | α ⟩ { \ displaystyle \ phi _ { \ alpha } ( x ) = \ langle x | \ alpha \ rangle } is the projection of the state into the | x ⟩ { \ displaystyle | x \ rangle } basis and is called the wave function of the particle. in both instances we notice that | α ⟩
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Quantum superposition
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x ) { \ displaystyle f _ { x \ mid y = y } ( x ) } is the posterior density of x { \ displaystyle x } given the data y = y { \ displaystyle y = y }. = = credible interval = = posterior probability is a conditional probability conditioned on randomly observed data. hence it is a random variable. for a random variable, it is important to summarize its amount of uncertainty. one way to achieve this goal is to provide a credible interval of the posterior probability. = = classification = = in classification, posterior probabilities reflect the uncertainty of assessing an observation to particular class, see also class - membership probabilities. while statistical classification methods by definition generate posterior probabilities, machine learners usually supply membership values which do not induce any probabilistic confidence. it is desirable to transform or rescale membership values to class - membership probabilities, since they are comparable and additionally more easily applicable for post - processing. = = see also = = prediction interval bernstein – von mises theorem probability of success bayesian epistemology metropolis – hastings algorithm = = references = = = = further reading = = lancaster, tony ( 2004 ). an introduction to modern bayesian econometrics. oxford : blackwell. isbn 1 - 4051 - 1720 - 6. lee, peter m. ( 2004 ). bayesian statistics : an introduction ( 3rd ed. ). wiley. isbn 0 - 340 - 81405 - 5.
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Posterior probability
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##ine ( lys, k ), methionine ( met, m ), phenylalanine ( phe, f ), proline ( pro, p ), serine ( ser, s ), threonine ( thr, t ), tryptophan ( trp, w ), tyrosine ( tyr, y ), valine ( val, v ). = = differences from the standard code = = note : the codon agg is absent in drosophila. = = = alternative initiation codons = = = ata / aua att / auu atc / auc : apis gtg / gug : polyplacophora ttg / uug : ascaris, caenorhabditis. = = systematic range = = nematoda : ascaris, caenorhabditis ; mollusca : bivalvia ) ; polyplacophora ; arthropoda / crustacea : artemia ; arthropoda / insecta : drosophila [ locusta migratoria ( migratory locust ), apis mellifera ( honeybee ) ]. = = other variations = = several arthropods translate the codon agg as lysine instead of serine ( as in the pterobranchia mitochondrial code ) or arginine ( as in the standard genetic code ). gug may possibly function as an initiator in drosophila. auu is not used as an initiator in mytilus " an exceptional mechanism must operate for initiation of translation of the cytochrome oxidase subunit i mrna in both d. melanogaster and d. yakuba, since its only plausible initiation codon, aua, is out of frame with the rest of the gene. initiation appears to require the " reading " of an auaa quadruplet, which would be equivalent to initiation at aua followed immediately by a specific ribosomal frameshift. another possible mechanism... is that the mrna is " edited " to bring the aua initiation into frame. " = = see also = = list of genetic codes = = references = = this article incorporates text from the united states national library of medicine, which is in the public domain.
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Invertebrate mitochondrial code
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translation ( specifically, english as source, and german as target ) in encoder - decoder transformer models, trained until convergence on the same datasets ( thus they did not fit scaling laws for computing cost c { \ displaystyle c } or dataset size d { \ displaystyle d } ). they varied n ∈ [ 10 8, 3. 5 × 10 9 ] { \ displaystyle n \ in [ 10 ^ { 8 }, 3. 5 \ times 10 ^ { 9 } ] } they found three results : l { \ displaystyle l } is a scaling law function of n e, n d { \ displaystyle n _ { e }, n _ { d } }, where n e, n d { \ displaystyle n _ { e }, n _ { d } } are encoder and decoder parameter count. it is not simply a function of total parameter count n = n e + n d { \ displaystyle n = n _ { e } + n _ { d } }. the function has form l ( n e, n d ) = α ( n e n e ) p e ( n d n d ) p d + l ∞ { \ displaystyle l \ left ( n _ { e }, n _ { d } \ right ) = \ alpha \ left ( { \ frac { { \ bar { n } } _ { e } } { n _ { e } } } \ right ) ^ { p _ { e } } \ left ( { \ frac { { \ bar { n } } _ { d } } { n _ { d } } } \ right ) ^ { p _ { d } } + l _ { \ infty } }, where α, p e, p d, l ∞, n e, n d { \ displaystyle \ alpha, p _ { e }, p _ { d }, l _ { \ infty }, { \ bar { n } } _ { e }, { \ bar { n } } _ { d } } are fitted parameters. they found that n d / n ≈ 0. 55 { \ displaystyle n _ { d } / n \ approx 0. 55 } minimizes loss if n { \ displaystyle n } is held fixed. l { \ displaystyle l } " saturates " ( that is, it reaches l ∞ { \ displaystyle l _ { \ infty }
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Neural scaling law
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of the target predictive function f t ( ⋅ ) { \ displaystyle f _ { t } ( \ cdot ) } in d t { \ displaystyle { \ mathcal { d } } _ { t } } using the knowledge in d s { \ displaystyle { \ mathcal { d } } _ { s } } and t s { \ displaystyle { \ mathcal { t } } _ { s } }. = = applications = = algorithms are available for transfer learning in markov logic networks and bayesian networks. transfer learning has been applied to cancer subtype discovery, building utilization, general game playing, text classification, digit recognition, medical imaging and spam filtering. in 2020, it was discovered that, due to their similar physical natures, transfer learning is possible between electromyographic ( emg ) signals from the muscles and classifying the behaviors of electroencephalographic ( eeg ) brainwaves, from the gesture recognition domain to the mental state recognition domain. it was noted that this relationship worked in both directions, showing that electroencephalographic can likewise be used to classify emg. the experiments noted that the accuracy of neural networks and convolutional neural networks were improved through transfer learning both prior to any learning ( compared to standard random weight distribution ) and at the end of the learning process ( asymptote ). that is, results are improved by exposure to another domain. moreover, the end - user of a pre - trained model can change the structure of fully - connected layers to improve performance. = = see also = = crossover ( genetic algorithm ) domain adaptation general game playing multi - task learning multitask optimization transfer of learning in educational psychology zero - shot learning feature learning external validity = = references = = = = sources = = thrun, sebastian ; pratt, lorien ( 6 december 2012 ). learning to learn. springer science & business media. isbn 978 - 1 - 4615 - 5529 - 2.
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Cost-sensitive machine learning
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in the same cell. = = = amoebozoa = = = cell fusion ( plasmogamy or syngamy ) is a stage in the amoebozoa sexual cycle. = = = bacteria = = = in escherichia coli spontaneous zygogenesis ( z - mating ) involves cell fusion, and appears to be a form of true sexuality in prokaryotes. bacteria that perform z - mating are called szp +. = = other uses = = to study the control of cell division and gene expression. to investigate malignant transformations. to obtain viral replication. for gene and chromosome mapping. for production of monoclonal antibodies by producing hybridoma. for production of induced stem cells. to assess protein shuttling in what is known as a heterokaryon fusion assay. = = see also = = cell - cell fusogens cellular differentiation fertilisation fusion mechanism fusion protein interbilayer forces in membrane fusion lipid bilayer fusion = = references = = = = further reading = = h. harris : cell fusion, 1970, harvard university press, mass. gordon, s ( 1975 ). " cell fusion and some subcellular properties of heterokaryons and hybrids ". the journal of cell biology. 67 ( 2 ) : 257 – 280. doi : 10. 1083 / jcb. 67. 2. 257. pmc 2109606. pmid 1104638.
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Cell fusion
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are functions that satisfy the pde, yet in other meanings than regular sense. the meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of distributions. an example for the definition of a weak solution is as follows : consider the boundary - value problem given by : l u = f in u, u = 0 on ∂ u, { \ displaystyle { \ begin { aligned } lu & = f \ quad { \ text { in } } u, \ \ u & = 0 \ quad { \ text { on } } \ partial u, \ end { aligned } } } where l u = − i, j ∂ j ( a i j ∂ i u ) + i b i ∂ i u + c u { \ displaystyle lu = - \ sum _ { i, j } \ partial _ { j } ( a ^ { ij } \ partial _ { i } u ) + \ sum _ { i } b ^ { i } \ partial _ { i } u + cu } denotes a second - order partial differential operator in divergence form. we say a u ∈ h 0 1 ( u ) { \ displaystyle u \ in h _ { 0 } ^ { 1 } ( u ) } is a weak solution if u [ i, j a i j ( ∂ i u ) ( ∂ j v ) + i b i ( ∂ i u ) v + c u v ] d x = u f v d x { \ displaystyle \ int _ { u } [ \ sum _ { i, j } a ^ { ij } ( \ partial _ { i } u ) ( \ partial _ { j } v ) + \ sum _ { i } b ^ { i } ( \ partial _ { i } u ) v + cuv ] dx = \ int _ { u } fvdx } for every v ∈ h 0 1 ( u ) { \ displaystyle v \ in h _ { 0 } ^ { 1 } ( u ) }, which can be derived by a formal integral by parts. an example for a weak solution is as follows : ( x ) = 1 4 π 1 | x | { \ displaystyle \ phi ( x ) = { \ frac { 1 } { 4 \ pi } } { \ frac { 1 } { | x | } } } is a weak solution satisfying ∇ 2 = δ in r 3 { \ displaystyle \
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Partial differential equation
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##egorical variables such as ethnic groups and experimental treatments as independent variables in moderated regression, one needs to code the variables so that each code variable represents a specific setting of the categorical variable. there are three basic ways of coding : dummy - variable coding, contrast coding and effects coding. below is an introduction to these coding systems. dummy coding compares a reference group ( one specific condition such as a control group in the experiment ) with each of the other experimental groups. in this case, the intercept b 0 = y reference { \ displaystyle b _ { 0 } = { \ bar { y } } _ { \ text { reference } } } is the mean of the reference group. each unstandardized regression coefficient b a = y a − y reference { \ displaystyle b _ { a } = { \ bar { y } } _ { a } - { \ bar { y } } _ { \ text { reference } } } is the difference in the dependent variable mean between a treatment group and the reference group. this coding system is similar to anova analysis. contrast coding investigates a series of orthogonal contrasts ( group comparisons ). the intercept b 0 = y 1 … n { \ displaystyle b _ { 0 } = { \ overline { { \ bar { y } } _ { 1 \ dots n } } } } is the unweighted mean of the individual group means. the unstandardized regression coefficient b a, b = y b − y a { \ displaystyle b _ { a, b } = { \ bar { y } } _ { b } - { \ bar { y } } _ { a } } represents the difference between two groups. this coding system is appropriate when researchers have an a priori hypothesis concerning the specific differences among the group means. effects coding is used when there is no reference group or orthogonal contrasts. the intercept b 0 { \ displaystyle b _ { 0 } } is the grand mean ( the mean of all the conditions ). the regression coefficient b a = y a − y 1 … n { \ displaystyle b _ { a } = { \ bar { y } } _ { a } - { \ overline { { \ bar { y } } _ { 1 \ dots n } } } } is the difference between one group mean and the mean of all the group means. this coding system is appropriate when the groups represent natural categories. = = = two continuous independent variables = = =
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Moderation (statistics)
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wikipedia
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shangmi 3 ( sm3 ) is a cryptographic hash function, standardised for use in commercial cryptography in china. it was published by the national cryptography administration ( chinese : 国 家 ) on 2010 - 12 - 17 as " gm / t 0004 - 2012 : sm3 cryptographic hash algorithm ". sm3 is used for implementing digital signatures, message authentication codes, and pseudorandom number generators. the algorithm is public and is considered similar to sha - 256 in security and efficiency. sm3 is used with transport layer security. = = definitive standards = = sm3 is defined in each of : gm / t 0004 - 2012 : sm3 cryptographic hash algorithm gb / t 32905 - 2016 : information security techniques — sm3 cryptographic hash algorithm iso / iec 10118 - 3 : 2018 — it security techniques — hash - functions — part 3 : dedicated hash - functions ietf rfc draft - sca - cfrg - sm3 - 02 = = references = = = = see also = = sm4 ( cipher )
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SM3 (hash function)
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wikipedia
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variations by alessio conte bron - kerbosch algorithm implementation visualized in javascript bron - kerbosch algorithm implementation in python bron - kerbosch algorithm with vertex ordering implementation in python bron - kerbosch algorithm implementation in c + + bron - kerbosch algorithm implementation in c + + 11 with unit tests c + + implementation of the algorithms presented in eppstein, strash ( 2011 ) by darren strash finding all cliques of an undirected graph. seminar notes by michaela regneri, january 11, 2007. bron - kerbosch, algorithms implementation in php, library with composer install by sergio zambrano
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Bron–Kerbosch algorithm
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et al. proposed to measure the effective degrees of freedom by counting the number of parameters that deviate from zero. the degrees of freedom approach was considered flawed by kaufman and rosset and janson et al., because a model's degrees of freedom might increase even when it is penalized harder by the regularization parameter. as an alternative, the relative simplicity measure defined above can be used to count the effective number of parameters. for the lasso, this measure is given by p ^ = i = 1 p | β i − β 0, i | 1 p l | b ols, l − β 0, l |, { \ displaystyle { \ hat { \ mathcal { p } } } = \ sum _ { i = 1 } ^ { p } { \ frac { | \ beta _ { i } - \ beta _ { 0, i } | } { { \ frac { 1 } { p } } \ sum _ { l } | b _ { { \ text { ols } }, l } - \ beta _ { 0, l } | } }, } which monotonically increases from zero to p { \ displaystyle p } as the regularization parameter decreases from ∞ { \ displaystyle \ infty } to zero. = = selected applications = = lasso has been applied in economics and finance, and was found to improve prediction and to select sometimes neglected variables, for example in corporate bankruptcy prediction literature, or high growth firms prediction. = = see also = = least absolute deviations model selection nonparametric regression tikhonov regularization = = references = =
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Lasso (statistics)
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wikipedia
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, html online version. spiking neuron models ( w. gerstner and w. kistler, cambridge university press, 2002 ) = = see also = = binding neuron bayesian approaches to brain function brain - computer interfaces free energy principle models of neural computation neural coding neural oscillation quantitative models of the action potential spiking neural network = = references = =
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Biological neuron model
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wikipedia
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##ar, hugh longbourne ( 1911 ). " thermodynamics ". encyclopædia britannica. vol. 26 ( 11th ed. ). pp. 808 – 814. thermodynamics data & property calculation websites thermodynamics educational websites biochemistry thermodynamics thermodynamics and statistical mechanics engineering thermodynamics – a graphical approach thermodynamics and statistical mechanics by richard fitzpatrick
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Axiomatics of thermodynamics
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wikipedia
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and densely defined operator - valued frames ". rocky mountain journal of mathematics. 51 ( 1 ). arxiv : 2004. 11729. doi : 10. 1216 / rmj. 2021. 51. 265. issn 0035 - 7596. young, robert m. ( 2001 ). an introduction to non - harmonic fourier series, revised edition, 93. academic press. isbn 978 - 0 - 12 - 772955 - 8.
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Frame (linear algebra)
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wikipedia
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varepsilon _ { f } } then, the true stress can be expressed as below : σ t = f a f = f a i × a i a f = σ e × l f l i = σ e × l i + δ l l i = σ e ( 1 + ε e ) { \ displaystyle { \ begin { aligned } \ sigma _ { t } = { \ frac { f } { a _ { f } } } & = { \ frac { f } { a _ { i } } } \ times { \ frac { a _ { i } } { a _ { f } } } \ \ & = \ sigma _ { e } \ times { \ frac { l _ { f } } { l _ { i } } } \ \ [ 2pt ] & = \ sigma _ { e } \ times { \ frac { l _ { i } + \ delta l } { l _ { i } } } \ \ [ 2pt ] & = \ sigma _ { e } ( 1 + \ varepsilon _ { e } ) \ end { aligned } } } additionally, the true strain εt can be expressed as below : ε t = d l l 0 + d l l 1 + d l l 2 + = i d l l i { \ displaystyle \ varepsilon _ { t } = { \ frac { dl } { l _ { 0 } } } + { \ frac { dl } { l _ { 1 } } } + { \ frac { dl } { l _ { 2 } } } + \ cdots = \ sum _ { i } { \ frac { dl } { l _ { i } } } } then, we can express the value as l 0 l i d l l d x = ln ( l i l 0 ) = ln ( 1 + ε e ) { \ displaystyle \ int _ { l _ { 0 } } ^ { l _ { i } } { \ frac { dl } { l } } \, dx = \ ln \ left ( { \ frac { l _ { i } } { l _ { 0 } } } \ right ) = \ ln ( 1 + \ varepsilon _ { e } ) } thus, we can induce the plot in terms of σ t { \ displaystyle \ sigma _ { t } } and ε e { \
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Deformation (engineering)
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cos ( a − b ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) { \ displaystyle \ cos ( a - b ) = \ cos ( a ) \ cos ( b ) + \ sin ( a ) \ sin ( b ) }, the spherical symmetry of gaussian distribution, then evaluate the integral − ∞ ∞ cos ( k x ) e − x 2 / 2 2 π d x = e − k 2 / 2. { \ displaystyle \ int _ { - \ infty } ^ { \ infty } { \ frac { \ cos ( kx ) e ^ { - x ^ { 2 } / 2 } } { \ sqrt { 2 \ pi } } } dx = e ^ { - k ^ { 2 } / 2 }. } theorem : var [ ⟨ φ ( x ), φ ( y ) ⟩ ] = o ( d − 1 ) { \ displaystyle \ operatorname { var } [ \ langle \ varphi ( x ), \ varphi ( y ) \ rangle ] = o ( d ^ { - 1 } ) }. ( appendix a. 2 ). = = = nystrom method = = = another approach uses the nystrom method to approximate the eigendecomposition of the gram matrix k, using only a random sample of the training set. = = see also = = gaussian function kernel ( statistics ) polynomial kernel radial basis function radial basis function network obst kernel network = = references = =
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Radial basis function kernel
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wikipedia
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##iker - pretre - thomas ( bpt ) formula which relates the geometric conductance to the s { \ displaystyle s } matrix of the pump. in the low temperature limit it yields g 3 j = 1 2 π i t r a c e ( p a ∂ s ∂ x j s † ) { \ displaystyle g ^ { 3j } = { \ frac { 1 } { 2 \ pi i } } \ mathrm { trace } \ left ( p _ { a } { \ frac { \ partial s } { \ partial x _ { j } } } s ^ { \ dagger } \ right ) } here p a { \ displaystyle p _ { a } } is a projector that restrict the trace operations to the open channels of the lead where the current is measured. this bpt formula has been originally derived using a scattering approach, but later its relation to the kubo formula has been worked out. = = the effect of interactions = = a very recent work considers the role of interactions in the stirring of bose condensed particles. otherwise the rest of the literature concerns primarily electronic devices. typically the pump is modeled as a quantum dot. the effect of electron – electron interactions within the dot region is taken into account in the coulomb blockade regime or in the kondo regime. in the former case charge transport is quantized even in the case of small backscattering. deviation from the exact quantized value is related to dissipation. in the kondo regime, as the temperature is lowered, the pumping effect is modified. there are also works that consider interactions over the whole system ( including the leads ) using the luttinger liquid model. = = quantum pumping in deformable mesoscopic systems = = a quantum pump, when coupled to classical mechanical degrees of freedom, may also induce cyclic variations of the mechanical degrees of freedom coupled to it. in such a configuration, the pump works similarly to an adiabatic quantum motor. a paradigmatic example of this class of systems is a quantum pump coupled to an elastically deformable quantum dot. the mentioned paradigm has been generalized to include non - linear effects and stochastic fluctuations. = = see also = = quantum mechanics brownian ratchet geometric phase § stochastic pump effect adiabatic quantum motor = = references = = = = = unsorted = = =
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Quantum stirring, ratchets, and pumping
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= 0 } ( the cauchy – riemann equations ). also, we have : d f = ∂ f ∂ z d z + ∂ f ∂ z d z. { \ displaystyle df = { \ frac { \ partial f } { \ partial z } } dz + { \ frac { \ partial f } { \ partial { \ bar { z } } } } d { \ bar { z } }. } let d = { z ∈ c < | z − z 0 | < r } { \ displaystyle d _ { \ epsilon } = \ { z \ in \ mathbb { c } \ mid \ epsilon < | z - z _ { 0 } | < r \ } } be a punctured disk with center z 0 { \ displaystyle z _ { 0 } }. since 1 / ( z − z 0 ) { \ displaystyle 1 / ( z - z _ { 0 } ) } is holomorphic on d { \ displaystyle d _ { \ epsilon } }, we have : d ( f z − z 0 d z ) = ∂ f ∂ z d z ∧ d z z − z 0 { \ displaystyle d \ left ( { \ frac { f } { z - z _ { 0 } } } dz \ right ) = { \ frac { \ partial f } { \ partial { \ bar { z } } } } { \ frac { d { \ bar { z } } \ wedge dz } { z - z _ { 0 } } } }. by stokes ’ formula, d ∂ f ∂ z d z ∧ d z z − z 0 = ( | z − z 0 | = r − | z − z 0 | = ) f z − z 0 d z. { \ displaystyle \ int _ { d _ { \ epsilon } } { \ frac { \ partial f } { \ partial { \ bar { z } } } } { \ frac { d { \ bar { z } } \ wedge dz } { z - z _ { 0 } } } = \ left ( \ int _ { | z - z _ { 0 } | = r } - \ int _ { | z - z _ { 0 } | = \ epsilon } \ right ) { \ frac { f } { z - z _ { 0 } } } dz. } letting
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Calculus on Euclidean space
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pressure system and one for the low - pressure system. these disks act as a pressure relief system that protects the system from over - pressurization pressure transducers : a pressure sensor for the low - and high - pressure systems. produces a 0 – 5 v output over their pressure range. pressure meters : digital displays connected to each pressure transducer and the plc system. vacuum pump and gauges : cleans the system ( by evacuation ) before loading. optical system : used visual observation ; allowing in situ observations of gasket deformation. ruby fluorescence system : pressure in the sample chamber can be measured during loading using an online ruby fluorescence system. not all systems have an online ruby fluorescence system for in situ measuring. however, being able to monitor the pressure within the chamber while the dac is being sealed is advantageous – ensuring the desired pressure is reached ( or not over - shot ). pressure is measured by the shift in the laser induced luminescence of rubies in the sample chamber. = = see also = = = = references = = = = external links = = " putting the squeeze on materials ". lawrence livermore national laboratory. december 2004. archived from the original on 20 november 2008. retrieved 5 may 2009.
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Diamond anvil cell
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wikipedia
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{ \ displaystyle \ lambda = l / l _ { 0 } } as determinant, the true stress and strain can be expressed with engineering stress and strain as below : σ t = σ e × λ, ε t = ln λ. { \ displaystyle \ sigma _ { t } = \ sigma _ { e } \ times \ lambda, \ qquad \ varepsilon _ { t } = \ ln \ lambda. } therefore, the value of engineering stress can be expressed by the secant line from made by true stress and λ { \ displaystyle \ lambda } value where λ = 0 { \ displaystyle \ lambda = 0 } to λ = 1 { \ displaystyle \ lambda = 1 }. by analyzing the shape of σ t − λ { \ displaystyle \ sigma _ { t } - \ lambda } diagram and secant line, we can determine whether the materials show drawing or necking. on the figure ( a ), there is only concave upward considere plot. it indicates that there is no yield drop so the material will be suffered from fracture before it yields. on the figure ( b ), there is specific point where the tangent matches with secant line at point where λ = λ y { \ displaystyle \ lambda = \ lambda _ { y } }. after this value, the slope becomes smaller than the secant line where necking starts to appear. on the figure ( c ), there is point where yielding starts to appear but when λ = λ d { \ displaystyle \ lambda = \ lambda _ { d } }, the drawing happens. after drawing, all the material will stretch and eventually show fracture. between λ y { \ displaystyle \ lambda _ { y } } and λ d { \ displaystyle \ lambda _ { d } }, the material itself does not stretch but rather, only the neck starts to stretch out. = = misconceptions = = a popular misconception is that all materials that bend are " weak " and those that do not are " strong ". in reality, many materials that undergo large elastic and plastic deformations, such as steel, are able to absorb stresses that would cause brittle materials, such as glass, with minimal plastic deformation ranges, to break. = = see also = = = = references = =
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Deformation (engineering)
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wikipedia
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of degree at most ( d + k − 3 ) / 2 { \ displaystyle \ lfloor ( d + k - 3 ) / 2 \ rfloor } and polynomials a ( x ), b ( x ) { \ displaystyle a ( x ), b ( x ) } such that r ( x ) = a ( x ) s ( x ) γ ( x ) + b ( x ) x d − 1. { \ displaystyle r ( x ) = a ( x ) s ( x ) \ gamma ( x ) + b ( x ) x ^ { d - 1 }. } low degree of r ( x ) { \ displaystyle r ( x ) } guarantees, that a ( x ) { \ displaystyle a ( x ) } would satisfy extended ( by γ { \ displaystyle \ gamma } ) defining conditions for λ. { \ displaystyle \ lambda. } defining ξ ( x ) = a ( x ) γ ( x ) { \ displaystyle \ xi ( x ) = a ( x ) \ gamma ( x ) } and using ξ { \ displaystyle \ xi } on the place of λ ( x ) { \ displaystyle \ lambda ( x ) } in the fourney formula will give us error values. the main advantage of the algorithm is that it meanwhile computes ω ( x ) = s ( x ) ξ ( x ) mod x d − 1 = r ( x ) { \ displaystyle \ omega ( x ) = s ( x ) \ xi ( x ) { \ bmod { x } } ^ { d - 1 } = r ( x ) } required in the forney formula. = = = = explanation of the decoding process = = = = the goal is to find a codeword which differs from the received word minimally as possible on readable positions. when expressing the received word as a sum of nearest codeword and error word, we are trying to find error word with minimal number of non - zeros on readable positions. syndrom s i { \ displaystyle s _ { i } } restricts error word by condition s i = j = 0 n − 1 e j α i j. { \ displaystyle s _ { i } = \ sum _ { j = 0 } ^ { n - 1 } e _ { j } \ alpha ^ { ij }. } we could write these conditions separately or we could create polynomial s ( x ) = i = 0 d − 2
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BCH code
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wikipedia
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functions f but not necessarily true for artificially constructed f. to formulate them precisely, it is necessary to have a precise definition for a natural function f for which the theorem is true. time - constructible functions are often used to provide such a definition. space - constructible functions are used similarly, for example in the space hierarchy theorem. = = references = = this article incorporates material from constructible on planetmath, which is licensed under the creative commons attribution / share - alike license.
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Constructible function
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wikipedia
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especially in the quasi - geostrophic framework ). therefore, the physical mechanism responsible for the propagation of these equatorial rossby waves is none other than the conservation of potential vorticity : ∂ ∂ t β y + ζ h = 0. { \ displaystyle { \ frac { \ partial } { \ partial t } } { \ frac { \ beta y + \ zeta } { h } } = 0. } thus, as a fluid parcel moves equatorward ( βy approaches zero ), the relative vorticity must increase and become more cyclonic in nature. conversely, if the same fluid parcel moves poleward, ( βy becomes larger ), the relative vorticity must decrease and become more anticyclonic in nature. as a side note, these equatorial rossby waves can also be vertically - propagating waves when the brunt – vaisala frequency ( buoyancy frequency ) is held constant, ultimately resulting in solutions proportional to e i ( k x + m z − ω t ) { \ displaystyle e ^ { i ( kx + mz - \ omega t ) } }, where m is the vertical wavenumber and k is the zonal wavenumber. equatorial rossby waves can also adjust to equilibrium under gravity in the tropics ; because the planetary waves have frequencies much lower than gravity waves. the adjustment process tends to take place in two distinct stages where the first stage is a rapid change due to the fast propagation of gravity waves, the same as that on an f - plane ( coriolis parameter held constant ), resulting in a flow that is close to geostrophic equilibrium. this stage could be thought of as the mass field adjusting to the wave field ( due to the wavelengths being smaller than the rossby deformation radius. the second stage is one where quasi - geostrophic adjustment takes place by means of planetary waves ; this process can be comparable to the wave field adjusting to the mass field ( due to the wavelengths being larger than the rossby deformation radius. = = see also = = equatorial waves = = references = =
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Equatorial Rossby wave
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wikipedia
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v in the chosen basis. under a change of basis, the column x is multiplied on the left by an n × n invertible matrix s, and the symmetric square matrix a is transformed into another symmetric square matrix b of the same size according to the formula a → b = s t a s. { \ displaystyle a \ to b = s ^ { \ mathsf { t } } as. } any symmetric matrix a can be transformed into a diagonal matrix b = ( λ 1 0 0 0 λ 2 0 0 0 0 λ n ) { \ displaystyle b = { \ begin { pmatrix } \ lambda _ { 1 } & 0 & \ cdots & 0 \ \ 0 & \ lambda _ { 2 } & \ cdots & 0 \ \ \ vdots & \ vdots & \ ddots & 0 \ \ 0 & 0 & \ cdots & \ lambda _ { n } \ end { pmatrix } } } by a suitable choice of an orthogonal matrix s, and the diagonal entries of b are uniquely determined – this is jacobi's theorem. if s is allowed to be any invertible matrix then b can be made to have only 0, 1, and −1 on the diagonal, and the number of the entries of each type ( n0 for 0, n + for 1, and n− for −1 ) depends only on a. this is one of the formulations of sylvester's law of inertia and the numbers n + and n− are called the positive and negative indices of inertia. although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix a, sylvester's law of inertia means that they are invariants of the quadratic form q. the quadratic form q is positive definite if q ( v ) > 0 ( similarly, negative definite if q ( v ) < 0 ) for every nonzero vector v. when q ( v ) assumes both positive and negative values, q is an isotropic quadratic form. the theorems of jacobi and sylvester show that any positive definite quadratic form in n variables can be brought to the sum of n squares by a suitable invertible linear transformation : geometrically, there is only one positive definite real quadratic form of every dimension. its isometry group is a compact orthogonal group o ( n ). this stands in contrast with the case of isotropic forms, when the corresponding group
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Quadratic-linear algebra
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wikipedia
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##ar, hugh longbourne ( 1911 ). " thermodynamics ". encyclopædia britannica. vol. 26 ( 11th ed. ). pp. 808 – 814. thermodynamics data & property calculation websites thermodynamics educational websites biochemistry thermodynamics thermodynamics and statistical mechanics engineering thermodynamics – a graphical approach thermodynamics and statistical mechanics by richard fitzpatrick
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Thermodynamics
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wikipedia
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a ) = p ( r = + | y = y, a = b ) y ∈ { +, − } a, b ∈ a { \ displaystyle p ( r = + \ | \ y = y, a = a ) = p ( r = + \ | \ y = y, a = b ) \ quad y \ in \ { +, - \ } \ quad \ forall a, b \ in a } conditional use accuracy equality. a classifier satisfies this definition if the subjects in the protected and unprotected groups have equal ppv and equal npv, satisfying the formula : p ( y = y | r = y, a = a ) = p ( y = y | r = y, a = b ) y ∈ { +, − } a, b ∈ a { \ displaystyle p ( y = y \ | \ r = y, a = a ) = p ( y = y \ | \ r = y, a = b ) \ quad y \ in \ { +, - \ } \ quad \ forall a, b \ in a } overall accuracy equality. a classifier satisfies this definition if the subject in the protected and unprotected groups have equal prediction accuracy, that is, the probability of a subject from one class to be assigned to it. this is, if it satisfies the following formula : p ( r = y | a = a ) = p ( r = y | a = b ) a, b ∈ a { \ displaystyle p ( r = y \ | \ a = a ) = p ( r = y \ | \ a = b ) \ quad \ forall a, b \ in a } treatment equality. a classifier satisfies this definition if the subjects in the protected and unprotected groups have an equal ratio of fn and fp, satisfying the formula : f n a = a f p a = a = f n a = b f p a = b { \ displaystyle { \ frac { fn _ { a = a } } { fp _ { a = a } } } = { \ frac { fn _ { a = b } } { fp _ { a = b } } } } = = = = definitions based on predicted probabilities and actual outcome = = = = these definitions are based in the actual outcome y { \ textstyle y } and the predicted probability score s
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Fairness (machine learning)
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active infections, whereas other reasons include iron - mediated infections. = = applications = = fenton's reagent is used as a sewage treatment agent. fenton's reagent can be used in different chemical processes that supply hydroxyl ion or oxidize certain compounds : the first stage of fenton's reaction ( oxidation of fe3 + with hydrogen peroxide ) is used in haber – weiss reaction fenton's reagent can be used in organic synthesis reactions : e. g. hydroxylation of arenes via a free radical substitution conversion of benzene into phenol by using fenton's reagent oxidation of barbituric acid into alloxan. coupling reactions of alkanes = = fenton - like reagent = = mixtures of fe2 + and h2o2 are called fenton reagent. if fe2 + is replaced by fe3 +, it is called fenton - like reagent. numerous transition metal ions and their complexes in their lower oxidation states ( lmmn + ) were found to have the oxidative features of the fenton reagent, and, therefore, the mixtures of these metal compounds with h2o2 were named " fenton - like " reagents. = = see also = = carbon monoxide - releasing molecules = = references = = = = further reading = = goldstein, sara ; meyerstein, dan ; czapski, gidon ( 1993 ). " the fenton reagents ". free radical biology and medicine. 15 ( 4 ) : 435 – 445. doi : 10. 1016 / 0891 - 5849 ( 93 ) 90043 - t. pmid 8225025. barbusinski, k. ( 2009 ). " fenton reaction – controversy concerning the chemistry " ( pdf ). ecological chemistry and engineering. 16 ( 3 ) : 347 – 358. = = external links = = reference library peroxide applications companies that use fenton's reagent for chemical remediation : orin
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Fenton's reaction
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varphi ( \ mathbf { x }') \ right ] \ cdot d { \ hat { \ boldsymbol { \ sigma } } } '. } using this expression, it is possible to solve laplace's equation ∇2φ ( x ) = 0 or poisson's equation ∇2φ ( x ) = −ρ ( x ), subject to either neumann or dirichlet boundary conditions. in other words, we can solve for φ ( x ) everywhere inside a volume where either ( 1 ) the value of φ ( x ) is specified on the bounding surface of the volume ( dirichlet boundary conditions ), or ( 2 ) the normal derivative of φ ( x ) is specified on the bounding surface ( neumann boundary conditions ). suppose the problem is to solve for φ ( x ) inside the region. then the integral v φ ( x ′ ) δ ( x − x ′ ) d 3 x ′ { \ displaystyle \ int _ { v } \ varphi ( \ mathbf { x }') \, \ delta ( \ mathbf { x } - \ mathbf { x }') \, d ^ { 3 } \ mathbf { x }'} reduces to simply φ ( x ) due to the defining property of the dirac delta function and we have φ ( x ) = − v g ( x, x ′ ) ρ ( x ′ ) d 3 x ′ + s [ φ ( x ′ ) ∇ ′ g ( x, x ′ ) − g ( x, x ′ ) ∇ ′ φ ( x ′ ) ] ⋅ d σ ^ ′. { \ displaystyle \ varphi ( \ mathbf { x } ) = - \ int _ { v } g ( \ mathbf { x }, \ mathbf { x }') \, \ rho ( \ mathbf { x }') \, d ^ { 3 } \ mathbf { x }'+ \ int _ { s } \ left [ \ varphi ( \ mathbf { x }') \, \ nabla'g ( \ mathbf { x }, \ mathbf { x }') - g ( \ mathbf { x }, \ mathbf { x }') \, \ nabla'\ varphi ( \ mathbf { x }') \ right ] \ cdot d { \ hat { \ boldsymbol { \ sigma } } } '. } this
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Green's function
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two mechanisms for protein folding, either the diffusion - collision model or nucleation condensation model, although recent findings have shown globular proteins, such as ptp - bl pdz2, that fold with characteristic features of both models. these new findings have shown that the transition states of proteins may affect the way they fold. the folding of globular proteins has also recently been connected to treatment of diseases, and anti - cancer ligands have been developed which bind to the folded but not the natural protein. these studies have shown that the folding of globular proteins affects its function. by the second law of thermodynamics, the free energy difference between unfolded and folded states is contributed by enthalpy and entropy changes. as the free energy difference in a globular protein that results from folding into its native conformation is small, it is marginally stable, thus providing a rapid turnover rate and effective control of protein degradation and synthesis. = = role = = unlike fibrous proteins which only play a structural function, globular proteins can act as : enzymes, by catalyzing organic reactions taking place in the organism in mild conditions and with a great specificity. different esterases fulfill this role. messengers, by transmitting messages to regulate biological processes. this function is done by hormones, i. e. insulin etc. transporters of other molecules through membranes stocks of amino acids. regulatory roles are also performed by globular proteins rather than fibrous proteins. structural proteins, e. g., actin and tubulin, which are globular and soluble as monomers, but polymerize to form long, stiff fibers = = members = = among the most known globular proteins is hemoglobin, a member of the globin protein family. other globular proteins are the alpha, beta and gamma ( iga, igd, ige, igg and igm ) globulin. see protein electrophoresis for more information on the different globulins. nearly all enzymes with major metabolic functions are globular in shape, as well as many signal transduction proteins. albumins are also globular proteins, although, unlike all of the other globular proteins, they are completely soluble in water. they are not soluble in oil. = = references = =
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Globular protein
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##hytes are ssu rdna, lsu rdna, rdna its ( leliaert et al. 2014 ). = = macroalgae = = macroalgae — a morphological rather than taxonomic grouping — can be very challenging to identify because of their simple morphology, phenotypic plasticity and alternate lifecycle stages. thus, algal systematics and identification have come to rely heavily on genetic / molecular tools such as dna barcoding. the ssu rdna gene is a common used barcode for phylogenetic studies on macroalgae. however, the ssu rdna is a highly conserved region and typically lack resolution for species identification. over the past 2 decades certain standards for dna barcoding with the aim of species identification have been developed for each of the main groups of macroalgae. the cytochrome c oxidase subunit i ( coi ) gene is commonly used as a barcode for red and brown algae, while tufa ( plastid elongation factor ), rbcl ( rubisco large subunit ) and its ( internal transcribe spacer ) are commonly used for green algae. these barcodes are typically 600 - 700 bp long. the barcodes typically differ between the 3 main groups of macroalgae ( red, green and brown ) because their evolutionary heritage is very diverse. macroalgae is a polyphyletic group, meaning that within the group they do not all share a recent common ancestor, making it challenging to find a gene that is conserved among all but variable enough for species identification. = = target regions = = adapted from = = see also = = detailed information on dna barcoding of different organisms can be found here : microbial dna barcoding dna barcoding fish dna barcoding dna barcoding in diet assessment = = references = =
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Algae DNA barcoding
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the vertebrate mitochondrial code ( translation table 2 ) is the genetic code found in the mitochondria of all vertebrata. = = evolution = = aga and agg were thought to have become mitochondrial stop codons early in vertebrate evolution. however, at least in humans it has now been shown that aga and agg sequences are not recognized as termination codons. a - 1 mitoribosome frameshift occurs at the aga and agg codons predicted to terminate the co1 and nd6 open reading frames ( orfs ), and consequently both orfs terminate in the standard uag codon. = = incomplete stop codons = = mitochondrial genes in some vertebrates ( including humans ) have incomplete stop codons ending in u or ua, which become complete termination codons ( uaa ) upon subsequent polyadenylation. = = translation table = = a the codon aug both codes for methionine and serves as an initiation site : the first aug in an mrna's coding region is where translation into protein begins. = = differences from the standard code = = = = = alternative initiation codons = = = bos : aua homo : aua, auu mus : aua, auu, auc coturnix, gallus : also gug = = see also = = list of genetic codes = = references = = this article contains public domain text from the ncbi page compiled by andrzej elzanowski and jim ostell.
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Vertebrate mitochondrial code
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##132. wye, kenneth f. the encyclopedia of shells. facts on file, 1991. murray, michael t. = = footnotes = = = = references = = guide to reference. [ 30 ] american library association. retrieved 5 december 2014. ( subscription required ). kister, kenneth f. ( 1994 ). kister's best encyclopedias ( 2nd ed. ). phoenix : oryx. isbn 0 - 89774 - 744 - 5. kroeger, alice bertha, isadore gilbert mudge. ( 1911 ). guide to the study and use of reference books. chicago : american library association. sheehy, eugene p., ed. ( 1986 ). guide to reference books ( tenth ed. ). chicago and london : american library association. isbn 978 - 0 - 8389 - 0390 - 2.
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Bibliography of encyclopedias: biology
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pmod { l } } }, for a given prime l = 2, p { \ displaystyle l \ neq 2, p }. if a point ( x, y ) { \ displaystyle ( x, y ) } is in the l { \ displaystyle l } - torsion subgroup e [ l ] = { p ∈ e ( f q ) l p = o } { \ displaystyle e [ l ] = \ { p \ in e ( { \ bar { \ mathbb { f } _ { q } } } ) \ mid lp = o \ } }, then q p = q p { \ displaystyle qp = { \ bar { q } } p } where q { \ displaystyle { \ bar { q } } } is the unique integer such that q ≡ q ( mod l ) { \ displaystyle q \ equiv { \ bar { q } } { \ pmod { l } } } and q < l / 2 { \ displaystyle \ mid { \ bar { q } } \ mid < l / 2 }. note that ( o ) = o { \ displaystyle \ phi ( o ) = o } and that for any integer r { \ displaystyle r } we have r ( p ) = ( r p ) { \ displaystyle r \ phi ( p ) = \ phi ( rp ) }. thus ( p ) { \ displaystyle \ phi ( p ) } will have the same order as p { \ displaystyle p }. thus for ( x, y ) { \ displaystyle ( x, y ) } belonging to e [ l ] { \ displaystyle e [ l ] }, we also have t ( x q, y q ) = t ( x q, y q ) { \ displaystyle t ( x ^ { q }, y ^ { q } ) = { \ bar { t } } ( x ^ { q }, y ^ { q } ) } if t ≡ t ( mod l ) { \ displaystyle t \ equiv { \ bar { t } } { \ pmod { l } } }. hence we have reduced our problem to solving the equation ( x q 2, y q 2 ) + q ( x, y ) ≡ t ( x q, y q ), { \ displaystyle ( x ^ { q ^ { 2 } }, y ^ { q ^ {
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Schoof's algorithm
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##ities = = = this kind of singularities occurs when for a particular configuration, both a { \ displaystyle { \ textbf { a } } } and b { \ displaystyle { \ textbf { b } } } become singular simultaneously. = = references = =
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Mechanical singularity
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displaystyle \ rho _ { t } = | \ psi _ { t } \ rangle \ langle \ psi _ { t } |. } the quantum angle ( fubini – study distance ) is then d b ( ρ 0, ρ t ) = arccos | ⟨ ψ 0 | ψ t ⟩ | { \ displaystyle d _ { b } ( \ rho _ { 0 }, \ rho _ { t } ) = \ arccos | \ langle \ psi _ { 0 } | \ psi _ { t } \ rangle | } and so one concludes d b = arccos 0 = π / 2 { \ displaystyle d _ { b } = \ arccos 0 = \ pi / 2 } when the initial and final states are orthogonal. = = margolus – levitin limit = = for the case of a pure state, margolus and levitin obtain a different limit, that τ ≥ h 4 ⟨ e ⟩, { \ displaystyle \ tau _ { \ perp } \ geq { \ frac { h } { 4 \ langle e \ rangle } }, } where ⟨ e ⟩ { \ displaystyle \ langle e \ rangle } is the average energy, ⟨ e ⟩ = e avg = ⟨ ψ | h | ψ ⟩ = n | c n | 2 e n. { \ displaystyle \ langle e \ rangle = e _ { \ text { avg } } = \ langle \ psi | h | \ psi \ rangle = \ sum _ { n } | c _ { n } | ^ { 2 } e _ { n }. } this form applies when the hamiltonian is not time - dependent, and the ground - state energy is defined to be zero. = = = for time - varying states = = = the margolus – levitin theorem can also be generalized to the case where the hamiltonian varies with time, and the system is described by a mixed state. in this form, it is given by 0 τ | tr ( ρ t h t ) | d t ≥ d b ( ρ 0, ρ τ ) { \ displaystyle \ int _ { 0 } ^ { \ tau } | { \ text { tr } } ( \ rho _ { t } h _ { t } ) | dt \ geq \ hbar d _ { b } ( \ rho _ { 0 }, \ rho _ { \ tau
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Quantum speed limit
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the compound lever is attributed to the birmingham inventor john wyatt in 1743, when he designed a weighing machine that used four compound levers to transfer a load from a weighing platform to a central lever from which the weight could be measured. = = references = =
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Compound lever
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_ { - } ) } } \ left ( 2e _ { + } ^ { 2 } + 2e _ { - } ^ { 2 } - c ^ { 2 } \ mathbf { q } ^ { 2 } \ right ) \ right ]. \ \ \ end { aligned } } } with d ω + = sin θ + d θ +, d ω − = sin θ − d θ −. { \ displaystyle { \ begin { aligned } d \ omega _ { + } & = \ sin \ theta _ { + } \ d \ theta _ { + }, \ \ d \ omega _ { - } & = \ sin \ theta _ { - } \ d \ theta _ { - }. \ end { aligned } } } this expression can be derived by using a quantum mechanical symmetry between pair production and bremsstrahlung. z { \ displaystyle z } is the atomic number, α f i n e ≈ 1 / 137 { \ displaystyle \ alpha _ { fine } \ approx 1 / 137 } the fine structure constant, { \ displaystyle \ hbar } the reduced planck constant and c { \ displaystyle c } the speed of light. the kinetic energies e k i n, + / − { \ displaystyle e _ { kin, + / - } } of the positron and electron relate to their total energies e +, − { \ displaystyle e _ { +, - } } and momenta p +, − { \ displaystyle \ mathbf { p } _ { +, - } } via e +, − = e k i n, + / − + m e c 2 = m e 2 c 4 + p +, − 2 c 2. { \ displaystyle e _ { +, - } = e _ { kin, + / - } + m _ { e } c ^ { 2 } = { \ sqrt { m _ { e } ^ { 2 } c ^ { 4 } + \ mathbf { p } _ { +, - } ^ { 2 } c ^ { 2 } } }. } conservation of energy yields ω = e + + e −. { \ displaystyle \ hbar \ omega = e _ { + } + e _ { - }. } the momentum q { \ displaystyle \ mathbf { q } } of the virtual photon between incident photon and nucleus is : − q 2 = − | p + | 2 − | p − |
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Quantum mechanical scattering of photon and nucleus
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driven models have become an essential topic of discussion and exploration within water resources management and research. the term " data - driven modelling " ( ddm ) refers to the overarching paradigm of using historical data in conjunction with advanced computational techniques, including machine learning and artificial intelligence, to create models that can reveal underlying trends, patterns, and, in some cases, make predictions data - driven models can be built with or without detailed knowledge of the underlying processes governing the system behavior, which makes them particularly useful when such knowledge is missing or fragmented. = = references = =
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Data-driven model
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). under this conformal identification, the mapping f becomes multiplication by λ, a dilation on u. = = = proof = = = uniqueness. if k is another solution then, by analyticity, it suffices to show that k = h near 0. let h = k ∘ h − 1 ( z ) { \ displaystyle h = k \ circ h ^ { - 1 } ( z ) } near 0. thus h ( 0 ) = 0, h'( 0 ) = 1 and, for | z | small, λ h ( z ) = λ h ( k − 1 ( z ) ) = h ( f ( k − 1 ( z ) ) = h ( k − 1 ( λ z ) = h ( λ z ). { \ displaystyle \ lambda h ( z ) = \ lambda h ( k ^ { - 1 } ( z ) ) = h ( f ( k ^ { - 1 } ( z ) ) = h ( k ^ { - 1 } ( \ lambda z ) = h ( \ lambda z ) ~. } substituting into the power series for h, it follows that h ( z ) = z near 0. hence h = k near 0. existence. if f ( z ) = f ( z ) / λ z, { \ displaystyle f ( z ) = f ( z ) / \ lambda z, } then by the schwarz lemma | f ( z ) − 1 | ≤ ( 1 + | λ | − 1 ) | z |. { \ displaystyle | f ( z ) - 1 | \ leq ( 1 + | \ lambda | ^ { - 1 } ) | z | ~. } on the other hand, g n ( z ) = z j = 0 n − 1 f ( f j ( z ) ). { \ displaystyle g _ { n } ( z ) = z \ prod _ { j = 0 } ^ { n - 1 } f ( f ^ { j } ( z ) ) ~. } hence gn converges uniformly for | z | ≤ r by the weierstrass m - test since sup | z | ≤ r | 1 − f ∘ f j ( z ) | ≤ ( 1 + | λ | − 1 ) m ( r ) j < ∞. { \ displaystyle \ sum \ sup _ { | z | \ leq r } | 1 - f \ circ f ^ { j } ( z )
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Koenigs function
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( 2001 ) [ 1994 ], " almost - periodic function ", encyclopedia of mathematics, ems press bredikhina, e. a. ( 2001 ) [ 1994 ], " besicovitch almost - periodic functions ", encyclopedia of mathematics, ems press bredikhina, e. a. ( 2001 ) [ 1994 ], " bohr almost - periodic functions ", encyclopedia of mathematics, ems press bredikhina, e. a. ( 2001 ) [ 1994 ], " stepanov almost - periodic functions ", encyclopedia of mathematics, ems press bredikhina, e. a. ( 2001 ) [ 1994 ], " weyl almost - periodic functions ", encyclopedia of mathematics, ems press j. von neumann, " almost periodic functions in a group i ", trans. amer. math. soc., 36 no. 3 ( 1934 ) pp. 445 – 492 = = external links = = " almost periodic function ( equivalent definition ) ". planetmath.
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Almost periodic function
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2 } - 4p _ { + } ^ { 2 } p _ { - } ^ { 2 } \ sin ^ { 2 } \ theta _ { + } ) } } \ end { aligned } } } and a = z 2 α f i n e 3 c 2 ( 2 π ) 2 | p + | | p − | ω 3, δ 1 ( p ) : = − | p + | 2 − | p − | 2 − ( c ω ) + 2 c ω | p + | cos θ +, δ 2 ( p ) : = 2 c ω | p i | − 2 | p + | | p − | cos θ + + 2. { \ displaystyle { \ begin { aligned } a & = { \ frac { z ^ { 2 } \ alpha _ { fine } ^ { 3 } c ^ { 2 } } { ( 2 \ pi ) ^ { 2 } \ hbar } } { \ frac { | \ mathbf { p } _ { + } | | \ mathbf { p } _ { - } | } { \ omega ^ { 3 } } }, \ \ \ delta _ { 1 } ^ { ( p ) } & : = - | \ mathbf { p } _ { + } | ^ { 2 } - | \ mathbf { p } _ { - } | ^ { 2 } - \ left ( { \ frac { \ hbar } { c } } \ omega \ right ) + 2 { \ frac { \ hbar } { c } } \ omega | \ mathbf { p } _ { + } | \ cos \ theta _ { + }, \ \ \ delta _ { 2 } ^ { ( p ) } & : = 2 { \ frac { \ hbar } { c } } \ omega | \ mathbf { p } _ { i } | - 2 | \ mathbf { p } _ { + } | | \ mathbf { p } _ { - } | \ cos \ theta _ { + } + 2. \ end { aligned } } } this cross section can be applied in monte carlo simulations. an analysis of this expression shows that positrons are mainly emitted in the direction of the incident photon. = = references = =
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Quantum mechanical scattering of photon and nucleus
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y _ { 2 }'\ \ \ vdots \ \ y _ { k }'\ end { bmatrix } } = g _ { t } { \ begin { bmatrix } w _ { 1 } \ \ w _ { 2 } \ \ \ vdots \ \ w _ { k } \ end { bmatrix } } } where the y i ′ { \ displaystyle y _ { i }'} are the vectors formed by removing the first k { \ displaystyle k } coordinates of the vector y i { \ displaystyle y _ { i } }. = = decoding at the receiver = = each receiver, t ∈ t { \ displaystyle t \ in t }, gets k { \ displaystyle k } vectors y 1, …, y k { \ displaystyle y _ { 1 }, \ ldots, y _ { k } } which are random linear combinations of the v i { \ displaystyle v _ { i } } ’ s. in fact, if y i = ( α i 1, …, α i k, a i 1, …, a i d ) { \ displaystyle y _ { i } = ( \ alpha _ { i _ { 1 } }, \ ldots, \ alpha _ { i _ { k } }, a _ { i _ { 1 } }, \ ldots, a _ { i _ { d } } ) } then y i = 1 ≤ j ≤ k ( α i j v j ). { \ displaystyle y _ { i } = \ sum _ { 1 \ leq j \ leq k } ( \ alpha _ { ij } v _ { j } ). } thus we can invert the linear transformation to find the v i { \ displaystyle v _ { i } } ’ s with high probability. = = history = = krohn, freedman and mazieres proposed a theory in 2004 that if we have a hash function h : v g { \ displaystyle h : v \ longrightarrow g } such that : h { \ displaystyle h } is collision resistant – it is hard to find x { \ displaystyle x } and y { \ displaystyle y } such that h ( x ) = h ( y ) { \ displaystyle h ( x ) = h ( y ) } ; h { \ displaystyle h } is a homomorphism – h ( x + y ) = h ( x
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Homomorphic signatures for network coding
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: = = = proprietary network operating systems = = = cisco ios, a family of network operating systems used on cisco systems routers and network switches. ( earlier switches ran the catalyst operating system, or catos ) routeros by mikrotik zynos, used in network devices made by zyxel = = = freebsd, netbsd, openbsd, and linux - based operating systems = = = cisco nx - os, ios xe, and ios xr ; families of network operating systems used across various cisco systems device including the cisco nexus and cisco asr platforms junos os ; a network operating system that runs on juniper networks platforms cumulus linux distribution, which uses the full tcp / ip stack of linux dd - wrt, a linux kernel - based firmware for wireless routers and access points as well as low - cost networking device platforms such as the linksys wrt54g dell networking operating system ; dnos9 is netbsd based, while os10 uses the linux kernel extensible operating system runs on switches from arista and uses an unmodified linux kernel extremexos ( exos ), used in network devices made by extreme networks ftos ( force10 operating system ), the firmware family used on force10 ethernet switches onos, an open source sdn operating system ( hosted by linux foundation ) for communications service providers that is designed for scalability, high performance and high availability. openbsd, an open source operating system which includes its own implementations of bgp, rpki, ospf, mpls, vxlan, and other ietf standardized networking protocols, as well as firewall ( pf ) and load - balancing functionality. openwrt used to route ip packets on embedded devices pfsense, a fork of m0n0wall, which uses pf opnsense, a fork of pfsense sonic, a linux - based network operating system developed by microsoft vyos, an open source fork of the vyatta routing package = = see also = = distributed operating system frrouting interruptible operating system network computer operating system network functions virtualization operating system projects sonic ( operating system ) = = references = =
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Network operating system
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transformation : geometrically, there is only one positive definite real quadratic form of every dimension. its isometry group is a compact orthogonal group o ( n ). this stands in contrast with the case of isotropic forms, when the corresponding group, the indefinite orthogonal group o ( p, q ), is non - compact. further, the isometry groups of q and −q are the same ( o ( p, q ) ≈ o ( q, p ) ), but the associated clifford algebras ( and hence pin groups ) are different. = = definitions = = a quadratic form over a field k is a map q : v → k from a finite - dimensional k - vector space to k such that q ( av ) = a2q ( v ) for all a ∈ k, v ∈ v and the function q ( u + v ) − q ( u ) − q ( v ) is bilinear. more concretely, an n - ary quadratic form over a field k is a homogeneous polynomial of degree 2 in n variables with coefficients in k : q ( x 1, …, x n ) = i = 1 n j = 1 n a i j x i x j, a i j ∈ k. { \ displaystyle q ( x _ { 1 }, \ ldots, x _ { n } ) = \ sum _ { i = 1 } ^ { n } \ sum _ { j = 1 } ^ { n } a _ { ij } { x _ { i } } { x _ { j } }, \ quad a _ { ij } \ in k. } this formula may be rewritten using matrices : let x be the column vector with components x1,..., xn and a = ( aij ) be the n × n matrix over k whose entries are the coefficients of q. then q ( x ) = x t a x. { \ displaystyle q ( x ) = x ^ { \ mathsf { t } } ax. } a vector v = ( x1,..., xn ) is a null vector if q ( v ) = 0. two n - ary quadratic forms φ and ψ over k are equivalent if there exists a nonsingular linear transformation c ∈ gl ( n, k ) such that ψ ( x ) = φ ( c x ). { \ displaystyle \ psi ( x ) = \ varphi ( cx ). } let the
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Quadratic-linear algebra
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a software bus is a software architecture model where a shared communication channel facilitates connections and communication between software modules. this makes software buses conceptually similar to the bus term used in computer hardware for interconnecting pathways. in the early microcomputer era of the 1970s, digital research's operating system cp / m was often described as a software bus. lifeboat associates, an early distributor of cp / m and later of ms - dos software, had a whole product line named software bus. d - bus is used in many modern desktop environments to allow multiple processes to communicate with one another. = = examples = = lifeboat associates software bus - 80 aka sb - 80, a version of cp / m - 80 for 8080 / z80 8 - bit computers lifeboat associates software bus - 86 aka sb - 86, a version of ms - dos for x86 16 - bit computers. component object model for in - process and interprocess communication. d - bus for interprocess communication. enterprise service bus for distributed communication. = = see also = = bus ( computing ) = = references = = = = external links = = microsoft msdn : microsoft on the enterprise service bus ( esb )
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Software bus
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based drug design, both in developing the computational methods used and in providing a large experimental dataset used by some methods to provide insights about the function of a protein. = = structural classifications of proteins = = protein structures can be grouped based on their structural similarity, topological class or a common evolutionary origin. the structural classification of proteins database and cath database provide two different structural classifications of proteins. when the structural similarity is large the two proteins have possibly diverged from a common ancestor, and shared structure between proteins is considered evidence of homology. structure similarity can then be used to group proteins together into protein superfamilies. if shared structure is significant but the fraction shared is small, the fragment shared may be the consequence of a more dramatic evolutionary event such as horizontal gene transfer, and joining proteins sharing these fragments into protein superfamilies is no longer justified. topology of a protein can be used to classify proteins as well. knot theory and circuit topology are two topology frameworks developed for classification of protein folds based on chain crossing and intrachain contacts respectively. = = computational prediction of protein structure = = the generation of a protein sequence is much easier than the determination of a protein structure. however, the structure of a protein gives much more insight in the function of the protein than its sequence. therefore, a number of methods for the computational prediction of protein structure from its sequence have been developed. ab initio prediction methods use just the sequence of the protein. threading and homology modeling methods can build a 3 - d model for a protein of unknown structure from experimental structures of evolutionarily - related proteins, called a protein family. = = see also = = biomolecular structure gene structure nucleic acid structure pcrpi - db ribbon diagram 3d schematic representation of proteins = = references = = = = further reading = = 50 years of protein structure determination timeline - html version - national institute of general medical sciences archived 29 october 2018 at the wayback machine at nih = = external links = = media related to protein structures at wikimedia commons protein structure drugdesign. org [ 1 ] method _ for _ the _ characterization _ of _ the _ three - dimensional _ structure _ of _ proteins _ employing _ mass _ spectrometric _ analysis _ and _ experimental - computational _ feedback _ modeling [ 2 ] a _ method _ for _ the _ determination _ of _ the _ conformation _ ( topology ) _ of _ proteins _ employing _ experimental - computational _ feedback _ modeling
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Protein structure
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##ing the treatment sum of squares into non - overlapping additive components that represent the variation due to each contrast. consider the numbers above : each of the rows sums up to zero ( hence they are contrasts ). if we multiply each element of the first row by the corresponding element of the second row and add those up, this again results in zero, thus the first and second contrast are orthogonal and so on. = = sets of contrast = = orthogonal contrasts are a set of contrasts in which, for any distinct pair, the sum of the cross - products of the coefficients is zero ( assume sample sizes are equal ). although there are potentially infinite sets of orthogonal contrasts, within any given set there will always be a maximum of exactly k – 1 possible orthogonal contrasts ( where k is the number of group means available ). polynomial contrasts are a special set of orthogonal contrasts that test polynomial patterns in data with more than two means ( e. g., linear, quadratic, cubic, quartic, etc. ). orthonormal contrasts are orthogonal contrasts which satisfy the additional condition that, for each contrast, the sum squares of the coefficients add up to one. = = background = = a contrast is defined as the sum of each group mean multiplied by a coefficient for each group ( i. e., a signed number, cj ). in equation form, l = c 1 x 1 + c 2 x 2 + + c k x k ≡ j c j x j { \ displaystyle l = c _ { 1 } { \ bar { x } } _ { 1 } + c _ { 2 } { \ bar { x } } _ { 2 } + \ cdots + c _ { k } { \ bar { x } } _ { k } \ equiv \ sum _ { j } c _ { j } { \ bar { x } } _ { j } }, where l is the weighted sum of group means, the cj coefficients represent the assigned weights of the means ( these must sum to 0 for orthogonal contrasts ), and x { \ displaystyle { \ bar { x } } } j represents the group means. coefficients can be positive or negative, and fractions or whole numbers, depending on the comparison of interest. linear contrasts are very useful and can be used to test complex hypotheses when used in conjunction with anova or multiple regression. in essence, each contrast defines and tests for a particular pattern of differences among the means
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Contrast (statistics)
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\ wedge \ alpha _ { 2 } \ wedge \ cdots \ wedge \ alpha _ { k } ) ^ { \ sharp } = \ alpha _ { 1 } ^ { \ sharp } \ wedge \ alpha _ { 2 } ^ { \ sharp } \ wedge \ cdots \ wedge \ alpha _ { k } ^ { \ sharp }. } = = = hodge star = = = for an n - manifold m, the hodge star operator : ω k ( m ) → ω n − k ( m ) { \ displaystyle { \ star } : \ omega ^ { k } ( m ) \ rightarrow \ omega ^ { n - k } ( m ) } is a duality mapping taking a k { \ displaystyle k } - form α ∈ ω k ( m ) { \ displaystyle \ alpha \ in \ omega ^ { k } ( m ) } to an ( n − k ) { \ displaystyle ( n { - } k ) } - form ( α ) ∈ ω n − k ( m ) { \ displaystyle ( { \ star } \ alpha ) \ in \ omega ^ { n - k } ( m ) }. it can be defined in terms of an oriented frame ( x 1, …, x n ) { \ displaystyle ( x _ { 1 }, \ ldots, x _ { n } ) } for t m { \ displaystyle tm }, orthonormal with respect to the given metric tensor g { \ displaystyle g } : ( α ) ( x 1, …, x n − k ) = α ( x n − k + 1, …, x n ). { \ displaystyle ( { \ star } \ alpha ) ( x _ { 1 }, \ ldots, x _ { n - k } ) = \ alpha ( x _ { n - k + 1 }, \ ldots, x _ { n } ). } = = = co - differential operator = = = the co - differential operator δ : ω k ( m ) → ω k − 1 ( m ) { \ displaystyle \ delta : \ omega ^ { k } ( m ) \ rightarrow \ omega ^ { k - 1 } ( m ) } on an n { \ displaystyle n } dimensional manifold m { \ displaystyle m } is defined by δ : = ( − 1 ) k − 1 d = ( − 1 ) n k + n + 1 d.
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Exterior calculus identities
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individual pairs of springs around each element. consequently, the developed stiffness matrix has total effects from all pairs of springs, according to the stress situation around the element. this technique can be used in both load and displacement control cases. the 3d stiffness matrix may be deduced similarly. = = applications = = the applied element method is currently being used in the following applications : structural vulnerability assessment progressive collapse blast analysis impact analysis seismic analysis forensic engineering performance based design demolition analysis glass performance analysis visual effects = = see also = = building implosion earthquake engineering extreme loading for structures failure analysis multidisciplinary design optimization physics engine progressive collapse shear modulus structural engineering young's modulus = = references = = = = further reading = = applied element method extreme loading for structures - applied element method
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Applied element method
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predictions, once an algorithm for this task only needs estimation of p ( x i = 1 | x − x i = 0 ) { \ displaystyle p ( x _ { i } = 1 | \ mathbf { x } - { x _ { i } } = 0 ) } to produce recommendations. in particular, these estimates may be obtained by a direct lookup in a dependency network. predicting what movies a person will like based on his or her ratings of movies seen ; predicting what web pages a person will access based on his or her history on the site ; predicting what news stories a person is interested in based on other stories he or she read ; predicting what product a person will buy based on products he or she has already purchased and / or dropped into his or her shopping basket. another class of useful applications for dependency networks is related to data visualization, that is, visualization of predictive relationships. = = see also = = relational dependency network = = references = =
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Dependency network (graphical model)
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3 )... ψ π ( x p ) { \ displaystyle \ psi ( \ mathbf { x } _ { 1 }, \ mathbf { x } _ { 2 }, \ mathbf { x } _ { 3 },..., \ mathbf { x } _ { p } ) = \ psi _ { \ alpha } ( \ mathbf { x } _ { 1 } ) \ psi _ { \ beta } ( \ mathbf { x } _ { 2 } ) \ psi _ { \ gamma } ( \ mathbf { x } _ { 3 } )... \ psi _ { \ pi } ( \ mathbf { x } _ { p } ) } this hartree product gives us the wavefunction of a system ( many - particle ) as a combination of wavefunctions of the individual particles. it is inherently mean - field ( assumes the particles are independent ) and is the unsymmetrized version of the slater determinant ansatz in the hartree – fock method. although it has the advantage of simplicity, the hartree product is not satisfactory for fermions, such as electrons, because the resulting wave function is not antisymmetric. an antisymmetric wave function can be mathematically described using the slater determinant. = = derivation = = let's start from a hamiltonian of one atom with z electrons. the same method with some modifications can be expanded to a monoatomic crystal using the born – von karman boundary condition and to a crystal with a basis. h ^ = − 2 2 m i ∇ r i 2 − i z e 2 4 π 0 | r i | + 1 2 i = j e 2 4 π 0 | r i − r j | { \ displaystyle { \ hat { h } } = - { \ frac { \ hbar ^ { 2 } } { 2m } } \ sum _ { i } \ nabla _ { \ mathbf { r } _ { i } } ^ { 2 } - \ sum _ { i } { \ frac { ze ^ { 2 } } { 4 \ pi \ epsilon _ { 0 } | \ mathbf { r } _ { i } | } } + { \ frac { 1 } { 2 } } \ sum _ { i \ neq j } { \ frac { e ^ { 2 } } { 4 \ pi \ epsilon _
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Hartree equation
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log ( n ) ) } steps, so the algorithm is robust. = = = more examples = = = learning augmented algorithms are known for : the ski rental problem the maximum weight matching problem the weighted paging problem = = see also = = machine learning = = references = = = = external links = = an overview of publications about learning augmented algorithms
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Learning augmented algorithm
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resource allocation is the process by which a computing system aims to meet the hardware requirements of an application run by it. computing, networking and energy resources must be optimised taking into account hardware, performance and environmental restrictions. this process may be undertaken by the hardware itself, an operating system, a distributed computing system, or as part of data center management. = = see also = = concurrency control = = references = =
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Resource allocation (computer)
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= = further reading = = knuth, donald e. ( 1998 ), sorting and searching, the art of computer programming, vol. 3 ( 2nd ed. ), boston : addison - wesley, isbn 0 - 201 - 89685 - 0 sedgewick, robert ( 1980 ), " efficient sorting by computer : an introduction ", computational probability, new york : academic press, pp. 101 – 130, isbn 0 - 12 - 394680 - 8 = = external links = = sorting algorithm animations at the wayback machine ( archived 3 march 2015 ). sequential and parallel sorting algorithms – explanations and analyses of many sorting algorithms. dictionary of algorithms, data structures, and problems – dictionary of algorithms, techniques, common functions, and problems. slightly skeptical view on sorting algorithms – discusses several classic algorithms and promotes alternatives to the quicksort algorithm. 15 sorting algorithms in 6 minutes ( youtube ) – visualization and " audibilization " of 15 sorting algorithms in 6 minutes. a036604 sequence in oeis database titled " sorting numbers : minimal number of comparisons needed to sort n elements " – performed by ford – johnson algorithm. xisort – external merge sort with symbolic key transformation – a variant of merge sort applied to large datasets using symbolic techniques. sorting algorithms used on famous paintings ( youtube ) – visualization of sorting algorithms on many famous paintings. a comparison of sorting algorithms – runs a series of tests of 9 of the main sorting algorithms using python timeit and google colab.
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Sorting algorithm
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= = = = citations = = = = = = sources = = = = = further reading = = pare, anthony. " genre and identity ". the rhetoric and ideology of genre : strategies for stability and change. eds. richard m. coe, lorelei lingard, and tatiana teslenko. creskill, n. j. : hampton press, 2002. isbn 978 - 1572733848. sullivan, ceri ( 2007 ) " disposable elements? indications of genre in early modern titles ", modern language review 102. 3, pp. 641 – 653. = = external links = = genres of film at the internet movie database helping children understand literary genres archived 2021 - 10 - 16 at the wayback machine rhetorica genre archived 2021 - 02 - 24 at the wayback machine museum of broadcast communications archived 2013 - 05 - 14 at the wayback machine dictionary. com
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Gene
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a transformer ). optical isolation works by using an electronic signal to modulate a signal encoded by light transmission ( optical encoding ). the decoded light transmission is then used for input for the next stage of processing. = = = surge protection = = = a surge protector absorbs voltage spikes to protect the next stage from damage. = = references = =
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Signal conditioning
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{ \ text { avg } } } } \ right ) } levitin and toffoli also provide a bound for the average energy in terms of the maximum. for any pure state | ψ ⟩, { \ displaystyle \ left | \ psi \ right \ rangle, } the average energy is bounded as e max 4 ≤ e avg ≤ e max 2 { \ displaystyle { \ frac { e _ { \ text { max } } } { 4 } } \ leq e _ { \ text { avg } } \ leq { \ frac { e _ { \ text { max } } } { 2 } } } where e max { \ displaystyle e _ { \ text { max } } } is the maximum energy eigenvalue appearing in | ψ ⟩. { \ displaystyle \ left | \ psi \ right \ rangle. } ( this is the quarter - pinched sphere theorem in disguise, transported to complex projective space. ) thus, one has the bound π e max ≤ t ≤ 2 π e max { \ displaystyle { \ frac { \ pi \ hbar } { e _ { \ text { max } } } } \ leq t _ { \ perp } \ leq { \ frac { 2 \ pi \ hbar } { e _ { \ text { max } } } } } the strict lower bound e max t = π { \ displaystyle e _ { \ text { max } } t _ { \ perp } = \ pi \ hbar } is again attained for the qubit state | ψ q ⟩ { \ displaystyle \ left | \ psi _ { q } \ right \ rangle } with e max = e 1 { \ displaystyle e _ { \ text { max } } = e _ { 1 } }. = = bremermann's limit = = the quantum speed limit bounds establish an upper bound at which computation can be performed. computational machinery is constructed out of physical matter that follows quantum mechanics, and each operation, if it is to be unambiguous, must be a transition of the system from one state to an orthogonal state. suppose the computing machinery is a physical system evolving under hamiltonian that does not change with time. then, according to the margolus – levitin theorem, the number of operations per unit time per unit energy is bounded above by 2 π = 6 × 10 33 s − 1 ⋅ j − 1 { \
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Quantum speed limit
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frac { e [ x - \ gamma ] ^ { + } } { x - \ gamma } } } where [ ⋅ ] + = max { ⋅, 0 } { \ displaystyle [ \ cdot ] ^ { + } = \ max \ { \ cdot, 0 \ } }. bpoe can be expressed as the inverse function of cvar : p x ( x ) = { 1 − α | q α ( x ) = x } { \ displaystyle { \ bar { p } } _ { x } ( x ) = \ { 1 - \ alpha | { \ bar { q } } _ { \ alpha } ( x ) = x \ } }, where q α ( x ) { \ displaystyle { \ bar { q } } _ { \ alpha } ( x ) } is the cvar of x { \ displaystyle x } with confidence level α { \ displaystyle \ alpha }. = = references = =
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Buffered probability of exceedance
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they can become passive partner, shareholder or investor ( isafe notes is an instrument to invest without becoming partner / shareholder ) and infuse capital. these people invest to make money ( the dirty but sole objective of business ) = = bank financing = = this is practically an impossible method as the bank requires security and personal loan guarantees. = = = government aid = = = governments usually give out non - repayable grants to encourage start - ups. there are also investment tax credits that can be claimed. one thing to be careful about government aid is the lengthy procedures that may be required before obtaining the funds. = = = venture capital = = = venture capital is risk capital invested into a start - up company at its early stages. venture capitalists usually invest in start - ups that already have a relatively developed software product and some early sales. they look for products that have a large potential in a growing market with a competitive edge. venture capital help offers a large sum of money and often assistance in managing the company. people with software start - up experiences are available for mentoring. there are also help available in assisting the process of going public. = = see also = = entrepreneurship entrepreneurs = = references = =
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Software entrepreneurship
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orthogonal signal correction ( osc ) is a spectral preprocessing technique that removes variation from a data matrix x that is orthogonal to the response matrix y. osc was introduced by researchers at the university of umea in 1998 and has since found applications in domains including metabolomics. = = references = =
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Orthogonal signal correction
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_ { 1 }, x _ { 2 }, \ dots, x _ { d } ) } can be expressed in terms of its marginals f i ( x i ) = pr [ x i ≤ x i ] { \ displaystyle f _ { i } ( x _ { i } ) = \ pr [ x _ { i } \ leq x _ { i } ] } and a copula c { \ displaystyle c }. indeed : h ( x 1, …, x d ) = c ( f 1 ( x 1 ), …, f d ( x d ) ). { \ displaystyle h ( x _ { 1 }, \ dots, x _ { d } ) = c \ left ( f _ { 1 } ( x _ { 1 } ), \ dots, f _ { d } ( x _ { d } ) \ right ). } if the multivariate distribution has a density h { \ displaystyle h }, and if this density is available, it also holds that h ( x 1, …, x d ) = c ( f 1 ( x 1 ), …, f d ( x d ) ) ⋅ f 1 ( x 1 ) ⋅ ⋅ f d ( x d ), { \ displaystyle h ( x _ { 1 }, \ dots, x _ { d } ) = c ( f _ { 1 } ( x _ { 1 } ), \ dots, f _ { d } ( x _ { d } ) ) \ cdot f _ { 1 } ( x _ { 1 } ) \ cdot \ dots \ cdot f _ { d } ( x _ { d } ), } where c { \ displaystyle c } is the density of the copula. the theorem also states that, given h { \ displaystyle h }, the copula is unique on ran ( f 1 ) × × ran ( f d ) { \ displaystyle \ operatorname { ran } ( f _ { 1 } ) \ times \ cdots \ times \ operatorname { ran } ( f _ { d } ) } which is the cartesian product of the ranges of the marginal cdf's. this implies that the copula is unique if the marginals f i { \ displaystyle f _ { i } } are continuous. the converse is also true : given a copula c : [ 0, 1 ] d → [ 0, 1 ] { \ displaystyle c : [ 0,
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Copula (statistics)
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the big data institute ( bdi ), part of the li ka shing centre for health information and discovery, is an interdisciplinary research institute at the university of oxford. the institute brings together researchers from both the nuffield department of population health and the nuffield department of medicine. the bdi building is on the old road campus in headington, east oxford, england. academics from the bdi are advising the british government about a mobile phone app to track the covid - 19 pandemic in the united kingdom. = = references = = = = external links = = bdi website big data institute on twitter
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Big Data Institute
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non - adhesive elastic contact = = the classical theory of contact focused primarily on non - adhesive contact where no tension force is allowed to occur within the contact area, i. e., contacting bodies can be separated without adhesion forces. several analytical and numerical approaches have been used to solve contact problems that satisfy the no - adhesion condition. complex forces and moments are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated. in addition, the contact stresses are usually a nonlinear function of the deformation. to simplify the solution procedure, a frame of reference is usually defined in which the objects ( possibly in motion relative to one another ) are static. they interact through surface tractions ( or pressures / stresses ) at their interface. as an example, consider two objects which meet at some surface s { \ displaystyle s } in the ( x { \ displaystyle x }, y { \ displaystyle y } ) - plane with the z { \ displaystyle z } - axis assumed normal to the surface. one of the bodies will experience a normally - directed pressure distribution p z = p ( x, y ) = q z ( x, y ) { \ displaystyle p _ { z } = p ( x, y ) = q _ { z } ( x, y ) } and in - plane surface traction distributions q x = q x ( x, y ) { \ displaystyle q _ { x } = q _ { x } ( x, y ) } and q y = q y ( x, y ) { \ displaystyle q _ { y } = q _ { y } ( x, y ) } over the region s { \ displaystyle s }. in terms of a newtonian force balance, the forces : p z = s p ( x, y ) d a ; q x = s q x ( x, y ) d a ; q y = s q y ( x, y ) d a { \ displaystyle p _ { z } = \ int _ { s } p ( x, y ) ~ \ mathrm { d } a ~ ; ~ ~ q _ { x } = \ int _ { s } q _ { x } ( x, y ) ~ \ mathrm { d } a ~ ; ~ ~ q _ { y } = \ int _ { s } q _ { y } ( x, y ) ~ \ mathrm { d } a } must be equal and opposite to the forces
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Contact mechanics
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of pnps, there remain several critical challenges to their clinical translation. only two pnps have been fda - approved, despite over 50 pnp formulations to date ( 2022 ). the two fda - approved drugs include abraxane, an albumin nanoparticle carrying paclitaxel used for breast cancer, non - small cell lung cancer, and pancreatic cancer treatment. the second fda - approved pnp is ontak, a protein conjugate carrying l - 2 and diphtheria toxin used for cutaneous t - cell lymphoma. the two approved formulations are summarized in table 5 below. the low approval rate of pnps is due to limited existing control over drug encapsulation and the pharmacokinetic variability between pnp batches. balancing both the repeatability of these two properties and their relative interactions is important because it ensures the predictability of their clinical outcomes, greater patient safety, and that protein loading does not interfere with the pnp's properties. another limitation surrounds the cost and ability of large - scale production. many synthesis methods that can deliver greater homogeneity between produced nanoparticles are also more costly options or cannot achieve mass production. this limitation is compounded by the lower yields of pnp manufacturing. this limits the availability of pnps to broad clinical adoption [ 20, 29 ]. = = references = =
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Protein nanoparticles
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therefore, we have the following result : corollary : the concatenated code c ∗ { \ displaystyle c ^ { * } } is an asymptotically good code ( that is, rate r { \ displaystyle r } > 0 and relative distance δ { \ displaystyle \ delta } > 0 for small q ) and has a strongly explicit construction. = = an example of a justesen code = = the following slightly different code is referred to as the justesen code in macwilliams / macwilliams. it is the particular case of the above - considered justesen code for a very particular wonzencraft ensemble : let r be a reed - solomon code of length n = 2m − 1, rank k and minimum weight n − k + 1. the symbols of r are elements of f = gf ( 2m ) and the codewords are obtained by taking every polynomial ƒ over f of degree less than k and listing the values of ƒ on the non - zero elements of f in some predetermined order. let α be a primitive element of f. for a codeword a = ( a1,..., an ) from r, let b be the vector of length 2n over f given by b = ( a 1, a 1, a 2, α 1 a 2, …, a n, α n − 1 a n ) { \ displaystyle \ mathbf { b } = \ left ( a _ { 1 }, a _ { 1 }, a _ { 2 }, \ alpha ^ { 1 } a _ { 2 }, \ ldots, a _ { n }, \ alpha ^ { n - 1 } a _ { n } \ right ) } and let c be the vector of length 2n m obtained from b by expressing each element of f as a binary vector of length m. the justesen code is the linear code containing all such c. the parameters of this code are length 2m n, dimension m k and minimum distance at least i = 1 ℓ i ( 2 m i ), { \ displaystyle \ sum _ { i = 1 } ^ { \ ell } i { \ binom { 2m } { i } }, } where ℓ { \ displaystyle \ ell } is the greatest integer satisfying i = 1 ℓ ( 2 m i ) ≤ n − k + 1 { \ displaystyle \ sum _ { i = 1 } ^ { \ el
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Justesen code
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is analogous to " blobs of liquid condensing spontaneously out of a cooling gas ". this not only alters the understanding of black holes, but has the potential to explain the dark energy and dark matter that are indirectly observed. = = see also = = = = references = = = = sources = = chapline, george ( 2005 ). " dark energy stars ". proceedings of the texas symposium on relativistic astrophysics. p. 101. arxiv : astro - ph / 0503200. bibcode : 2005tsra. conf.. 101c. barbieri, j. ; chapline, g. ″ ( 2004 ). " have nucleon decays already been seen? ". physics letters b. 590 ( 1 – 2 ) : 8 – 12. bibcode : 2004phlb.. 590.... 8b. doi : 10. 1016 / j. physletb. 2004. 03. 054. chapline, george ; hohlfeld, e. ; laughlin, r. b. ; santiago, d. i. ( 2003 ). " quantum phase transitions and the failure of classical general relativity ". international journal of modern physics a. 18 ( 21 ) : 3587 – 3590. arxiv : gr - qc / 0012094. bibcode : 2003ijmpa.. 18. 3587c. doi : 10. 1142 / s0217751x03016380. s2cid 119456781. = = external links = = mpie galactic center research george chapline ( 28 march 2005 ). " black holes'do not exist'". nature news. ( subscription only )
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Dark-energy star
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1 ed. ). wiley. isbn 978 - 0 - 47194148 - 4. ( 404 pages ) schorr, herbert ; waite, william m. ( august 1967 ). " an efficient machine - independent procedure for garbage collection in various list structures " ( pdf ). communications of the acm. 10 ( 8 ) : 501 – 506. doi : 10. 1145 / 363534. 363554. s2cid 5684388. archived ( pdf ) from the original on 2021 - 01 - 22. wilson, paul r. ( 1992 ). " uniprocessor garbage collection techniques ". memory management. lecture notes in computer science. vol. 637. springer - verlag. pp. 1 – 42. citeseerx 10. 1. 1. 47. 2438. doi : 10. 1007 / bfb0017182. isbn 3 - 540 - 55940 - x. { { cite book } } : | journal = ignored ( help ) wilson, paul r. ; johnstone, mark s. ; neely, michael ; boles, david ( 1995 ). " dynamic storage allocation : a survey and critical review ". memory management. lecture notes in computer science. vol. 986 ( 1 ed. ). pp. 1 – 116. citeseerx 10. 1. 1. 47. 275. doi : 10. 1007 / 3 - 540 - 60368 - 9 _ 19. isbn 978 - 3 - 540 - 60368 - 9. { { cite book } } : | journal = ignored ( help ) = = external links = = the memory management reference the very basics of garbage collection java se 6 hotspot virtual machine garbage collection tuning tinygc - an independent implementation of the boehmgc api conservative garbage collection implementation for c language meixnergc - an incremental mark and sweep garbage collector for c + + using smart pointers
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Garbage collection (computer science)
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theorem extends to this context. positive - definite functions on groups occur naturally in the representation theory of groups on hilbert spaces ( i. e. the theory of unitary representations ). = = definition 2 = = alternatively, a function f : r n → r { \ displaystyle f : \ mathbb { r } ^ { n } \ to \ mathbb { r } } is called positive - definite on a neighborhood d of the origin if f ( 0 ) = 0 { \ displaystyle f ( 0 ) = 0 } and f ( x ) > 0 { \ displaystyle f ( x ) > 0 } for every non - zero x ∈ d { \ displaystyle x \ in d }. note that this definition conflicts with definition 1, given above. in physics, the requirement that f ( 0 ) = 0 { \ displaystyle f ( 0 ) = 0 } is sometimes dropped ( see, e. g., corney and olsen ). = = see also = = positive definiteness positive - definite kernel = = references = = christian berg, christensen, paul ressel. harmonic analysis on semigroups, gtm, springer verlag. z. sasvari, positive definite and definitizable functions, akademie verlag, 1994 wells, j. h. ; williams, l. r. embeddings and extensions in analysis. ergebnisse der mathematik und ihrer grenzgebiete, band 84. springer - verlag, new york - heidelberg, 1975. vii + 108 pp. = = notes = = = = external links = = " positive - definite function ", encyclopedia of mathematics, ems press, 2001 [ 1994 ]
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Positive-definite function
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protocol that has built its own networking stack is nostr, which has been designed to be simple for implementors to build as it has no dependencies on any existing standards. the protocol has gained some traction among newer snses, particularly within the cryptocurrency community. while many of these standards have been in use for both early and modern projects, some older projects typically used standards such as ostatus, xrds, portable contacts, the wave federation protocol, xmpp, opensocial, microformats like xfn and hcard, and atom web feeds. some of these standards were referred to as the open stack, due to their status as open standards. = = see also = = at protocol comparison of software and protocols for distributed social networking fediverse nostr = = references = = = = further reading = = = = external links = = w3c social activity w3c social web working group w3c social interest group federated social web conference 2011 video by henry story demonstrating foaf in an android environment buddycloud mastodon
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Distributed social network
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of a pendulum of length 1 m on earth ( g = 9. 80665 m / s2 ) at an initial angle of 10 degrees is 4 1 m g k ( sin 10 ∘ 2 ) ≈ 2. 0102 s. { \ displaystyle 4 { \ sqrt { \ frac { 1 { \ text { m } } } { g } } } \ k \ left ( \ sin { \ frac { 10 ^ { \ circ } } { 2 } } \ right ) \ approx 2. 0102 { \ text { s } }. } the linear approximation gives 2 π 1 m g ≈ 2. 0064 s. { \ displaystyle 2 \ pi { \ sqrt { \ frac { 1 { \ text { m } } } { g } } } \ approx 2. 0064 { \ text { s } }. } the difference between the two values, less than 0. 2 %, is much less than that caused by the variation of g with geographical location. from here there are many ways to proceed to calculate the elliptic integral. = = = legendre polynomial solution for the elliptic integral = = = given eq. 3 and the legendre polynomial solution for the elliptic integral : k ( k ) = π 2 n = 0 ∞ ( ( 2 n − 1 )!! ( 2 n )!! k n ) 2 { \ displaystyle k ( k ) = { \ frac { \ pi } { 2 } } \ sum _ { n = 0 } ^ { \ infty } \ left ( { \ frac { ( 2n - 1 )!! } { ( 2n )!! } } k ^ { n } \ right ) ^ { 2 } } where n!! denotes the double factorial, an exact solution to the period of a simple pendulum is : t = 2 π ℓ g ( 1 + ( 1 2 ) 2 sin 2 θ 0 2 + ( 1 ⋅ 3 2 ⋅ 4 ) 2 sin 4 θ 0 2 + ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) 2 sin 6 θ 0 2 + ) = 2 π ℓ g ⋅ n = 0 ∞ ( ( ( 2 n )! ( 2 n ⋅ n! ) 2 ) 2 ⋅ sin 2 n θ 0 2 ). { \ displaystyle { \ begin { alignedat } { 2 } t & = 2 \ pi { \ sqrt { \ frac { \ ell } { g } }
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Pendulum (mechanics)
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3 = = = this is another example of using the hilbert transform method to remove negative frequency components. nothing prevents us from computing s a ( t ) { \ displaystyle s _ { \ mathrm { a } } ( t ) } for a complex - valued s ( t ) { \ displaystyle s ( t ) }. but it might not be a reversible representation, because the original spectrum is not symmetrical in general. so except for this example, the general discussion assumes real - valued s ( t ) { \ displaystyle s ( t ) }. s ( t ) = e − j ω t { \ displaystyle s ( t ) = e ^ { - j \ omega t } }, where ω > 0 { \ displaystyle \ omega > 0 }. then : s ^ ( t ) = j e − j ω t, s a ( t ) = e − j ω t + j 2 e − j ω t = e − j ω t − e − j ω t = 0. { \ displaystyle { \ begin { aligned } { \ hat { s } } ( t ) & = je ^ { - j \ omega t }, \ \ s _ { \ mathrm { a } } ( t ) & = e ^ { - j \ omega t } + j ^ { 2 } e ^ { - j \ omega t } = e ^ { - j \ omega t } - e ^ { - j \ omega t } = 0. \ end { aligned } } } = = properties = = = = = instantaneous amplitude and phase = = = an analytic signal can also be expressed in polar coordinates : s a ( t ) = s m ( t ) e j ( t ), { \ displaystyle s _ { \ mathrm { a } } ( t ) = s _ { \ mathrm { m } } ( t ) e ^ { j \ phi ( t ) }, } where the following time - variant quantities are introduced : s m ( t ) | s a ( t ) | { \ displaystyle s _ { \ mathrm { m } } ( t ) \ triangleq | s _ { \ mathrm { a } } ( t ) | } is called the instantaneous amplitude or the envelope ; ( t ) arg [ s a ( t ) ] { \ displaystyle \ phi ( t ) \ triangleq \ arg \! \ left [ s _ { \ mathrm { a } }
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Analytic signal
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specified precisely by any given quantum state. in particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well, for then its position and momentum would both be completely determined to arbitrarily great precision. therefore, the lowest - energy state ( the ground state ) of the system must have a distribution in position and momentum that satisfies the uncertainty principle, which implies its energy must be greater than the minimum of the potential well. near the bottom of a potential well, the hamiltonian of a general system ( the quantum - mechanical operator giving its energy ) can be approximated as a quantum harmonic oscillator, h ^ = v 0 + 1 2 k ( x ^ − x 0 ) 2 + 1 2 m p ^ 2, { \ displaystyle { \ hat { h } } = v _ { 0 } + { \ tfrac { 1 } { 2 } } k \ left ( { \ hat { x } } - x _ { 0 } \ right ) ^ { 2 } + { \ frac { 1 } { 2m } } { \ hat { p } } ^ { 2 } \,, } where v0 is the minimum of the classical potential well. the uncertainty principle tells us that ⟨ ( x ^ − x 0 ) 2 ⟩ ⟨ p ^ 2 ⟩ ≥ 2, { \ displaystyle { \ sqrt { \ left \ langle \ left ( { \ hat { x } } - x _ { 0 } \ right ) ^ { 2 } \ right \ rangle } } { \ sqrt { \ left \ langle { \ hat { p } } ^ { 2 } \ right \ rangle } } \ geq { \ frac { \ hbar } { 2 } } \,, } making the expectation values of the kinetic and potential terms above satisfy ⟨ 1 2 k ( x ^ − x 0 ) 2 ⟩ ⟨ 1 2 m p ^ 2 ⟩ ≥ ( 4 ) 2 k m. { \ displaystyle \ left \ langle { \ tfrac { 1 } { 2 } } k \ left ( { \ hat { x } } - x _ { 0 } \ right ) ^ { 2 } \ right \ rangle \ left \ langle { \ frac { 1 } { 2m } } { \ hat { p } } ^ { 2 } \ right \ rangle \ geq \ left ( { \ frac { \ hbar } { 4
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Zero-point energy
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wikipedia
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) = p ( r = + | l = l, a = b ) a, b ∈ a l ∈ l { \ displaystyle p ( r = + \ | \ l = l, a = a ) = p ( r = + \ | \ l = l, a = b ) \ quad \ forall a, b \ in a \ quad \ forall l \ in l } = = = = definitions based on predicted and actual outcomes = = = = these definitions not only considers the predicted outcome r { \ textstyle r } but also compare it to the actual outcome y { \ textstyle y }. predictive parity, also referred to as outcome test. a classifier satisfies this definition if the subjects in the protected and unprotected groups have equal ppv. this is, if the following formula is satisfied : p ( y = + | r = +, a = a ) = p ( y = + | r = +, a = b ) a, b ∈ a { \ displaystyle p ( y = + \ | \ r = +, a = a ) = p ( y = + \ | \ r = +, a = b ) \ quad \ forall a, b \ in a } mathematically, if a classifier has equal ppv for both groups, it will also have equal fdr, satisfying the formula : p ( y = − | r = +, a = a ) = p ( y = − | r = +, a = b ) a, b ∈ a { \ displaystyle p ( y = - \ | \ r = +, a = a ) = p ( y = - \ | \ r = +, a = b ) \ quad \ forall a, b \ in a } false positive error rate balance, also referred to as predictive equality. a classifier satisfies this definition if the subjects in the protected and unprotected groups have equal fpr. this is, if the following formula is satisfied : p ( r = + | y = −, a = a ) = p ( r = + | y = −, a = b ) a, b ∈ a { \ displaystyle p ( r = + \ | \ y = -, a = a ) = p ( r = + \ | \ y = -, a = b ) \ quad \ forall a, b \ in a } mathematically, if a classifier
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Fairness (machine learning)
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wikipedia
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., and lewis, c. e., 1999. acoustic tree and wooden member imaging apparatus. patent wo 99 / 44050 ( 1999. 02. 27 ) rinn, f. ( 1999 ) : vorrichtung zur materialuntersuchung. / device for investigation materials. international patent pct / de00 / 01467 ( 1999. 05. 11 ). rust s. ; gocke, l. ( 2000 ) : a new tomographic device for the non - destructive testing of standing trees. in : proceedings of the 12th international symposium on nondestructive testing of wood. university of western hungary, sopron, 13 – 15 september 2000, 233 – 238. rust, s. ( 2001 ) : baumdiagnose ohne bohren. afz – der wald 56 : 924 – 925. rinn, f. ( 2003 ) : technische grundlagen der impuls - tomographie, baumzeitung ( 8 ) : 29 – 31. rabe, c., ferner, d., fink, s., schwarze, f. ( 2004 ) : detection of decay in trees with stress waves and interpretation of acoustic tomograms. arborcultural journal 28 ( 1 / 2 ) : 3 – 19 haaben, c., sander, c., hapla, f., 2006 : untersuchung der stammqualitat verschiedener laubholzarten mittels schallimpuls - tomographie. holztechnologie 47 ( 6 ) : 2 – 5
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Stress wave tomography
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wikipedia
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/ 20081203005457 / http : / / coe. berkeley. edu / students / engann08. pdf >. = = external links = = engineering engineering and science program at stanford university [ 1 ] what people go on to do in engineering science at uc berkeley [ 2 ] curriculum at university of florida [ 3 ] curriculum at mit [ 4 ] curriculum at university of illinois [ 5 ]
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Environmental engineering science
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wikipedia
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contexts containing one of these defining words. a publicly available database wordnet can be used as an automatic source for such defining terms. in addition, words that occur near the target word in great frequency can be selected as seed collocations representative. this approach is not fully automatic, a human judge must decide which word will be selected for each target word ’ s sense, the outputs will be reliable indicators of the senses. a decision list algorithm is then used to identify other reliable collocations. this training algorithm calculates the probability pr ( sense | collocation ), and the decision list is ranked by the log - likelihood ratio : log ( pr ( sense a collocation i ) pr ( sense b collocation i ) ) { \ displaystyle \ log \ left ( { \ frac { \ pr ( { \ text { sense } } _ { a } \ mid { \ text { collocation } } _ { i } ) } { \ pr ( { \ text { sense } } _ { b } \ mid { \ text { collocation } } _ { i } ) } } \ right ) } a smoothing algorithm will then be used to avoid 0 values. the decision - list algorithm resolves many problems in a large set of non - independent evidence source by using only the most reliable piece of evidence rather than the whole matching collocation set. the new resulting classifier will then be applied to the whole sample set. add those examples in the residual that are tagged as a or b with probability above a reasonable threshold to the seed sets. the decision - list algorithm and the above adding step are applied iteratively. as more newly - learned collocations are added to the seed sets, the sense a or sense b set will grow, and the original residual will shrink. however, these collocations stay in the seed sets only if their probability of classification remains above the threshold, otherwise they are returned to the residual for later classification. at the end of each iteration, the " one sense per discourse " property can be used to help preventing initially mistagged collocates and hence improving the purity of the seed sets. in order to avoid strong collocates becoming indicators for the wrong class, the class - inclusion threshold needs to be randomly altered. for the same purpose, after intermediate convergence the algorithm will also need to increase the width of the context window. the algorithm will continue to iterate until no more reliable collocations are found
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Yarowsky algorithm
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wikipedia
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). this sensing information is sufficient to allow algorithms based on learning automata to find a proper graph coloring with probability one. = = = computational complexity = = = graph coloring is computationally hard. it is np - complete to decide if a given graph admits a k - coloring for a given k except for the cases k ∈ { 0, 1, 2 }. in particular, it is np - hard to compute the chromatic number. the 3 - coloring problem remains np - complete even on 4 - regular planar graphs. on graphs with maximal degree 3 or less, however, brooks'theorem implies that the 3 - coloring problem can be solved in linear time. further, for every k > 3, a k - coloring of a planar graph exists by the four color theorem, and it is possible to find such a coloring in polynomial time. however, finding the lexicographically smallest 4 - coloring of a planar graph is np - complete. the best known approximation algorithm computes a coloring of size at most within a factor o ( n ( log log n ) 2 ( log n ) −3 ) of the chromatic number. for all ε > 0, approximating the chromatic number within n1−ε is np - hard. it is also np - hard to color a 3 - colorable graph with 5 colors, 4 - colorable graph with 7 colours, and a k - colorable graph with ( k k / 2 ) − 1 { \ displaystyle \ textstyle { \ binom { k } { \ lfloor k / 2 \ rfloor } } - 1 } colors for k ≥ 5. computing the coefficients of the chromatic polynomial is ♯p - hard. in fact, even computing the value of χ ( g, k ) { \ displaystyle \ chi ( g, k ) } is ♯p - hard at any rational point k except for k = 1 and k = 2. there is no fpras for evaluating the chromatic polynomial at any rational point k ≥ 1. 5 except for k = 2 unless np = rp. for edge coloring, the proof of vizing's result gives an algorithm that uses at most δ + 1 colors. however, deciding between the two candidate values for the edge chromatic number is np - complete. in terms of approximation algorithms, vizing's algorithm shows that the edge chromatic number can be approximated to within 4 / 3, and the hardness result shows that no ( 4 / 3 − ε
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Cole–Vishkin algorithm
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wikipedia
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phi } } { 2i _ { 1 } ^ { 2 } } } \,, \ quad k = { \ frac { mg \ ell } { i _ { 1 } } } \,, } a simpler form of the equation is obtained θ ¨ = cos θ sin 3 θ ( a cos 2 θ + b − c cos θ ) + 1 2 sin θ ( 2 a cos θ − c ) + k sin θ. { \ displaystyle { \ ddot { \ theta } } = { \ frac { \ cos \ theta } { \ sin ^ { 3 } \ theta } } ( a \ cos ^ { 2 } \ theta + b - c \ cos \ theta ) + { \ frac { 1 } { 2 \ sin \ theta } } ( 2a \ cos \ theta - c ) + k \ sin \ theta \,. } although the equation is highly nonlinear, there is only one equation to solve for, it was obtained directly, and the cyclic coordinates are not involved. by contrast, the lagrangian approach leads to three nonlinear coupled equations to solve, despite the absence of the coordinates ψ and φ in the lagrangian. the θ equation is d d t ∂ l ∂ θ = ∂ l ∂ θ ⇒ i 1 θ ¨ = ( i 1 − i 3 ) 2 sin θ cos θ − i 3 ψ sin θ + m g ℓ sin θ, { \ displaystyle { \ frac { d } { dt } } { \ frac { \ partial l } { \ partial { \ dot { \ theta } } } } = { \ frac { \ partial l } { \ partial \ theta } } \ quad \ rightarrow \ quad i _ { 1 } { \ ddot { \ theta } } = ( i _ { 1 } - i _ { 3 } ) { \ dot { \ phi } } ^ { 2 } \ sin \ theta \ cos \ theta - i _ { 3 } { \ dot { \ psi } } { \ dot { \ phi } } \ sin \ theta + mg \ ell \ sin \ theta \,, } the ψ equation is d d t ∂ l ∂ ψ = ∂ l ∂ ψ ⇒ ψ ¨ + ¨ cos θ − θ sin θ = 0, { \ displaystyle { \ frac { d } { dt } } { \ frac { \
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Routhian mechanics
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wikipedia
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credence or degree of belief is a statistical term that expresses how much a person believes that a proposition is true. as an example, a reasonable person will believe with close to 50 % credence that a fair coin will land on heads the next time it is flipped ( minus the probability that the coin lands on its edge ). if the prize for correctly predicting the coin flip is $ 100, then a reasonable risk - neutral person will wager $ 49 on heads, but will not wager $ 51 on heads. credence is a measure of belief strength, expressed as a percentage. credence values range from 0 % to 100 %. credence is closely related to odds, and a person's level of credence is directly related to the odds at which they will place a bet. credence is especially important in bayesian statistics. if a bag contains 4 red marbles and 1 blue marble, and a person withdraws one marble at random, then they should believe with 80 % credence that the random marble will be red. in this example, the probability of drawing a red marble is 80 %. credence values can be based entirely on subjective feelings. for example, if alice is fairly certain that she saw bob at the grocery store on monday, then she might say, " i believe with 90 % credence that bob was at the grocery store on monday. " if the prize for being correct is $ 100, then alice will wager $ 89 that her memory is accurate, but she would not be willing to wager $ 91 or more. given that alice is 90 % credent, this level of belief can be expressed as gambling odds in the following ways : 90 % credence 1 / 9 fractional odds ( 1 to 9 ) 1. 11 decimal odds −900 moneyline odds the return on a $ 100 wager is $ 11. 11 ( plus the $ 100 initial wager ). see the article odds for conversion equations. = = references = =
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Credence (statistics)
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wikipedia
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difference between the number of messages received and the number of signals sent in reply. when a node wishes to terminate, it waits until it has received signals from outgoing edges reducing their deficits to zero. then it sends enough signals to ensure that the deficit is zero on each incoming edge. since the graph is acyclic, some nodes will have no outgoing edges and these nodes will be the first to terminate after sending enough signals to their incoming edges. after that the nodes at higher levels will terminate level by level. = = dijkstra – scholten algorithm for cyclic directed graphs = = if cycles are allowed, the previous algorithm does not work. this is because, there may not be any node with zero outgoing edges. so, potentially there is no node which can terminate without consulting other nodes. the dijkstra – scholten algorithm solves this problem by implicitly creating a spanning tree of the graph. a spanning - tree is a tree which includes each node of the underlying graph once and the edge - set is a subset of the original set of edges. the tree will be directed ( i. e., the channels will be directed ) with the source node ( which initiates the computation ) as the root. the spanning - tree is created in the following way. a variable first _ edge is added to each node. when a node receives a message for the first time, it initializes first _ edge with the edge through which it received the message. first _ edge is never changed afterwards. note that, the spanning tree is not unique and it depends on the order of messages in the system. termination is handled by each node in three steps : send signals on all incoming edges except the first edge. ( each node will send signals which reduces the deficit on each incoming edge to zero. ) wait for signals from all outgoing edges. ( the number of signals received on each outgoing edge should reduce each of their deficits to zero. ) send signals on first _ edge. ( once steps 1 and 2 are complete, a node informs its parent in the spanning tree about its intention of terminating. ) = = see also = = huang's algorithm = = references = =
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Dijkstra–Scholten algorithm
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wikipedia
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y = y. more generally, because mag is algebraic, it is a complete category. an important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the ( constant sequence of the ) endomorphism. = = see also = = universal algebra magma computer algebra system, named after the object of this article. commutative magma algebraic structures whose axioms are all identities groupoid algebra hall set = = references = = = = further reading = = bruck, richard hubert ( 1971 ), a survey of binary systems ( 3rd ed. ), springer, isbn 978 - 0 - 387 - 03497 - 3
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Magma (algebra)
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wikipedia
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. several " linear delta " arrangements have been developed where the motors drive linear actuators rather than rotating an arm. such linear delta arrangements can have much larger working volumes than rotational delta arrangements. the majority of delta robots use rotary actuators. vertical linear actuators have recently been used ( using a linear delta design ) to produce a novel design of 3d printer. these offer advantages over conventional leadscrew - based 3d printers of quicker access to a larger build volume for a comparable investment in hardware. = = applications = = industries that take advantage of the high speed of delta robots are the food, pharmaceutical and electronics industry. for its stiffness it is also used for surgery, in particular, the surgiscope is a delta robot used as a microscopic holder system. the structure of a delta robot can also be used to create haptic controllers. more recently, the technology has been adapted to 3d printers. = = references = =
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Delta robot
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wikipedia
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