text stringlengths 11 320k | source stringlengths 26 161 |
|---|---|
Q-Chem is a general-purpose electronic structure package [ 1 ] [ 2 ] [ 3 ] [ 4 ] featuring a variety of established and new methods implemented using innovative algorithms that enable fast calculations of large systems on various computer architectures, from laptops and regular lab workstations to midsize clusters, HPCC, and cloud computing using density functional and wave-function based approaches. It offers an integrated graphical interface and input generator; a large selection of functionals and correlation methods, including methods for electronically excited states and open-shell systems; solvation models; and wave-function analysis tools. In addition to serving the computational chemistry [ 5 ] community, Q-Chem also provides a versatile code development platform.
Q-Chem software is maintained and distributed by Q-Chem, Inc., [ 6 ] located in Pleasanton, California, USA. It was founded in 1993 as a result of disagreements within the Gaussian company that led to the departure (and subsequent "banning") of John Pople and a number of his students and postdocs (see Gaussian License Controversy [ 7 ] ). [ 6 ] [ 8 ]
The first lines of the Q-Chem code were written by Peter Gill , at that time a postdoc of Pople , during a winter vacation (December 1992) in Australia. Gill was soon joined by Benny Johnson (a Pople graduate student) and Carlos Gonzalez (another Pople postdoc), but the latter left the company shortly thereafter. In mid-1993, Martin Head-Gordon , formerly a Pople student, but at that time on the Berkeley tenure track, joined the growing team of academic developers. [ 6 ] [ 8 ]
In preparation for the first commercial release, the company hired Eugene Fleischmann as marketing director and acquired its URL www.q-chem.com in January 1997. The first commercial product, Q-Chem 1.0, was released in March 1997. Advertising postcards celebrated the release with the proud headline, "Problems which were once impossible are now routine"; however, version 1.0 had many shortcomings, and a wit once remarked that the words "impossible" and "routine" should probably be interchanged! [ 8 ] However, vigorous code development continued, and by the following year Q-Chem 1.1 was able to offer most of the basic quantum chemical functionality as well as a growing list of features (the continuous fast multipole method, J-matrix engine, COLD PRISM for integrals, and G96 density functional , for example) that were not available in any other package. [ 6 ] [ 8 ]
Following a setback when Johnson left, the company became more decentralized, establishing and cultivating relationships with an ever-increasing circle of research groups in universities around the world. In 1998, Fritz Schaefer accepted an invitation to join the Board of Directors and, early in 1999, as soon as his non-compete agreement with Gaussian had expired, John Pople joined as both a Director and code developer. [ 6 ] [ 8 ]
In 2000, Q-Chem established a collaboration with Wavefunction Inc., which led to the incorporation of Q-Chem as the ab initio engine in all subsequent versions of the Spartan package. The Q-Chem Board was expanded in March 2003 with the addition of Anna Krylov and Jing Kong. In 2012, John Herbert joined the Board and Fritz Schaefer became a Member Emeritus. The following year, Shirin Faraji joined the Board; Peter Gill , who had been President of Q-Chem since 1988, stepped down; and Anna Krylov became the new president.
In 2022-23 Yuezhi Mao and Joonho Lee joined the board.
The active Board of Directors currently consists of Lee, Mao, Faraji, Gill (past-President), Herbert, Krylov (President), and Hilary Pople ( John's daughter). Martin Head-Gordon remains a Scientific Advisor to the Board. [ 6 ] [ 8 ]
Currently, there are thousands of Q-Chem licenses in use, and Q-Chem's user base is expanding, as illustrated by citation records for releases 2.0, 3.0, and 4.0, which reached 400 per year in 2016 (see Figure 2). [ 8 ]
Q-Chem has been used as an engine in high-throughput studies, such as the Harvard Clean Energy Project , [ 9 ] in which about 350,000 calculations were performed daily on the IBM World Community Grid .
Innovative algorithms and new approaches to electronic structure have been enabling cutting-edge scientific discoveries. This transition, from in-house code to major electronic structure engine, has become possible due to contributions from numerous scientific collaborators; the Q-Chem business model encourages broad developer participation. Q-Chem defines its genre as open-teamware: [ 8 ] its source code is open to a large group of developers. In addition, some Q-Chem modules are distributed as open source. [ 8 ] Since 1992, over 400 man- (and woman-) years have been devoted to code development. Q-Chem 5.2.2, released in December 2019, consists of 7.5 million lines of code, which includes contributions by more than 300 active developers (current estimate is 312). [ 6 ] [ 8 ] See Figure 3.
Q-Chem can perform a number of general quantum chemistry calculations, such as Hartree–Fock , density functional theory (DFT) including time-dependent DFT ( TDDFT ), Møller–Plesset perturbation theory (MP2), coupled cluster (CC), equation-of-motion coupled-cluster (EOM-CC), [ 10 ] [ 11 ] [ 12 ] configuration interaction (CI), algebraic diagrammatic construction (ADC), and other advanced electronic structure methods. Q-Chem also includes QM/MM functionality. Q-Chem 4.0 and higher releases come with the graphical user interface, IQMol, which includes a hierarchical input generator, a molecular builder, and general visualization capabilities (MOs, densities, molecular vibrations, reaction pathways, etc.). IQMol is developed by Andrew Gilbert (in coordination with Q-Chem) and is distributed as free open-source software. IQmol is written using the Qt libraries, enabling it to run on a range of platforms, including OS X, Widows, and Linux. It provides an intuitive environment to set up, run, and analyze Q-Chem calculations. It can also read and display a variety of file formats, including the widely available formatted checkpoint format. A complete, up-to-date list of features is published on the Q-Chem website and in the user manual. [ 6 ]
In addition, Q-Chem is interfaced with WebMO and is used as the computing engine in Spartan , or as a back-end to CHARMM , GROMACS , NAMD , and ChemShell. Other popular visualization programs such as Jmol and Molden can also be used.
In 2018, Q-Chem established a partnership with BrianQC, produced by StreamNovation, Ltd., a new integral engine exploiting the computational power of GPUs. The BrianQC plug-in speeds up Q-Chem calculations by taking advantage of GPUs on mixed architectures, which is highly efficient for simulating large molecules and extended systems. BrianQC is the first GPU Quantum Chemistry software capable of calculating high angular momentum orbitals.
Beginning with Q-Chem 2.0 only major releases versions are shown. | https://en.wikipedia.org/wiki/Q-Chem |
The q -Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution . The q -Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy . [ 1 ] The normal distribution is recovered as q → 1.
The q -Gaussian has been applied to problems in the fields of statistical mechanics , geology , anatomy , astronomy , economics , finance , and machine learning . [ citation needed ] The distribution is often favored for its heavy tails in comparison to the Gaussian for 1 < q < 3. For q < 1 {\displaystyle q<1} the q -Gaussian distribution is the PDF of a bounded random variable . This makes in biology and other domains [ 2 ] the q -Gaussian distribution more suitable than Gaussian distribution to model the effect of external stochasticity. A generalized q -analog of the classical central limit theorem [ 3 ] was proposed in 2008, in which the independence constraint for the i.i.d. variables is relaxed to an extent defined by the q parameter, with independence being recovered as q → 1. However, a proof of such a theorem is still lacking. [ 4 ]
In the heavy tail regions, the distribution is equivalent to the Student's t -distribution with a direct mapping between q and the degrees of freedom . A practitioner using one of these distributions can therefore parameterize the same distribution in two different ways. The choice of the q -Gaussian form may arise if the system is non-extensive , or if there is lack of a connection to small samples sizes.
The standard q -Gaussian has the probability density function [ 3 ]
where
is the q -exponential and the normalization factor C q {\displaystyle C_{q}} is given by
Note that for q < 1 {\displaystyle q<1} the q -Gaussian distribution is the PDF of a bounded random variable .
For 1 < q < 3 {\displaystyle 1<q<3} cumulative density function is [ 5 ]
where 2 F 1 ( a , b ; c ; z ) {\displaystyle {}_{2}F_{1}(a,b;c;z)} is the hypergeometric function . As the hypergeometric function is defined for | z | < 1 but x is unbounded, Pfaff transformation could be used.
For q < 1 {\displaystyle q<1} , F ( x ) = { 0 x < − 1 β ( 1 − q ) , 1 2 + 1 − q Γ ( 5 − 3 q 2 ( 1 − q ) ) x β 2 F 1 ( 1 2 , 1 q − 1 ; 3 2 ; − ( q − 1 ) β x 2 ) π Γ ( 2 − q 1 − q ) − 1 β ( 1 − q ) < x < 1 β ( 1 − q ) , 1 x > 1 β ( 1 − q ) . {\displaystyle F(x)={\begin{cases}0&x<-{\frac {1}{\sqrt {\beta (1-q)}}},\\{\frac {1}{2}}+{\frac {{\sqrt {1-q}}\,\Gamma \left({5-3q \over 2(1-q)}\right)x{\sqrt {\beta }}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{q-1}};{\tfrac {3}{2}};-(q-1)\beta x^{2}\right)}{{\sqrt {\pi }}\,\Gamma \left({2-q \over 1-q}\right)}}&-{\frac {1}{\sqrt {\beta (1-q)}}}<x<{\frac {1}{\sqrt {\beta (1-q)}}},\\1&x>{\frac {1}{\sqrt {\beta (1-q)}}}.\end{cases}}}
Just as the normal distribution is the maximum information entropy distribution for fixed values of the first moment E ( X ) {\displaystyle \operatorname {E} (X)} and second moment E ( X 2 ) {\displaystyle \operatorname {E} (X^{2})} (with the fixed zeroth moment E ( X 0 ) = 1 {\displaystyle \operatorname {E} (X^{0})=1} corresponding to the normalization condition), the q -Gaussian distribution is the maximum Tsallis entropy distribution for fixed values of these three moments.
While it can be justified by an interesting alternative form of entropy, statistically it is a scaled reparametrization of the Student's t -distribution introduced by W. Gosset in 1908 to describe small-sample statistics. In Gosset's original presentation the degrees of freedom parameter ν was constrained to be a positive integer related to the sample size, but it is readily observed that Gosset's density function is valid for all real values of ν . [ citation needed ] The scaled reparametrization introduces the alternative parameters q and β which are related to ν .
Given a Student's t -distribution with ν degrees of freedom, the equivalent q -Gaussian has
with inverse
Whenever β ≠ 1 3 − q {\displaystyle \beta \neq {1 \over {3-q}}} , the function is simply a scaled version of Student's t -distribution.
It is sometimes argued that the distribution is a generalization of Student's t -distribution to negative and or non-integer degrees of freedom. However, the theory of Student's t -distribution extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of ν < 0. [ citation needed ]
As with many distributions centered on zero, the q -Gaussian can be trivially extended to include a location parameter μ . The density then becomes defined by
The Box–Muller transform has been generalized to allow random sampling from q -Gaussians. [ 6 ] The standard Box–Muller technique generates pairs of independent normally distributed variables from equations of the following form.
The generalized Box–Muller technique can generates pairs of q -Gaussian deviates that are not independent. In practice, only a single deviate will be generated from a pair of uniformly distributed variables. The following formula will generate deviates from a q -Gaussian with specified parameter q and β = 1 3 − q {\displaystyle \beta ={1 \over {3-q}}}
where ln q {\displaystyle {\text{ ln}}_{q}} is the q -logarithm and q ′ = 1 + q 3 − q {\displaystyle q'={{1+q} \over {3-q}}}
These deviates can be transformed to generate deviates from an arbitrary q -Gaussian by
It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a q -Gaussian. [ 7 ]
The q -Gaussian distribution is also obtained as the asymptotic probability density function of the position of the unidimensional motion of a mass subject to two forces: a deterministic force of the type F 1 ( x ) = − 2 x / ( 1 − x 2 ) {\textstyle F_{1}(x)=-2x/(1-x^{2})} (determining an infinite potential well) and a stochastic white noise force F 2 ( t ) = 2 ( 1 − q ) ξ ( t ) {\textstyle F_{2}(t)={\sqrt {2(1-q)}}\xi (t)} , where ξ ( t ) {\displaystyle \xi (t)} is a white noise . Note that in the overdamped/small mass approximation the above-mentioned convergence fails for q < 0 {\displaystyle q<0} , as recently shown. [ 8 ]
Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere have been interpreted as q -Gaussians. [ 9 ] [ 10 ] | https://en.wikipedia.org/wiki/Q-Gaussian_distribution |
CubeSat Particle Aggregation and Collision Experiment ( Q-PACE ) or Cu-PACE , [ 4 ] was an orbital spacecraft mission that would have studied the early stages of proto-planetary accretion by observing particle dynamical aggregation for several years. [ 5 ]
Current hypotheses have trouble explaining how particles can grow larger than a few centimeters. This is called the meter size barrier . This mission was selected in 2015 as part of NASA's ELaNa program, and it was launched on 17 January 2021. [ 6 ] As of March 2021, however, contact has yet to be established with the satellite, and the mission was feared to be lost. The mission was eventually terminated.
Q-PACE was led by Joshua Colwell at the University of Central Florida and was selected NASA's CubeSat Launch Initiative (CSLI) which placed it on Educational Launch of Nanosatellites ELaNa XX. [ 7 ] The development of the mission was funded through NASA's Small Innovative Missions for Planetary Exploration (SIMPLEx) program. [ 5 ] [ 8 ]
Observations of the collisional evolution and accretion of particles in a microgravity environment are necessary to elucidate the processes that lead to the formation of planetesimals (the building blocks of planets), km-size, and larger bodies, within the protoplanetary disk . The current hypotheses of planetesimal formation have difficulties in explaining how particles grow beyond one centimeter in size, so repeated experimentation in relevant conditions is necessary. [ 9 ]
Q-PACE was to explore the fundamental properties of low‐velocity (< 10 cm/s (3.9 in/s)) particle collisions in a microgravity environment in an effort to better understand accretion in the protoplanetary disk . [ 10 ] Several precursor tests and flight missions were performed in suborbital flights as well as in the International Space Station . [ 1 ] [ 11 ] The small spacecraft does not need accurate pointing or propulsion, which simplified the design.
On 17 January 2021, Q-PACE launched on a Virgin Orbit Launcher One , an air launch to orbit rocket that was dropped from the Cosmic Girl airplane over the Pacific Ocean . [ 12 ] As of March 2021, however, contact was not established with the satellite after it reached orbit, [ 13 ] and the spacecraft was declared lost and the mission ended.
The main objective of Q-PACE was to understand protoplanetary growth from pebbles to boulders by performing long-duration microgravity collision experiments. The specific goals are: [ 1 ]
Q-PACE was a 3U CubeSat with a collision test chamber and several particle reservoirs that contain meteoritic chondrules , dust particles, dust aggregates, and larger spherical particles. Particles will be introduced into the test chamber for a series of separate experimental runs.
The scientists designed a series of experiments involving a broad range of particle size, density, surface properties, and collision velocities to observe collisional outcomes from bouncing to sticking as well as aggregate disruption in tens of thousands of collisions. [ 9 ] [ 14 ] The test chamber will be mechanically agitated to induce collisions that will be recorded by on‐board video for downlink and analysis. [ 10 ] Long duration microgravity allows a very large number of collisions to be studied and produce statistically significant data. [ 1 ] | https://en.wikipedia.org/wiki/Q-PACE |
In the mathematical field of combinatorics , the q -Pochhammer symbol , also called the q -shifted factorial , is the product ( a ; q ) n = ∏ k = 0 n − 1 ( 1 − a q k ) = ( 1 − a ) ( 1 − a q ) ( 1 − a q 2 ) ⋯ ( 1 − a q n − 1 ) , {\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}),} with ( a ; q ) 0 = 1. {\displaystyle (a;q)_{0}=1.} It is a q -analog of the Pochhammer symbol ( x ) n = x ( x + 1 ) … ( x + n − 1 ) {\displaystyle (x)_{n}=x(x+1)\dots (x+n-1)} , in the sense that lim q → 1 ( q x ; q ) n ( 1 − q ) n = ( x ) n . {\displaystyle \lim _{q\to 1}{\frac {(q^{x};q)_{n}}{(1-q)^{n}}}=(x)_{n}.} The q -Pochhammer symbol is a major building block in the construction of q -analogs; for instance, in the theory of basic hypergeometric series , it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series .
Unlike the ordinary Pochhammer symbol, the q -Pochhammer symbol can be extended to an infinite product: ( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 − a q k ) . {\displaystyle (a;q)_{\infty }=\prod _{k=0}^{\infty }(1-aq^{k}).} This is an analytic function of q in the interior of the unit disk , and can also be considered as a formal power series in q . The special case ϕ ( q ) = ( q ; q ) ∞ = ∏ k = 1 ∞ ( 1 − q k ) {\displaystyle \phi (q)=(q;q)_{\infty }=\prod _{k=1}^{\infty }(1-q^{k})} is known as Euler's function , and is important in combinatorics , number theory , and the theory of modular forms .
The finite product can be expressed in terms of the infinite product: ( a ; q ) n = ( a ; q ) ∞ ( a q n ; q ) ∞ , {\displaystyle (a;q)_{n}={\frac {(a;q)_{\infty }}{(aq^{n};q)_{\infty }}},} which extends the definition to negative integers n . Thus, for nonnegative n , one has ( a ; q ) − n = 1 ( a q − n ; q ) n = ∏ k = 1 n 1 ( 1 − a / q k ) {\displaystyle (a;q)_{-n}={\frac {1}{(aq^{-n};q)_{n}}}=\prod _{k=1}^{n}{\frac {1}{(1-a/q^{k})}}} and ( a ; q ) − n = ( − q / a ) n q n ( n − 1 ) / 2 ( q / a ; q ) n . {\displaystyle (a;q)_{-n}={\frac {(-q/a)^{n}q^{n(n-1)/2}}{(q/a;q)_{n}}}.} Alternatively, ∏ k = n ∞ ( 1 − a q k ) = ( a q n ; q ) ∞ = ( a ; q ) ∞ ( a ; q ) n , {\displaystyle \prod _{k=n}^{\infty }(1-aq^{k})=(aq^{n};q)_{\infty }={\frac {(a;q)_{\infty }}{(a;q)_{n}}},} which is useful for some of the generating functions of partition functions.
The q -Pochhammer symbol is the subject of a number of q -series identities, particularly the infinite series expansions ( x ; q ) ∞ = ∑ n = 0 ∞ ( − 1 ) n q n ( n − 1 ) / 2 ( q ; q ) n x n {\displaystyle (x;q)_{\infty }=\sum _{n=0}^{\infty }{\frac {(-1)^{n}q^{n(n-1)/2}}{(q;q)_{n}}}x^{n}} and 1 ( x ; q ) ∞ = ∑ n = 0 ∞ x n ( q ; q ) n , {\displaystyle {\frac {1}{(x;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {x^{n}}{(q;q)_{n}}},} which are both special cases of the q -binomial theorem : ( a x ; q ) ∞ ( x ; q ) ∞ = ∑ n = 0 ∞ ( a ; q ) n ( q ; q ) n x n . {\displaystyle {\frac {(ax;q)_{\infty }}{(x;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}x^{n}.} Fridrikh Karpelevich found the following identity (see Olshanetsky and Rogov ( 1995 ) for the proof): ( q ; q ) ∞ ( z ; q ) ∞ = ∑ n = 0 ∞ ( − 1 ) n q n ( n + 1 ) / 2 ( q ; q ) n ( 1 − z q − n ) , | z | < 1. {\displaystyle {\frac {(q;q)_{\infty }}{(z;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}q^{n(n+1)/2}}{(q;q)_{n}(1-zq^{-n})}},\ |z|<1.}
The q -Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of q m a n {\displaystyle q^{m}a^{n}} in ( a ; q ) ∞ − 1 = ∏ k = 0 ∞ ( 1 − a q k ) − 1 {\displaystyle (a;q)_{\infty }^{-1}=\prod _{k=0}^{\infty }(1-aq^{k})^{-1}} is the number of partitions of m into at most n parts.
Since, by conjugation of partitions, this is the same as the number of partitions of m into parts of size at most n , by identification of generating series we obtain the identity ( a ; q ) ∞ − 1 = ∑ k = 0 ∞ ( ∏ j = 1 k 1 1 − q j ) a k = ∑ k = 0 ∞ a k ( q ; q ) k {\displaystyle (a;q)_{\infty }^{-1}=\sum _{k=0}^{\infty }\left(\prod _{j=1}^{k}{\frac {1}{1-q^{j}}}\right)a^{k}=\sum _{k=0}^{\infty }{\frac {a^{k}}{(q;q)_{k}}}} as in the above section.
We also have that the coefficient of q m a n {\displaystyle q^{m}a^{n}} in ( − a ; q ) ∞ = ∏ k = 0 ∞ ( 1 + a q k ) {\displaystyle (-a;q)_{\infty }=\prod _{k=0}^{\infty }(1+aq^{k})} is the number of partitions of m into n or n -1 distinct parts.
By removing a triangular partition with n − 1 parts from such a partition, we are left with an arbitrary partition with at most n parts. This gives a weight-preserving bijection between the set of partitions into n or n − 1 distinct parts and the set of pairs consisting of a triangular partition having n − 1 parts and a partition with at most n parts. By identifying generating series, this leads to the identity ( − a ; q ) ∞ = ∏ k = 0 ∞ ( 1 + a q k ) = ∑ k = 0 ∞ ( q ( k 2 ) ∏ j = 1 k 1 1 − q j ) a k = ∑ k = 0 ∞ q ( k 2 ) ( q ; q ) k a k {\displaystyle (-a;q)_{\infty }=\prod _{k=0}^{\infty }(1+aq^{k})=\sum _{k=0}^{\infty }\left(q^{k \choose 2}\prod _{j=1}^{k}{\frac {1}{1-q^{j}}}\right)a^{k}=\sum _{k=0}^{\infty }{\frac {q^{k \choose 2}}{(q;q)_{k}}}a^{k}} also described in the above section.
The reciprocal of the function ( q ) ∞ := ( q ; q ) ∞ {\displaystyle (q)_{\infty }:=(q;q)_{\infty }} similarly arises as the generating function for the partition function , p ( n ) {\displaystyle p(n)} , which is also expanded by the second two q-series expansions given below: [ 1 ] 1 ( q ; q ) ∞ = ∑ n ≥ 0 p ( n ) q n = ∑ n ≥ 0 q n ( q ; q ) n = ∑ n ≥ 0 q n 2 ( q ; q ) n 2 . {\displaystyle {\frac {1}{(q;q)_{\infty }}}=\sum _{n\geq 0}p(n)q^{n}=\sum _{n\geq 0}{\frac {q^{n}}{(q;q)_{n}}}=\sum _{n\geq 0}{\frac {q^{n^{2}}}{(q;q)_{n}^{2}}}.}
The q -binomial theorem itself can also be handled by a slightly more involved combinatorial argument of a similar flavor (see also the expansions given in the next subsection ).
Similarly, ( q ; q ) ∞ = 1 − ∑ n ≥ 0 q n + 1 ( q ; q ) n = ∑ n ≥ 0 q n ( n + 1 ) 2 ( − 1 ) n ( q ; q ) n . {\displaystyle (q;q)_{\infty }=1-\sum _{n\geq 0}q^{n+1}(q;q)_{n}=\sum _{n\geq 0}q^{\frac {n(n+1)}{2}}{\frac {(-1)^{n}}{(q;q)_{n}}}.}
Since identities involving q -Pochhammer symbols so frequently involve products of many symbols, the standard convention is to write a product as a single symbol of multiple arguments: ( a 1 , a 2 , … , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n … ( a m ; q ) n . {\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.}
A q -series is a series in which the coefficients are functions of q , typically expressions of ( a ; q ) n {\displaystyle (a;q)_{n}} . [ 2 ] Early results are due to Euler , Gauss , and Cauchy . The systematic study begins with Eduard Heine (1843). [ 3 ]
The q -analog of n , also known as the q -bracket or q -number of n , is defined to be [ n ] q = 1 − q n 1 − q . {\displaystyle [n]_{q}={\frac {1-q^{n}}{1-q}}.} From this one can define the q -analog of the factorial , the q -factorial , as
[ n ] ! q = ∏ k = 1 n [ k ] q = [ 1 ] q ⋅ [ 2 ] q ⋯ [ n − 1 ] q ⋅ [ n ] q = 1 − q 1 − q 1 − q 2 1 − q ⋯ 1 − q n − 1 1 − q 1 − q n 1 − q = 1 ⋅ ( 1 + q ) ⋯ ( 1 + q + ⋯ + q n − 2 ) ⋅ ( 1 + q + ⋯ + q n − 1 ) = ( q ; q ) n ( 1 − q ) n {\displaystyle {\begin{aligned}\left[n\right]!_{q}&=\prod _{k=1}^{n}[k]_{q}=[1]_{q}\cdot [2]_{q}\cdots [n-1]_{q}\cdot [n]_{q}\\&={\frac {1-q}{1-q}}{\frac {1-q^{2}}{1-q}}\cdots {\frac {1-q^{n-1}}{1-q}}{\frac {1-q^{n}}{1-q}}\\&=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1})\\&={\frac {(q;q)_{n}}{(1-q)^{n}}}\\\end{aligned}}}
These numbers are analogues in the sense that lim q → 1 [ n ] q = n , {\displaystyle \lim _{q\rightarrow 1}[n]_{q}=n,} and so also lim q → 1 [ n ] ! q = n ! . {\displaystyle \lim _{q\rightarrow 1}[n]!_{q}=n!.}
The limit value n ! counts permutations of an n -element set S . Equivalently, it counts the number of sequences of nested sets E 1 ⊂ E 2 ⊂ ⋯ ⊂ E n = S {\displaystyle E_{1}\subset E_{2}\subset \cdots \subset E_{n}=S} such that E i {\displaystyle E_{i}} contains exactly i elements. [ 4 ] By comparison, when q is a prime power and V is an n -dimensional vector space over the field with q elements, the q -analogue [ n ] ! q {\displaystyle [n]!_{q}} is the number of complete flags in V , that is, it is the number of sequences V 1 ⊂ V 2 ⊂ ⋯ ⊂ V n = V {\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V} of subspaces such that V i {\displaystyle V_{i}} has dimension i . [ 4 ] The preceding considerations suggest that one can regard a sequence of nested sets as a flag over a conjectural field with one element .
A product of negative integer q -brackets can be expressed in terms of the q -factorial as ∏ k = 1 n [ − k ] q = ( − 1 ) n [ n ] ! q q n ( n + 1 ) / 2 {\displaystyle \prod _{k=1}^{n}[-k]_{q}={\frac {(-1)^{n}\,[n]!_{q}}{q^{n(n+1)/2}}}}
From the q -factorials, one can move on to define the q -binomial coefficients, also known as the Gaussian binomial coefficients , as [ n k ] q = [ n ] ! q [ n − k ] ! q [ k ] ! q , {\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}_{q}={\frac {[n]!_{q}}{[n-k]!_{q}[k]!_{q}}},}
where it is easy to see that the triangle of these coefficients is symmetric in the sense that
for all 0 ≤ m ≤ n {\displaystyle 0\leq m\leq n} . One can check that
[ n + 1 k ] q = [ n k ] q + q n − k + 1 [ n k − 1 ] q = [ n k − 1 ] q + q k [ n k ] q . {\displaystyle {\begin{aligned}{\begin{bmatrix}n+1\\k\end{bmatrix}}_{q}&={\begin{bmatrix}n\\k\end{bmatrix}}_{q}+q^{n-k+1}{\begin{bmatrix}n\\k-1\end{bmatrix}}_{q}\\&={\begin{bmatrix}n\\k-1\end{bmatrix}}_{q}+q^{k}{\begin{bmatrix}n\\k\end{bmatrix}}_{q}.\end{aligned}}}
One can also see from the previous recurrence relations that the next variants of the q {\displaystyle q} -binomial theorem are expanded in terms of these coefficients as follows: [ 5 ] ( z ; q ) n = ∑ j = 0 n [ n j ] q ( − z ) j q ( j 2 ) = ( 1 − z ) ( 1 − q z ) ⋯ ( 1 − z q n − 1 ) ( − q ; q ) n = ∑ j = 0 n [ n j ] q 2 q j ( q ; q 2 ) n = ∑ j = 0 2 n [ 2 n j ] q ( − 1 ) j 1 ( z ; q ) m + 1 = ∑ n ≥ 0 [ n + m n ] q z n . {\displaystyle {\begin{aligned}(z;q)_{n}&=\sum _{j=0}^{n}{\begin{bmatrix}n\\j\end{bmatrix}}_{q}(-z)^{j}q^{\binom {j}{2}}=(1-z)(1-qz)\cdots (1-zq^{n-1})\\(-q;q)_{n}&=\sum _{j=0}^{n}{\begin{bmatrix}n\\j\end{bmatrix}}_{q^{2}}q^{j}\\(q;q^{2})_{n}&=\sum _{j=0}^{2n}{\begin{bmatrix}2n\\j\end{bmatrix}}_{q}(-1)^{j}\\{\frac {1}{(z;q)_{m+1}}}&=\sum _{n\geq 0}{\begin{bmatrix}n+m\\n\end{bmatrix}}_{q}z^{n}.\end{aligned}}}
One may further define the q -multinomial coefficients [ n k 1 , … , k m ] q = [ n ] ! q [ k 1 ] ! q ⋯ [ k m ] ! q , {\displaystyle {\begin{bmatrix}n\\k_{1},\ldots ,k_{m}\end{bmatrix}}_{q}={\frac {[n]!_{q}}{[k_{1}]!_{q}\cdots [k_{m}]!_{q}}},} where the arguments k 1 , … , k m {\displaystyle k_{1},\ldots ,k_{m}} are nonnegative integers that satisfy ∑ i = 1 m k i = n {\displaystyle \sum _{i=1}^{m}k_{i}=n} . The coefficient above counts the number of flags V 1 ⊂ ⋯ ⊂ V m {\displaystyle V_{1}\subset \dots \subset V_{m}} of subspaces in an n -dimensional vector space over the field with q elements such that dim V i = ∑ j = 1 i k j {\displaystyle \dim V_{i}=\sum _{j=1}^{i}k_{j}} .
The limit q → 1 {\displaystyle q\to 1} gives the usual multinomial coefficient ( n k 1 , … , k m ) {\displaystyle {n \choose k_{1},\dots ,k_{m}}} , which counts words in n different symbols { s 1 , … , s m } {\displaystyle \{s_{1},\dots ,s_{m}\}} such that each s i {\displaystyle s_{i}} appears k i {\displaystyle k_{i}} times.
One also obtains a q -analog of the gamma function , called the q-gamma function , and defined as Γ q ( x ) = ( 1 − q ) 1 − x ( q ; q ) ∞ ( q x ; q ) ∞ {\displaystyle \Gamma _{q}(x)={\frac {(1-q)^{1-x}(q;q)_{\infty }}{(q^{x};q)_{\infty }}}} This converges to the usual gamma function as q approaches 1 from inside the unit disc. Note that Γ q ( x + 1 ) = [ x ] q Γ q ( x ) {\displaystyle \Gamma _{q}(x+1)=[x]_{q}\Gamma _{q}(x)} for any x and Γ q ( n + 1 ) = [ n ] ! q {\displaystyle \Gamma _{q}(n+1)=[n]!_{q}} for non-negative integer values of n . Alternatively, this may be taken as an extension of the q -factorial function to the real number system. | https://en.wikipedia.org/wiki/Q-Pochhammer_symbol |
In mathematics , in the field of combinatorics , the q -Vandermonde identity is a q -analogue of the Chu–Vandermonde identity . Using standard notation for q -binomial coefficients , the identity states that
The nonzero contributions to this sum come from values of j such that the q -binomial coefficients on the right side are nonzero, that is, max(0, k − m ) ≤ j ≤ min( n , k ).
As is typical for q -analogues, the q -Vandermonde identity can be rewritten in a number of ways. In the conventions common in applications to quantum groups , a different q -binomial coefficient is used. This q -binomial coefficient, which we denote here by B q ( n , k ) {\displaystyle B_{q}(n,k)} , is defined by
In particular, it is the unique shift of the "usual" q -binomial coefficient by a power of q such that the result is symmetric in q and q − 1 {\displaystyle q^{-1}} . Using this q -binomial coefficient, the q -Vandermonde identity can be written in the form
As with the (non- q ) Chu–Vandermonde identity, there are several possible proofs of the q -Vandermonde identity. The following proof uses the q -binomial theorem .
One standard proof of the Chu–Vandermonde identity is to expand the product ( 1 + x ) m ( 1 + x ) n {\displaystyle (1+x)^{m}(1+x)^{n}} in two different ways. Following Stanley, [ 1 ] we can tweak this proof to prove the q -Vandermonde identity, as well. First, observe that the product
can be expanded by the q -binomial theorem as
Less obviously, we can write
and we may expand both subproducts separately using the q -binomial theorem. This yields
Multiplying this latter product out and combining like terms gives
Finally, equating powers of x {\displaystyle x} between the two expressions yields the desired result.
This argument may also be phrased in terms of expanding the product ( A + B ) m ( A + B ) n {\displaystyle (A+B)^{m}(A+B)^{n}} in two different ways, where A and B are operators (for example, a pair of matrices) that " q -commute," that is, that satisfy BA = qAB . | https://en.wikipedia.org/wiki/Q-Vandermonde_identity |
Q-vectors are used in atmospheric dynamics to understand physical processes such as vertical motion and frontogenesis . Q-vectors are not physical quantities that can be measured in the atmosphere but are derived from the quasi-geostrophic equations and can be used in the previous diagnostic situations. On meteorological charts, Q-vectors point toward upward motion and away from downward motion. Q-vectors are an alternative to the omega equation for diagnosing vertical motion in the quasi-geostrophic equations.
First derived in 1978, [ 1 ] Q-vector derivation can be simplified for the midlatitudes, using the midlatitude β-plane quasi-geostrophic prediction equations: [ 2 ]
And the thermal wind equations:
f 0 ∂ u g ∂ p = R p ∂ T ∂ y {\displaystyle f_{0}{\frac {\partial u_{g}}{\partial p}}={\frac {R}{p}}{\frac {\partial T}{\partial y}}} (x component of thermal wind equation)
f 0 ∂ v g ∂ p = − R p ∂ T ∂ x {\displaystyle f_{0}{\frac {\partial v_{g}}{\partial p}}=-{\frac {R}{p}}{\frac {\partial T}{\partial x}}} (y component of thermal wind equation)
where f 0 {\displaystyle f_{0}} is the Coriolis parameter , approximated by the constant 1e −4 s −1 ; R {\displaystyle R} is the atmospheric ideal gas constant ; β {\displaystyle \beta } is the latitudinal change in the Coriolis parameter β = ∂ f ∂ y {\displaystyle \beta ={\frac {\partial f}{\partial y}}} ; σ {\displaystyle \sigma } is a static stability parameter; c p {\displaystyle c_{p}} is the specific heat at constant pressure; p {\displaystyle p} is pressure; T {\displaystyle T} is temperature; anything with a subscript g {\displaystyle g} indicates geostrophic ; anything with a subscript a {\displaystyle a} indicates ageostrophic ; J {\displaystyle J} is a diabatic heating rate; and ω {\displaystyle \omega } is the Lagrangian rate change of pressure with time. ω = D p D t {\displaystyle \omega ={\frac {Dp}{Dt}}} . Note that because pressure decreases with height in the atmosphere, a negative value of ω {\displaystyle \omega } is upward vertical motion, analogous to + w = D z D t {\displaystyle +w={\frac {Dz}{Dt}}} .
From these equations we can get expressions for the Q-vector:
Q i = − R σ p [ ∂ u g ∂ x ∂ T ∂ x + ∂ v g ∂ x ∂ T ∂ y ] {\displaystyle Q_{i}=-{\frac {R}{\sigma p}}\left[{\frac {\partial u_{g}}{\partial x}}{\frac {\partial T}{\partial x}}+{\frac {\partial v_{g}}{\partial x}}{\frac {\partial T}{\partial y}}\right]}
Q j = − R σ p [ ∂ u g ∂ y ∂ T ∂ x + ∂ v g ∂ y ∂ T ∂ y ] {\displaystyle Q_{j}=-{\frac {R}{\sigma p}}\left[{\frac {\partial u_{g}}{\partial y}}{\frac {\partial T}{\partial x}}+{\frac {\partial v_{g}}{\partial y}}{\frac {\partial T}{\partial y}}\right]}
And in vector form:
Q i = − R σ p ∂ V g → ∂ x ⋅ ∇ → T {\displaystyle Q_{i}=-{\frac {R}{\sigma p}}{\frac {\partial {\vec {V_{g}}}}{\partial x}}\cdot {\vec {\nabla }}T}
Q j = − R σ p ∂ V g → ∂ y ⋅ ∇ → T {\displaystyle Q_{j}=-{\frac {R}{\sigma p}}{\frac {\partial {\vec {V_{g}}}}{\partial y}}\cdot {\vec {\nabla }}T}
Plugging these Q-vector equations into the quasi-geostrophic omega equation gives:
( σ ∇ 2 → + f ∘ 2 ∂ 2 ∂ p 2 ) ω = − 2 ∇ → ⋅ Q → + f ∘ β ∂ v g ∂ p − κ p ∇ 2 → J {\displaystyle \left(\sigma {\overrightarrow {\nabla ^{2}}}+f_{\circ }^{2}{\frac {\partial ^{2}}{\partial p^{2}}}\right)\omega =-2{\vec {\nabla }}\cdot {\vec {Q}}+f_{\circ }\beta {\frac {\partial v_{g}}{\partial p}}-{\frac {\kappa }{p}}{\overrightarrow {\nabla ^{2}}}J}
If second derivatives are approximated as a negative sign, as is true for a sinusoidal function, the above in an adiabatic setting may be viewed as a statement about upward motion:
− ω ∝ − 2 ∇ → ⋅ Q → {\displaystyle -\omega \propto -2{\vec {\nabla }}\cdot {\vec {Q}}}
Expanding the left-hand side of the quasi-geostrophic omega equation in a Fourier Series gives the − ω {\displaystyle -\omega } above, implying that a − ω {\displaystyle -\omega } relationship with the right-hand side of the quasi-geostrophic omega equation can be assumed.
This expression shows that the divergence of the Q-vector ( ∇ → ⋅ Q → {\displaystyle {\vec {\nabla }}\cdot {\vec {Q}}} ) is associated with downward motion. Therefore, convergent Q → {\displaystyle {\vec {Q}}} forces ascent and divergent Q → {\displaystyle {\vec {Q}}} forces descend. [ 3 ] Q-vectors and all ageostrophic flow exist to preserve thermal wind balance. Therefore, low level Q-vectors tend to point in the direction of low-level ageostrophic winds. [ 4 ]
Q-vectors can be determined wholly with: geopotential height ( Φ {\displaystyle \Phi } ) and temperature on a constant pressure surface. Q-vectors always point in the direction of ascending air. For an idealized cyclone and anticyclone in the Northern Hemisphere (where ∂ T ∂ y < 0 {\displaystyle {\frac {\partial T}{\partial y}}<0} ), cyclones have Q-vectors which point parallel to the thermal wind and anticyclones have Q-vectors that point antiparallel to the thermal wind. [ 5 ] This means upward motion in the area of warm air advection and downward motion in the area of cold air advection.
In frontogenesis , temperature gradients need to tighten for initiation. For those situations Q-vectors point toward ascending air and the tightening thermal gradients. [ 6 ] In areas of convergent Q-vectors, cyclonic vorticity is created, and in divergent areas, anticyclonic vorticity is created. [ 1 ] | https://en.wikipedia.org/wiki/Q-Vectors |
In statistics, the q -Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution (Pareto Type II). It is one example of a Tsallis distribution .
The probability density function of a q -Weibull random variable is: [ 1 ]
where q < 2, κ {\displaystyle \kappa } > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and
is the q -exponential [ 1 ] [ 2 ] [ 3 ]
The cumulative distribution function of a q -Weibull random variable is:
where
The mean of the q -Weibull distribution is
where B ( ) {\displaystyle B()} is the Beta function and Γ ( ) {\displaystyle \Gamma ()} is the Gamma function . The expression for the mean is a continuous function of q over the range of definition for which it is finite.
The q -Weibull is equivalent to the Weibull distribution when q = 1 and equivalent to the q -exponential when κ = 1 {\displaystyle \kappa =1}
The q -Weibull is a generalization of the Weibull, as it extends this distribution to the cases of finite support ( q < 1) and to include heavy-tailed distributions ( q ≥ 1 + κ κ + 1 ) {\displaystyle (q\geq 1+{\frac {\kappa }{\kappa +1}})} .
The q -Weibull is a generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support and adds the κ {\displaystyle \kappa } parameter. The Lomax parameters are:
As the Lomax distribution is a shifted version of the Pareto distribution , the q -Weibull for κ = 1 {\displaystyle \kappa =1} is a shifted reparameterized generalization of the Pareto. When q > 1, the q -exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically: | https://en.wikipedia.org/wiki/Q-Weibull_distribution |
In mathematics , a q -analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1 . Typically, mathematicians are interested in q -analogs that arise naturally, rather than in arbitrarily contriving q -analogs of known results. The earliest q -analog studied in detail is the basic hypergeometric series , which was introduced in the 19th century. [ 1 ]
q -analogs are most frequently studied in the mathematical fields of combinatorics and special functions . In these settings, the limit q → 1 is often formal, as q is often discrete-valued (for example, it may represent a prime power ). q -analogs find applications in a number of areas, including the study of fractals and multi-fractal measures , and expressions for the entropy of chaotic dynamical systems . The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket ) and the modular group in particular. The connection passes through hyperbolic geometry and ergodic theory , where the elliptic integrals and modular forms play a prominent role; the q -series themselves are closely related to elliptic integrals.
q -analogs also appear in the study of quantum groups and in q -deformed superalgebras . The connection here is similar, in that much of string theory is set in the language of Riemann surfaces , resulting in connections to elliptic curves , which in turn relate to q -series.
Classical q -theory begins with the q -analogs of the nonnegative integers. [ 2 ] The equality
suggests that we define the q -analog of n , also known as the q -bracket or q -number of n , to be
By itself, the choice of this particular q -analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use [ n ] q as the q -analog of n , one may define the q -analog of the factorial , known as the q -factorial , by
This q -analog appears naturally in several contexts. Notably, while n ! counts the number of permutations of length n , [ n ] q ! counts permutations while keeping track of the number of inversions . That is, if inv( w ) denotes the number of inversions of the permutation w and S n denotes the set of permutations of length n , we have
In particular, one recovers the usual factorial by taking the limit as q → 1 {\displaystyle q\rightarrow 1} .
The q -factorial also has a concise definition in terms of the q -Pochhammer symbol , a basic building-block of all q -theories:
From the q -factorials, one can move on to define the q -binomial coefficients , also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients :
The q -exponential is defined as:
q -trigonometric functions, along with a q -Fourier transform, have been defined in this context.
The Gaussian coefficients count subspaces of a finite vector space . Let q be the number of elements in a finite field . (The number q is then a power of a prime number , q = p e , so using the letter q is especially appropriate.) Then the number of k -dimensional subspaces of the n -dimensional vector space over the q -element field equals
Letting q approach 1, we get the binomial coefficient
or in other words, the number of k -element subsets of an n -element set.
Thus, one can regard a finite vector space as a q -generalization of a set, and the subspaces as the q -generalization of the subsets of the set. As another example, the number of flags is [ n ] q ! {\displaystyle [n]_{q}!} as the order in which we build the flag matters, and after taking the limit we get n ! {\displaystyle n!} . This has been a fruitful point of view in finding interesting new theorems. For example, there are q -analogs of Sperner's theorem [ 3 ] and Ramsey theory . [ citation needed ]
Let q = ( e 2 π i / n ) d be the d -th power of a primitive n -th root of unity. Let C be a cyclic group of order n generated by an element c . Let X be the set of k -element subsets of the n -element set {1, 2, ..., n }. The group C has a canonical action on X given by sending c to the cyclic permutation (1, 2, ..., n ). Then the number of fixed points of c d on X is equal to
Conversely, by letting q vary and seeing q -analogs as deformations, one can consider the combinatorial case of q = 1 as a limit of q -analogs as q → 1 (often one cannot simply let q = 1 in the formulae, hence the need to take a limit).
This can be formalized in the field with one element , which recovers combinatorics as linear algebra over the field with one element: for example, Weyl groups are simple algebraic groups over the field with one element.
q -analogs are often found in exact solutions of many-body problems. [ citation needed ] In such cases, the q → 1 limit usually corresponds to relatively simple dynamics, e.g., without nonlinear interactions, while q < 1 gives insight into the complex nonlinear regime with feedbacks.
An example from atomic physics is the model of molecular condensate creation from an ultra cold fermionic atomic gas during a sweep of an external magnetic field through the Feshbach resonance . [ 4 ] This process is described by a model with a q -deformed version of the SU(2) algebra of operators, and its solution is described by q -deformed exponential and binomial distributions. | https://en.wikipedia.org/wiki/Q-analog |
In mathematics , in the area of combinatorics and quantum calculus , the q -derivative , or Jackson derivative , is a q -analog of the ordinary derivative , introduced by Frank Hilton Jackson . It is the inverse of Jackson's q -integration . For other forms of q-derivative, see Chung et al. (1994) .
The q -derivative of a function f ( x ) is defined as [ 1 ] [ 2 ] [ 3 ]
It is also often written as D q f ( x ) {\displaystyle D_{q}f(x)} . The q -derivative is also known as the Jackson derivative .
Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator
which goes to the plain derivative, D q → d d x {\displaystyle D_{q}\to {\frac {d}{dx}}} as q → 1 {\displaystyle q\to 1} .
It is manifestly linear,
It has a product rule analogous to the ordinary derivative product rule, with two equivalent forms
Similarly, it satisfies a quotient rule,
There is also a rule similar to the chain rule for ordinary derivatives. Let g ( x ) = c x k {\displaystyle g(x)=cx^{k}} . Then
The eigenfunction of the q -derivative is the q -exponential e q ( x ).
Q -differentiation resembles ordinary differentiation, with curious differences. For example, the q -derivative of the monomial is: [ 2 ]
where [ n ] q {\displaystyle [n]_{q}} is the q -bracket of n . Note that lim q → 1 [ n ] q = n {\displaystyle \lim _{q\to 1}[n]_{q}=n} so the ordinary derivative is regained in this limit.
The n -th q -derivative of a function may be given as: [ 3 ]
provided that the ordinary n -th derivative of f exists at x = 0. Here, ( q ; q ) n {\displaystyle (q;q)_{n}} is the q -Pochhammer symbol , and [ n ] ! q {\displaystyle [n]!_{q}} is the q -factorial . If f ( x ) {\displaystyle f(x)} is analytic we can apply the Taylor formula to the definition of D q ( f ( x ) ) {\displaystyle D_{q}(f(x))} to get
A q -analog of the Taylor expansion of a function about zero follows: [ 2 ]
The following representation for higher order q {\displaystyle q} -derivatives is known: [ 4 ] [ 5 ]
( n k ) q {\displaystyle {\binom {n}{k}}_{q}} is the q {\displaystyle q} -binomial coefficient. By changing the order of summation as r = n − k {\displaystyle r=n-k} , we obtain the next formula: [ 4 ] [ 6 ]
Higher order q {\displaystyle q} -derivatives are used to q {\displaystyle q} -Taylor formula and the q {\displaystyle q} - Rodrigues' formula (the formula used to construct q {\displaystyle q} - orthogonal polynomials [ 4 ] ).
Post quantum calculus is a generalization of the theory of quantum calculus , and it uses the following operator: [ 7 ] [ 8 ]
Wolfgang Hahn introduced the following operator (Hahn difference): [ 9 ] [ 10 ]
When ω → 0 {\displaystyle \omega \to 0} this operator reduces to q {\displaystyle q} -derivative, and when q → 1 {\displaystyle q\to 1} it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems. [ 11 ] [ 12 ] [ 13 ]
β {\displaystyle \beta } -derivative is an operator defined as follows: [ 14 ] [ 15 ]
In the definition, I {\displaystyle I} is a given interval, and β ( t ) {\displaystyle \beta (t)} is any continuous function that strictly monotonically increases (i.e. t > s → β ( t ) > β ( s ) {\displaystyle t>s\rightarrow \beta (t)>\beta (s)} ). When β ( t ) = q t {\displaystyle \beta (t)=qt} then this operator is q {\displaystyle q} -derivative, and when β ( t ) = q t + ω {\displaystyle \beta (t)=qt+\omega } this operator is Hahn difference.
The q-calculus has been used in machine learning for designing stochastic activation functions. [ 16 ] | https://en.wikipedia.org/wiki/Q-derivative |
In combinatorial mathematics , the q -difference polynomials or q -harmonic polynomials are a polynomial sequence defined in terms of the q -derivative . They are a generalized type of Brenke polynomial , and generalize the Appell polynomials . See also Sheffer sequence .
The q-difference polynomials satisfy the relation
where the derivative symbol on the left is the q-derivative. In the limit of q → 1 {\displaystyle q\to 1} , this becomes the definition of the Appell polynomials:
The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely
where e q ( t ) {\displaystyle e_{q}(t)} is the q-exponential :
Here, [ n ] q ! {\displaystyle [n]_{q}!} is the q-factorial and
is the q-Pochhammer symbol . The function A ( w ) {\displaystyle A(w)} is arbitrary but assumed to have an expansion
Any such A ( w ) {\displaystyle A(w)} gives a sequence of q-difference polynomials. | https://en.wikipedia.org/wiki/Q-difference_polynomial |
The q -exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution . The q -exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy . [ 1 ] The exponential distribution is recovered as q → 1. {\displaystyle q\rightarrow 1.}
Originally proposed by the statisticians George Box and David Cox in 1964, [ 2 ] and known as the reverse Box–Cox transformation for q = 1 − λ , {\displaystyle q=1-\lambda ,} a particular case of power transform in statistics.
The q -exponential distribution has the probability density function
where
is the q -exponential if q ≠ 1 . When q = 1 , e q (x) is just exp( x ).
In a similar procedure to how the exponential distribution can be derived (using the standard Boltzmann–Gibbs entropy or Shannon entropy and constraining the domain of the variable to be positive), the q -exponential distribution can be derived from a maximization of the Tsallis Entropy subject to the appropriate constraints.
The q -exponential is a special case of the generalized Pareto distribution where
The q -exponential is the generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support. The Lomax parameters are:
As the Lomax distribution is a shifted version of the Pareto distribution , the q -exponential is a shifted reparameterized generalization of the Pareto. When q > 1 , the q -exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, if
then X ∼ Y . {\displaystyle X\sim Y.}
Random deviates can be drawn using inverse transform sampling . Given a variable U that is uniformly distributed on the interval (0,1), then
where ln q ′ {\displaystyle \ln _{q'}} is the q -logarithm and q ′ = 1 2 − q . {\displaystyle q'={\frac {1}{2-q}}.}
Being a power transform , it is a usual technique in statistics for stabilizing the variance, making the data more normal distribution-like and improving the validity of measures of association such as the Pearson correlation between variables.
It has been found to be an accurate model for train delays. [ 3 ] It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance. [ 4 ] | https://en.wikipedia.org/wiki/Q-exponential_distribution |
In statistics , the Q-function is the tail distribution function of the standard normal distribution . [ 1 ] [ 2 ] In other words, Q ( x ) {\displaystyle Q(x)} is the probability that a normal (Gaussian) random variable will obtain a value larger than x {\displaystyle x} standard deviations. Equivalently, Q ( x ) {\displaystyle Q(x)} is the probability that a standard normal random variable takes a value larger than x {\displaystyle x} .
If Y {\displaystyle Y} is a Gaussian random variable with mean μ {\displaystyle \mu } and variance σ 2 {\displaystyle \sigma ^{2}} , then X = Y − μ σ {\displaystyle X={\frac {Y-\mu }{\sigma }}} is standard normal and
where x = y − μ σ {\displaystyle x={\frac {y-\mu }{\sigma }}} .
Other definitions of the Q -function, all of which are simple transformations of the normal cumulative distribution function , are also used occasionally. [ 3 ]
Because of its relation to the cumulative distribution function of the normal distribution, the Q -function can also be expressed in terms of the error function , which is an important function in applied mathematics and physics.
Formally, the Q -function is defined as
Thus,
where Φ ( x ) {\displaystyle \Phi (x)} is the cumulative distribution function of the standard normal Gaussian distribution .
The Q -function can be expressed in terms of the error function , or the complementary error function, as [ 2 ]
An alternative form of the Q -function known as Craig's formula, after its discoverer, is expressed as: [ 4 ]
This expression is valid only for positive values of x , but it can be used in conjunction with Q ( x ) = 1 − Q (− x ) to obtain Q ( x ) for negative values. This form is advantageous in that the range of integration is fixed and finite.
Craig's formula was later extended by Behnad (2020) [ 5 ] for the Q -function of the sum of two non-negative variables, as follows:
These bounds are derived from a unified form Q B ( x ; a , b ) = exp ( − x 2 ) a + exp ( − x 2 / 2 ) b ( x + 1 ) {\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}}} , where the parameters a {\displaystyle a} and b {\displaystyle b} are chosen to satisfy specific conditions ensuring the lower ( a L = 12 {\displaystyle a_{\mathrm {L} }=12} , b L = 2 π {\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}} ) and upper ( a U = 50 {\displaystyle a_{\mathrm {U} }=50} , b U = 2 {\displaystyle b_{\mathrm {U} }=2} ) bounding properties. The resulting expressions are notable for their simplicity and tightness, offering a favorable trade-off between accuracy and mathematical tractability. These bounds are particularly useful in theoretical analysis, such as in communication theory over fading channels. Additionally, they can be extended to bound Q n ( x ) {\displaystyle Q^{n}(x)} for positive integers n {\displaystyle n} using the binomial theorem, maintaining their simplicity and effectiveness.
The inverse Q -function can be related to the inverse error functions :
The function Q − 1 ( y ) {\displaystyle Q^{-1}(y)} finds application in digital communications. It is usually expressed in dB and generally called Q-factor :
where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for quadrature phase-shift keying (QPSK) in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y .
The Q -function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python , MATLAB and Mathematica . Some values of the Q -function are given below for reference.
The Q -function can be generalized to higher dimensions: [ 15 ]
where X ∼ N ( 0 , Σ ) {\displaystyle \mathbf {X} \sim {\mathcal {N}}(\mathbf {0} ,\,\Sigma )} follows the multivariate normal distribution with covariance Σ {\displaystyle \Sigma } and the threshold is of the form x = γ Σ l ∗ {\displaystyle \mathbf {x} =\gamma \Sigma \mathbf {l} ^{*}} for some positive vector l ∗ > 0 {\displaystyle \mathbf {l} ^{*}>\mathbf {0} } and positive constant γ > 0 {\displaystyle \gamma >0} . As in the one dimensional case, there is no simple analytical formula for the Q -function. Nevertheless, the Q -function can be approximated arbitrarily well as γ {\displaystyle \gamma } becomes larger and larger. [ 16 ] [ 17 ] | https://en.wikipedia.org/wiki/Q-function |
Q-system is a genetic tool that allows to express transgenes in a living organism . [ 1 ] Originally the Q-system was developed [ 2 ] [ 3 ] for use in the vinegar fly Drosophila melanogaster , and was rapidly adapted for use in cultured mammalian cells , [ 2 ] zebrafish , [ 4 ] worms [ 5 ] and mosquitoes . [ 6 ] The Q-system utilizes genes from the qa cluster [ 7 ] of the bread fungus Neurospora crassa , and consists of four components: the transcriptional activator (QF/QF2/QF2 w ), the enhancer QUAS, the repressor QS, and the chemical de-repressor quinic acid . Similarly to GAL4/UAS [ 8 ] and LexA/LexAop, [ 9 ] the Q-system is a binary expression system that allows to express reporters or effectors (e.g. fluorescent proteins , ion channels , toxins and other genes) in a defined subpopulation of cells with the purpose of visualising these cells or altering their function. In addition, GAL4/UAS, LexA/LexAop and the Q-system function independently of each other and can be used simultaneously to achieve a desired pattern of reporter expression, or to express several reporters in different subsets of cells.
The Q-system is based on two out of the seven genes of the qa gene cluster of the bread fungus Neurospora crassa . [ 7 ] The genes of the qa cluster are responsible for the catabolism of quinic acid, which is used by the fungus as a carbon source in conditions of low glucose. [ 7 ] The cluster contains a transcriptional activator qa-1F , a transcriptional repressor qa-1S , and five structural genes. The qa-1F binds to a specific DNA sequence, found upstream of the qa genes. The presence of quinic acid disrupts interaction between qa-1F and qa-1S , thus disinhibiting the transcriptional activity of qa-1F .
Genes qa-1F , qa-1S and the DNA binding sequence of qa-1F form the basis of the Q-system. The genes were renamed to simplify their use as follows: transcriptional activator qa-1F as QF, repressor qa-1S as QS, and the DNA binding sequence as QUAS. [ 2 ] The quinic acid represents the fourth component of the Q-system.
The original transactivator QF appeared to be toxic when expressed broadly in Drosophila . To overcome this problem, two new transactivators were developed: QF2 and QF2 w. [ 3 ]
The Q-system functions similarly to, and independently of, the GAL4/UAS [ 8 ] and the LexA/LexAop [ 9 ] systems. QF, QF2 and QF2 w are analogous to GAL4 and LexA, and their expression is usually under the control of cell-type specific promoter, such as nsyb (to target neurons) or tubulin (to target all cells). QUAS is analogous to UAS and LexAop, and is placed upstream of an effector gene, such as GFP . QS is analogous to GAL80, and may be driven by any promoter (e.g. tubulin-QS ). Quinic acid is a unique feature of the Q-system, and it must be fed to the flies or maggots in order to alleviate the QS-induced repression. In some ways, quinic acid is analogous to temperature in the case of GAL80 ts .
In its basic form, two transgenic fly lines, one containing a QF transgene and the other one containing a QUAS transgene, are crossed together. Their progeny that had both a QF transgene and a QUAS transgene will be expressing a reporter gene in a subset of cells (e.g. nsyb-QF2, QUAS-GFP flies express GFP in all neurons). If a fly also expresses QS in some of the cells, the activity of QF will be repressed in these cells, but it may be restored of a fly is fed quinic acid (e.g. a nsyb-QF2, QUAS-GFP, tub-QS fly expresses no GFP when its diet doesn't contain quinic acid, and expresses GFP in its neurons when fed quinic acid). [ 2 ] [ 3 ] The use of QS repressor and quinic acid allows to fine-tune the temporal control of transgene expression.
Chimeric transactivators GAL4QF [ 3 ] and LexAQF [ 3 ] allow to combine the use of all three binary expression systems. GAL4QF binds to UAS, and may be repressed by QS while being unaffected by GAL80. Similarly, LexAQF binds to LexAop, and may be repressed by QS. LexAQF represents a useful extension of the LexA/LexAop system that doesn't have its own repressor.
A variety of expression patterns may be achieved by combination of the three binary expression systems and the FLP/FRT or other recombinases. [ 10 ] Expression patterns may be constructed as AND, OR, NOR etc. logic gates [ 1 ] [ 2 ] to e.g. narrow down expression patterns of available GAL4 lines. The resulting expression pattern somewhat depends on the developmental timing of activation of the transcription factors (discussed in [ 1 ] ).
Q-system appeared to be working successfully in a variety of organisms. It has been used to drive expression of luciferase, as a proof of principle, in cultured mammalian cells . [ 2 ] In zebrafish [ 4 ] the Q-system has been successfully used with several tissue-specific promoters, and was shown to work independently of the GAL4/UAS system when expressed in the same cell. In C. elegans [ 5 ] the Q-system has been shown to work in muscles and in neuronal tissue. In 2016, the Q-system was used to target, for the first time, the olfactory neurons of malaria mosquitoes Anopheles gambiae . [ 6 ] In 2019, the Q-system in Anopheles mosquitoes was used to examine the functional responses of olfactory neurons to odors. [ 11 ] In 2019, the Q-system was introduced into the Aedes aegypti mosquito to capture tissue specific expression patterns. [ 12 ] These successes make the Q-system the system of choice when developing genetic tools for other organisms. Currently the main shortcoming of the Q-system is the low number of available transgenic lines, but it will be overcome as the scientific community creates and shares these resources, such as by the use of the GAL4>QF2 HACK system to convert existing GAL4 transgenic insertions to QF2. [ 13 ] DNA binding domain of QF2 fused with VP16 transcriptional activator domain was successfully applied in Penicillium to gain control over the penicillin producing secondary metabolite gene cluster in a scalable manner. [ 14 ] | https://en.wikipedia.org/wiki/Q-system_(genetics) |
The Q 10 temperature coefficient is a measure of temperature sensitivity based on the chemical reactions.
The Q 10 is calculated as:
where;
Rewriting this equation, the assumption behind Q 10 is that the reaction rate R depends exponentially on temperature:
Q 10 is a unitless quantity, as it is the factor by which a rate changes, and is a useful way to express the temperature dependence of a process.
For most biological systems, the Q 10 value is ~ 2 to 3. [ 1 ]
The temperature of a muscle has a significant effect on the velocity and power of the muscle contraction, with performance generally declining with decreasing temperatures and increasing with rising temperatures. The Q 10 coefficient represents the degree of temperature dependence a muscle exhibits as measured by contraction rates. [ 2 ] A Q 10 of 1.0 indicates thermal independence of a muscle whereas an increasing Q 10 value indicates increasing thermal dependence. Values less than 1.0 indicate a negative or inverse thermal dependence, i.e., a decrease in muscle performance as temperature increases. [ 3 ]
Q 10 values for biological processes vary with temperature. Decreasing muscle temperature results in a substantial decline of muscle performance such that a 10 degree Celsius temperature decrease results in at least a 50% decline in muscle performance. [ 4 ] Persons who have fallen into icy water may gradually lose the ability to swim or grasp safety lines due to this effect, although other effects such as atrial fibrillation are a more immediate cause of drowning deaths. At some minimum temperature biological systems do not function at all, but performance increases with rising temperature ( Q 10 of 2-4) to a maximum performance level and thermal independence ( Q 10 of 1.0-1.5). With continued increase in temperature, performance decreases rapidly ( Q 10 of 0.2-0.8) up to a maximum temperature at which all biological function again ceases. [ 5 ]
Within vertebrates, different skeletal muscle activity has correspondingly different thermal dependencies. The rate of muscle twitch contractions and relaxations are thermally dependent ( Q 10 of 2.0-2.5), whereas maximum contraction, e.g., tetanic contraction, is thermally independent. [ 6 ]
Muscles of some ectothermic species. e.g., sharks, show less thermal dependence at lower temperatures than endothermic species [ 4 ] [ 7 ] | https://en.wikipedia.org/wiki/Q10_(temperature_coefficient) |
The QAMA Calculator is a calculator that requires users to provide a reasonable estimate of the answer before the precise answer is delivered. [ 1 ] [ 2 ] QAMA stands for Quick Approximate Mental Arithmetic .
Invented by Ilan Samson, it aims to get users to think first by estimating before they get the correct answer. [ 3 ] Estimation is seen by many as an essential part of mathematics , and some believe that the presence and popularity of calculators could inhibit the use of estimation skills. [ 4 ] [ 5 ]
A physical version of the calculator was released for sale in 2014, with apps for smartphones and tablets developed in 2016.
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/QAMA_Calculator |
A QA & UX Manager work with both Quality Assurance (QA) and User Experience (UX) in relation to video game and software development . QA & UX Manager can work independently or in co-operation with other QA & UX Managers, like in larger QA & UX teams with other QA & UX Managers and game testers. In the larger QA & UX teams, there is usually a lead QA & UX Group Manager that works as the daily leader of this team of QA & UX Managers and game testers. QA & UX Manager also usually work very close with the project managers and the QA Programmer as part of video game development. [ 1 ] [ 2 ]
In terms of work assignments that QA & UX Managers do, this can include planning and management , technical testing, User Experience, metrics and communication . In terms of planning and management done by the QA & UX Manager, this may include making test plans, test cycles and test cases for a video game as well as being responsible for the overall testing and recruiting of game testers in a video game development . In terms of technical testing that the QA & UX Managers do, this can be very alternating and can be everything from testing correctness and regression to check an optimization in a video game. In terms of User Experience that the QA & UX Manager work with, this is particularly centered around making interviews and doing monitoring in relation to User Experience of a video game by the player. In terms of Metrics that the QA & UX Manager work with, this task mainly is about collecting and analyze data an about game and gamer activity in relation to a particular game. In terms of the communication characterized parts of the work the QA & UX Manager do, this is often done in relation with meetings with the development team and where the QA & UX Manager pass on his or her knowledge and data to the development team. [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/QA_&_UX_Manager |
Quark matter or QCD matter ( quantum chromodynamic ) refers to any of a number of hypothetical phases of matter whose degrees of freedom include quarks and gluons , of which the prominent example is quark-gluon plasma . [ 1 ] Several series of conferences in 2019, 2020, and 2021 were devoted to this topic. [ 2 ] [ 3 ] [ 4 ]
Quarks are liberated into quark matter at extremely high temperatures and/or densities, and some of them are still only theoretical as they require conditions so extreme that they cannot be produced in any laboratory, especially not at equilibrium conditions. Under these extreme conditions, the familiar structure of matter , where the basic constituents are nuclei (consisting of nucleons which are bound states of quarks) and electrons, is disrupted. In quark matter it is more appropriate to treat the quarks themselves as the basic degrees of freedom.
In the standard model of particle physics, the strong force is described by the theory of QCD . At ordinary temperatures or densities this force just confines the quarks into composite particles ( hadrons ) of size around 10 −15 m = 1 femtometer = 1 fm (corresponding to the QCD energy scale Λ QCD ≈ 200 MeV ) and its effects are not noticeable at longer distances.
However, when the temperature reaches the QCD energy scale ( T of order 10 12 kelvins ) or the density rises to the point where the average inter-quark separation is less than 1 fm (quark chemical potential μ around 400 MeV), the hadrons are melted into their constituent quarks, and the strong interaction becomes the dominant feature of the physics. Such phases are called quark matter or QCD matter.
The strength of the color force makes the properties of quark matter unlike gas or plasma, instead leading to a state of matter more reminiscent of a liquid. At high densities, quark matter is a Fermi liquid , but is predicted to exhibit color superconductivity at high densities and temperatures below 10 12 K.
At this time no star with properties expected of these objects has been observed, although some evidence has been provided for quark matter in the cores of large neutron stars. [ 7 ]
Laboratory experiments suggests that the inevitable interaction with heavy noble gas nuclei in the upper atmosphere would lead to quark–gluon plasma formation.
Even though quark-gluon plasma can only occur under quite extreme conditions of temperature and/or pressure, it is being actively studied at particle colliders , such as the Large Hadron Collider LHC at CERN and the Relativistic Heavy Ion Collider RHIC at Brookhaven National Laboratory .
In these collisions, the plasma only occurs for a very short time before it spontaneously disintegrates. The plasma's physical characteristics are studied by detecting the debris emanating from the collision region with large particle detectors [ 11 ] [ 12 ]
Heavy-ion collisions at very high energies can produce small short-lived regions of space whose energy density is comparable to that of the 20-micro-second-old universe . This has been achieved by colliding heavy nuclei such as lead nuclei at high speeds, and a first time claim of formation of quark–gluon plasma came from the SPS accelerator at CERN in February 2000. [ 13 ]
This work has been continued at more powerful accelerators, such as RHIC in the US, and as of 2010 at the European LHC at CERN located in the border area of Switzerland and France. There is good evidence that the quark–gluon plasma has also been produced at RHIC. [ 14 ]
The context for understanding the thermodynamics of quark matter is the standard model of particle physics, which contains six different flavors of quarks, as well as leptons like electrons and neutrinos . These interact via the strong interaction , electromagnetism , and also the weak interaction which allows one flavor of quark to turn into another. Electromagnetic interactions occur between particles that carry electrical charge; strong interactions occur between particles that carry color charge .
The correct thermodynamic treatment of quark matter depends on the physical context. For large quantities that exist for long periods of time (the "thermodynamic limit"), we must take into account the fact that the only conserved charges in the standard model are quark number (equivalent to baryon number), electric charge, the eight color charges, and lepton number. Each of these can have an associated chemical potential. However, large volumes of matter must be electrically and color-neutral, which determines the electric and color charge chemical potentials. This leaves a three-dimensional phase space , parameterized by quark chemical potential, lepton chemical potential, and temperature.
In compact stars quark matter would occupy cubic kilometers and exist for millions of years, so the thermodynamic limit is appropriate. However, the neutrinos escape, violating lepton number, so the phase space for quark matter in compact stars only has two dimensions, temperature ( T ) and quark number chemical potential μ. A strangelet is not in the thermodynamic limit of large volume, so it is like an exotic nucleus: it may carry electric charge.
A heavy-ion collision is in neither the thermodynamic limit of large volumes nor long times. Putting aside questions of whether it is sufficiently equilibrated for thermodynamics to be applicable, there is certainly not enough time for weak interactions to occur, so flavor is conserved, and there are independent chemical potentials for all six quark flavors. The initial conditions (the impact parameter of the collision, the number of up and down quarks in the colliding nuclei, and the fact that they contain no quarks of other flavors) determine the chemical potentials. (Reference for this section: [ 15 ] [ 16 ] ).
Based on rigorous theoretical calculations valid at ultrahigh density and a few experimental ultrarelativistic heavy ion collision experiments, an outline of the phase diagram of quark matter has been worked out as shown in the figure to the right. It is relevant for the understanding the core of neutron stars, where the only relevant thermodynamic potentials are quark chemical potential μ and temperature T. [ 15 ]
For guidance it also shows the typical values of μ and T in heavy-ion collisions and in the early universe. For readers who are not familiar with the concept of a chemical potential, it is helpful to think of μ as a measure of the imbalance between quarks and antiquarks in the system. Higher μ means a stronger bias favoring quarks over antiquarks. At low temperatures there are no antiquarks, and then higher μ generally means a higher density of quarks.
Ordinary atomic matter as we know it is really a mixed phase, droplets of nuclear matter (nuclei) surrounded by vacuum, which exists at the low-temperature phase boundary between vacuum and nuclear matter, at μ = 310 MeV and T close to zero. If we increase the quark density (i.e. increase μ) keeping the temperature low, we move into a phase of more and more compressed nuclear matter. Following this path corresponds to burrowing more and more deeply into a neutron star .
Eventually, at an unknown critical value of μ, there is a transition to quark matter. At ultra-high densities we expect to find the color-flavor-locked (CFL) phase of color-superconducting quark matter. At intermediate densities we expect some other phases (labelled "non-CFL quark liquid" in the figure) whose nature is presently unknown. [ 15 ] [ 16 ] They might be other forms of color-superconducting quark matter, or something different.
Now, imagine starting at the bottom left corner of the phase diagram, in the vacuum where μ = T = 0. If we heat up the system without introducing any preference for quarks over antiquarks, this corresponds to moving vertically upwards along the T axis. At first, quarks are still confined and we create a gas of hadrons ( pions , mostly). Then around T = 150 MeV there is a crossover to the quark gluon plasma: thermal fluctuations break up the pions, and we find a gas of quarks, antiquarks, and gluons, as well as lighter particles such as photons, electrons, positrons, etc. Following this path corresponds to travelling far back in time (so to say), to the state of the universe shortly after the big bang (where there was a very tiny preference for quarks over antiquarks).
The line that rises up from the nuclear/quark matter transition and then bends back towards the T axis, with its end marked by a star, is the conjectured boundary between confined and unconfined phases. Until recently it was also believed to be a boundary between phases where chiral symmetry is broken (low temperature and density) and phases where it is unbroken (high temperature and density). It is now known that the CFL phase exhibits chiral symmetry breaking, and other quark matter phases may also break chiral symmetry, so it is not clear whether this is really a chiral transition line. The line ends at the "chiral critical point ", marked by a star in this figure, which is a special temperature and density at which striking physical phenomena, analogous to critical opalescence , are expected. (Reference for this section: [ 15 ] [ 16 ] [ 17 ] ).
For a complete description of phase diagram it is required that one must have complete understanding of dense, strongly interacting hadronic matter and strongly interacting quark matter from some underlying theory e.g. quantum chromodynamics (QCD). However, because such a description requires the proper understanding of QCD in its non-perturbative regime, which is still far from being completely understood, any theoretical advance remains very challenging.
The phase structure of quark matter remains mostly conjectural because it is difficult to perform calculations predicting the properties of quark matter. The reason is that QCD, the theory describing the dominant interaction between quarks, is strongly coupled at the densities and temperatures of greatest physical interest, and hence it is very hard to obtain any predictions from it. Here are brief descriptions of some of the standard approaches.
The only first-principles calculational tool currently available is lattice QCD , i.e. brute-force computer calculations. Because of a technical obstacle known as the fermion sign problem , this method can only be used at low density and high temperature (μ < T ), and it predicts that the crossover to the quark–gluon plasma will occur around T = 150 MeV [ 18 ] However, it cannot be used to investigate the interesting color-superconducting phase structure at high density and low temperature. [ 19 ]
Because QCD is asymptotically free it becomes weakly coupled at unrealistically high densities, and diagrammatic
methods can be used. [ 16 ] Such methods show that the CFL phase occurs at very high density. At high temperatures, however, diagrammatic methods are still not under full control.
To obtain a rough idea of what phases might occur, one can use a model that has some of the same properties as QCD, but is easier to manipulate. Many physicists use Nambu–Jona-Lasinio models , which contain no gluons, and replace the strong interaction with a four-fermion interaction . Mean-field methods are commonly used to analyse the phases. Another approach is the bag model , in which the effects of confinement are simulated by an additive energy density that penalizes unconfined quark matter.
Many physicists simply give up on a microscopic approach, and make informed guesses of the expected phases (perhaps based on NJL model results). For each phase, they then write down an effective theory for the low-energy excitations, in terms of a small number of parameters, and use it to make predictions that could allow those parameters to be fixed by experimental observations. [ 17 ]
There are other methods that are sometimes used to shed light on QCD, but for various reasons have not yet yielded useful results in studying quark matter.
Treat the number of colors N , which is actually 3, as a large number, and expand in powers of 1/ N . It turns out that at high density the higher-order corrections are large, and the expansion gives misleading results. [ 15 ]
Adding scalar quarks (squarks) and fermionic gluons (gluinos) to the theory makes it more tractable, but the thermodynamics of quark matter depends crucially on the fact that only fermions can carry quark number, and on the number of degrees of freedom in general.
Experimentally, it is hard to map the phase diagram of quark matter because it has been rather difficult to learn how to tune to high enough temperatures and density in the laboratory experiment using collisions of relativistic heavy ions as experimental tools. However, these collisions ultimately will provide information about the crossover from hadronic matter to QGP. It has been suggested that the observations of compact stars may also constrain the information about the high-density low-temperature region. Models of the cooling, spin-down, and precession of these stars offer information about the relevant properties of their interior. As observations become more precise, physicists hope to learn more. [ 15 ]
One of the natural subjects for future research is the search for the exact location of the chiral critical point. Some ambitious lattice QCD calculations may have found evidence for it, and future calculations will clarify the situation. Heavy-ion collisions might be able to measure its position experimentally, but this will require scanning across a range of values of μ and T. [ 20 ]
In 2020, evidence was provided that the cores of neutron stars with mass ~2 M ⊙ were likely composed of quark matter. [ 7 ] [ 21 ] Their result was based on neutron-star tidal deformability during a neutron star merger as measured by gravitational-wave observatories , leading to an estimate of star radius, combined with calculations of the equation of state relating the pressure and energy density of the star's core. The evidence was strongly suggestive but did not conclusively prove the existence of quark matter. | https://en.wikipedia.org/wiki/QCD_matter |
QDGC - Quarter Degree Grid Cells (or QDS - Quarter degree Squares) are a way of dividing the longitude latitude degree square cells into smaller squares, forming in effect a system of geocodes . Historically QDGC has been used in a lot of African atlases. Several African biodiversity projects uses QDGC, among which The atlas of Southern African Birds [ 1 ] is the most prominent one. In 2009 a paper by Larsen et al. [ 2 ] describes the QDGC standard in detail.
The squares themselves are based on the degree squares covering earth. QDGC represents a way of making approximately equal area squares covering a specific area to represent specific qualities of the area covered. However, differences in area between 'squares' enlarge along with longitudinal distance and this can violate assumptions of many statistical analyses requiring truly equal-area grids. For instance species range modelling or estimates of ecological niche could be substantially affected if data were not appropriately transformed, e.g. projected onto a plane using a special projection. [ 3 ]
Around the equator we have 360 longitudinal lines, and from the north to the south pole we have 180 latitudinal lines. Together this gives us 64800 segments or tiles covering earth. The form of the squares becomes more rectangular the longer north we come. At the poles they are not square or even rectangular at all, but end up in elongated triangles.
Each degree square is designated by a full reference to the main degree square. S01E010 is a reference to a square in Tanzania. S means the square is south of equator, and E means it is East of the zero meridian. The numbers refer to longitudinal and latitudinal degree.
A square with no sublevel reference is also called QDGC level 0. This is square based on a full degree longitude by a full degree latitude. The QDGC level 0 squares are themselves divided into four.
To get smaller squares the above squares are again divided in four - giving us a total of 16 squares within a degree square. The names for the new level of squares are named the same way. The full reference of a square could then be:
The number of squares for each QDGC level can be calculated with this formula:
number of squares = (2 d ) 2
(where d is QDGC level)
Table showing level, number of squares and an example reference:
To decide which name a specific longitude latitude value belongs to it is possible to use the code provided on this GitHub project:
Download shapefiles datasets here:
Related websites | https://en.wikipedia.org/wiki/QDGC |
The QED vacuum or quantum electrodynamic vacuum is the field-theoretic vacuum of quantum electrodynamics . It is the lowest energy state (the ground state ) of the electromagnetic field when the fields are quantized . [ 1 ] When the Planck constant is hypothetically allowed to approach zero, QED vacuum is converted to classical vacuum , which is to say, the vacuum of classical electromagnetism. [ 2 ] [ 3 ]
Another field-theoretic vacuum is the QCD vacuum of the Standard Model .
The QED vacuum is subject to fluctuations about a dormant zero average-field condition; [ 4 ] Here is a description of the quantum vacuum:
The quantum theory asserts that a vacuum, even the most perfect vacuum devoid of any matter, is not really empty. Rather the quantum vacuum can be depicted as a sea of continuously appearing and disappearing [pairs of] particles that manifest themselves in the apparent jostling of particles that is quite distinct from their thermal motions. These particles are ‘virtual’, as opposed to real, particles. ...At any given instant, the vacuum is full of such virtual pairs, which leave their signature behind, by affecting the energy levels of atoms.
It is sometimes attempted to provide an intuitive picture of virtual particles based upon the Heisenberg energy-time uncertainty principle : Δ E Δ t ≥ ℏ 2 , {\displaystyle \Delta E\Delta t\geq {\frac {\hbar }{2}}\,,} (where Δ E and Δ t are energy and time variations, and ħ the Planck constant divided by 2 π ) arguing along the lines that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum and thus permits particle generation for short times. [ 6 ]
This interpretation of the energy-time uncertainty relation is not universally accepted, however. [ 7 ] [ 8 ] One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty Δ t determines a "budget" for borrowing energy Δ E . Another issue is the meaning of "time" in this relation, because energy and time (unlike position q and momentum p , for example) do not satisfy a canonical commutation relation (such as [ q , p ] = iħ ). [ 9 ] Various schemes have been advanced to construct an observable that has some kind of time interpretation, and yet does satisfy a canonical commutation relation with energy. [ 10 ] [ 11 ] The many approaches to the energy-time uncertainty principle are a continuing subject of study. [ 11 ]
The Heisenberg uncertainty principle does not allow a particle to exist in a state in which the particle is simultaneously at a fixed location, say the origin of coordinates, and has also zero momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations; if confined, it has a zero-point energy . [ 12 ]
An uncertainty principle applies to all quantum mechanical operators that do not commute . [ 13 ] In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field. [ 14 ]
Because of the non-commutation of field variables, the variances of the fields cannot be zero, although their averages are zero. [ 17 ] The electromagnetic field has therefore a zero-point energy, and a lowest quantum state. The interaction of an excited atom with this lowest quantum state of the electromagnetic field is what leads to spontaneous emission , the transition of an excited atom to a state of lower energy by emission of a photon even when no external perturbation of the atom is present. [ 18 ]
As a result of quantization, the quantum electrodynamic vacuum can be considered as a material medium. [ 20 ] It is capable of vacuum polarization . [ 21 ] [ 22 ] In particular, the force law between charged particles is affected. [ 23 ] [ 24 ] The electrical permittivity of quantum electrodynamic vacuum can be calculated, and it differs slightly from the simple ε 0 of the classical vacuum . Likewise, its permeability can be calculated and differs slightly from μ 0 . This medium is a dielectric with relative dielectric constant > 1, and is diamagnetic, with relative magnetic permeability < 1. [ 25 ] [ 26 ] Under some extreme circumstances in which the field exceeds the Schwinger limit (for example, in the very high fields found in the exterior regions of pulsars [ 27 ] ), the quantum electrodynamic vacuum is thought to exhibit nonlinearity in the fields. [ 28 ] Calculations also indicate birefringence and dichroism at high fields. [ 29 ] Many of electromagnetic effects of the vacuum are small, and only recently have experiments been designed to enable the observation of nonlinear effects. [ 30 ] PVLAS and other teams are working towards the needed sensitivity to detect QED effects.
A perfect vacuum is itself only attainable in principle. [ 31 ] [ 32 ] It is an idealization, like absolute zero for temperature, that can be approached, but never actually realized:
One reason [a vacuum is not empty] is that the walls of a vacuum chamber emit light in the form of black-body radiation...If this soup of photons is in thermodynamic equilibrium with the walls, it can be said to have a particular temperature, as well as a pressure. Another reason that perfect vacuum is impossible is the Heisenberg uncertainty principle which states that no particles can ever have an exact position ...Each atom exists as a probability function of space, which has a certain nonzero value everywhere in a given volume. ...More fundamentally, quantum mechanics predicts ...a correction to the energy called the zero-point energy [that] consists of energies of virtual particles that have a brief existence. This is called vacuum fluctuation .
Virtual particles make a perfect vacuum unrealizable, but leave open the question of attainability of a quantum electrodynamic vacuum or QED vacuum. Predictions of QED vacuum such as spontaneous emission , the Casimir effect and the Lamb shift have been experimentally verified, suggesting QED vacuum is a good model for a high quality realizable vacuum. There are competing theoretical models for vacuum, however. For example, quantum chromodynamic vacuum includes many virtual particles not treated in quantum electrodynamics. The vacuum of quantum gravity treats gravitational effects not included in the Standard Model. [ 33 ] It remains an open question whether further refinements in experimental technique ultimately will support another model for realizable vacuum.
This article incorporates material from the Citizendium article " Vacuum (quantum electrodynamic) ", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL . | https://en.wikipedia.org/wiki/QED_vacuum |
QEMSCAN is the name for an integrated automated mineralogy and petrography system providing quantitative analysis of minerals , rocks and man-made materials. QEMSCAN is an abbreviation standing for quantitative evaluation of minerals by scanning electron microscopy , and a registered trademark owned by FEI Company since 2009. Prior to 2009, QEMSCAN was sold by LEO, a company jointly owned by Leica and ZEISS. The integrated system comprises a scanning electron microscope (SEM) with a large specimen chamber, up to four light-element energy-dispersive X-ray spectroscopy (EDS) detectors, and proprietary software controlling automated data acquisition. The offline software package iDiscover provides data processing and reporting functionality.
QEMSCAN creates phase assemblage maps of a specimen surface scanned by a high-energy accelerated electron beam along a predefined raster scan pattern. Low-count energy-dispersive X-ray spectra (EDX) are generated and provide information on the elemental composition at each measurement point. The elemental composition in combination with back-scattered electron (BSE) brightness and x-ray count rate information is converted into mineral phases. [ 1 ] QEMSCAN data includes bulk mineralogy and calculated chemical assays. By mapping the sample surface, textural properties and contextual information such as particle and mineral grain size and shape, mineral associations, mineral liberation, elemental deportment, porosity , and matrix density can be calculated, visualized, and reported numerically. Data processing capabilities include combining multiple phases into mineral groups, resolving mixed spectra (boundary phase processing), image-based filtering, and particle-based classification. Quantitative reports can be generated for any selected number of samples, individual particles, and for particle classes sharing similar compositional and/or textural attributes, such as size fractions or rock types.
QEMSCAN is routinely employed in the analysis of rock- and ore-forming minerals. Sample preparation requirements include a level, dry specimen surface, coated with a thin electrically conductive layer (e.g. carbon). The sample must be stable under high vacuum conditions and the electron beam, typically 15 to 25 kV. Common sample types include 30 mm resin-impregnated blocks of drill cuttings and ore , thin sections of drill core and rocks , as well as soil samples. Very small particles such as atmospheric dust have been measured on carbon tape or filter paper. Coal samples are generally mounted in carnauba wax , providing sufficient contrast to allow for separation of the sample from the mounting medium, and subsequent measurement of coal and macerals .
QEMSCAN consists of proprietary software package iDiscover which consists of four software modules:
QEMSCAN consists of five customisable measurement modes:
QEMSCAN measurements can be applied in quantitative mineral characterisation of rocks , weathering products such as regolith and soils , and most man-made materials. As a result, it has commercial and scientific applications in mining and mineral processing ; [ 2 ] O&G ; [ 3 ] coal ; [ 4 ] environmental sciences ;, [ 5 ] [ 6 ] forensic geosciences ; [ 7 ] archaeology ; [ 8 ] agribusiness ; built environment and planetary geology . [ 9 ] | https://en.wikipedia.org/wiki/QEMSCAN |
Qivicon is an alliance of companies from different industries [ 1 ] that was founded in 2011 by Deutsche Telekom . [ 2 ] These companies collaborate on a cross-vendor wireless home automation solution that has been available in the German market since the fall of 2013. [ 3 ] It includes products in the areas of energy, security, and comfort. It connects and combines controllable devices made by different manufacturers such as motion detectors , smoke detectors , water detectors, wireless adapters for power outlets, door and window contact, temperature and humidity sensors, wireless switches, carbon monoxide sensors, thermostats , cameras, household appliances (e. g. washing machines , dryers , coffee machines ), weather stations , sound systems, and lighting controls . [ 4 ]
Qivicon has stated that it would like to take the "Smart Home" further forward around the world. [ 5 ] [ 6 ] [ 7 ] The alliance uses Smart Home optimized wireless protocols to make solutions easy to install in any home without needing to lay cables. The technical platform is international and open for companies of all sizes and in all industries. [ 8 ]
It currently consists of over 43 companies in different industries such as energy, electrical and household appliances, security and telecommunications. [ 9 ] [ 10 ] Qivicon partners include Deutsche Telekom , E wie Einfach, eQ-3, Miele , Samsung and Philips . [ 11 ] In March 2018, Deutsche Telekom announced that it had integrated the Home Connect platform, which works with Bosch and Siemens connected devices, into Qivicon to enable greater functunality between the two. For example, as well as being able to control connected Bosch and Siemens appliances directly via the Home Connect app. DT also announced a number of new compatible devices broaden the Qivicon portfolio, such as the Nest Protect smoke and CO alarm.
The Qivicon platform has been around in the German market since the fall of 2013.
The platform's technical control unit, its home base, is connected to the Internet via a broadband connection in the house or apartment. [ 12 ] In August 2016, Qivicon launched a new generation of the home base focusing on international markets. The range of different models will keep up with the diverse range of wireless protocols found throughout the international market. The models all have an identical outward appearance. But they differ in terms of their pre-installed protocols. For example, the model designed for the German market, and several other markets, already includes the protocols HomeMatic, ZigBee Pro and the inclusion of HomeMatic IP and DECT ULE has also been completed. Another model includes the ZigBee Pro and Z-Wave radio modules. All versions of the new home base can be connected to home DSL routers either by cable, wirelessly, via Wi-Fi or via Deutsche Telekom's Speedport Smart router. [ 13 ] [ 14 ]
The system can be expanded to include other wireless standards by means of USB sticks for which there are four corresponding slots in the home base of the first generation and two slots in the second generation. [ 15 ] Qivicon partners’ devices can be controlled and monitored via various partner apps for the smartphone, the tablet or the PC. [ 16 ] Since November 2017 Qivicon is compatible with Alexa from Amazon . Users can control lights, blinds or alarm systems with their voice via Amazon Echo or Google Home . [ 17 ]
In March 2017, Deutsche Telekom launched a White Label Smart Home portfolio [ 18 ] that includes platform, gateways, applications, compatible devices and services. The portfolio is designed to help telecommunications service providers, utility providers, hardware manufacturers and other enterprises create and offer smart home services.
Deutsche Telekom extended its international footprint within the smart home sector by partnering with Cosmote, the largest mobile operator in Greece and part of the OTE group, as well as Hitch in Norway , adding Greece and Norway to Qivicon's current footprint of Germany, Slovakia , the Netherlands , Austria , and Italy . [ 19 ] [ 20 ]
AV-Test, an IT security test institution, rates Qivicon as “secure”. [ 21 ] It found that the Smart Home platform used encryption for communication and provided protection from unauthorized access. [ 22 ]
Qivicon has won repeat awards from the international management consulting company Frost & Sullivan ’s. In 2016, Frost & Sullivan has awarded Qivicon with the European Connected Home New Product Innovation Award. In 2014, the smart home platform has been awarded with the European Visionary Innovation Leadership Award in recognition of what the management consulting company saw as the most innovative Smart Home solution of the year. [ 23 ]
Ohland, Günther. Smart-Living. Books on Demand, Norderstedt 2013. ISBN 978-3-7322923-0-1 . | https://en.wikipedia.org/wiki/QIVICON |
In mathematics , the QM-AM-GM-HM inequalities , also known as the mean inequality chain , state the relationship between the harmonic mean (HM), geometric mean (GM), arithmetic mean (AM), and quadratic mean (QM; also known as root mean square). Suppose that x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} are positive real numbers . Then
In other words, QM≥AM≥GM≥HM. These inequalities often appear in mathematical competitions and have applications in many fields of science. [ citation needed ]
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction , the Cauchy–Schwarz inequality , Lagrange multipliers , and Jensen's inequality . For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means .
From the Cauchy–Schwarz inequality on real numbers , setting one vector to (1, 1, ...) :
Jensen's inequality states that the value of a concave function of an arithmetic mean is greater than or equal to the arithmetic mean of the function's values. Since the logarithm function is concave, we have
The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals 1 / x 1 , … , 1 / x n {\displaystyle 1/x_{1},\dots ,1/x_{n}} , and it exceeds 1 / x 1 … x n n {\displaystyle 1/{\sqrt[{n}]{x_{1}\dots x_{n}}}} by the AM-GM inequality. x i > 0 {\displaystyle x_{i}>0} implies the inequality:
When n = 2, the inequalities become
which can be visualized in a semi-circle whose diameter is [ AB ] and center D .
Suppose AC = x 1 and BC = x 2 . Construct perpendiculars to [ AB ] at D and C respectively. Join [ CE ] and [ DF ] and further construct a perpendicular [ CG ] to [ DF ] at G . Then the length of GF can be calculated to be the harmonic mean, CF to be the geometric mean, DE to be the arithmetic mean, and CE to be the quadratic mean. The inequalities then follow easily by the Pythagorean theorem . | https://en.wikipedia.org/wiki/QM-AM-GM-HM_inequalities |
The hybrid QM/MM ( quantum mechanics / molecular mechanics ) approach is a molecular simulation method that combines the strengths of ab initio QM calculations (accuracy) and MM (speed) approaches, thus allowing for the study of chemical processes in solution and in proteins. The QM/MM approach was introduced in the 1976 paper of Warshel and Levitt . [ 1 ] They, along with Martin Karplus , won the 2013 Nobel Prize in Chemistry for "the development of multiscale models for complex chemical systems". [ 2 ] [ 3 ]
An important advantage of QM/MM methods is their efficiency. The cost of doing classical molecular mechanics (MM) simulations in the most straightforward case scales as O ( N 2 ), where N is the number of atoms in the system. This is mainly due to electrostatic interactions term (every particle interacts with everything else). However, use of cutoff radius, periodic pair-list updates and more recently the variations of the particle mesh Ewald (PME) method has reduced this to between O ( N ) to O ( N 2 ). In other words, if a system with twice as many atoms is simulated then it would take between twice to four times as much computing power. On the other hand, the simplest ab initio calculations formally scale as O ( N 3 ) or worse (restricted Hartree–Fock calculations have been suggested to scale ~ O ( N 2.7 )). Here in the ab initio calculations, N stands for the number of basis functions rather than the number of atoms. Each atom has at least as many basis functions as is the number of electrons. To overcome the limitation, a small part of the system that is of major interest is treated quantum-mechanically (for instance, the active site of an enzyme) and the remaining system is treated classically. [ 4 ] [ 5 ]
The energy of the combined system may be calculated in two different ways. The simplest is referred to as the 'subtractive scheme' which was proposed by Maseras and Morokuma in 1995. In the subtractive scheme the energy of the entire system is calculated using a molecular mechanics force field , then the energy of the QM system is added (calculated using a QM method), finally the MM energy of the QM system is subtracted.
E = E Q M ( Q M ) + E M M ( Q M + M M ) − E M M ( Q M ) {\displaystyle E=E^{QM}(QM)+E^{MM}(QM+MM)-E^{MM}(QM)}
In this equation E M M ( Q M ) {\displaystyle E^{MM}(QM)} would refer to the energy of the QM region as calculated using molecular mechanics. In this scheme, the interaction between the two regions will only be considered at a MM level of theory.
In practice, a more widely used approach is a more accurate, additive method. The equation for this consists of three terms: E ( Q M / M M ) = ∑ I ′ = 1 M M [ ∫ d r q I ′ ρ ( r ) | R I ′ − r | + ∑ I = 1 Q M q I ′ q I | R I ′ − R I | ] + ∑ non-bonded pairs ( A I I ′ R I I ′ 12 − B I I ′ R I I ′ 6 ) + ∑ bonds k r ( R I I ′ − r 0 ) 2 + ∑ angles k θ ( θ − θ 0 ) 2 + ∑ torsions ∑ n k ϕ , n [ cos ( n ϕ + δ n ) + 1 ] {\displaystyle {\begin{aligned}E(QM/MM)&=\sum _{I'=1}^{MM}\left[\int \operatorname {d} \mathbf {r} {q_{I'}\rho (\mathbf {r} ) \over \left\vert \mathbf {R} _{I'}-\mathbf {r} \right\vert }+\sum _{I=1}^{QM}{\frac {q_{I'}q_{I}}{\left\vert \mathbf {R} _{I'}-\mathbf {R} _{I}\right\vert }}\right]\\&+\sum _{\text{non-bonded pairs}}\left({\frac {A_{II'}}{R_{II'}^{12}}}-{\frac {B_{II'}}{R_{II'}^{6}}}\right)+\sum _{\text{bonds}}k_{r}(R_{II'}-r_{0})^{2}\\&+\sum _{\text{angles}}k_{\theta }(\theta -\theta _{0})^{2}+\sum _{\text{torsions}}\sum _{n}k_{\phi ,n}[\cos(n\phi +\delta _{n})+1]\end{aligned}}}
The index I {\displaystyle I} labels the nuclei in the QM region whereas I ′ {\displaystyle I'} labels the MM nuclei. The first two terms represent the interaction between the total charge density (due to electrons and cores) in the QM region and classical charges of the MM region. The third term accounts for dispersion interactions across the QM/MM boundary. Any covalent bond -stretching potentials that cross the boundary are accounted for by the fourth term. The final two terms account for the energy across the boundary that arises from bending covalent bonds and torsional potentials. At least one of the atoms in the angles θ {\displaystyle \theta } or ϕ {\displaystyle \phi } will be a QM atom with the others being MM atoms. [ 6 ] : 422–3
Evaluating the charge-charge term in the QM/MM interaction equation given previously can be very computationally expensive (consider the number of evaluations required a system with 10 6 grid points for the electron density of the QM system and 10 4 MM atoms). A method by which this issue can be mitigated is to construct three concentric spheres around the QM region and evaluate which one of these spheres the MM atoms lie within. If the MM atoms reside within the innermost sphere their interactions with the QM system are treated as per the equation for E ( Q M / M M ) {\displaystyle E(QM/MM)} . The MM charges that lie within the second sphere (but not the first) interact with the QM region by giving the QM nuclei constructed charges. These charges are determined by the RESP approach in an attempt to mimic electron density. Using this approach the changing charges on the QM nuclei during the course of a simulation are accounted for.
In the third outermost region the classical charges interact with the multipole moments of the quantum charge distribution. By calculating charge-charge interactions by using successively more approximate methods it is possible to obtain a very significant reduction in computational cost whilst not suffering a significant loss in accuracy. [ 6 ] : 423–4
Electrostatic interactions between the QM and MM region may be considered at different levels of sophistication. These methods can be classified as either mechanical embedding, electrostatic embedding or polarized embedding.
Mechanical embedding treats the electrostatic interactions at the MM level, though simpler than the other methods, certain issues may occur, in part due to the extra difficulty in assigning appropriate MM properties such as atom centered point charges to the QM region. The QM region being simulated is the site of the reaction thus it is likely that during the course of the reaction the charge distribution will change resulting in a high level of error if a single set of MM electrostatic parameters is used to describe it. Another problem is the fact that mechanical embedding will not consider the effects of electrostatic interactions with the MM system on the electronic structure of the QM system. [ 7 ]
Electrostatic embedding does not require the MM electrostatic parameters for the QM. This is due to it considering the effects of the electrostatic interactions by including certain one electron terms in the QM regions Hamiltonian . This means that polarization of the QM system by the electrostatic interactions with the MM system will now be accounted for. Though an improvement on the mechanical embedding scheme it comes at the cost of increased complexity hence requiring more computational effort. Another issue is it neglects the effects of the QM system on the MM system whereas in reality both systems would polarize each other until an equilibrium is met.
In order to construct the required one electron terms for the MM region it is possible to utilize the partial charges described by the MM calculation. This is the most popular method for constructing the QM Hamiltonian however it may not be suitable for all systems. [ 7 ]
Whereas electrostatic embedding accounts for the polarisation of the QM system by the MM system, neglecting the polarization of the MM system by the QM system, polarized embedding accounts for both the polarization of the MM system by the QM. These models allow for flexible MM charges and fall into two categories. In the first category, the MM region is polarized by the QM electric field but then does not act back on the QM system. In the second category are fully self-consistent formulations which allow for mutual polarization between the QM and the MM systems. Polarized embedding schemes have scarcely been applied to bio-molecular simulations and have essentially been restricted to explicit solvation models where the solute will be treated as a QM system and the solvent a polarizable force field. [ 7 ]
Even though QM/MM methods are often very efficient, they are still rather tricky to handle. A researcher has to limit the regions (atomic sites) which are simulated by QM, however methods have been developed that allow particles to move between the QM and MM region. [ 8 ] Moving the limitation borders can both affect the results and the time computing the results. The way the QM and MM systems are coupled can differ substantially depending on the arrangement of particles in the system and their deviations from equilibrium positions in time. Usually, limits are set at carbon-carbon bonds and avoided in regions that are associated with charged groups, since such an electronically variant limit can influence the quality of the model. [ 9 ]
Directly connected atoms, where one is described by QM and the other by MM are referred to as Junction atoms. Having the boundary between the QM region and MM region pass through a covalent bond may prove problematic however this is sometimes unavoidable. When it does occur it is important that the bond of the QM atom be capped in order to prevent the appearance of bond cleavage in the QM system. [ 9 ]
In systems where the QM/MM boundary cuts a bond three issues must be dealt with. First, the dangling bond of the QM system must be capped, this is because it is undesirable to truncate the QM system (treating the bond as if it has been cleaved will yield very unrealistic calculations). The second issue relates to polarisation, more specifically for electrostatic or polarized embedding it is important to ensure that the proximity of the MM charges near the boundary does not cause over-polarisation of the QM density. The final issue is the bonding MM terms must be carefully selected in order to prevent double counting of interactions when looking at bonds across the boundary. [ 9 ]
Overall the goal is to obtain a good description of QM-MM interactions at the boundary between the QM and the MM system and there are three schemes by which this can be achieved. [ 9 ]
Link atom schemes introduce an additional atomic centre (usually a hydrogen atom). This atom is not part of the real system. It is covalently bonded to the atom being described by quantum mechanics which serves to saturate its valency (by replacing the bond that has been broken). [ 9 ]
In boundary atom schemes, the MM atom which is bonded across the boundary to a QM atom is replaced with a special boundary atom which appears in both the QM and the MM calculation. In the MM calculation, it simply behaves as an MM atom but in the QM system it mimics the electronic character of the MM atom bounded across the boundary to the QM atom. [ 9 ]
These schemes place hybrid orbitals at the boundary and keep some of them frozen. These orbitals cap the QM region and replace the cut bond. [ 9 ] | https://en.wikipedia.org/wiki/QM/MM |
QMC@Home was a volunteer computing project for the BOINC client aimed at further developing and testing Quantum Monte Carlo (QMC) for use in quantum chemistry . [ 1 ] It is hosted by the University of Münster with participation by the Cavendish Laboratory . QMC@Home allows volunteers from around the world to donate idle computer cycles to help calculate molecular geometry using Diffusion Monte Carlo .
The project is developing a new application using density functional theory .
The project began its Beta testing on 23 May 2006. As of February 2010 [update] , QMC@Home has about 7,500 active participants from 102 countries, contributing about 5 teraFLOPS of computation power. [ 2 ]
In order to get results from home computers the work is split into "workunits". The time it takes to complete a workunit depends on the size of the calculated system and the speed of the user's computer. The target time is between 4 and 48 hours on a 2.4 GHz system.
This is a list of molecules recently [ when? ] tested:
1a Ammonia ; 1 Ammonia dimer; 2a Water ; 2 Water dimer; 3a Formic acid ; 3 Formic acid dimer; 4a Formamide ; 4 Formamide dimer; 5a Uracil ; 5 Uracil dimer; 6a 2- pyridoxine ; 6b 2- aminopyridine ; 6 2-pyridoxine/2-aminopyridine; 7a Adenine ; 7b Thymine ; 7 Adenine/thymine WC; 8a Methane ; 8 Methane dimer; 9a Ethene ; 9 Ethene dimer; 10 Benzene /methane; 11a Benzene; 11 Benzene dimer; 12a Pyrazine ; 12 Pyrazine dimer; 13 Uracil dimer; 14a Indole ; 14 Indole/benzene; 15 Adenine/thymine stack; 16b Ethyne ; 16 Ethene/ethyne; 17 Benzene/water; 18 Benzene/ammonia; 19b Hydrogen cyanide ; 19 Benzene/hydrogen cyanide; 20 Benzene dimer; 21 Indole/benzene; 22a Phenol; 22 Phenol dimer
This quantum chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/QMC@Home |
QMCF Technology is an episomal protein production system that uses genetically modified mammalian cells and specially designed plasmids . QMCF plasmids carry a combination of regulatory sequences from mouse polyomavirus (Py) DNA replication origin which in combination with Epstein-Barr virus (EBV) EBNA-1 protein binding site as nuclear retention elements ensure stable propagation of plasmids in mammalian cells. In addition the vectors carry the selection marker operational for selection of plasmid carrying bacteria and QMCF cells, bacterial ColE1 origin of replication, and cassette for expression of protein of interest. QMCF cell lines express Large-T antigen and EBNA-1 proteins which bind the viral sequences on the QMCF plasmid and hence support plasmid replication and maintenance in the cells. QMCF Technology has several important differences compared to commonly known transient expression and stable cell line expression systems. Unlike in transient expression system, QMCF Technology enables to maintain episomally replicating QMCF plasmids inside the cells for up to 50 days thus providing an option for production phase of 2–3 weeks. Therefore, the production levels of QMCF Technology are higher (up to 1g/L). Another difference is the option of establishing expression cell banks within one week, which is not feasible with transient system. Compared to usage of stable cell line, QMCF technology is a rapid method leaving out time-consuming clone selection step during cell line development.
QMCF Technology can be used for the production of proteins, antibodies and virus-like particles for basic research and pharmaceutical development. [ 1 ]
Protein production process starts with cloning cDNA of a target molecule into QMCF expression vector . Thereafter 1-6 μg of plasmidial DNA is transfected by electroporation into QMCF cells ( CHO or HEK293 based cells) for expression in serum-free suspension. Since QMCF plasmids contain antibiotic resistance gene and are able to stably replicate and remain inside dividing cells, a selection and growth of the cell culture takes place. This allows to upscale the production into desired volume. Finally, the process is switched to production phase by changing media temperature to 30 ̊C. | https://en.wikipedia.org/wiki/QMCF_Technology |
QNX ( / ˌ k juː ˌ ɛ n ˈ ɛ k s / or / ˈ k juː n ɪ k s / ) is a commercial Unix-like real-time operating system , aimed primarily at the embedded systems market.
The product was originally developed in the early 1980s by Canadian company Quantum Software Systems, founded March 30, 1980, and later renamed QNX Software Systems.
As of 2022 [update] , it is used in a variety of devices including automobiles , [ 1 ] medical devices , program logic controllers , automated manufacturing , trains , and more.
Gordon Bell and Dan Dodge , both students at the University of Waterloo in 1980, took a course in real-time operating systems, in which the students constructed a basic real-time microkernel and user programs. Both were convinced there was a commercial need for such a system, and moved to the high-tech planned community Kanata, Ontario , to start Quantum Software Systems that year. In 1982, the first version of QUNIX was released for the Intel 8088 CPU. In 1984, Quantum Software Systems renamed QUNIX to QNX (Quantum's Network eXecutive) in an effort to avoid any trademark infringement challenges.
One of the first widespread uses of the QNX real-time OS (RTOS) was in the nonembedded world when it was selected as the operating system for the Ontario education system's own computer design, the Unisys ICON . Over the years QNX was used mostly for larger projects, as its 44k kernel was too large to fit inside the one-chip computers of the era. The system garnered a reputation for reliability [ citation needed ] and became used in running machinery in many industrial applications.
In the late-1980s, Quantum realized that the market was rapidly moving towards the Portable Operating System Interface ( POSIX ) model and decided to rewrite the kernel to be much more compatible at a low level. The result was QNX 4. During this time Patrick Hayden , while working as an intern, along with Robin Burgener (a full-time employee at the time), developed a new windowing system. This patented [ 2 ] concept was developed into the embeddable graphical user interface (GUI) named the QNX Photon microGUI. QNX also provided a version of the X Window System .
To demonstrate the OS's capability and relatively small size, in the late 1990s QNX released a demo image that included the POSIX-compliant QNX 4 OS, a full graphical user interface, graphical text editor, TCP/IP networking, web browser and web server that all fit on a bootable 1.44 MB floppy disk for the 386 PC. [ 3 ] [ 4 ]
Toward the end of the 1990s, the company, then named QNX Software Systems, began work on a new version of QNX, designed from the ground up to be symmetric multiprocessing (SMP) capable, and to support all current POSIX application programming interfaces (APIs) and any new POSIX APIs that could be anticipated while still retaining the microkernel architecture. This resulted in QNX Neutrino, released in 2001.
Along with the Neutrino kernel, QNX Software Systems became a founding member of the Eclipse ( integrated development environment ) consortium. The company released a suite of Eclipse plug-ins packaged with the Eclipse workbench in 2002, and named QNX Momentics Tool Suite.
In 2004, the company announced it had been sold to Harman International Industries. Before this acquisition, QNX software was already widely used in the automotive industry for telematics systems. Since the purchase by Harman, QNX software has been designed into over 200 different automobile makes and models, in telematics systems, and in infotainment and navigation units. [ citation needed ] The QNX CAR Application Platform was running in over 20 million vehicles as of mid-2011. [ 5 ] The company has since released several middleware products including the QNX Aviage Multimedia Suite, the QNX Aviage Acoustic Processing Suite and the QNX HMI Suite.
The microkernels of Cisco Systems ' IOS-XR (ultra high availability IOS, introduced 2004) [ 6 ] [ 7 ] and IOS Software Modularity (introduced 2006) [ 8 ] were based on QNX. IOS Software Modularity never gained traction and was limited only to small run for Catalyst 6500, while IOS XR moved to Linux as of release 6.x .
In September 2007, QNX Software Systems announced the availability of some of its source code . [ 9 ]
On April 9, 2010, Research In Motion (later renamed to BlackBerry Limited ) announced they would acquire QNX Software Systems from Harman International Industries. [ 10 ] On the same day, QNX source code access was restricted from the public and hobbyists. [ 11 ]
In September 2010, the company announced a tablet computer , the BlackBerry PlayBook , and a new operating system BlackBerry Tablet OS based on QNX to run on the tablet. [ 12 ]
On October 18, 2011, Research In Motion announced "BBX", [ 13 ] which was later renamed BlackBerry 10 , in December 2011. [ 14 ] Blackberry 10 devices build upon the BlackBerry PlayBook QNX based operating system for touch devices, but adapt the user interface for smartphones using the Qt based Cascades Native User-Interface framework.
At the Geneva Motor Show, Apple demonstrated CarPlay which provides an iOS -like user interface to head units in compatible vehicles. Once configured by the automaker, QNX can be programmed to hand off its display and some functions to an Apple CarPlay device. [ 15 ] [ 16 ]
On December 11, 2014, Ford Motor Company stated that it would replace Microsoft Auto with QNX. [ 1 ]
In January 2017, QNX announced the upcoming release of its SDP 7.0, with support for Intel and ARM 32- and 64-bit platforms, and support for C++14 . It was released in March 2017. [ 17 ]
In December 2023, QNX released QNX SDP 8.0 which is powered by a next generation microkernel with support for the latest Intel and ARM [v8 and v9] 64 bit platforms, GCC12 based toolchain and a QNX toolkit for Visual Studio Code . [ 18 ]
As a microkernel -based OS, QNX is based on the idea of running most of the operating system kernel in the form of a number of small tasks, named Resource Managers. This differs from the more traditional monolithic kernel , in which the operating system kernel is one very large program composed of a huge number of parts, with special abilities. In the case of QNX, the use of a microkernel allows users (developers) to turn off any functions they do not need without having to change the OS. Instead, such services will simply not run.
The QNX kernel, procnto (also name of the binary executable program for the QNX Neutrino ('nto') process ('proc') itself), contains only CPU scheduling , interprocess communication , interrupt redirection and timers. Everything else runs as a user process, including a special process known as proc which performs process creation and memory management by operating in conjunction with the microkernel . This is made possible by two key mechanisms: subroutine-call type interprocess communication, and a boot loader which can load an image containing the kernel and any desired set of user programs and shared libraries. There are no device drivers in the kernel. The network stack is based on NetBSD code. [ 19 ] Along with its support for its own, native, device drivers, QNX supports its legacy, io-net manager server, and the network drivers ported from NetBSD. [ 20 ]
QNX interprocess communication consists of sending a message from one process to another and waiting for a reply. This is a single operation, called MsgSend . The message is copied, by the kernel, [ citation needed ] from the address space of the sending process to that of the receiving process. If the receiving process is waiting for the message, control of the CPU is transferred at the same time, without a pass through the CPU scheduler. Thus, sending a message to another process and waiting for a reply does not result in "losing one's turn" for the CPU. This tight integration between message passing and CPU scheduling is one of the key mechanisms that makes QNX message passing broadly usable. Most Unix and Linux interprocess communication mechanisms lack this tight integration, although a user space implementation of QNX-type messaging for Linux does exist . Mishandling of this subtle issue is a primary reason for the disappointing performance of some other microkernel systems such as early versions of Mach . [ citation needed ] The recipient process need not be on the same physical machine.
All I/O operations, file system operations, and network operations were meant to work through this mechanism, and the data transferred was copied during message passing. Later versions of QNX reduce the number of separate processes and integrate the network stack and other function blocks into single applications for performance reasons.
Message handling is prioritized by thread priority. Since I/O requests are performed using message passing, high priority threads receive I/O service before low priority threads, an essential feature in a hard real-time system.
The boot loader is the other key component of the minimal microkernel system. Because user programs can be built into the boot image, the set of device drivers and support libraries needed for startup need not be, and are not, in the kernel. Even such functions as program loading are not in the kernel, but instead are in shared user-space libraries loaded as part of the boot image. It is possible to put an entire boot image into ROM , which is used for diskless embedded systems.
Neutrino supports symmetric multiprocessing and processor affinity , called bound multiprocessing (BMP) in QNX terminology. BMP is used to improve cache hitting and to ease the migration of non-SMP safe applications to multi-processor computers.
Neutrino supports strict priority-preemptive scheduling and adaptive partition scheduling (APS). APS guarantees minimum CPU percentages to selected groups of threads, even though others may have higher priority. The adaptive partition scheduler is still strictly priority-preemptive when the system is underloaded. It can also be configured to run a selected set of critical threads strictly real time , even when the system is overloaded.
The QNX operating system also contained a web browser known as 'Voyager'. [ 21 ]
Due to its microkernel architecture QNX is also a distributed operating system . Dan Dodge and Peter van der Veen hold U.S. Patent 6,697,876: Distributed kernel operating system based on the QNX operating system's distributed processing features known commercially as Transparent Distributed Processing. This allows the QNX kernels on separate devices to access each other's system services using effectively the same communication mechanism as is used to access local services. [ non-primary source needed ]
The BlackBerry PlayBook tablet computer designed by BlackBerry uses a version of QNX as the primary operating system. The BlackBerry 10 operating system is also based on QNX.
QNX is also used in car infotainment systems with many major car makers offering variants that include an embedded QNX architecture. It is supported by popular SSL/TLS libraries such as wolfSSL . [ 25 ]
Since the introduction of its "Safe Kernel 1.0" in 2010, QNX was projected and used subsequently in automated drive or ADAS systems for automotive projects that require a functional safety certified RTOS. QNX provides this with its QNX OS for Safety products. [ 26 ]
QNX Neutrino (2001) has been ported to a number of platforms and now runs on practically any modern central processing unit (CPU) family that is used in the embedded market. This includes the PowerPC , x86 , MIPS , SH-4 , and the closely interrelated group of ARM , StrongARM , and XScale .
As of June 26, 2023, QNX software is now embedded in over 255 million [ 27 ] vehicles worldwide, including most leading OEMs and Tier 1s, such as BMW, Bosch, Continental, Dongfeng Motor, Geely, Ford, Honda, Mercedes-Benz, Subaru, Toyota, Volkswagen, Volvo, and more. [ 28 ]
On Jan 06, 2005, QNX made its software development platform available for non-commercial use. [ 29 ]
QNX offers a license for noncommercial and academic users. [ 30 ] In January 2024, BlackBerry introduced QNX Everywhere to make QNX more accessible to Hobbyists. QNX Everywhere was made publicly accessible in early 2024. [ 31 ]
QNX Standard Support is available for a BSP that is listed below as available on QNX Software Center. For other BSPs, alternative forms of support (e.g., custom support plans, etc.) may be available or required from the “BSP Supplier” or “Board Vendor” indicated below. [ 34 ]
BlackBerry QNX has worked with a network of partner organizations to provide complementary technologies. These important relationships have ability to provide the foundational software, middleware, and services behind the world's most critical embedded systems. [ 35 ] | https://en.wikipedia.org/wiki/QNX |
QPNC-PAGE , or Quantitative Preparative Native Continuous Polyacrylamide Gel Electrophoresis , is a bioanalytical , high-resolution and high-precision zone electrophoresis technique applied in biochemistry and bioinorganic chemistry to separate proteins or protein isoforms by isoelectric point and by continuous elution from a gel column for further characterization. [ 1 ] [ 2 ]
This standardized 1-D hybrid variant of native gel electrophoresis and preparative polyacrylamide gel electrophoresis is used to quantitatively resolve physiological concentrations of macromolecules with high recovery, for example, into active or native metalloproteins in biological samples or into properly and improperly folded metal cofactor -containing proteins in complex protein mixtures . [ 1 ] [ 3 ]
Proteins perform several functions in living organisms , including catalytic reactions and transport of molecules or ions within the cells , the organs or the whole body . The understanding of the processes in human organisms, which are mainly driven by biochemical reactions and protein-protein interactions , depends to a great extent on the ability to isolate active proteins in biological extracts or fluids for more detailed examination of chemical structure and physiological function . This essential information can imply an important indication of a patient's state of health. [ 4 ]
As about 30–40% of all known proteins contain one or more metal ion cofactors (e.g., ceruloplasmin , ferritin , amyloid-beta precursor protein , matrix metalloproteinase , or metallochaperones ), especially native and denatured metalloproteins have to be isolated, identified and quantified after liquid biopsy . Many of these cofactors (e.g., iron , copper , or zinc ) play a key role in vital enzymatic catalytic processes, such as electron transport and redox transformations, or stabilize globular protein molecules driven by protein-cofactor interactions. [ 5 ] Therefore, the high-precision gel electrophoresis and other complementary separation techniques are highly relevant as initial step of protein and trace metal speciation analysis of biological specimen , subsequently, followed by modern mass spectrometric and magnetic resonance methods for quantifying and identifying the soluble proteins of interest. [ 6 ]
In gel electrophoresis, zwitterions , such as proteins and peptides , are normally separated by charge , size , or shape . [ 8 ] The aim of isoelectric focusing (IEF), for example, is to separate proteins according to their isoelectric point (pI), thus, according to their charge at different pH values using immobilized pH gradient gels. [ 9 ] Here, a similar mechanism is accomplished in a commercially available electrophoresis chamber with integrated elution chamber for separating charged biomolecules , for example, superoxide dismutase (SOD) [ 10 ] or allergens , [ 11 ] at constantly higher pH conditions and different migration rates depending on different isoelectric points in an optimized resolving gel . The separated (metal) proteins elute sequentially, almost independently of their molecular mass or hydrodynamic volume , starting with the lowest (pI > 2–4) and ending with the highest isoelectric point (pI < 10.0) of the dissolved protein molecules to be analyzed. [ 12 ]
Due to the specific properties of the prepared gel and electrophoresis buffer solution which is basic and contains Tris - HCl and NaN 3 , [ 7 ] most proteins of a biological system (e.g., Helicobacter pylori [ 13 ] ) are charged negatively in the solution, and will migrate from the cathode to the anode due to the electric field . In general, reaction equation (1) shows that the carboxyl side group of a proteinogenic amino acid is negatively charged, equation (2) that the amino side groups are electrically neutral under these conditions:
(1) R-COOH + OH − → R-COO − + H 2 O
(2) R-NH 3 + + OH − → R-NH 2 + H 2 O
At the anode, electrochemically - generated hydrogen ions react with Tris molecules to form monovalent Tris ions (3). The positively charged Tris ions migrate through the gel to the cathode where they neutralise hydroxide ions to form Tris molecules and water (4):
(3) (HOCH 2 ) 3 CNH 2 + H + → [(HOCH 2 ) 3 CNH 3 ] +
(4) [(HOCH 2 ) 3 CNH 3 ] + + OH − → (HOCH 2 ) 3 CNH 2 + H 2 O
Thus, the Tris-based buffering mechanism causes a constant pH in the continuous buffer system with a high buffer capacity . [ 14 ]
At 25 °C Tris buffer has an effective pH range between 7.5 and 9.0. Under the conditions given here (addressing the concentration of buffer components, buffering mechanism, pH and temperature) the effective pH is shifted in the range of about 10.0 to 10.5. Native buffer systems all have low conductivity and range in pH from 3.8 to 10.2. Continuous native buffer systems are thus used to separate proteins according their pI. [ 15 ]
Although the constant pH value (10.00) of the electrophoresis buffer does not correspond to a physiological pH value within a cell or tissue type, the separated ring-shaped protein bands (zones) are continuously eluted from the gel column into a physiological buffer solution (pH 8.00) and isolated in different fractions . [ 7 ] Provided that irreversible denaturation cannot be demonstrated by an independent procedure, most protein molecules are stable in aqueous solution , at pH values from 3 to 10 if the temperature is below 50 °C. [ 16 ] As the Joule heat and temperature generated during electrophoresis may exceed 50 °C, [ 17 ] and thus, have a negative impact on the stability and migration behavior of proteins in the gel, the separation system, consisting of the electrophoresis chamber, fraction collector and other devices, is cooled in a refrigerator at 4 °C, thus, greatly reducing the risk of heat convection currents. [ 18 ] Overheating of the gel is prevented by an additional internal cooling circuit as an integrated part of the electrophoresis chamber and by generating a constant power by the power supply. [ 19 ]
Best polymerization conditions for acrylamide gels are obtained at 25–30 °C [ 20 ] and polymerization seems terminated after 20–30 min of reaction although residual monomers (10–30%) are detected after this time. [ 21 ] The co-polymerization of acrylamide (AA) monomer/ N,N'-Methylenebisacrylamide (Bis-AA) cross-linker initiated by ammonium persulfate (APS)/ tetramethylethylenediamine (TEMED) reactions, is most efficient at alkaline pH of the acrylamide solution. Thereby, acrylamide chains are created and cross-linked at a time. Due to the properties of the electrophoresis buffer, the gel polymerization is conducted at pH 10.00 making sure an efficient use of TEMED and APS as catalysts of the polymerization reaction, and concurrently, suppressing a competitive hydrolysis of the produced acrylamide polymer network. Polymer networks are three-dimensionally linked polymer chains. Otherwise, proteins could be modified by reaction with unpolymerized monomers of acrylamide, forming covalent acrylamide adduction products that may result in multiple bands. [ 22 ]
Additionally, the time of polymerization of a gel can directly affect the peak-elution times of separated metalloproteins in the electropherogram due to the compression and dilatation of the gels and their pores if the incubation times for the reaction mixture used to prepare a gel are not optimized. In order to ensure maximum reproducibility in gel pore size and to obtain a fully polymerized and non-restrictive large pore gel for a PAGE run, the polyacrylamide gel is polymerized for a time period of 69 hr at room temperature (RT) in a graduated glass column (gel tube) enclosing the cooling core on the casting stand. [ 19 ] The exothermic heat generated by the polymerization processes is dissipated constantly while the temperature may rise rapidly to over 75 °C in the first minutes, after which it falls slowly. [ 23 ] After 69 hr, the gel has reached room temperature and is in its lowest energy state , as the basic chemical reactions and gelation are complete. [ 19 ] Gelation means that the solvent (water) gets immobilized within the polymer network by means of hydrogen bonds and also van der Waals forces . As a result, the prepared gel is homogeneous (in terms of homogeneous distribution of cross-links throughout the gel sample [ 24 ] ), inherently stable and free of monomers or radicals . Fresh polyacrylamide gels are further hydrophilic , electrically neutral and do not bind proteins. [ 25 ] Sieving effects due to gravity -induced compression of the gel column can be excluded for the same reasons. Thus, in a medium without molecular sieving properties a high-resolution can be expected. [ 26 ]
Before an electrophoretic run is started the prepared 4% T (total polymer content (T)), 2.67% C (cross-linker concentration (C)) resolving gel, which has the shape of a hollow cylinder (gel column) and encapsulates the cooling core, is pre-run to equilibrate it. [ 7 ] It is essentially non-sieving and optimal for electrophoresis of proteins greater than or equal to 200 k u . Proteins migrate in it more or less on the basis of their free mobility. [ 27 ] For these reasons interactions of the gel with the biomolecules are negligibly low, and thus, the proteins separate cleanly and predictably at a polymerization time of 69 hr. [ 7 ] The separated metalloproteins including biomolecules ranging from approximately < 1 ku to greater than 30 ku (e.g., metal chaperones , prions , metal transport proteins , amyloids , metalloenzymes , metallopeptides , metallothionein , phytochelatins ) are not dissociated into apoproteins and metal cofactors. [ 28 ]
The bioactive structures (native or 3D conformation or shape) of the isolated protein molecules do not undergo any significant conformational changes . Thus, active metal cofactor-containing proteins can be isolated reproducibly in the same fractions after a PAGE run. [ 12 ] A shifting peak in the respective electropherogram indicates that the standardized time of gel polymerization (69 hr, RT) is not implemented in a PAGE experiment . A lower deviation of the standardized polymerization time (< 69 hr) stands for incomplete polymerization, and thus, for inherent instability due to gel softening during the cross-linking of polymers as the material reaches swelling equilibrium, [ 29 ] whereas exceeding this time limit (> 69 hr) is an indicator of gel aging. [ 30 ] The phenomenon of gel aging is closely connected to long-term viscosity decrease of aqueous polyacrylamide solutions [ 31 ] and increased swelling of hydrogels. [ 32 ]
Under standard conditions, metalloproteins with different molecular mass ranges and isoelectric points have been recovered in biologically active form at a quantitative yield of more than 95%. [ 19 ] By preparative SDS-PAGE standard proteins ( cytochrome c , aldolase , ovalbumin and bovine serum albumin ) with molecular masses of 14–66 ku can be recovered with an average yield of about 73.6%. [ 33 ] Preparative isotachophoresis (ITP) is applied for isolating palladium -containing proteins with molecular masses of 362 ku (recovery: 67%) and 158 ku (recovery: 97%). [ 34 ]
Physiological concentrations ( ppb -range) of Fe, Cu, Zn, Ni , Mo , Pd, Co , Mn , Pt , Cr , Cd and other metal cofactors can be identified and absolutely quantified in an aliquot of a fraction by inductively coupled plasma mass spectrometry (ICP-MS) [ 35 ] or total reflection X-ray fluorescence (TXRF), [ 36 ] for example. In case of ICP-MS the structural information of the associated metallobiomolecules is irreversibly lost due to ionization of the sample with plasma . [ 37 ] [ 38 ] Another established high sensitive detection method for the determination of trace elements in biological samples is graphite furnace atomic absorption spectrometry (GF-AAS). [ 39 ] Because of high purity and optimized concentration of the separated properly folded metalloproteins, for example, therapeutic recombinant plant-made pharmaceuticals such as copper chaperone for superoxide dismutase (CCS) from medicinal plants , in a few specific PAGE fractions, the related structures of these bioactive analytes can be elucidated quantitatively by using solution NMR spectroscopy under non-denaturing conditions. [ 40 ]
Improperly folded metal proteins, for example, CCS or Cu-Zn-superoxide dismutase (SOD1) present in brain , blood or other clinical samples, are indicative of neurodegenerative diseases like Alzheimer's disease (AD) or amyotrophic lateral sclerosis (ALS). [ 41 ] Active CCS or SOD molecules contribute to intracellular homeostatic control of essential metal ion species (e.g., Cu 1+/2+ , Zn 2+ , Fe 2+/3+ , Mn 2+ , Ni 3+ ) in organisms, and thus, these biomolecules can balance pro-oxidative and antioxidative processes in the cytoplasm . [ 42 ] Otherwise, free or loosely bound transition metal ions take part in Fenton-like reactions in which deleterious hydroxyl radical is formed, which unrestrained would be destructive to proteins. [ 43 ] The loss of active CCS increases the amyloid-β production in neurons which, in turn, is a major pathological hallmark of AD. [ 44 ] Therefore, copper chaperone for superoxide dismutase is proposed to be one of the most promising biomarkers of Cu toxicity in these diseases. [ 45 ] CCS should be analysed primarily in blood because a meta-analysis of serum data showed that AD patients have higher levels of serum Cu than healthy controls . [ 46 ]
In the late 1990s, QPNC-PAGE was developed at Jülich Research Centre to optimize the chemical composition and physico- chemical properties of the acrylamide gel matrix, as dynamic compression and decompression effects in the gel during electrophoresis influence the result of protein purification . [ 47 ]
This unique electrophoretic method was significantly influenced by the pioneering work of David E. Garfin and several other specialist authors . | https://en.wikipedia.org/wiki/QPNC-PAGE |
A QR code, quick-response code , [ 1 ] is a type of two-dimensional matrix barcode invented in 1994 by Masahiro Hara of Japanese company Denso Wave for labelling automobile parts. [ 2 ] [ 3 ] It features black squares on a white background with fiducial markers , readable by imaging devices like cameras, and processed using Reed–Solomon error correction until the image can be appropriately interpreted. The required data is then extracted from patterns that are present in both the horizontal and the vertical components of the QR image. [ 4 ]
Whereas a barcode is a machine-readable optical image that contains information specific to the labeled item, the QR code contains the data for a locator, an identifier, and web-tracking. To store data efficiently, QR codes use four standardized modes of encoding: numeric , alphanumeric , byte or binary , and kanji . [ 5 ] Compared to standard UPC barcodes , the QR labeling system was applied beyond the automobile industry because of faster reading of the optical image and greater data-storage capacity in applications such as product tracking, item identification, time tracking, document management, and general marketing. [ 4 ]
The QR code system was invented in 1994, at the Denso Wave automotive products company, in Japan. [ 6 ] [ 7 ] [ 8 ] The initial alternating-square design presented by the team of researchers, headed by Masahiro Hara , was influenced by the black counters and the white counters played on a Go board ; [ 9 ] the pattern of the position detection markers was determined by finding the least-used sequence of alternating black-white areas on printed matter, which was found to be (1:1:3:1:1). [ 10 ] [ 6 ] The functional purpose of the QR code system was to facilitate keeping track of the types and numbers of automobile parts, by replacing individually-scanned bar-code labels on each box of auto parts with a single label that contained the data of each label. The quadrangular configuration of the QR code system consolidated the data of the various bar-code labels with Kanji, Kana , and alphanumeric codes printed onto a single label. [ 11 ] [ 10 ] [ 6 ]
As of 2024, [update] QR codes are used in a much broader context, including both commercial tracking applications and convenience-oriented applications aimed at mobile phone users (termed mobile tagging). QR codes may be used to display text to the user, to open a webpage on the user's device, to add a vCard contact to the user's device, to open a Uniform Resource Identifier (URI), to connect to a wireless network, or to compose an email or text message. There are a great many QR code generators available as software or as online tools that are either free or require a paid subscription. [ 12 ] The QR code has become one of the most-used types of two-dimensional code. [ 13 ]
During June 2011, 14 million American mobile users scanned a QR code or a barcode. Some 58% of those users scanned a QR or barcode from their homes, while 39% scanned from retail stores; 53% of the 14 million users were men between the ages of 18 and 34. [ 14 ]
In 2022, 89 million people in the United States scanned a QR code using their mobile devices, up by 26 percent compared to 2020. The majority of QR code users used them to make payments or to access product and menu information. [ 15 ]
In September 2020, a survey found that 18.8 percent of consumers in the United States and the United Kingdom strongly agreed that they had noticed an increase in QR code use since the then-active COVID-19 -related restrictions had begun several months prior. [ 16 ]
Several standards cover the encoding of data as QR codes: [ 17 ]
At the application layer , there is some variation between most of the implementations. Japan's NTT DoCoMo has established de facto standards for the encoding of URLs, contact information, and several other data types. [ 22 ] The open-source " ZXing " project maintains a list of QR code data types. [ 23 ]
QR codes have become common in consumer advertising. Typically, a smartphone is used as a QR code scanner, displaying the code and converting it to some useful form (such as a standard URL for a website, thereby obviating the need for a user to type it into a Web browser ).
QR codes have become a focus of advertising strategy to provide a way to access a brand's website more quickly than by manually entering a URL. [ 24 ] [ 25 ] Beyond mere convenience to the consumer, the importance of this capability is the belief that it increases the conversion rate : the chance that contact with the advertisement will convert to a sale. It coaxes interested prospects further down the conversion funnel with little delay or effort, bringing the viewer to the advertiser's website immediately, whereas a longer and more targeted sales pitch may lose the viewer's interest.
Although initially used to track parts in vehicle manufacturing, QR codes are used over a much wider range of applications. These include commercial tracking, warehouse stock control, entertainment and transport ticketing, product and loyalty marketing, and in-store product labeling. [ citation needed ] Examples of marketing include where a company's discounted and percent discount can be captured using a QR code decoder that is a mobile app, or storing a company's information such as address and related information alongside its alpha-numeric text data as can be seen in telephone directory yellow pages . [ citation needed ]
They can also be used to store personal information for organizations. An example of this is the Philippines National Bureau of Investigation (NBI) where NBI clearances now come with a QR code. Many of these applications target mobile-phone users (via mobile tagging ). Users may receive text, add a vCard contact to their device, open a URL, or compose an e-mail or text message after scanning QR codes. They can generate and print their own QR codes for others to scan and use by visiting one of several pay or free QR code-generating sites or apps. Google had an API , now deprecated, to generate QR codes, [ 26 ] and apps for scanning QR codes can be found on nearly all smartphone devices. [ 27 ]
QR codes storing addresses and URLs may appear in magazines, on signs, on buses, on business cards, or on almost any object about which users might want information. Users with a camera phone equipped with the correct reader application can scan the image of the QR code to display text and contact information, connect to a wireless network , or open a web page in the phone's browser. This act of linking from physical world objects is termed hardlinking or object hyperlinking . QR codes also may be linked to a location to track where a code has been scanned. Either the application that scans the QR code retrieves the geo information by using GPS and cell tower triangulation (aGPS) or the URL encoded in the QR code itself is associated with a location. In 2008, a Japanese stonemason announced plans to engrave QR codes on gravestones, allowing visitors to view information about the deceased, and family members to keep track of visits. [ 29 ] Psychologist Richard Wiseman was one of the first authors to include QR codes in a book, in Paranormality: Why We See What Isn't There (2011). [ 30 ] [ failed verification ] Microsoft Office and LibreOffice have a functionality to insert QR code into documents. [ 31 ] [ 32 ]
QR codes have been incorporated into currency. In June 2011, The Royal Dutch Mint ( Koninklijke Nederlandse Munt ) issued the world's first official coin with a QR code to celebrate the centenary of its current building and premises. The coin can be scanned by a smartphone and originally linked to a special website with content about the historical event and design of the coin. [ 33 ] In 2014, the Central Bank of Nigeria issued a 100-naira banknote to commemorate its centennial, the first banknote to incorporate a QR code in its design. When scanned with an internet-enabled mobile device, the code goes to a website that tells the centenary story of Nigeria. [ 34 ]
In 2015, the Central Bank of the Russian Federation issued a 100- rubles note to commemorate the annexation of Crimea by the Russian Federation . [ 35 ] It contains a QR code into its design, and when scanned with an internet-enabled mobile device, the code goes to a website that details the historical and technical background of the commemorative note. In 2017, the Bank of Ghana issued a 5- cedis banknote to commemorate 60 years of central banking in Ghana . It contains a QR code in its design which, when scanned with an internet-enabled mobile device, goes to the official Bank of Ghana website.
Credit card functionality is under development. In September 2016, the Reserve Bank of India (RBI) launched the eponymously named BharatQR , a common QR code jointly developed by all the four major card payment companies – National Payments Corporation of India that runs RuPay cards along with Mastercard , Visa , and American Express . It will also have the capability of accepting payments on the Unified Payments Interface (UPI) platform. [ 36 ] [ 37 ]
QR codes are used in some augmented reality systems to determine the positions of objects in 3-dimensional space. [ 11 ]
QR codes can be used on various mobile device operating systems. While initially requiring the installation and use of third-party apps, both Android and iOS (since iOS 11 [ 38 ] [ 39 ] ) devices can now natively scan QR codes, without requiring an external app to be used. [ 40 ] The camera app can scan and display the kind of QR code along with the link. These devices support URL redirection , which allows QR codes to send metadata to existing applications on the device.
QR codes have been used to establish "virtual stores", where a gallery of product information and QR codes is presented to the customer, e.g. on a train station wall. The customers scan the QR codes, and the products are delivered to their homes. This use started in South Korea , [ 41 ] and Argentina, [ 42 ] but is currently expanding globally. [ 43 ] Walmart, Procter & Gamble and Woolworths have already adopted the Virtual Store concept. [ 44 ]
QR codes can be used to store bank account information or credit card information, or they can be specifically designed to work with particular payment provider applications. There are several trial applications of QR code payments across the world. [ 45 ] [ 46 ] In developing countries including China, [ 47 ] [ 48 ] India [ 49 ] QR code payment is a very popular and convenient method of making payments. Since Alipay designed a QR code payment method in 2011, [ 50 ] mobile payment has been quickly adopted in China. As of 2018, around 83% of all payments were made via mobile payment. [ 51 ]
In November 2012, QR code payments were deployed on a larger scale in the Czech Republic when an open format for payment information exchange – a Short Payment Descriptor – was introduced and endorsed by the Czech Banking Association as the official local solution for QR payments. [ 52 ] [ 53 ] In 2013, the European Payment Council provided guidelines for the EPC QR code enabling SCT initiation within the Eurozone .
In 2017, Singapore created a task force including government agencies such as the Monetary Authority of Singapore and Infocomm Media Development Authority to spearhead a system for e-payments using standardized QR code specifications. These specific dimensions are specialized for Singapore. [ 54 ]
The e-payment system, Singapore Quick Response Code (SGQR), essentially merges various QR codes into one label that can be used by both parties in the payment system. This allows for various banking apps to facilitate payments between multiple customers and a merchant that displays a single QR code. The SGQR scheme is co-owned by MAS and IMDA. [ 55 ] A single SDQR label contains e-payments and combines multiple payment options. People making purchases can scan the code and see which payment options the merchant accepts. [ 55 ]
QR codes can be used to log into websites: a QR code is shown on the login page on a computer screen , and when a registered user scans it with a verified smartphone, they will automatically be logged in. Authentication is performed by the smartphone, which contacts the server. Google deployed such a login scheme in 2012. [ 56 ]
There is a system whereby a QR code can be displayed on a device such as a smartphone and used as an admission ticket . [ 57 ] [ 58 ] Its use is common for J1 League and Nippon Professional Baseball tickets in Japan. [ 59 ] [ 60 ] In some cases, rights can be transferred via the Internet. In Latvia , QR codes can be scanned in Riga public transport to validate Rīgas Satiksme e-tickets. [ 61 ]
Restaurants can present a QR code near the front door or at the table allowing guests to view an online menu, or even redirect them to an online ordering website or app, allowing them to order and/or possibly pay for their meal without having to use a cashier or waiter. QR codes can also link to daily or weekly specials that are not printed on the standardized menus, [ 62 ] and enable the establishment to update the entire menu without needing to print copies. At table-serve restaurants, QR codes enable guests to order and pay for their meals without a waiter involved – the QR code contains the table number so servers know where to bring the food. [ 63 ] This application has grown especially since the need for social distancing during the 2020 COVID-19 pandemic prompted reduced contact between service staff and customers. [ 63 ]
By specifying the SSID, encryption type, password/passphrase, and if the SSID is hidden or not, mobile device users can quickly scan and join networks without having to manually enter the data. [ 64 ] A MeCard -like format is supported by Android and iOS 11+. [ 65 ]
A QR code can link to an obituary and can be placed on a headstone . In 2008, Ishinokoe in Yamanashi Prefecture, Japan began to sell tombstones with QR codes produced by IT DeSign, where the code leads to a virtual grave site of the deceased. [ 66 ] [ 67 ] [ 68 ] Other companies, such as Wisconsin-based Interactive Headstones, have also begun implementing QR codes into tombstones. [ 69 ] In 2014, the Jewish Cemetery of La Paz in Uruguay began implementing QR codes for tombstones. [ 70 ]
QR codes can be used to generate time-based one-time passwords for electronic authentication .
QR codes have been used by various retail outlets that have loyalty programs . Sometimes these programs are accessed with an app that is loaded onto a phone and includes a process triggered by a QR code scan. The QR codes for loyalty programs tend to be found printed on the receipt for a purchase or on the products themselves. Users in these schemes collect award points by scanning a code.
Serialised QR codes have been used by brands [ 71 ] and governments [ 72 ] to let consumers, retailers and distributors verify the authenticity of the products and help with detecting counterfeit products, as part of a brand protection program. [ 73 ] However, the security level of a regular QR code is limited since QR codes printed on original products are easily reproduced on fake products, even though the analysis of data generated as a result of QR code scanning can be used to detect counterfeiting and illicit activity. [ 74 ] A higher security level can be attained by embedding a digital watermark or copy detection pattern into the image of the QR code. This makes the QR code more secure against counterfeiting attempts; products that display a code which is counterfeit, although valid as a QR code, can be detected by scanning the secure QR code with the appropriate app. [ 75 ]
The treaty regulating apostilles (documents bearing a seal of authenticity), has been updated to allow the issuance of digital apostilles by countries; a digital apostille is a PDF document with a cryptographic signature containing a QR code for a canonical URL of the original document, allowing users to verify the apostille from a printed version of the document.
Different studies have been conducted to assess the effectiveness of QR codes as a means of conveying labelling information and their use as part of a food traceability system. In a field experiment, it was found that when provided free access to a smartphone with a QR code scanning app, 52.6% of participants would use it to access labelling information. [ 76 ] A study made in South Korea showed that consumers appreciate QR code used in food traceability system, as they provide detailed information about food, as well as information that helps them in their purchasing decision. [ 77 ] If QR codes are serialised, consumers can access a web page showing the supply chain for each ingredient, as well as information specific to each related batch, including meat processors and manufacturers, which helps address the concerns they have about the origin of their food. [ 78 ]
After the COVID-19 pandemic began spreading, QR codes began to be used as a "touchless" system to display information, show menus, or provide updated consumer information, especially in the hospitality industry. Restaurants replaced paper or laminated plastic menus with QR code decals on the table, which opened an online version of the menu. This prevented the need to dispose of single-use paper menus, or institute cleaning and sanitizing procedures for permanent menus after each use. [ 79 ] Local television stations have also begun to utilize codes on local newscasts to allow viewers quicker access to stories or information involving the pandemic, including testing and immunization scheduling websites, or for links within stories mentioned in the newscasts overall.
In Australia , patrons were required to scan QR codes at shops, clubs, supermarkets, and other service and retail establishments on entry to assist contact tracing . Singapore, Taiwan , the United Kingdom, and New Zealand used similar systems. [ 80 ]
QR codes are also present on COVID-19 vaccination certificates in places such as Canada and the EU ( EU Digital COVID certificate ), where they can be scanned to verify the information on the certificate. [ 81 ]
Unlike the older, one-dimensional barcodes that were designed to be mechanically scanned by a narrow beam of light, a QR code is detected by a two-dimensional digital image sensor and then digitally analyzed by a programmed processor. The processor locates the three distinctive squares at the corners of the QR code image, using a smaller square (or multiple squares) near the fourth corner to normalize the image for size, orientation, and angle of viewing. The small dots throughout the QR code are then converted to binary numbers and validated with an error-correcting algorithm.
The amount of data that can be represented by a QR code symbol depends on the data type ( mode , or input character set), version (1, ..., 40, indicating the overall dimensions of the symbol, i.e. 4 × version number + 17 dots on each side), and error correction level. The maximum storage capacities occur for version 40 and error correction level L (low), denoted by 40-L: [ 13 ] [ 82 ]
Here are some samples of QR codes:
QR codes use Reed–Solomon error correction over the finite field F 256 {\displaystyle \mathbb {F} _{256}} or GF(2 8 ) , the elements of which are encoded as bytes of 8 bits ; the byte b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 {\displaystyle b_{7}b_{6}b_{5}b_{4}b_{3}b_{2}b_{1}b_{0}} with a standard numerical value ∑ i = 0 7 b i 2 i {\displaystyle \textstyle \sum _{i=0}^{7}b_{i}2^{i}} encodes the field element ∑ i = 0 7 b i α i {\displaystyle \textstyle \sum _{i=0}^{7}b_{i}\alpha ^{i}} where α ∈ F 256 {\displaystyle \alpha \in \mathbb {F} _{256}} is taken to be a primitive element satisfying α 8 + α 4 + α 3 + α 2 + 1 = 0 {\displaystyle \alpha ^{8}+\alpha ^{4}+\alpha ^{3}+\alpha ^{2}+1=0} . The primitive polynomial is x 8 + x 4 + x 3 + x 2 + 1 {\displaystyle x^{8}+x^{4}+x^{3}+x^{2}+1} , corresponding to the polynomial number 285, with initial root = 0 to obtain generator polynomials.
The Reed–Solomon code uses one of 37 different polynomials over F 256 {\displaystyle \mathbb {F} _{256}} , with degrees ranging from 7 to 68, depending on how many error correction bytes the code adds. It is implied by the form of Reed–Solomon used ( systematic BCH view ) that these polynomials are all on the form ∏ i = 0 n − 1 ( x − α i ) {\textstyle \prod _{i=0}^{n-1}(x-\alpha ^{i})} . However, the rules for selecting the degree n {\displaystyle n} are specific to the QR standard.
For example, the generator polynomial used for the Version 1 QR code (21×21), when 7 error correction bytes are used, is: g ( x ) = x 7 + α 87 x 6 + α 229 x 5 + α 146 x 4 + α 149 x 3 + α 238 x 2 + α 102 x + α 21 {\displaystyle g(x)=x^{7}+\alpha ^{87}x^{6}+\alpha ^{229}x^{5}+\alpha ^{146}x^{4}+\alpha ^{149}x^{3}+\alpha ^{238}x^{2}+\alpha ^{102}x+\alpha ^{21}} .
This is obtained by multiplying the first seven terms: g ( x ) = ( x + 1 ) ( x + α ) ( x + α 2 ) ( x + α 3 ) ( x + α 4 ) ( x + α 5 ) ( x + α 6 ) {\displaystyle g(x)=(x+1)(x+\alpha )(x+\alpha ^{2})(x+\alpha ^{3})(x+\alpha ^{4})(x+\alpha ^{5})(x+\alpha ^{6})} .
The same may also be expressed using decimal coefficients (over F 256 {\displaystyle \mathbb {F} _{256}} ), as: g ( x ) = x 7 + 127 x 6 + 122 x 5 + 154 x 4 + 164 x 3 + 11 x 2 + 68 x + 117 {\displaystyle g(x)=x^{7}+127x^{6}+122x^{5}+154x^{4}+164x^{3}+11x^{2}+68x+117} .
The highest power of x {\displaystyle x} in the polynomial (the degree n {\displaystyle n} , of the polynomial) determines the number of error correction bytes. In this case, the degree is 7.
When discussing the Reed–Solomon code phase there is some risk for confusion, in that the QR ISO/IEC standard uses the term codeword for the elements of F 256 {\displaystyle \mathbb {F} _{256}} , which with respect to the Reed–Solomon code are symbols , whereas it uses the term block for what with respect to the Reed–Solomon code are the codewords. The number of data versus error correction bytes within each block depends on (i) the version (side length) of the QR symbol and (ii) the error correction level, of which there are four. The higher the error correction level, the less storage capacity. The following table lists the approximate error correction capability at each of the four levels:
In larger QR symbols, the message is broken up into several Reed–Solomon code blocks. The block size is chosen so that no attempt is made at correcting more than 15 errors per block; this limits the complexity of the decoding algorithm. The code blocks are then interleaved together, making it less likely that localized damage to a QR symbol will overwhelm the capacity of any single block.
The Version 1 QR symbol with level L error correction, for example, consists of a single error correction block with a total of 26 code bytes (made of 19 message bytes and seven error correction bytes). It can correct up to 2 byte errors. Hence, this code is known as a (26,19,2) error correction code over GF(2 8 ) . It is also sometimes represented in short, as (26,19) code.
Due to error correction, it is possible to create artistic QR codes with embellishments to make them more readable or attractive to the human eye, and to incorporate colors, logos, and other features into the QR code block; the embellishments are treated as errors, but the codes still scan correctly. [ 84 ] [ 85 ]
It is also possible to design artistic QR codes without reducing the error correction capacity by manipulating the underlying mathematical constructs. [ 86 ] [ 87 ] Image processing algorithms are also used to reduce errors in QR-code. [ 88 ]
The format information records two things: the error correction level and the mask pattern used for the symbol. Masking is used to break up patterns in the data area that might confuse a scanner, such as large blank areas or misleading features that look like the locator marks. The mask patterns are defined on a grid that is repeated as necessary to cover the whole symbol. Modules corresponding to the dark areas of the mask are inverted. The 5-bit format information is protected from errors with a BCH code , and two complete copies are included in each QR symbol. [ 4 ] A (15,5) triple error-correcting BCH code over GF(2 4 ) is used, having the generator polynomial g ( x ) = x 10 + x 8 + x 5 + x 4 + x 2 + x + 1 {\displaystyle g(x)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1} . It can correct at most 3 bit-errors out of the 5 data bits. There are a total of 15 bits in this BCH code (10 bits are added for error correction). This 15-bit code is itself X-ORed with a fixed 15-bit mask pattern ( 101010000010010 ) to prevent an all-zero string.
To obtain the error correction (EC) bytes for a message "www.wikipedia.org", the following procedure may be carried out:
The message is 17 bytes long, hence it can be encoded using a (26,19,2) Reed-Solomon code to fit in a Ver1 (21×21) symbol, which has a maximum capacity of 19 bytes (for L level error correction).
The generator polynomial specified for the (26,19,2) code, is: g ( x ) = x 7 + 127 x 6 + 122 x 5 + 154 x 4 + 164 x 3 + 11 x 2 + 68 x + 117 {\displaystyle g(x)=x^{7}+127x^{6}+122x^{5}+154x^{4}+164x^{3}+11x^{2}+68x+117} ,
which may also be written in the form of a matrix of decimal coefficients:
The 17-byte long message "www.wikipedia.org" as hexadecimal coefficients (ASCII values), denoted by M1 through M17 is:
The encoding mode is "Byte encoding". Hence the 'Enc' field is [0100] (4 bits). The length of the above message is 17 bytes hence 'Len' field is [00010001] (8 bits). The 'End' field is End of message marker [0000] (4 bits).
The message code word (without EC bytes) is of the form:
Substituting the hexadecimal values, it can be expressed as:
This is rearranged as 19-byte blocks of 8 bits each:
Using the procedure for Reed-Solomon systematic encoding , the 7 EC bytes obtained (E1 through E7, as shown in the symbol) which are the coefficients (in decimal) of the remainder after polynomial division are:
or in hexadecimal values:
These 7 EC bytes are then appended to the 19-byte message. The resulting coded message has 26 bytes (in hexadecimal):
Note: The bit values shown in the Ver1 QR symbol below do not match with the above values, as the symbol has been masked using a mask pattern (001).
The message dataset is placed from right to left in a zigzag pattern, as shown below. In larger symbols, this is complicated by the presence of the alignment patterns and the use of multiple interleaved error-correction blocks.
The general structure of a QR encoding is as a sequence of 4 bit indicators with payload length dependent on the indicator mode (e.g. byte encoding payload length is dependent on the first byte). [ 89 ]
Four-bit indicators are used to select the encoding mode and convey other information.
Encoding modes can be mixed as needed within a QR symbol. (e.g., a url with a long string of alphanumeric characters )
After every indicator that selects an encoding mode is a length field that tells how many characters are encoded in that mode. The number of bits in the length field depends on the encoding and the symbol version.
Alphanumeric encoding mode stores a message more compactly than the byte mode can, but cannot store lower-case letters and has only a limited selection of punctuation marks, which are sufficient for rudimentary web addresses . Two characters are coded in an 11-bit value by this formula:
This has the exception that the last character in an alphanumeric string with an odd length is read as a 6-bit value instead.
The following images offer more information about the QR code.
Model 1 QR code is an older version of the specification. It is visually similar to the widely seen model 2 codes, but lacks alignment patterns. Differences are in the bottom right corner, and in the midsections of the bottom and right edges are additional functional regions.
Micro QR code is a smaller version of the QR code standard for applications where symbol size is limited. There are four different versions (sizes) of Micro QR codes: the smallest is 11×11 modules; the largest can hold 35 numeric characters, [ 90 ] or 21 ASCII alphanumeric characters, or 15 bytes (128 bits).
Rectangular Micro QR Code (also known as rMQR Code) is a two-dimensional (2D) matrix barcode invented and standardized in 2022 by Denso Wave as ISO/IEC 23941. rMQR Code is designed as a rectangular variation of the QR code and has the same parameters and applications as original QR codes; however, rMQR Code is more suitable for rectangular areas, and has a difference between width and height up to 19 in the R7x139 version.
iQR code is an alternative to existing square QR codes developed by Denso Wave. iQR codes can be created in square or rectangular formations; this is intended for situations where a longer and narrower rectangular shape is more suitable, such as on cylindrical objects. iQR codes can fit the same amount of information in 30% less space. There are 61 versions of square iQR codes, and 15 versions of rectangular codes. For squares, the minimum size is 9 × 9 modules; rectangles have a minimum of 19 × 5 modules. iQR codes add error correction level S, which allows for 50% error correction. [ 91 ] iQR Codes had not been given an ISO/IEC specification as of 2015, and only proprietary Denso Wave products could create or read iQR codes. [ 92 ]
Secure Quick Response (SQR) code is a QR code that contains a "private data" segment after the terminator instead of the specified filler bytes "ec 11". [ 93 ] This private data segment must be deciphered with an encryption key. This can be used to store private information and to manage a company's internal information. [ 94 ]
Frame QR is a QR code with a "canvas area" that can be flexibly used. In the center of this code is the canvas area, where graphics, letters, and more can be flexibly arranged, making it possible to lay out the code without losing the design of illustrations, photos, etc. [ 95 ]
Researchers have proposed a new High Capacity Colored 2-Dimensional (HCC2D) Code, which builds upon a QR code basis for preserving the QR robustness to distortions and uses colors for increasing data density (as of 2014 it is still in the prototyping phase). The HCC2D code specification is described in details in Querini et al. (2011), [ 96 ] while techniques for color classification of HCC2D code cells are described in detail in Querini and Italiano (2014), [ 97 ] which is an extended version of Querini and Italiano (2013). [ 98 ]
Introducing colors into QR codes requires addressing additional issues. In particular, during QR code reading only the brightness information is taken into account, while HCC2D codes have to cope with chromatic distortions during the decoding phase. In order to ensure adaptation to chromatic distortions that arise in each scanned code, HCC2D codes make use of an additional field: the Color Palette Pattern. This is because color cells of a Color Palette Pattern are supposed to be distorted in the same way as color cells of the Encoding Region. Replicated color palettes are used for training machine-learning classifiers.
Accessible QR is a type of QR code that combines a standard QR code with a dot-dash pattern positioned around one corner of the code to provide product information for people who are blind and partially sighted. The codes, announce product categories and product details such as instructions, ingredients, safety warnings, and recycling information. The data is structured for the needs of users who are blind or partially sighted and offers larger text or audio output. It can read QR codes from a metre away, activating the smartphone's accessibility features like VoiceOver to announce product details.
The use of QR code technology is freely licensed as long as users follow the standards for QR code documented with JIS or ISO / IEC . Non-standardized codes may require special licensing. [ 99 ]
Denso Wave owns a number of patents on QR code technology, but has chosen to exercise them in a limited fashion. [ 99 ] In order to promote widespread usage of the technology Denso Wave chose to waive its rights to a key patent in its possession for standardized codes only. [ 17 ] In the US, the granted QR code patent, 5726435, expired on March 14, 2015. In Japan, the corresponding patent, 2938338, expired on March 14, 2014. The European Patent Office granted patent 0672994 to Denso Wave, which was then validated into French , UK, and German patents, all of which expired in March 2015. [ 100 ]
The text QR Code itself is a registered trademark and wordmark of Denso Wave Incorporated. [ 101 ] In UK, the trademark is registered as E921775, the term QR Code , with a filing date of 3 September 1998. [ 102 ] The UK version of the trademark is based on the Kabushiki Kaisha Denso (DENSO CORPORATION) trademark, filed as Trademark 000921775, the term QR Code , on 3 September 1998 and registered on 16 December 1999 with the European Union OHIM (Office for Harmonization in the Internal Market). [ 103 ] The U.S. Trademark for the term QR Code is Trademark 2435991 and was filed on 29 September 1998 with an amended registration date of 13 March 2001, assigned to Denso Corporation. [ 104 ] In South Korea, trademark application filed on 18 November 2011 was refused at 20 March 2012, because the Korean Intellectual Property Office viewed that the phrase was genericized among South Korean people to refer to matrix barcodes in general. [ 105 ]
The only context in which common QR codes can carry executable data is the URL data type. These URLs may host JavaScript code, which can be used to exploit vulnerabilities in applications on the host system, such as the reader, the web browser, or the image viewer, since a reader will typically send the data to the application associated with the data type used by the QR code.
In the case of no software exploits, malicious QR codes combined with a permissive reader can still put a computer's contents and user's privacy at risk. This practice is known as "attagging", a portmanteau of "attack tagging". [ 106 ] They are easily created and can be affixed over legitimate QR codes. [ 107 ] [ failed verification ] [ 108 ] On a smartphone, the reader's permissions may allow use of the camera, full Internet access, read/write contact data, GPS , read browser history, read/write local storage, and global system changes. [ 109 ] [ 110 ] [ 111 ] [ improper synthesis? ]
Risks include linking to dangerous web sites with browser exploits, enabling the microphone/camera/GPS, and then streaming those feeds to a remote server, analysis of sensitive data (passwords, files, contacts, transactions), [ 112 ] and sending email/ SMS /IM messages or packets for DDoS as part of a botnet , corrupting privacy settings, stealing identity, [ 113 ] and even containing malicious logic themselves such as JavaScript [ 114 ] or a virus. [ 115 ] [ 116 ] These actions could occur in the background while the user is only seeing the reader opening a seemingly harmless web page. [ 117 ] In Russia, a malicious QR code caused phones that scanned it to send premium texts at a fee of $ 6 each. [ 106 ] QR codes have also been linked to scams in which stickers are placed on parking meters and other devices, posing as quick payment options, as seen in Austin , San Antonio and Boston , among other cities across the United States and Australia. [ 118 ] [ 119 ] [ 120 ] | https://en.wikipedia.org/wiki/QR_code |
Qatar Sustainability Assessment System (QSAS) is a green building certification system developed for the State of Qatar . The primary objective of Qatar Sustainability Assessment System [QSAS] is to create a sustainable built environment that minimizes ecological impact while addressing the specific regional needs and environment of Qatar. [ 1 ]
QSAS was developed by the Gulf Organisation for Research and Development (GORD) in collaboration with the T.C. Chan Center for Building Simulation and Energy Studies at the University of Pennsylvania . [ 2 ] [ 3 ] [ 4 ] Since its deployment in 2009, over 128 buildings in Qatar have been certified through QSAS. [ 5 ] In December 2010, QSAS was adopted into the curriculum of the environmental design faculty at King Fahd University and Qatar University . [ 6 ] In March, 2011 the State of Qatar integrated QSAS into the Qatar Construction Specifications [QCS] making the implementation of certain criteria mandatory for buildings developed in Qatar. [ 7 ]
The development of the rating system took advantage of a comprehensive review of combined best practices employed by a mix of established international and regional rating systems. This review has been performed while taking into consideration the needs that are specific to Qatar’s local environment, culture, and policies. This has led to adaptations and additions to sustainability criteria. Measurements for the rating system are designed to be performance-based and quantifiable. The result is a performance-based sustainable building rating system customized to the unique conditions and requirements of the State of Qatar. [ 8 ]
QSAS consists of a series of sustainable categories and criteria, each with a direct impact on environmental stress mitigation. Each category measures a different aspect of the project’s environmental impact. The categories define these broad impacts and address ways in which a project can mitigate the negative environmental effects. These categories are then broken down into specific criteria that measure and define individual issues. These issues range from a thorough review of water consumption to an assessment of light quality. Each criterion specifies a process for measuring individual aspects of environmental impact and for documenting the degree to which the requirements have been met. A score is then awarded to each criterion based on the degree of compliance. [ 9 ]
The eight categories of QSAS are the following:
Urban Connectivity [UC]
The Urban Connectivity category consists of factors associated with the urban environment such as zoning, transportation networks and loadings. Loadings on the urban environment include traffic congestion and pollution.
Site [S]
The Site category consists of factors associated with land use such as land conservation or remediation and site selection, planning and development.
Energy [E]
The Energy category consists of factors associated with energy demand of buildings, the efficiency of energy delivery, and the use of fossil energy sources that result in harmful emissions.
Water [W]
The Water category consists of factors associated with water consumption and its associated burden on municipal supply and treatment systems.
Materials [M]
The Materials category consists of factors associated with material extraction, processing, manufacturing, distribution, use/re-use, and disposal.
Indoor Environment [IE]
The Indoor Environment category consists of factors associated with indoor environmental quality such as thermal comfort, air quality, acoustic quality, and light quality.
Cultural & Economic Value [CE]
The Cultural and Economic Value category consists of factors associated with cultural conservation and support of the national economy.
Management & Operations [MO]
The Management and Operations category consists of factors associated with building design management and operations.
QSAS consists of several resources used to facilitate the assessment of sustainability performance for buildings in Qatar. Target users for the resources are building planners, developers, owners, designers, engineers and environmentalists. The major resources available to QSAS members include the following:
QSAS has been developed to rate different building types as well as projects on the Neighborhood scale for each phase of a building’s lifespan. The building schemes that can be rated with QSAS include:
Phases that can be rated using QSAS include:
The various schemes use similar categories and criteria across each system to facilitate use. The measurements, calculations, simulations, scoring ranges, and weights change accordingly for each building type and project phase.
The aim for all QSAS criteria and their associated measurements is to be performance-based and quantifiable on the scoring scale of -1 to 3 (-1, 0, 1, 2, 3) or 0 to 3, depending on the criterion’s level of impact. Each category and criterion has an associated weight based on its relative environmental, social, and economic impact. Once a score is assigned to each criterion in the assessment system, the values are multiplied by the weight and a cumulative final score is determined.
QSAS consists of six certification levels to measure the project’s impact. A building that obtains a cumulative final score below 0 does not meet the baseline and will be denied certification. Certification can only be achieved when the final score is greater than or equal to 0, earning a rating of 1, 2, 3, 4, 5, or 6 stars. The highest score a building can achieve is 3.0 and the highest certification level is 6 stars. [ 11 ]
Professional Accreditation is granted to those qualified individuals who demonstrate in‐depth knowledge of the Qatar Sustainability Assessment System (QSAS) and sustainable building practices. The role of the accredited professional or QSAS Certified Green Professional (CGP) is to facilitate the submittal of projects for assessment under QSAS. The candidate must meet the following requirements to receive a QSAS CGP certificate:
The T.C. Chan Center and GORD are currently working on further developing and expanding QSAS to other regions in the Middle East. [ 15 ] | https://en.wikipedia.org/wiki/QSAS |
In quantitative genetics , Q ST is a statistic intended to measure the degree of genetic differentiation among populations with regard to a quantitative trait . It was developed by Ken Spitze in 1993. [ 1 ] Its name reflects that Q ST was intended to be analogous to the fixation index for a single genetic locus (F ST ). [ 2 ] [ 3 ] Q ST is often compared with F ST of neutral loci to test if variation in a quantitative trait is a result of divergent selection or genetic drift, an analysis known as Q ST –F ST comparisons .
Q ST represents the proportion of variance among subpopulations, and its calculation is synonymous to F ST developed by Sewall Wright . [ 4 ] However, instead of using genetic differentiation, Q ST is calculated by finding the variance of a quantitative trait within and among subpopulations, and for the total population. [ 1 ] Variance of a quantitative trait among populations (σ 2 GB ) is described as:
σ G B 2 = ( 1 − Q S T ) σ T 2 {\displaystyle \sigma _{GB}^{2}=(1-Q_{ST})\sigma _{T}^{2}}
And the variance of a quantitative trait within populations (σ 2 GW ) is described as:
σ G W 2 = 2 Q S T σ T 2 {\displaystyle \sigma _{GW}^{2}=2Q_{ST}\sigma _{T}^{2}}
Where σ 2 T is the total genetic variance in all populations. Therefore, Q ST can be calculated with the following equation:
Q S T = σ G B 2 σ G B 2 + 2 σ G W 2 {\displaystyle Q_{ST}={\frac {\sigma _{GB}^{2}}{\sigma _{GB}^{2}+2\sigma _{GW}^{2}}}}
Calculation of Q ST is subject to several assumptions: populations must be in Hardy–Weinberg equilibrium , observed variation is assumed to be due to additive genetic effects only, selection and linkage disequilibrium are not present, [ 5 ] and the subpopulations exist within an island model. [ 6 ]
Q ST –F ST analyses often involve culturing organisms in consistent environmental conditions, known as common garden experiments , [ 7 ] and comparing the phenotypic variance to genetic variance. If Q ST is found to exceed F ST , this is interpreted as evidence of divergent selection, because it indicates more differentiation in the trait than could be produced solely by genetic drift . If Q ST is less than F ST , balancing selection is expected to be present. If the values of Q ST and F ST are equivalent, the observed trait differentiation could be due to genetic drift. [ 6 ]
Suitable comparison of Q ST and F ST is subject to multiple ecological and evolutionary assumptions, [ 8 ] [ 9 ] [ 10 ] and since the development of Q ST , multiple studies have examined the limitations and constrictions of Q ST –F ST analyses. Leinonen et al. notes F ST must be calculated with neutral loci, however over filtering of non-neutral loci can artificially reduce F ST values. [ 7 ] Cubry et al. found Q ST is reduced in the presence of dominance , resulting in conservative estimates of divergent selection when Q ST is high, and inconclusive results of balancing selection when Q ST is low. [ 5 ] Additionally, population structure can significantly impact Q ST –F ST ratios. Stepping stone models, which can generate more evolutionary noise than island models, are more likely to experience type 1 errors . [ 6 ] If a subset of populations act as sources, such as during invasion , weighting the genetic contributions of each population can increase detection of adaptation. [ 11 ] In order to improve precision of Q ST analyses, more populations (>20) should be included in analyses. [ 12 ]
Multiple studies have incorporated Q ST to separate effects of natural selection and genetic drift, and Q ST is often observed to exceed F ST , indicating local adaptation. [ 13 ] In an ecological restoration study, Bower and Aitken used Q ST to evaluate suitable populations for seed transfer of whitebark pine . They found high Q ST values in many populations, suggesting local adaptation for cold-adapted characteristics. [ 14 ] During an assessment of the invasive species, Brachypodium sylvaticum , Marchini et al. found divergence between native and invasive populations during initial establishment in the invaded range, but minimal divergence during range expansion . [ 11 ] In an examination of the common snapdragon ( Antirrhinum majus ) along an elevation gradient, Q ST –F ST analyses revealed different adaptation trends between two subspecies ( A. m. pseudomajus and A. m. striatum ). While both subspecies occur at all elevations, A. m. striatum had high Q ST values for traits associated with altitude adaptation: plant height, number of branches, and internode length. A. m. pseudomajus had lower Q ST than F ST values for germination time. [ 15 ] | https://en.wikipedia.org/wiki/QST_(genetics) |
The QTY Code is a design method to transform membrane proteins that are intrinsically insoluble in water into variants with water solubility , while retaining their structure and function.
The QTY Code is based on two key molecular structural facts: 1) all 20 natural amino acids are found in alpha-helices regardless of their chemical properties , although some amino acids have a higher propensity to form an alpha-helix ; and, 2) several amino acids share striking structural similarities despite their very different chemical properties. These may be paired as: Glutamine (Q) vs Leucine (L); Threonine (T) vs Valine (V) and Isoleucine (I); and Tyrosine (Y) vs Phenylalanine (F). [ 1 ] [ 2 ]
The QTY Code systematically replaces water-insoluble amino acids (L, V, I and F) with water-soluble amino acids (Q, T and Y) in transmembrane alpha-helices. [ 3 ] Thus, its application to membrane proteins changes the water-insoluble form of membrane proteins into water-soluble variants. [ 3 ] [ 4 ] The QTY Code was specifically conceived to render G protein-coupled receptors (GPCRs) into a water-soluble form. Despite substantial transmembrane domain changes, the QTY variants of GPCRs maintain stable structure and ligand binding activities. [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ]
The side chain of glutamine (Q) can form 4 hydrogen bonds with 4 water molecules. There are 2 hydrogen donors from nitrogen and 2 hydrogen acceptors for oxygen . The –OH group of threonine (T) and tyrosine (Y) can form 3 hydrogen bonds with 3 water molecules (2 H-acceptors and 1 H-donor). [ 1 ] Color code: Green = carbon, red = oxygen, blue = nitrogen, gray = hydrogen, yellow disks = hydrogen bonds.
There are 3 types of alpha-helices and with nearly identical molecular structure, namely: a) 1.5Å per amino acid rise, b) 100˚ per amino acid turn, c) 3.6 amino acids and 360˚ per helical turn, and d) 5.4Å per helical turn. The 3 types of alpha-helices are: 1) mostly hydrophobic amino acids including Leucine (L), Isoleucine (I), Valine (V), Phenylalanine (F), Methionine (M) and Alanine (A) that are commonly found as the helical transmembrane segments in membrane proteins; 2) mostly hydrophilic amino acids including Aspartic acid (D), Glutamic acid (E), Glutamine (Q), Lysine (K), Arginine (R), Serine (S), Threonine (T), Tyrosine (Y) that are commonly found on the out layer in water-soluble globular proteins ; 3) mixed hydrophobic and hydrophilic amino acids that are partitioned in 2 faces: hydrophobic face and hydrophilic face, in an analogy, like our fingers with front and back. These alpha-helices sometimes attach to surface of membrane lipid bilayer , or partially buried to the hydrophobic core and partially close to the surface of water-soluble globular proteins. [ 2 ]
The QTY Code is likely universally applicable and also reversible, namely, Q changes to L, T changes to V and I, and Y changes to F. The QTY Code has been successful in designing many water-soluble variants of chemokine receptors and cytokine receptors . The QTY Code may likely be successfully applied to other water-insoluble aggregated proteins. The QTY Code is robust and straightforward: it is the simplest tool to carry out membrane protein design without sophisticated computer algorithms. Thus, it can be used broadly. The QTY Code has implications for designing additional GPCRs and other membrane proteins including cytokine receptors that are directly involved in cytokine storm syndrome . [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ]
The QTY Code has also been applied to cytokine receptor water-soluble variants with the aim of combatting the cytokine storm syndrome (also called cytokine release syndrome ) suffered by cancer patients receiving CAR-T therapy . This therapeutic application may be equally applicable to severely infected COVID-19 patients, for whom cytokine storms often lead to death. [ 7 ] | https://en.wikipedia.org/wiki/QTY_Code |
QVT ( Query/View/Transformation ) is a standard set of languages for model transformation defined by the Object Management Group . [ 1 ]
Model transformation is a key technique used in model-driven architecture . As the name QVT indicates, the OMG standard covers transformations, views and queries together. Model queries and model views can be seen as special kinds of model transformation, provided that we use a suitably broad definition of model transformation: a model transformation is a program which operates on models.
The QVT standard defines three model transformation languages. All of them operate on models which conform to Meta-Object Facility (MOF) 2.0 metamodels; the transformation states which metamodels are used. A transformation in any of the three QVT languages can itself be regarded as a model, conforming to one of the metamodels specified in the standard. The QVT standard integrates the OCL 2.0 standard and also extends it with imperative features.
Finally, QVT-BlackBox is a mechanism to invoke transformation facilities expressed in other languages (for example XSLT or XQuery ).
Although QVT has a broad scope, it does not cover everything that has been considered as a model transformation, view or query. For example, the QVT languages do not permit transformations to or from textual models, since each model must conform to some MOF 2.0 metamodel. Model-to-text transformations are being standardised separately by OMG (see MOFM2T ).
In 2002, OMG issued a Request for proposal (RFP) on MOF Query/View/Transformation to seek a standard compatible with the Model Driven Architecture (MDA) recommendation suite (UML, MOF, OCL, etc.).
Several replies were given by a number of companies and research institutions that evolved during three years to produce a common proposal, based on a draft by UK research Dr Laurence Tratt. The first version was submitted and approved in 2005. [ 3 ] QVT Version 1.1 was released in January 2011. [ 1 ]
QVT-Operational:
QVT-Core:
QVT-Relations:
QVT-Like: | https://en.wikipedia.org/wiki/QVT |
The Q cycle (named for quinol ) describes a series of sequential oxidation and reduction of the lipophilic electron carrier Coenzyme Q (CoQ) between the ubiquinol and ubiquinone forms. These reactions can result in the net movement of protons across a lipid bilayer (in the case of the mitochondria, the inner mitochondrial membrane ).
The Q cycle was first proposed by Peter D. Mitchell , though a modified version of Mitchell's original scheme is now accepted as the mechanism by which Complex III moves protons (i.e. how complex III contributes to the biochemical generation of the proton or pH, gradient, which is used for the biochemical generation of ATP ).
The first reaction of Q cycle is the 2-electron oxidation of ubiquinol by two oxidants, c 1 (Fe 3+ ) and ubiquinone:
The second reaction of the cycle involves the 2-electron oxidation of a second ubiquinol by two oxidants, a fresh c 1 (Fe 3+ ) and the CoQ' −• produced in the first step:
These net reactions are mediated by electron-transfer mediators including a Rieske 2Fe-2S cluster (shunt to c 1 ) and c b (shunt to CoQ' and later to CoQ' −• )
In chloroplasts, a similar reaction is done with plastoquinone by cytochrome b6f complex .
Operation of the modified Q cycle in Complex III results in the reduction of Cytochrome c , oxidation of ubiquinol to ubiquinone , and the transfer of four protons into the intermembrane space, per two-cycle process.
Ubiquinol (QH 2 ) binds to the Q o site of complex III via hydrogen bonding to His182 of the Rieske iron-sulfur protein and Glu272 of Cytochrome b . Ubiquinone (Q), in turn, binds the Q i site of complex III. Ubiquinol is divergently oxidized (gives up one electron each) to the Rieske iron-sulfur '(FeS) protein' and to the b L heme . This oxidation reaction produces a transient semiquinone before complete oxidation to ubiquinone, which then leaves the Q o site of complex III.
Having acquired one electron from ubiquinol, the 'FeS protein' is freed from its electron donor and is able to migrate to the Cytochrome c 1 subunit. 'FeS protein' then donates its electron to Cytochrome c 1 , reducing its bound heme group. [ 1 ] [ 2 ] The electron is from there transferred to an oxidized molecule of Cytochrome c externally bound to complex III, which then dissociates from the complex. In addition, the reoxidation of the 'FeS protein' releases the proton bound to His181 into the intermembrane space.
The other electron, which was transferred to the b L heme, is used to reduce the b H heme, which in turn transfers the electron to the ubiquinone bound at the Q i site. The movement of this electron is energetically unfavourable, as the electron is moving towards the negatively charged side of the membrane. This is offset by a favourable change in E M from −100 mV in B L to +50mV in the B H heme. [ citation needed ] The attached ubiquinone is thus reduced to a semiquinone radical. The proton taken up by Glu272 is subsequently transferred to a hydrogen-bonded water chain as Glu272 rotates 170° to hydrogen bond a water molecule, in turn hydrogen-bonded to a propionate of the b L heme. [ 3 ]
Because the last step leaves an unstable semiquinone at the Q i site, the reaction is not yet fully completed. A second Q cycle is necessary, with the second electron transfer from cytochrome b H reducing the semiquinone to ubiquinol. The ultimate products of the Q cycle are four protons entering the intermembrane space, two from the matrix and two from the reduction of two molecules of cytochrome c. The reduced cytochrome c is eventually reoxidized by complex IV . The process is cyclic as the ubiquinol created at the Q i site can be reused by binding to the Q o site of complex III. | https://en.wikipedia.org/wiki/Q_cycle |
In physics and engineering , the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. [ 1 ] Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. [ 2 ] Higher Q indicates a lower rate of energy loss and the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q , while a pendulum immersed in oil has a low one. Resonators with high quality factors have low damping , so that they ring or vibrate longer.
The Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the bandwidth. Thus, a high- Q tuned circuit in a radio receiver would be more difficult to tune, but would have more selectivity ; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High- Q oscillators oscillate with a smaller range of frequencies and are more stable.
The quality factor of oscillators varies substantially from system to system, depending on their construction. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q near 1 ⁄ 2 . Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability have high quality factors. Tuning forks have quality factors around 1000. The quality factor of atomic clocks , superconducting RF cavities used in accelerators, and some high- Q lasers can reach as high as 10 11 [ 3 ] and higher. [ 4 ]
There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include: the damping ratio , relative bandwidth , linewidth and bandwidth measured in octaves .
The concept of Q originated with K. S. Johnson of Western Electric Company 's Engineering Department while evaluating the quality of coils (inductors). His choice of the symbol Q was only because, at the time, all other letters of the alphabet were taken. The term was not intended as an abbreviation for "quality" or "quality factor", although these terms have grown to be associated with it. [ 5 ] [ 6 ] [ 7 ]
The definition of Q since its first use in 1914 has been generalized to apply to coils and condensers, resonant circuits, resonant devices, resonant transmission lines, cavity resonators, [ 5 ] and has expanded beyond the electronics field to apply to dynamical systems in general: mechanical and acoustic resonators, material Q and quantum systems such as spectral lines and particle resonances.
In the context of resonators, there are two common definitions for Q , which are not exactly equivalent. They become approximately equivalent as Q becomes larger, meaning the resonator becomes less damped. One of these definitions is the frequency-to-bandwidth ratio of the resonator: [ 5 ]
Q = def f r Δ f = ω r Δ ω , {\displaystyle Q\mathrel {\stackrel {\text{def}}{=}} {\frac {f_{\mathrm {r} }}{\Delta f}}={\frac {\omega _{\mathrm {r} }}{\Delta \omega }},}
where f r is the resonant frequency, Δ f is the resonance width or full width at half maximum (FWHM) i.e. the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ω r = 2 πf r is the angular resonant frequency, and Δ ω is the angular half-power bandwidth.
Under this definition, Q is the reciprocal of fractional bandwidth .
The other common nearly equivalent definition for Q is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes: [ 8 ] [ 9 ] [ 5 ]
Q = def 2 π × energy stored energy dissipated per cycle = 2 π f r × energy stored power loss . {\displaystyle Q\mathrel {\stackrel {\text{def}}{=}} 2\pi \times {\frac {\text{energy stored}}{\text{energy dissipated per cycle}}}=2\pi f_{\mathrm {r} }\times {\frac {\text{energy stored}}{\text{power loss}}}.}
The factor 2 π makes Q expressible in simpler terms, involving only the coefficients of the second-order differential equation describing most resonant systems, electrical or mechanical. In electrical systems, the stored energy is the sum of energies stored in lossless inductors and capacitors ; the lost energy is the sum of the energies dissipated in resistors per cycle. In mechanical systems, the stored energy is the sum of the potential and kinetic energies at some point in time; the lost energy is the work done by an external force , per cycle, to maintain amplitude.
More generally and in the context of reactive component specification (especially inductors), the frequency-dependent definition of Q is used: [ 8 ] [ 10 ] [ failed verification – see discussion ] [ 9 ]
Q ( ω ) = ω × maximum energy stored power loss , {\displaystyle Q(\omega )=\omega \times {\frac {\text{maximum energy stored}}{\text{power loss}}},}
where ω is the angular frequency at which the stored energy and power loss are measured. This definition is consistent with its usage in describing circuits with a single reactive element (capacitor or inductor), where it can be shown to be equal to the ratio of reactive power to real power . ( See Individual reactive components .)
The Q -factor determines the qualitative behavior of simple damped oscillators. (For mathematical details about these systems and their behavior see harmonic oscillator and linear time invariant (LTI) system .)
Starting from the stored energy definition for, it can be shown that Q = 1 2 ζ {\displaystyle Q={\frac {1}{2\zeta }}} , where ζ {\displaystyle \zeta } is the damping ratio . There are three key distinct cases:
In negative feedback systems, the dominant closed-loop response is often well-modeled by a second-order system. The phase margin of the open-loop system sets the quality factor Q of the closed-loop system; as the phase margin decreases, the approximate second-order closed-loop system is made more oscillatory (i.e., has a higher quality factor).
Q Ω = R w δ 1 − m 2 v m , p 2 , {\displaystyle Q_{\Omega }={\frac {R_{\mathrm {w} }}{\delta }}{\frac {1-m^{2}}{v_{m,p}^{2}}},}
Physically speaking, Q is approximately the ratio of the stored energy to the energy dissipated over one radian of the oscillation; or nearly equivalently, at high enough Q values, 2 π times the ratio of the total energy stored and the energy lost in a single cycle. [ 14 ]
It is a dimensionless parameter that compares the exponential time constant τ for decay of an oscillating physical system's amplitude to its oscillation period . Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy. More precisely, the frequency and period used should be based on the system's natural frequency, which at low Q values is somewhat higher than the oscillation frequency as measured by zero crossings.
Equivalently (for large values of Q ), the Q factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off to e −2 π , or about 1 ⁄ 535 or 0.2%, of its original energy. [ 15 ] This means the amplitude falls off to approximately e − π or 4% of its original amplitude. [ 16 ]
The width (bandwidth) of the resonance is given by (approximately): Δ f = f N Q , {\displaystyle \Delta f={\frac {f_{\mathrm {N} }}{Q}},\,} where f N is the natural frequency , and Δ f , the bandwidth , is the width of the range of frequencies for which the energy is at least half its peak value.
The resonant frequency is often expressed in natural units (radians per second), rather than using the f N in hertz , as ω N = 2 π f N . {\displaystyle \omega _{\mathrm {N} }=2\pi f_{\mathrm {N} }.}
The factors Q , damping ratio ζ , natural frequency ω N , attenuation rate α , and exponential time constant τ are related such that: [ 17 ] [ page needed ]
Q = 1 2 ζ = ω N 2 α = τ ω N 2 , {\displaystyle Q={\frac {1}{2\zeta }}={\frac {\omega _{\mathrm {N} }}{2\alpha }}={\frac {\tau \omega _{\mathrm {N} }}{2}},}
and the damping ratio can be expressed as:
ζ = 1 2 Q = α ω N = 1 τ ω N . {\displaystyle \zeta ={\frac {1}{2Q}}={\alpha \over \omega _{\mathrm {N} }}={1 \over \tau \omega _{\mathrm {N} }}.}
The envelope of oscillation decays proportional to e − αt or e − t/τ , where α and τ can be expressed as:
α = ω N 2 Q = ζ ω N = 1 τ {\displaystyle \alpha ={\omega _{\mathrm {N} } \over 2Q}=\zeta \omega _{\mathrm {N} }={1 \over \tau }} and τ = 2 Q ω N = 1 ζ ω N = 1 α . {\displaystyle \tau ={2Q \over \omega _{\mathrm {N} }}={1 \over \zeta \omega _{\mathrm {N} }}={\frac {1}{\alpha }}.}
The energy of oscillation, or the power dissipation, decays twice as fast, that is, as the square of the amplitude, as e −2 αt or e −2 t/τ .
For a two-pole lowpass filter, the transfer function of the filter is [ 17 ]
H ( s ) = ω N 2 s 2 + ω N Q ⏟ 2 ζ ω N = 2 α s + ω N 2 {\displaystyle H(s)={\frac {\omega _{\mathrm {N} }^{2}}{s^{2}+\underbrace {\frac {\omega _{\mathrm {N} }}{Q}} _{2\zeta \omega _{\mathrm {N} }=2\alpha }s+\omega _{\mathrm {N} }^{2}}}\,}
For this system, when Q > 1 / 2 (i.e., when the system is underdamped), it has two complex conjugate poles that each have a real part of −α . That is, the attenuation parameter α represents the rate of exponential decay of the oscillations (that is, of the output after an impulse ) into the system. A higher quality factor implies a lower attenuation rate, and so high- Q systems oscillate for many cycles. For example, high-quality bells have an approximately pure sinusoidal tone for a long time after being struck by a hammer.
For an electrically resonant system, the Q factor represents the effect of electrical resistance and, for electromechanical resonators such as quartz crystals , mechanical friction .
The 2-sided bandwidth relative to a resonant frequency of F 0 (Hz) is F 0 Q {\displaystyle {\frac {F_{0}}{Q}}} .
For example, an antenna tuned to have a Q value of 10 and a centre frequency of 100 kHz would have a 3 dB bandwidth of 10 kHz.
In audio, bandwidth is often expressed in terms of octaves . Then the relationship between Q and bandwidth is
Q = 2 B W 2 2 B W − 1 = 1 2 sinh ( 1 2 ln ( 2 ) B W ) , {\displaystyle Q={\frac {2^{\frac {BW}{2}}}{2^{BW}-1}}={\frac {1}{2\sinh \left({\frac {1}{2}}\ln(2)BW\right)}},} where BW is the bandwidth in octaves. [ 19 ]
In an ideal series RLC circuit , and in a tuned radio frequency receiver (TRF) the Q factor is: [ 20 ]
Q = 1 R L C = ω 0 L R = 1 ω 0 R C {\displaystyle Q={\frac {1}{R}}{\sqrt {\frac {L}{C}}}={\frac {\omega _{0}L}{R}}={\frac {1}{\omega _{0}RC}}}
where R , L , and C are the resistance , inductance and capacitance of the tuned circuit, respectively. Larger series resistances correspond to lower circuit Q values.
For a parallel RLC circuit, the Q factor is the inverse of the series case: [ 21 ] [ 20 ]
Q = R C L = R ω 0 L = ω 0 R C {\displaystyle Q=R{\sqrt {\frac {C}{L}}}={\frac {R}{\omega _{0}L}}=\omega _{0}RC} [ 22 ]
Consider a circuit where R , L , and C are all in parallel. The lower the parallel resistance is, the more effect it will have in damping the circuit and thus result in lower Q . This is useful in filter design to determine the bandwidth.
In a parallel LC circuit where the main loss is the resistance of the inductor, R , in series with the inductance, L , Q is as in the series circuit. This is a common circumstance for resonators, where limiting the resistance of the inductor to improve Q and narrow the bandwidth is the desired result.
The Q of an individual reactive component depends on the frequency at which it is evaluated, which is typically the resonant frequency of the circuit that it is used in. The Q of an inductor with a series loss resistance is the Q of a resonant circuit using that inductor (including its series loss) and a perfect capacitor. [ 23 ]
Q L = X L R L = ω 0 L R L {\displaystyle Q_{L}={\frac {X_{L}}{R_{L}}}={\frac {\omega _{0}L}{R_{L}}}}
where:
The Q of a capacitor with a series loss resistance is the same as the Q of a resonant circuit using that capacitor with a perfect inductor: [ 23 ]
Q C = − X C R C = 1 ω 0 C R C {\displaystyle Q_{C}={\frac {-X_{C}}{R_{C}}}={\frac {1}{\omega _{0}CR_{C}}}}
where:
In general, the Q of a resonator involving a series combination of a capacitor and an inductor can be determined from the Q values of the components, whether their losses come from series resistance or otherwise: [ 23 ]
Q = 1 1 Q L + 1 Q C {\displaystyle Q={\frac {1}{{\frac {1}{Q_{L}}}+{\frac {1}{Q_{C}}}}}}
For a single damped mass-spring system, the Q factor represents the effect of simplified viscous damping or drag , where the damping force or drag force is proportional to velocity. The formula for the Q factor is: Q = M k D , {\displaystyle Q={\frac {\sqrt {Mk}}{D}},\,} where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation F damping = − Dv , where v is the velocity. [ 24 ]
The Q of a musical instrument is critical; an excessively high Q in a resonator will not evenly amplify the multiple frequencies an instrument produces. For this reason, string instruments often have bodies with complex shapes, so that they produce a wide range of frequencies fairly evenly.
The Q of a brass instrument or wind instrument needs to be high enough to pick one frequency out of the broader-spectrum buzzing of the lips or reed.
By contrast, a vuvuzela is made of flexible plastic, and therefore has a very low Q for a brass instrument, giving it a muddy, breathy tone. Instruments made of stiffer plastic, brass, or wood have higher Q values. An excessively high Q can make it harder to hit a note. Q in an instrument may vary across frequencies, but this may not be desirable.
Helmholtz resonators have a very high Q , as they are designed for picking out a very narrow range of frequencies.
In optics , the Q factor of a resonant cavity is given by Q = 2 π f o E P , {\displaystyle Q={\frac {2\pi f_{o}\,E}{P}},\,} where f o is the resonant frequency, E is the stored energy in the cavity, and P = − dE / dt is the power dissipated. The optical Q is equal to the ratio of the resonant frequency to the bandwidth of the cavity resonance. The average lifetime of a resonant photon in the cavity is proportional to the cavity's Q . If the Q factor of a laser 's cavity is abruptly changed from a low value to a high one, the laser will emit a pulse of light that is much more intense than the laser's normal continuous output. This technique is known as Q -switching . Q factor is of particular importance in plasmonics , where loss is linked to the damping of the surface plasmon resonance . [ 25 ] While loss is normally considered a hindrance in the development of plasmonic devices, it is possible to leverage this property to present new enhanced functionalities. [ 26 ] | https://en.wikipedia.org/wiki/Q_factor |
A Q meter is a piece of equipment used in the testing of radio frequency circuits. It has been largely replaced in professional laboratories by other types of impedance measuring devices, though it is still in use among radio amateurs. It was developed at Boonton Radio Corporation in Boonton, New Jersey in 1934 by William D. Loughlin . [ 1 ]
A Q meter measures the quality factor of a circuit, Q , which expresses how much energy is dissipated per cycle in a non-ideal reactive circuit:
This expression applies to an RF and microwave filter , bandpass LC filter , or any resonator. It also can be applied to an inductor or capacitor at a chosen frequency. For inductors
Where X L {\displaystyle X_{L}} is the reactance of the inductor, L {\displaystyle L} is the inductance, ω {\displaystyle \omega } is the angular frequency and R {\displaystyle R} is the resistance of the inductor. The resistance R {\displaystyle R} represents the loss in the inductor, mainly due to the resistance of the wire. A Q meter works on the principle of series resonance.
For LC band pass circuits and filters:
Where F {\displaystyle F} is the resonant frequency (center frequency) and B W {\displaystyle BW} is the filter bandwidth. In a band pass filter using an LC resonant circuit , when the loss (resistance) of the inductor increases, its Q factor is reduced, and so the bandwidth of the filter is increased. In a coaxial cavity filter, there are no inductors and capacitors, but the cavity has an equivalent LC model with losses (resistance) and the Q factor can be applied as well.
Internally, a minimal Q meter consists of a tuneable RF generator with a very low (pass) impedance output and a detector with a very high impedance input. There is usually provision to add a calibrated amount of high Q capacitance across the component under test to allow inductors to be measured in isolation. The generator is effectively placed in series with the tuned circuit formed by the components under test, and having negligible output resistance, does not materially affect the Q factor, while the detector measures the voltage developed across one element (usually the capacitor) and being high impedance in shunt does not affect the Q factor significantly either.
The ratio of the developed RF voltage to the applied RF current, coupled with knowledge of the reactive impedance from the resonant frequency, and the source impedance, allows the Q factor to be directly read by scaling the detected voltage. | https://en.wikipedia.org/wiki/Q_meter |
In electronics , a Q multiplier is a circuit added to a radio receiver to improve its selectivity and sensitivity. It is a regenerative amplifier adjusted to provide positive feedback within the receiver. This has the effect of narrowing the receiver's bandwidth , as if the Q factor of its tuned circuits had been increased. The Q multiplier was a common accessory in shortwave receivers of the vacuum tube era as either a factory installation or an add-on device. In use, the Q multiplier had to be adjusted to a point just short of oscillation to provide maximum sensitivity and rejection of interfering signals. [ 1 ]
A Q multiplier could also be adjusted to act as a notch filter , useful for reducing the interfering effect of signals on frequencies near to the desired signal. In some receiver designs, the Q multiplier was made to also serve as a beat frequency oscillator by adjusting it to oscillate. This could be used for reception of single sideband or Morse radiotelegraphy , but in that case the circuit no longer provided improved selectivity. [ 2 ]
The principle of regeneration applied to radio receivers was developed by Edwin Armstrong , who patented a regenerative receiver in 1914. At least one console-model broadcast superheterodyne receiver used positive feedback to improve selectivity in a 1926 design. [ 3 ] Q-multipliers were common on shortwave general-coverage and communications receivers of the 1950s. With the advent of crystal and ceramic intermediate frequency filters, the Q-multiplier was no longer popular.
This electronics-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Q_multiplier |
In physics, Q {\displaystyle \mathbf {Q} } tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase. [ 1 ] The Q {\displaystyle \mathbf {Q} } tensor is a second-order, traceless, symmetric tensor and is defined by [ 2 ] [ 3 ] [ 4 ]
where S = S ( T ) {\displaystyle S=S(T)} and R = R ( T ) {\displaystyle R=R(T)} are scalar order parameters, ( n , m ) {\displaystyle (\mathbf {n} ,\mathbf {m} )} are the two directors of the nematic phase and T {\displaystyle T} is the temperature ; in uniaxial liquid crystals, P = 0 {\displaystyle P=0} . The components of the tensor are
The states with directors n {\displaystyle \mathbf {n} } and − n {\displaystyle -\mathbf {n} } are physically equivalent and similarly the states with directors m {\displaystyle \mathbf {m} } and − m {\displaystyle -\mathbf {m} } are physically equivalent.
The Q {\displaystyle \mathbf {Q} } tensor can always be diagonalized,
The following are the two invariants of the Q {\displaystyle \mathbf {Q} } tensor,
the first-order invariant t r Q = Q i i = 0 {\displaystyle \mathrm {tr} \,\mathbf {Q} =Q_{ii}=0} is trivial here. It can be shown that ( t r Q 2 ) 3 ≥ 6 ( t r Q 3 ) 2 . {\displaystyle (\mathrm {tr} \,\mathbf {Q} ^{2})^{3}\geq 6(\mathrm {tr} \,\mathbf {Q} ^{3})^{2}.} The measure of biaxiality of the liquid crystal is commonly measured through the parameter
In uniaxial nematic liquid crystals, P = 0 {\displaystyle P=0} and therefore the Q {\displaystyle \mathbf {Q} } tensor reduces to
The scalar order parameter is defined as follows. If θ m o l {\displaystyle \theta _{\mathrm {mol} }} represents the angle between the axis of a nematic molecular and the director axis n {\displaystyle \mathbf {n} } , then [ 2 ]
where ⟨ ⋅ ⟩ {\displaystyle \langle \cdot \rangle } denotes the ensemble average of the orientational angles calculated with respect to the distribution function f ( θ m o l ) {\displaystyle f(\theta _{\mathrm {mol} })} and d Ω = sin θ m o l d θ m o l d ϕ m o l {\displaystyle d\Omega =\sin \theta _{\mathrm {mol} }d\theta _{\mathrm {mol} }d\phi _{\mathrm {mol} }} is the solid angle . The distribution function must necessarily satisfy the condition f ( θ m o l + π ) = f ( θ m o l ) {\displaystyle f(\theta _{\mathrm {mol} }+\pi )=f(\theta _{\mathrm {mol} })} since the directors n {\displaystyle \mathbf {n} } and − n {\displaystyle -\mathbf {n} } are physically equivalent.
The range for S {\displaystyle S} is given by − 1 / 2 ≤ S ≤ 1 {\displaystyle -1/2\leq S\leq 1} , with S = 1 {\displaystyle S=1} representing the perfect alignment of all molecules along the director and S = 0 {\displaystyle S=0} representing the complete random alignment (isotropic) of all molecules with respect to the director; the S = − 1 / 2 {\displaystyle S=-1/2} case indicates that all molecules are aligned perpendicular to the director axis although such nematics are rare or hard to synthesize. | https://en.wikipedia.org/wiki/Q_tensor |
In nuclear physics and chemistry , the Q value for a nuclear reaction is the amount of energy absorbed or released during the reaction. The value relates to the enthalpy of a chemical reaction or the energy of radioactive decay products. It can be determined from the masses of reactants and products:
where m r {\displaystyle m_{\text{r}}} and m p {\displaystyle m_{\text{p}}} are the sums of the reactant and product masses in atomic mass units .
Q values affect reaction rates . In general, the larger the positive Q value for the reaction, the faster the reaction proceeds, and the more likely the reaction is to "favor" the products.
The conservation of energy, between the initial and final energy of a nuclear process ( E i = E f ) , {\displaystyle (E_{\text{i}}=E_{\text{f}}),} enables the general definition of Q based on the mass–energy equivalence . For any radioactive particle decay, the kinetic energy difference will be given by
where K denotes the kinetic energy of the mass m .
A reaction with a positive Q value is exothermic , i.e. has a net release of energy, since the kinetic energy of the final state is greater than the kinetic energy of the initial state.
A reaction with a negative Q value is endothermic , i.e. requires a net energy input, since the kinetic energy of the final state is less than the kinetic energy of the initial state. [ 1 ] Observe that a chemical reaction is exothermic when it has a negative enthalpy of reaction, in contrast a positive Q value in a nuclear reaction.
The Q value can also be expressed in terms of the Mass excess Δ M {\displaystyle \Delta M} of the nuclear species as
Chemical Q values are measurement in calorimetry . Exothermic chemical reactions tend to be more spontaneous and can emit light or heat, resulting in runaway feedback(i.e. explosions).
Q values are also featured in particle physics . For example, Sargent's rule states that weak reaction rates are proportional to Q 5 . The Q value is the kinetic energy released in the decay at rest. For neutron decay, some mass disappears as neutrons convert to a proton, electron and antineutrino: [ 2 ]
where m n is the mass of the neutron , m p is the mass of the proton , m ν is the mass of the electron antineutrino , m e is the mass of the electron , and the K are the corresponding kinetic energies. The neutron has no initial kinetic energy since it is at rest. In beta decay, a typical Q is around 1 MeV.
The decay energy is divided among the products in a continuous distribution for more than two products. Measuring this spectrum allows one to find the mass of a product. Experiments are studying emission spectra to search for neutrinoless decay and neutrino mass; this is the principle of the ongoing KATRIN experiment. | https://en.wikipedia.org/wiki/Q_value_(nuclear_science) |
Within molecular and cell biology , Qa-1b is a MHC class I molecule and is the functional homolog of HLA-E in humans. Qa-1b is characterised by its limited polymorphisms and small peptide repertoire. Qa-1b binds to peptides derived from signal peptides of MHC class Ia molecule and interact with the CD94/ NKG2 receptors on natural killer cells . The Qa-1b-peptide complex signals natural killer cells not to engage in cell lysis. Despite its homology with HLA-E, it seems that Qa-1b evolved a similar function to HLA-E coincidentally. [ citation needed ] | https://en.wikipedia.org/wiki/Qa-1b |
Qadad ( Arabic : قضاض qaḍāḍ ) or qudad is a waterproof plaster surface, made of a lime plaster treated with slaked lime and oils and fats. The technique is over a thousand years old, [ 1 ] [ 2 ] with the remains of this early plaster still seen on the standing sluices of the ancient Marib Dam . [ 3 ]
Volcanic ash , pumice , scoria ( Arabic : شاش ), in the Yemeni dialect, or other crushed volcanic aggregate are often used as pozzolanic agents, reminiscent of ancient Roman lime plaster which incorporated pozzolanic volcanic ash.
Due to the slowness of some of the chemical reactions, qadad mortar can take over a hundred days to prepare, from quarrying of raw materials to the beginning of application to the building. It can also take over a year to set fully. [ 4 ]
In 2004, a documentary film Qudad, Re-inventing a Tradition [ 5 ] was made by the filmmaker Caterina Borelli . [ 6 ] [ 7 ] It documents the restoration of the Amiriya Complex , which was awarded the Aga Khan Award for Architecture in 2007. [ 8 ]
After collecting blocks of lime stone, they were fired in a kiln for 4 days, after which the fire and baked lime were extinguished with water, and allowed to cool for 2-3 days more. The baked lime ( Arabic : nūreh ) was then crushed and mixed with soft, black volcanic cinders known as scoria ( Arabic : shāsh ), a pumice having the consistency of gravel. [ 9 ] The scoria and lime were pounded with a stone to break them down into finer particles and thoroughly mixed together without water (the two ingredients being mixed together in a ratio of two parts of aggregate to one part of lime), [ 3 ] and then allowed to rest 3-4 days until settled. Afterwards, the two elements were mixed together with water (usually 1 volume of water to 3 volumes of lime/aggregate), during which time the batch is continuously agitated in a tedious process known as slaking and which required many long hours of manual labour (as much as 4-5 weeks), before a finer lime water solution was added thereto for 1-2 months so as to convert it to a paste. The more that it was pounded with a long shovel or wooden paddle, the more the qadad became adhesive. [ 9 ]
With the now ready mixture of lime and volcanic cinders, they would apply three-layers of qadad -plaster to the walls of cisterns to make them impermeable; the first layer having the largest particles of volcanic cinders (scoria) and the least amount of lime was applied to rough stone, the plaster being added to a thickness of about two inches. They took a sharp-edged stone and, for several days, pounded and rubbed the first layer of qadad firmly onto the wall, all the while sprinkling it with lime-water to keep it wet. [ 9 ] The second layer was applied after fully working the first layer by beating. The first process was repeated, this time the wall being plastered with a mixture of qadad containing smaller particles of volcanic cinders and more lime. A sharp-edged stone was again used to pound the qadad firmly onto the wall, all the while sprinkling it with lime-water to keep it wet. Finally, the third layer was applied containing the smallest particles of volcanic cinders and the largest quantity of lime and worked with a sharp-edged stone (one part aggregate to two parts lime, and pounded to a fine paste), and lime-water spattered on the wall to maintain its wetness. [ 3 ] After the final application, the wall was treated with a very finely-ground consistency of qadad which was allowed to dry, and when dried, an application of animal fat ( suet ) was then smeared on the wall for smoothing and burnishing. [ 9 ] The end result is that of a wall that is as hard as smooth-marble with beating. [ 10 ]
According to archaeologist Selma Al-Radi , qadad can only be used as a plaster on buildings constructed of stone and baked brick, but it will not adhere to mudbrick, cement blocks or concrete. [ 3 ] In Yemen it was traditionally made with two basic ingredients, baked lime and volcanic scoria , other countries have traditionally made-use of fine riverbed sand or pebbles instead of scoria, and which were mixed together with lime for use as a common mortar , [ 3 ] or to be used as an impervious wall plaster.
In Sana'a of the early 20th century, qadad -plaster was used to line pools, reservoirs, drainage pipes, and cesspits , and to make them impermeable. [ 9 ] After applying the qadad , the coating was burnished with a stone. [ 9 ] Often its use extended unto the main kitchen room and to gutters and sinks, wherever water was likely to be used extensively [ 13 ] (see also tadelakt ). The walls of store-rooms where grain was kept and which required being impervious to water were also frequently painted-over with qadad and which gave to the rooms an appearance of being painted with oil paint. [ 13 ] Carl Rathjens , who visited Yemen in the first half of the 20th century, mentions seeing in Sana'a "the houses of well-to-do people" where the entrance halls were often painted with qadad up to a certain height. [ 13 ] The interior walls of public baths were sometimes brick, sometimes stone. If brick, they were protected with a thick layer of hard gypsum plaster which were then oil-painted.
In Islamic architecture, different consistencies of qadad were made for different usages: domes, flat ceilings, vertical walls and decorations in the geometric interlace. [ 14 ] | https://en.wikipedia.org/wiki/Qadad |
Qbox is an open-source software package for atomic-scale simulations of molecules , liquids and solids . It implements first principles (or ab initio ) molecular dynamics , a simulation method in which inter-atomic forces are derived from quantum mechanics . Qbox is released under a GNU General Public License (GPL) with documentation provided at http://qboxcode.org . It is available as a FreeBSD port. [ 1 ]
Qbox computes molecular dynamics trajectories of atoms using Newton 's equations of motion, with forces derived from electronic structure calculations performed using Density Functional Theory . Simulations can be performed either within the Born–Oppenheimer approximation or using Car-Parrinello molecular dynamics . The electronic ground state is computed at each time step by solving the Kohn-Sham equations . Various levels of Density Functional Theory approximations can be used, including the local-density approximation (LDA), the generalized gradient approximation (GGA), or hybrid functionals that incorporate a fraction of Hartree-Fock exchange energy . Electronic wave functions are expanded using the plane wave basis set . The electron-ion interaction is represented by pseudopotentials .
Qbox is written in C++ and implements parallelism using both the message passing interface (MPI) and the OpenMP application programming interface. It makes use of the BLAS , LAPACK , ScaLAPACK , FFTW and Apache Xerces libraries. Qbox was designed [ 7 ] for operation on massively parallel computers such as the IBM Blue Gene supercomputer, or the Cray XC40 supercomputer.
In 2006 it was used to establish a performance record [ 8 ] on the BlueGene/L computer installed at the Lawrence Livermore National Laboratory .
The functionality of Qbox can be enhanced by coupling it with other simulation software using a client-server paradigm. Examples of Qbox coupled operation include: | https://en.wikipedia.org/wiki/Qbox |
Qianfan ( Chinese : 千帆星座 ; pinyin : Qiānfān xīngzuò ; lit. 'Thousand Sails Constellation'), [ 1 ] officially known as the Spacesail Constellation [ 2 ] and also referred to as G60 Starlink , [ 3 ] is a planned Chinese low-Earth orbit satellite internet megaconstellation to create a system of worldwide internet coverage . It was created by Shanghai Spacecom Satellite Technology (SSST), a firm backed by the Shanghai Municipal People's Government and the Chinese Academy of Sciences . [ 4 ] The project was started in 2024 as a rival to the Starlink satellite constellation installed by SpaceX , and plans to be constituted of over 15,000 satellites by the project's end. [ 5 ] [ 6 ]
The "Thousand Sails" program began with the creation of the "Shanghai Action Plan to Promote Commercial Aerospace Development and Create a Space Information Industry Highland (2023-2025)" program first announced on 20 November. [ 7 ] The government of Shanghai raised 6.7 billion Chinese Yuan ($943 million) in funds for the construction of the project, which was initially dubbed the G60 Starlink. [ 4 ]
The first flat panel satellite for the megaconstellation was assembled in December 2023. The satellite's facilities were allocated to the state owned Shanghai Gesi Aerospace Technology (Genesat). [ 8 ]
On 6 August 2024 at 06:42 UTC, China launched its first set of eighteen flat panel satellites associated with the project using the Long March 6A launch vehicle , China's 35th orbital launch in the year 2024. The rocket launched from the Taiyuan Launch Complex located in the north of Shanxi Province , and brought the satellites into a polar orbit . The Chinese Academy of Sciences and the China Aerospace Science and Technology Corporation both reported that the space mission was "a complete success". [ 5 ] [ 6 ] However, the United States Space Command reported that soon after the delivery of 18 satellites, the upper stage of Long March 6A broke apart and created a cloud of debris of “over 300 pieces of trackable debris in low-Earth orbit”. [ 9 ] [ 10 ]
On 15 October 2024 at 11:06 UTC, a Long March 6A rocket launched the second group of eighteen Qianfan satellites into a polar orbit. [ 11 ]
On 5 December 2024 at 04:41 UTC, a third group of eighteen Qianfan satellites were launched into a polar orbit by a Long March 6A rocket. [ 12 ]
Based on Chinese state media China Central Television coverage, China has planned to launch and establish 648 satellites by the end of 2025 as part of the 1,296 satellites in the first phase of construction of the constellation, with the finished broadband multimedia satellite megaconstellation consisting of over 15,000 internet satellites. [ 5 ] Of these, 108 satellites were planned to be deployed in 2024 in separate launches of 36 and 54 internet satellites each, and would operate in "Ku, Q and V" bands. [ 8 ]
The system also planned to annex finite frequencies and orbital slots , and also provide data security . [ 6 ] The People's Liberation Army expressed intentions of potentially using the megaconstellation for military uses akin to Starlink's utility for Ukrainian Armed Forces communications while fighting against Russia during the Russian invasion of Ukraine . [ 6 ] [ 13 ]
Qianfan satellites are bright, and their light pollution poses a threat to observational astronomy . At their current luminosity the spacecraft will leave streaks in photographic research images that cannot be removed by software. They would also interfere with aesthetic appreciation of the night sky because they are visible to the unaided eye. Other spacecraft operators have mitigated the brightness of the satellites to reduce their impact on astronomy. [ 16 ] This is a known problem with satellite constellations , which can be partly mitigated. [ 17 ] | https://en.wikipedia.org/wiki/Qianfan |
Twice every year, the Sun culminates at the zenith of the Kaaba in Mecca , the holiest site in Islam , at local solar noon , allowing the qibla (the direction towards the Kaaba ) to be ascertained in other parts of the world by observing the shadows cast by vertical objects. This phenomenon occurs at 12:18 Saudi Arabia Standard Time (SAST; 09:18 UTC ) on 27 or 28 May (depending on the year), and at 12:27 SAST (09:27 UTC) on 15 or 16 July (depending on the year). At these times, the Sun appears in the direction of Mecca, and shadows cast by vertical objects determine the qibla. At two other moments in the year, the Sun passes through the nadir (the antipodal zenith) of the Kaaba , casting shadows that point in the opposite direction, and thus also determine the qibla. These occur on 12, 13, or 14 January at 00:30 SAST (21:30 UTC on the preceding day), and 28 or 29 November at 00:09 SAST (21:09 UTC on the preceding day).
The shadow points towards Mecca because the Sun path makes the subsolar point travel through every latitude between the Tropic of Cancer and the Tropic of Capricorn every year, including the latitude of the Kaaba (21°25′N), and because the Sun crosses the local meridian once a day. This observation has been known since at least the 13th century, when it was noted by the astronomers Jaghmini and Nasir al-Din al-Tusi , but their timings could not be fixed to a particular date because the Islamic calendar is lunar rather than solar ; the solar date on which the Sun culminates at the zenith of Mecca is constant, but the lunar date varies from year to year.
The qibla is the direction of the Kaaba , a cube-shaped building at the centre of the Great Mosque of Mecca ( al-Masjid al-Haram ) in the Hejaz region of Saudi Arabia. [ 1 ] This direction is special in Islamic rituals and religious law because Muslims must face it during daily prayers ( salat ) and in other religious contexts. [ 2 ] The determination of qibla was an important problem for Muslim communities because Muslims are required to know the qibla to perform their daily prayers and because it is needed to determine the orientation of mosques. [ 3 ] When Muhammad lived among the Muslims in Medina , which is also in the Hejaz region, he prayed due south, the known direction of Mecca. Within a few generations of Muhammad's death in 632, Muslims had reached places far distant from Mecca, making the determination of the qibla in these new locations problematic. [ 4 ] Initially, Muslims relied on traditional folk knowledge methods, [ 5 ] but after the introduction of astronomy into the Islamic world , solutions based on mathematical and astronomical knowledge began to be developed in the early 9th century. [ 6 ] The shadow-observation method has been attested since at least the 13th century CE. [ 7 ] [ 8 ]
Places on Earth experience the apparent diurnal motion of the Sun from the east to the west, during which it culminates, or reaches its highest point of the day and crosses the local meridian . The Sun also appears to move seasonally between the Tropic of Cancer (approximately 23.5°N) and the Tropic of Capricorn (approximately 23.5°S); therefore, the solar culmination usually occurs to the north or south of the zenith. For locations between the tropics, at certain times of the year, the Sun crosses the local latitude and then culminates at or near the zenith; this location is known as the subsolar point . The Kaaba is located at a latitude of 21°25′N, inside the zone that experiences this phenomenon. [ 9 ] [ 10 ] In the terminology of Islamic astronomy (' ilm al-falak ), these events are called the "great culmination" ( al-istiwa al-a'dham ). [ 11 ]
The great culmination, when the Sun appears directly over the Kaaba , occurs on 27 or 28 May at approximately 12:18 SAST (09:18 UTC ), and on 15 or 16 July at 12:27 SAST (09:27 UTC), [ 9 ] [ 10 ] coinciding with the solar noon and the Zuhr adhan (midday call to prayer) in Mecca. [ 12 ] As the sun crosses almost directly above the Kaaba , any shadow cast by vertical objects on earth will point directly away from the Kaaba , [ 10 ] which casts nearly no shadow. [ 12 ] This phenomenon allows the direction of the qibla to be determined without needing to perform calculations or to use sophisticated instruments. This observation is called rasd al-qibla ('observing the qibla'). [ 13 ]
This observation is not observable in the hemisphere opposite the Kaaba , since the phenomenon occurs when the Sun is below the horizon. [ 14 ] This hemisphere includes most of the Americas, the Pacific Ocean, Australia, and Eastern Indonesia . [ 15 ] People in these places can observe a comparable event when the Sun passes directly above the antipodal point of the Kaaba – the point directly opposite on the other side of the Earth. [ 8 ] [ 16 ] The shadows cast during these times point in the exact opposite direction shown during the rasd al-qibla . [ 10 ] The antipodal events occur on 12, 13, or 14 January at 00:30 SAST (21:30 UTC on the previous day), and again on 28 or 29 November 00:09 SAST (21:09 UTC on the previous day). [ 8 ] [ 16 ] [ a ] During any of these events, observations made within a five-minute interval, and at the same time one or two days before or after the prescribed date, are accurate with negligible deviation. [ 10 ] [ 9 ]
A practical problem occurs in locations whose angular distances to Mecca are almost equal to 90 degrees at the edge of the hemisphere centred in Mecca. In these locations, the rasd al-qibla events always occur close to sunrise or sunset. This is the case for several places in the east coast of North America; for instance, the first rasd al-qibla (28 May at 12:18 SAST) occurs six minutes after sunrise in Boston and Montreal , two minutes before sunrise in Ottawa , and eleven minutes before sunrise in New York City . The phenomenon cannot be viewed in New York City and Ottawa, while in Boston and Montreal, the Sun appears so low that the place of observation must be completely unobstructed by buildings or terrain. [ 17 ]
In addition to the twice yearly rasd al-qibla , in most locations the Sun crosses the direct path between the location and the Kaaba each day; at the instant this happens, the Sun's shadow points in the direction of the qibla or its antipodal point. The time of this daily event depends upon the location and the day of the year, and can be determined using geographical data and calculations, but this is more complex than the yearly rasd al-qibla , the times of which are the same globally, with no calculations needed. [ 18 ] [ 8 ] [ b ]
The method of observing the qibla by shadows was attested by the Central Asian astronomer Jaghmini , who wrote c. 1221 it can be done twice a year when the Sun's position in the ecliptic is at 7°21′, in the constellation Gemini , and 22°39′, in Cancer . Subsequently, Nasir al-Din al-Tusi (1201–1276) also related this method in his work al-Tadhkira al-Nasīriyya fī ʿilm al-Hayʾa ("Memoir on the Science of Astronomy"), although with less precision than Jaghmini: [ 7 ] [ 8 ]
The sun transits the zenith of Mecca when it is in degree 8 of Gemini and in [degree] 23 of Cancer at noontime there. The difference between its noon and the noon of other localities is measured by the difference between the two longitudes. Let this [latter] difference be taken and let an hour be assumed for each 15 degrees and 4 minutes for each degree. The resulting total is the interval in hours from noon [for that locality]. Let an observation be made on that day at that time – before noon if Mecca is to the east or after if [Mecca] is to the west; the direction of the shadow [of the sun] at that time is [opposite to that of] the qibla bearing.
Al-Tusi stated the two rasd al-qibla days by specifying the Sun's position on the ecliptic (8° Gemini and 23° Cancer), rather than giving specific dates. This is because during their time, the Muslim world used the lunar Islamic calendar rather than a solar one, therefore the two days could not be specified on a fixed day and month. [ 7 ] [ 8 ] Because the obliquity of the ecliptic is slowly decreasing, the values during the lives of Jaghmini and al-Tusi's differ from modern values. As of 2000, the appropriate solar positions are 6°40′ Gemini and 23°20′ Cancer. [ 8 ] Other than specifying the sun's position, the passage by al-Tusi describes how to convert the noontime in Mecca to the local time. [ 7 ] | https://en.wikipedia.org/wiki/Qibla_observation_by_shadows |
In the Baháʼí Faith , the Qiblih ( Arabic : قبلة , "direction") is the location to which Baháʼís face when saying their daily obligatory prayers . The Qiblih is fixed at the Shrine of Baháʼu'lláh , near Acre , in present-day Israel ; approximately at 32°56′37″N 35°5′31″E / 32.94361°N 35.09194°E / 32.94361; 35.09194 .
In Bábism the Qiblih was originally identified by the Báb with " the One Whom God will make manifest ", a messianic figure predicted by the Báb. Baháʼu'lláh , the Prophet-founder of the Baháʼí Faith claimed to be the figure predicted by the Báb. In the Kitáb-i-Aqdas , Baháʼu'lláh confirms the Báb's ordinance and further ordains his final resting-place as the Qiblih for his followers. [ 1 ] ʻAbdu'l-Bahá describes that spot as the "luminous Shrine", "the place around which circumambulate the Concourse on High". The concept exists in other religions. Jews face Jerusalem, more specifically the site of the former Temple of Jerusalem . Muslims face the Kaaba in Mecca, which they also call the Qibla (another transliteration of Qiblih).
Baháʼís do not worship the Shrine of Baháʼu'lláh or its contents, the Qiblih is simply a focal point for the obligatory prayers . When praying obligatory prayers the members of the Baháʼí Faith face in the direction of the Qiblih. It is a fixed requirement for the recitation of an obligatory prayer, but for other prayers and devotions one may follow what is written in the Qurʼan: "Whichever way ye turn, there is the face of God." [ 2 ]
"The dead should be buried with their face turned towards the Qiblih. This also is in accordance with what is practiced in Islam . There is also a congregational prayer to be recited. Besides this there is no other ceremony to be performed" (From a letter written on behalf of Shoghi Effendi to an individual believer, July 6, 1935). | https://en.wikipedia.org/wiki/Qiblih |
Qilin Li is a Chinese environmental engineer who is a professor of Civil and Environmental Engineering at Rice University . She develops new technologies to analyze and treat contaminated water. Li is a Fellow of the International Water Association .
Li is from China . [ 1 ] She earned her undergraduate degree at Tsinghua University . She moved to the United States for her graduate studies, joining the University of Illinois at Urbana–Champaign as a doctoral student. [ 1 ] She worked on membrane filtration systems with Vernon Snoeyink. [ 2 ] After completing her PhD Li moved to Yale University , where she worked as a postdoctoral fellow. [ 3 ]
Li studies water contamination and treatment. She joined the faculty at Rice University in 2006. [ 4 ] The research of the Li laboratory for Advanced Water Treatment Technologies includes analysis of how membranes become fouled during filtration, how to remove salt from seawater and how to use nanotechnology to clean water. [ 5 ]
She is the Associate Director of the Center for Nanotechnology Enabled Water Treatment (NEWT). Here she was awarded a $1.7 million United States Department of Energy grant to develop new technologies that make use of sunlight and nanoparticles to treat water. [ 6 ] [ 7 ] Li created the Nanophotonics Enabled Solar Membrane Distillation (NESMD), which combines a traditional porous membrane with low-cost light-capturing nanoparticles. [ 6 ] [ 8 ] Membrane distillation is cheap and can be operated in low temperature, low pressure environments. In NESMD, highly localized solar illumination and photothermal heating drives the process of membrane distillation. [ 9 ] She was elected Fellow of the International Water Association in 2018. [ 10 ] At the 2019 American Water Summit Li and her NESMD devices were awarded first place in the Tech Idol competition. [ 4 ]
In 2019 Li was awarded one of Rice University's InterDisciplinary Excellence Awards, using which she combined resource management and efforts to encourage more sustainable resource consumption on the Rice campus. [ 11 ] Li serves as Co-Chair of the International Water Association Nano & Water Specialty Group. [ 4 ] | https://en.wikipedia.org/wiki/Qilin_Li |
Qiu Dahong ( Chinese : 邱大洪 ; pinyin : Qiū Dàhóng ; 6 April 1930 – 11 January 2025) was a Chinese coastal and offshore engineer . He served as chief engineer of the Dalian Fishing Port, the New Dalian Port , the Qinhuangdao Petroleum Port, and many other projects. He was a professor of the Dalian University of Technology and directed the State Key Laboratory of Coastal and Offshore Engineering. He was elected an academician of the Chinese Academy of Sciences in 1991.
Qiu was born on 6 April 1930 in Shanghai , Republic of China , with his ancestral home in Huzhou , Zhejiang . [ 1 ] After graduating from the Department of Civil Engineering of Tsinghua University in 1951, he joined the Dalian University of Technology (DUT), where he worked under Professor Qian Lingxi and helped create China's first port and harbour engineering program. [ 2 ]
In 1958, Qiu was appointed chief engineer for the construction of Dalian Fishing Port at the age of 28. [ 2 ] [ 3 ] The port, designed to occupy 50,000 square metres (540,000 sq ft) of open water with docks for 300 fishing boats, was unprecedented in China in both scale and difficulty. [ 3 ] When completed in 1966, it was Asia's largest fishing port. [ 2 ] In 1987, Qiu again served as chief engineer for the port's expansion project, which was completed in 1989. [ 3 ]
In 1973, Qiu became chief engineer of the New Dalian Port , the first port in China capable of handling oil tankers with a displacement of 100,000 tons. It was opened in 1976, and won a national gold medal for its design. [ 2 ] [ 3 ]
Qiu later led or participated in the design of the Qinhuangdao Petroleum Port, the Lianyungang Container Port, the Shenzhen Chiwan Port, the Hainan Petroleum Port, the Yamen Shipping Channel of the Pearl River estuary , [ 2 ] and the Yangshan Port of Shanghai. [ 3 ]
Qiu served as Director of the State Key Laboratory of Coastal and Offshore Engineering at DUT [ 4 ] and was elected an academician of the Chinese Academy of Sciences in 1991. [ 1 ] In 1992, he was elected a Central Committee member of the Jiusan Society . [ 2 ]
On 11 January 2025, Qiu died in Dalian at the age of 94. [ 5 ]
Qiu published more than 100 scientific papers and multiple monographs and textbooks. [ 2 ] In addition to coastal and offshore engineering, he researched wave theory and conducted experiments to calculate the forces that sea waves exert on engineering structures. [ 4 ] In 2011, China Ocean Press published The Collective Writings of Qiu Dahong ( 邱大洪文集 ), which includes 93 scientific papers, 15 published articles on port construction projects, and 20 other articles. [ 6 ] | https://en.wikipedia.org/wiki/Qiu_Dahong |
The Qiu Shi Prizes ( simplified Chinese : 求是奖 ; traditional Chinese : 求是獎 ) are awarded on an annual basis in recognition of advances in science and technology. The Qiu Shi Science and Technology Foundation was established by Cha Chi Ming ( 查濟民 ) (1914–2007) in 1994 in Hong Kong, with the intention of promoting science and technology research in China, and to encourage and reward successful Chinese scientists and scholars. [ 1 ] Prizes are awarded each year Prize categories include Physics, Chemistry, Physiology or Medicine, Mathematics or Information Technology. [ 2 ]
Qiu Shi (Chinese: 求是 ; pronounced ch-OO/sh-ER) means "seeking truth". The Qiu Shi Foundation was named after Qiu Shi Academy ( Chinese : 求是书院 ) in Hangzhou, which was subsequently renamed Zhejiang University ( 浙江大学 ). Qiu Shi Science and Technology Foundation is not related to the Qiushi journal, the political theory periodical.
Cha Chi-ming, GBM, JP, was born in 1914, in Haining County, Jiaxing, Zhejiang province. He studied textile technology and graduated from Zhejiang University in 1931. Cha built a multinational textile conglomerate. He was the chairman of CDW International Limited, The Mingly Corporation Limited and Hong Kong Resort International Limited. With his family, he donated US$20 million in 1994 to establish the award. [ 3 ]
Members:
Chen Ning Yang (a Nobel laureate in physics), Zhou Guangzhao (physicist, and honorary chairman of China Association for Science and Technology), Yuan T. Lee (a Nobel laureate in chemistry), Yuet Wai Kan (genetic researcher), David Ho (physician and innovator of the "cocktail" therapy for HIV), Andrew Yao (computer scientist and the first Asian A.M. Turing Award recipient).
Alumni | https://en.wikipedia.org/wiki/Qiu_Shi_Science_and_Technology_Prize |
The QosCosGrid is a quasi-opportunistic supercomputing system using grid computing . [ 1 ] [ 2 ]
QosCosGrid acts as middleware resource management facilities which provide end-users with supercomputer-like performance by connecting many computing clusters together. By using QosCosGrid large-scale computing models in existing programming languages such as Fortran or C can be distributed among multiple computing resources.
CORDIS
This supercomputer-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Qoscos_Grid |
QuEChERS is a solid phase extraction method for detection of biocide residues in food. The name is a portmanteau word formed from " quick, easy, cheap, effective, rugged, and safe ". [ 1 ]
The sample ( fruits , vegetables , tobacco , etc.) is homogenized and centrifuged with a reagent and agitated for 1 minute. The reagents used depend on the type of sample to be analyzed. Following this, the sample is put through a dispersive solid phase extraction cleanup prior to analysis by gas-liquid chromatography or liquid-liquid chromatography .
Samples prepared using the QuEChERS method can be processed more quickly using a homogenization instrument. [ citation needed ] Such instruments can homogenize the food sample in a centrifuge tube, then agitate the sample with the reagent of choice, before moving the extracted sample for centrifuging. By using such an instrument, the samples can be moved through the QuEChERS method more quickly. [ citation needed ]
Some modifications to the original QuEChERS method had to be introduced to ensure efficient extraction of pH-dependent compounds (e.g., phenoxyalkanoic acids), to minimize degradation of susceptible compounds (e.g., base and acid labile pesticides) and to expand the spectrum of matrices covered. [ citation needed ]
The QuEChERS method has been readily accepted by many pesticide residue analysts. [ 2 ] [ 3 ] | https://en.wikipedia.org/wiki/QuEChERS |
QuTiP , short for the Quantum Toolbox in Python , is an open-source computational physics software library for simulating quantum systems , particularly open quantum systems . [ 1 ] [ 2 ] QuTiP allows simulation of Hamiltonians with arbitrary time-dependence, allowing simulation of situations of interest in quantum optics , ion trapping , superconducting circuits and quantum nanomechanical resonators . The library includes extensive visualization facilities for content under simulations.
QuTiP's API provides a Python interface and uses Cython to allow run-time compilation and extensions via C and C++ . QuTiP is built to work well with popular Python packages NumPy , SciPy , Matplotlib and IPython .
The idea for the QuTip project was conceived in 2010 by PhD student Paul Nation, who was using the quantum optics toolbox for MATLAB in his research. According to Paul Nation, he wanted to create a python package similar to qotoolbox because he "was not a big fan of MATLAB" and then decided to "just write it [him]self". [ 3 ] As a postdoctoral fellow, at the RIKEN Institute in Japan, he met Robert Johansson and the two worked together on the package.
In contrast to its predecessor qotoolbox, which relies on the proprietary MATLAB environment, it was published in 2012 under an open source license. [ 2 ]
The Version created by Nation and Johansson already contained the most important features of the package, but QuTips scope and features are constantly being extended by a large community of contributors. [ 4 ] It has grown in popularity amongst physicists, with over 250.000 downloads in the year 2021. [ 5 ]
Simulating a non-unitary time evolution according to the Lindblad Master Equation is possible with the qutip.mesolve function [ 6 ] | https://en.wikipedia.org/wiki/QuTiP |
Qu Geping is a Chinese environmental scientist. He was Director of the Chinese State Environmental Protection Agency from 1987 to June 1993. [ 1 ] [ 2 ]
In 1999, he was the winner of the Blue Planet Prize along with Paul R. Ehrlich .
In 2001, he was awarded the Duke of Edinburgh Conservation Medal , the highest award of the World Wildlife Fund , calling him the father of environmental protection in China. [ 3 ]
This biographical article about a scientist is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Qu_Geping |
A quad chart is a form of technical documentation used to briefly describe an invention or other innovation through writing, illustration and/or photographs. [ 1 ] Such documents are described as "quad" charts because they are divided into four quadrants laid out on a landscape perspective. [ 2 ] [ 3 ] [ 4 ] They are typically one-page only; their succinctness facilitates rapid decision-making. [ 5 ] Though shorter, quad charts often serve in a similar capacity to white papers and the two documents are often requested alongside one another.
Quad charts as a genre were developed by the United States Department of Commerce 's National Oceanic and Atmospheric Administration in an attempt to improve budgeting and planning systems, and became widely used in the Administration's National Weather Service . [ 5 ] The genre's development was parallel to that of display boards , also an early tool used by the NWS for staff communication.
In the early 2000s, software was developed to allow automated creation of quad charts as a means of saving time for technical writers who would otherwise spend long periods of time drafting them. [ citation needed ]
Both government agencies and large businesses often require submission of a quad chart on the part of potential contractors as part of the contract bidding process. [ 6 ] [ 7 ] NASA , for example, uses quad charts to document the process of all Small Business Innovation Research projects. [ 8 ] Because decision makers often review a large volume of both solicited and unsolicited proposals, the quad chart may be the only submission from a potential contractor which the decision maker actually reads. [ 4 ]
Due to the nature of quad charts as relatively short documents, there are opportunities for misuse. While quad charts are intended for brief overviews of a topic, they can also be misconstrued to influence public policy and budgeting decisions, as was the case with the politicization of the National Defense Strategy 's 2005 edition. [ 9 ]
While there are no industry-wide standards for quad charts, there are a number of common elements. In addition to the title of the invention or idea and the name of the developer, the technical approach and the need which the invention or idea addresses are often included. [ 4 ] Decision makers often look to operational needs first, though including the cost and projected schedule are also often required elements. | https://en.wikipedia.org/wiki/Quad_chart |
A unit of information is any unit of measure of digital data size. In digital computing , a unit of information is used to describe the capacity of a digital data storage device. In telecommunications , a unit of information is used to describe the throughput of a communication channel . In information theory , a unit of information is used to measure information contained in messages and the entropy of random variables.
Due to the need to work with data sizes that range from very small to very large, units of information cover a wide range of data sizes. Units are defined as multiples of a smaller unit except for the smallest unit which is based on convention and hardware design. Multiplier prefixes are used to describe relatively large sizes.
For binary hardware , by far the most common hardware today, the smallest unit is the bit , a portmanteau of binary digit, [ 1 ] which represents a value that is one of two possible values; typically shown as 0 and 1. The nibble , 4 bits, represents the value of a single hexadecimal digit. The byte , 8 bits, 2 nibbles, is possibly the most commonly known and used base unit to describe data size. The word is a size that varies by and has a special importance for a particular hardware context. On modern hardware, a word is typically 2, 4 or 8 bytes, but the size varies dramatically on older hardware. Larger sizes can be expressed as multiples of a base unit via SI metric prefixes (powers of ten) or the newer and generally more accurate IEC binary prefixes (powers of two).
In 1928, Ralph Hartley observed a fundamental storage principle, [ 2 ] which was further formalized by Claude Shannon in 1945: the information that can be stored in a system is proportional to the logarithm of N possible states of that system, denoted log b N . Changing the base of the logarithm from b to a different number c has the effect of multiplying the value of the logarithm by a fixed constant, namely log c N = (log c b ) log b N .
Therefore, the choice of the base b determines the unit used to measure information. In particular, if b is a positive integer, then the unit is the amount of information that can be stored in a system with b possible states.
When b is 2, the unit is the shannon , equal to the information content of one "bit". A system with 8 possible states, for example, can store up to log 2 8 = 3 bits of information. Other units that have been named include:
The trit, ban, and nat are rarely used to measure storage capacity; but the nat, in particular, is often used in information theory, because natural logarithms are mathematically more convenient than logarithms in other bases.
Several conventional names are used for collections or groups of bits.
Historically, a byte was the number of bits used to encode a character of text in the computer, which depended on computer hardware architecture, but today it almost always means eight bits – that is, an octet . An 8-bit byte can represent 256 (2 8 ) distinct values, such as non-negative integers from 0 to 255, or signed integers from −128 to 127. The IEEE 1541-2002 standard specifies "B" (upper case) as the symbol for byte ( IEC 80000-13 uses "o" for octet in French, but also allows "B" in English). Bytes, or multiples thereof, are almost always used to specify the sizes of computer files and the capacity of storage units. Most modern computers and peripheral devices are designed to manipulate data in whole bytes or groups of bytes, rather than individual bits.
A group of four bits, or half a byte, is sometimes called a nibble , nybble or nyble. This unit is most often used in the context of hexadecimal number representations, since a nibble has the same number of possible values as one hexadecimal digit has. [ 7 ]
Computers usually manipulate bits in groups of a fixed size, conventionally called words . The number of bits in a word is usually defined by the size of the registers in the computer's CPU , or by the number of data bits that are fetched from its main memory in a single operation. In the IA-32 architecture more commonly known as x86-32, a word is 32 bits, but other past and current architectures use words with 4, 8, 9, 12, 13, 16, 18, 20, 21, 22, 24, 25, 29, 30, 31, 32, 33, 35, 36, 38, 39, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 72 [ 8 ] bits or others.
Some machine instructions and computer number formats use two words (a "double word" or "dword"), or four words (a "quad word" or "quad").
Computer memory caches usually operate on blocks of memory that consist of several consecutive words. These units are customarily called cache blocks , or, in CPU caches , cache lines .
Virtual memory systems partition the computer's main storage into even larger units, traditionally called pages .
A unit for a large amount of data can be formed using either a metric or binary prefix with a base unit. For storage, the base unit is typically byte. For communication throughput, a base unit of bit is common. For example, using the metric kilo prefix, a kilobyte is 1000 bytes and a kilobit is 1000 bits.
Use of metric prefixes is common, but often inaccurate since binary storage hardware is organized with capacity that is a power of 2 – not 10 as the metric prefixes are. In the context of computing, the metric prefixes are often intended to mean something other than their normal meaning. For example, 'kilobyte' often refers to 1024 bytes even though the standard meaning of kilo is 1000. Also, 'mega' normally means one million, but in computing is often used to mean 2 20 = 1 048 576 . The table below illustrates the differences between normal metric sizes and the intended size – the binary size.
The International Electrotechnical Commission (IEC) issued a standard that introduces binary prefixes that accurately represent binary sizes without changing the meaning of the standard metric terms. Rather than based on powers of 1000, these are based on powers of 1024 which is a power of 2. [ 9 ]
The JEDEC memory standard JESD88F notes that the definitions of kilo (K), giga (G), and mega (M) based on powers of two are included only to reflect common usage, but are otherwise deprecated. [ 10 ]
Some notable unit names that are today obsolete or only used in limited contexts. | https://en.wikipedia.org/wiki/Quadlet |
Divine Proportions: Rational Trigonometry to Universal Geometry is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry , called rational trigonometry . The book advocates replacing the usual basic quantities of trigonometry, Euclidean distance and angle measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa). The author claims his approach holds some advantages, such as avoiding the need for irrational numbers .
The book was "essentially self-published" [ 1 ] by Wildberger through his publishing company Wild Egg. The formulas and theorems in the book are regarded as correct mathematics but the claims about practical or pedagogical superiority are primarily promoted by Wildberger himself and have received mixed reviews.
The main idea of Divine Proportions is to replace distances by the squared Euclidean distance , which Wildberger calls the quadrance , and to replace angle measures by the squares of their sines, which Wildberger calls the spread between two lines. Divine Proportions defines both of these concepts directly from the Cartesian coordinates of points that determine a line segment or a pair of crossing lines. Defined in this way, they are rational functions of those coordinates, and can be calculated directly without the need to take the square roots or inverse trigonometric functions required when computing distances or angle measures. [ 1 ]
For Wildberger, a finitist , this replacement has the purported advantage of avoiding the concepts of limits and actual infinity used in defining the real numbers , which Wildberger claims to be unfounded. [ 2 ] [ 1 ] It also allows analogous concepts to be extended directly from the rational numbers to other number systems such as finite fields using the same formulas for quadrance and spread. [ 1 ] Additionally, this method avoids the ambiguity of the two supplementary angles formed by a pair of lines, as both angles have the same spread. This system is claimed to be more intuitive, and to extend more easily from two to three dimensions. [ 3 ] However, in exchange for these benefits, one loses the additivity of distances and angles: for instance, if a line segment is divided in two, its length is the sum of the lengths of the two pieces, but combining the quadrances of the pieces is more complicated and requires square roots. [ 1 ]
Divine Proportions is divided into four parts. Part I presents an overview of the use of quadrance and spread to replace distance and angle, and makes the argument for their advantages. Part II formalizes the claims made in part I, and proves them rigorously. [ 1 ] Rather than defining lines as infinite sets of points, they are defined by their homogeneous coordinates , which may be used in formulas for testing the incidence of points and lines. Like the sine, the cosine and tangent are replaced with rational equivalents, called the "cross" and "twist", and Divine Proportions develops various analogues of trigonometric identities involving these quantities, [ 3 ] including versions of the Pythagorean theorem , law of sines and law of cosines . [ 4 ]
Part III develops the geometry of triangles and conic sections using the tools developed in the two previous parts. [ 1 ] Well known results such as Heron's formula for calculating the area of a triangle from its side lengths, or the inscribed angle theorem in the form that the angles subtended by a chord of a circle from other points on the circle are equal, are reformulated in terms of quadrance and spread, and thereby generalized to arbitrary fields of numbers. [ 3 ] [ 5 ] Finally, Part IV considers practical applications in physics and surveying, and develops extensions to higher-dimensional Euclidean space and to polar coordinates . [ 1 ]
Divine Proportions does not assume much in the way of mathematical background in its readers, but its many long formulas, frequent consideration of finite fields, and (after part I) emphasis on mathematical rigour are likely to be obstacles to a popular mathematics audience. Instead, it is mainly written for mathematics teachers and researchers. However, it may also be readable by mathematics students, and contains exercises making it possible to use as the basis for a mathematics course. [ 1 ] [ 6 ]
The feature of the book that was most positively received by reviewers was its work extending results in distance and angle geometry to finite fields. Reviewer Laura Wiswell found this work impressive, and was charmed by the result that the smallest finite field containing a regular pentagon is F 19 {\displaystyle \mathbb {F} _{19}} . [ 1 ] Michael Henle calls the extension of triangle and conic section geometry to finite fields, in part III of the book, "an elegant theory of great generality", [ 4 ] and William Barker also writes approvingly of this aspect of the book, calling it "particularly novel" and possibly opening up new research directions. [ 6 ]
Wiswell raises the question of how many of the detailed results presented without attribution in this work are actually novel. [ 1 ] In this light, Michael Henle notes that the use of squared Euclidean distance "has often been found convenient elsewhere"; [ 4 ] for instance it is used in distance geometry , least squares statistics, and convex optimization . James Franklin points out that for spaces of three or more dimensions, modelled conventionally using linear algebra , the use of spread by Divine Proportions is not very different from standard methods involving dot products in place of trigonometric functions. [ 5 ]
An advantage of Wildberger's methods noted by Henle is that, because they involve only simple algebra, the proofs are both easy to follow and easy for a computer to verify. However, he suggests that the book's claims of greater simplicity in its overall theory rest on a false comparison in which quadrance and spread are weighed not against the corresponding classical concepts of distances, angles, and sines, but the much wider set of tools from classical trigonometry. He also points out that, to a student with a scientific calculator, formulas that avoid square roots and trigonometric functions are a non-issue, [ 4 ] and Barker adds that the new formulas often involve a greater number of individual calculation steps. [ 6 ] Although multiple reviewers felt that a reduction in the amount of time needed to teach students trigonometry would be very welcome, [ 3 ] [ 5 ] [ 7 ] Paul Campbell is skeptical that these methods would actually speed learning. [ 7 ] Gerry Leversha keeps an open mind, writing that "It will be interesting to see some of the textbooks aimed at school pupils [that Wildberger] has promised to produce, and ... controlled experiments involving student guinea pigs." [ 3 ] However, these textbooks and experiments have not been published.
Wiswell is unconvinced by the claim that conventional geometry has foundational flaws that these methods avoid. [ 1 ] While agreeing with Wiswell, Barker points out that there may be other mathematicians who share Wildberger's philosophical suspicions of the infinite, and that this work should be of great interest to them. [ 6 ]
A final issue raised by multiple reviewers is inertia: supposing for the sake of argument that these methods are better, are they sufficiently better to make worthwhile the large individual effort of re-learning geometry and trigonometry in these terms, and the institutional effort of re-working the school curriculum to use them in place of classical geometry and trigonometry? Henle, Barker, and Leversha conclude that the book has not made its case for this, [ 3 ] [ 4 ] [ 6 ] but Sandra Arlinghaus sees this work as an opportunity for fields such as her mathematical geography "that have relatively little invested in traditional institutional rigidity" to demonstrate the promise of such a replacement. [ 8 ] | https://en.wikipedia.org/wiki/Quadrance |
The Quadrans Vetus is a medieval astronomical instrument.
Known as the quadrans vetus ["old quadrant"], the three surviving medieval examples are in the Museo Galileo in Florence , [ 1 ] the Museum of the History of Science in Oxford, [ 2 ] and the British Museum in London. [ 3 ]
There are two sights on one of the straight sides. The front carries the shadow square, the hour lines, and a mobile zodiacal cursor in its guide, to be positioned for the desired latitude. The back is inscribed with the zodiacal calendar. The instrument displays Gothic characters. Designed to measure heights, distances, and depths, the instrument could also be used as a universal dial. A similar quadrant is documented in a drawing by Antonio da Sangallo the Younger (c. 1520?) at the Gabinetto dei Disegni e delle Stampe (Department of Drawings and Prints) of the Uffizi .
Mara Miniati, ed. (1991). Museo di storia della scienza: catalogo (in Italian). Firenze: Giunti. p. 8, board n. 10. ISBN 88-09-20036-5 .
Anthony J. Turner, ed. (2007). Catalogue of sun-dials, nocturnals and related instruments (in Italian). Firenze: Giunti. pp. 34– 38, board n. 3. ISBN 978-88-09-04999-4 . | https://en.wikipedia.org/wiki/Quadrans_Vetus |
A quadrant is an instrument used to measure angles up to 90° . Different versions of this instrument could be used to calculate various readings, such as longitude , latitude , and time of day . It was first proposed by Ptolemy as a better kind of astrolabe . [ 1 ] Several different variations of the instrument were later produced by medieval Muslim astronomers . Mural quadrants were important astronomical instruments in 18th-century European observatories , establishing a use for positional astronomy .
The term quadrant , meaning one fourth , refers to the fact that early versions of the instrument were derived from astrolabes. The quadrant condensed the workings of the astrolabe into an area one fourth the size of the astrolabe face; it was essentially a quarter of an astrolabe.
During Rigvedic times in ancient India, quadrants called 'Tureeyam's were used to measure the extent of a great solar eclipse . The use of a Tureeyam for observing a solar eclipse by Rishi Atri is described in the fifth mandala of the Rigveda , [ 2 ] [ 3 ] most likely between c. 1500 and 1000 BC. [ 4 ]
Early accounts of a quadrant also come from Ptolemy 's Almagest around AD 150. He described a "plinth" that could measure the altitude of the noon sun by projecting the shadow of a peg on a graduated arc of 90 degrees. [ 5 ] This quadrant was unlike later versions of the instrument; it was larger and consisted of several moving parts. Ptolemy's version was a derivative of the astrolabe and the purpose of this rudimentary device was to measure the meridian angle of the sun.
Islamic astronomers in the Middle Ages improved upon these ideas and constructed quadrants throughout the Middle East, in observatories such as Marageh , Rey and Samarkand . At first these quadrants were usually very large and stationary, and could be rotated to any bearing to give both the altitude and azimuth for any celestial body. [ 5 ] As Islamic astronomers made advancements in astronomical theory and observational accuracy they are credited with developing four different types of quadrants during the Middle Ages and beyond. The first of these, the sine quadrant , was invented by Muhammad ibn Musa al-Khwarizmi in the 9th century at the House of Wisdom in Baghdad. [ 6 ] : 128 The other types were the universal quadrant, the horary quadrant and the astrolabe quadrant.
During the Middle Ages the knowledge of these instruments spread to Europe. In the 13th century Jewish astronomer Jacob ben Machir ibn Tibbon was crucial in further developing the quadrant. [ 7 ] He was a skilled astronomer and wrote several volumes on the topic, including an influential book detailing how to build and use an improved version of the quadrant. The quadrant that he invented came to be known as the novus quadrans , or new quadrant. [ 8 ] This device was revolutionary because it was the first quadrant to be built that did not involve several moving parts and thus could be much smaller and more portable.
Tibbon's Hebrew manuscripts were translated into Latin and improved upon by Danish scholar Peter Nightingale several years later. [ 9 ] [ 10 ] Because of the translation, Tibbon, or Prophatius Judaeus as he was known in Latin, became an influential name in astronomy. His new quadrant was based upon the idea that the stereographic projection that defines a planispheric astrolabe can still work if the astrolabe parts are folded into a single quadrant. [ 11 ] The result was a device that was far cheaper, easier to use and more portable than a standard astrolabe. Tibbon's work had a far reach and influenced Copernicus , Christopher Clavius and Erasmus Reinhold ; and his manuscript was referenced in Dante's Divine Comedy . [ 7 ]
As the quadrant became smaller and thus more portable, its value for navigation was soon realized. The first documented use of the quadrant to navigate at sea is in 1461, by Diogo Gomes . [ 12 ] Sailors began by measuring the height of Polaris to ascertain their latitude. This application of quadrants is generally attributed to Arab sailors who traded along the east coast of Africa and often travelled out of sight of land. It soon became more common to take the height of the sun at a given time due to the fact that Polaris is not visible south of the equator.
In 1618, the English mathematician Edmund Gunter further adapted the quadrant with an invention that came to be known as the Gunter quadrant. [ 13 ] This pocket sized quadrant was revolutionary because it was inscribed with projections of the tropics, the equator, the horizon and the ecliptic. With the correct tables one could use the quadrant to find the time, the date, the length of the day or night, the time of sunrise and sunset and the meridian. The Gunter quadrant was extremely useful but it had its drawbacks; the scales only applied to a certain latitude so the instrument's use was limited at sea.
There are several types of quadrants:
They can also be classified as: [ 15 ]
The geometric quadrant is a quarter-circle panel usually of wood or brass. Markings on the surface might be printed on paper and pasted to the wood or painted directly on the surface. Brass instruments had their markings scribed directly into the brass.
For marine navigation, the earliest examples were found around 1460. They were not graduated in degrees but rather had the latitudes of the most common destinations directly scribed on the limb . When in use, the navigator would sail north or south until the quadrant indicated he was at the destination's latitude, turn in the direction of the destination and sail to the destination maintaining a course of constant latitude. After 1480, more of the instruments were made with limbs graduated in degrees. [ 21 ]
Along one edge there were two sights forming an alidade . A plumb bob was suspended by a line from the centre of the arc at the top.
In order to measure the altitude of a star, the observer would view the star through the sights and hold the quadrant so that the plane of the instrument was vertical. The plumb bob was allowed to hang vertical and the line indicated the reading on the arc's graduations . It was not uncommon for a second person to take the reading while the first concentrated on observing and holding the instrument in proper position.
The accuracy of the instrument was limited by its size and by the effect the wind or observer's motion would have on the plumb bob. For navigators on the deck of a moving ship, these limitations could be difficult to overcome.
In order to avoid staring into the sun to measure its altitude, navigators could hold the instrument in front of them with the sun to their side. By having the sunward sighting vane cast its shadow on the lower sighting vane, it was possible to align the instrument to the sun. Care would have to be taken to ensure that the altitude of the centre of the sun was determined. This could be done by averaging the elevations of the upper and lower umbra in the shadow.
In order to perform measurements of the altitude of the sun, a back observation quadrant was developed. [ 21 ]
With such a quadrant, the observer viewed the horizon from a sight vane (C in the figure on the right) through a slit in the horizon vane (B). This ensured the instrument was level. The observer moved the shadow vane (A) to a position on the graduated scale so as to cause its shadow to appear coincident with the level of the horizon on the horizon vane. This angle was the elevation of the sun.
Large frame quadrants were used for astronomical measurements, notably determining the altitude of celestial objects. They could be permanent installations, such as mural quadrants . Smaller quadrants could be moved. Like the similar astronomical sextants , they could be used in a vertical plane or made adjustable for any plane.
When set on a pedestal or other mount, they could be used to measure the angular distance between any two celestial objects.
The details on their construction and use are essentially the same as those of the astronomical sextants ; refer to that article for details.
Navy: Used to gauge elevation on ships cannon, the quadrant had to be placed on each gun's trunnion in order to judge range, after the loading. The reading was taken at the top of the ship's roll, the gun adjusted, and checked, again at the top of the roll, and he went to the next gun, until all that were going to be fired were ready. The ship's Gunner was informed, who in turn informed the captain...You may fire when ready...at the next high roll, the cannon would be fired.
In more modern applications, the quadrant is attached to the trunnion ring or of a large naval gun to align it to benchmarks welded to the ship's deck. This is done to ensure firing of the gun hasn't "warped the deck." A flat surface on the mount gunhouse or turret is also checked against benchmarks, also, to ensure large bearings and/or bearing races haven't changed... to "calibrate" the gun.
During the Middle Ages, makers often added customization to impress the person for whom the quadrant was intended. In large, unused spaces on the instrument, a sigil or badge would often be added to denote the ownership by an important person or the allegiance of the owner. [ 22 ] | https://en.wikipedia.org/wiki/Quadrant_(instrument) |
A quadrat is a frame used in ecology , geography , and biology to isolate a standard unit of area for study of the distribution of an item over a large area. Quadrats typically occupy an area of 0.25 m 2 and are traditionally square, but modern quadrats can be rectangular, circular, or irregular. [ 1 ] [ 2 ] A quadrat is suitable for sampling or observing plants , slow-moving animals, and some aquatic organisms.
A photo-quadrat is a photographic record of the area framed by a quadrat. It may use a physical frame to indicate the area, or may rely on fixed camera distance and lens field of view to automatically cover the specified area of substrate. [ 3 ] Parallel laser pointers mounted on the camera can also be used as scale indicators. The photo is taken perpendicular to the surface, or as close as possible to perpendicular for uneven surfaces.
The systematic use of quadrats was developed by the pioneering plant ecologists Roscoe Pound and Frederic Clements between 1898 [ 4 ] and 1900. [ 5 ] The method was then swiftly applied for many purposes in ecology , such as the study of plant succession . [ 6 ] Botanists and ecologists such as Arthur Tansley soon took up and modified the method. [ 7 ] [ 8 ]
The ecologist John Ernst Weaver applied the use of quadrats to the teaching of ecology in 1918. [ 9 ]
A quadrat can be used by researchers to methodically count organisms within a smaller, representative area in order to extrapolate to a larger habitat when comprehensive sampling is impossible or not practical. The quadrat's size corresponds to the size of the organism being sampled and the overall sampling area. To avoid selection bias , researchers randomly distribute quadrats throughout the sampling area. [ 10 ]
For long-term studies, the same quadrats can be revisited after their initial sampling. Methods of precisely relocating the area of study vary widely in accuracy and include measurement from nearby permanent markers, use of total station theodolites , consumer-grade GPS , and differential GPS . [ 11 ] | https://en.wikipedia.org/wiki/Quadrat |
In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by Polishchuk and Positselski ( 2005 , p.101). An example is the universal enveloping algebra of a Lie algebra , with generators a basis of the Lie algebra and relations of the form XY – YX – [ X , Y ] = 0.
This abstract algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quadratic-linear_algebra |
A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form . Compatibility means that it is invariant under the adjoint representation . Examples of such are semisimple Lie algebras , such as su( n ) and sl( n , R ) .
A quadratic Lie algebra is a Lie algebra ( g ,[.,.]) together with a non-degenerate symmetric bilinear form ( . , . ) : g ⊗ g → R {\displaystyle (.,.)\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R} } that is invariant under the adjoint action, i.e.
where X,Y,Z are elements of the Lie algebra g .
A localization/ generalization is the concept of Courant algebroid where the vector space g is replaced by (sections of) a vector bundle .
As a first example, consider R n with zero-bracket and standard inner product
Since the bracket is trivial the invariance is trivially fulfilled.
As a more elaborate example consider so(3) , i.e. R 3 with base X,Y,Z , standard inner product, and Lie bracket
Straightforward computation shows that the inner product is indeed preserved. A generalization is the following.
A big group of examples fits into the category of semisimple Lie algebras, i.e. Lie algebras whose adjoint representation is faithful. Examples are sl(n,R) and su(n) , as well as direct sums of them. Let thus g be a semi-simple Lie algebra with adjoint representation ad , i.e.
Define now the Killing form
Due to the Cartan criterion , the Killing form is non-degenerate if and only if the Lie algebra is semisimple.
If g is in addition a simple Lie algebra , then the Killing form is up to rescaling the only invariant symmetric bilinear form.
This article incorporates material from Quadratic Lie algebra on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
This article about theoretical physics is a stub . You can help Wikipedia by expanding it .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quadratic_Lie_algebra |
In mathematics , a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups . The most important class of graded quadratic algebras is Koszul algebras .
A graded quadratic algebra A is determined by a vector space of generators V = A 1 and a subspace of homogeneous quadratic relations S ⊂ V ⊗ V . [ 1 ] Thus
and inherits its grading from the tensor algebra T ( V ).
If the subspace of relations is instead allowed to also contain inhomogeneous degree 2 elements, i.e. S ⊂ k ⊕ V ⊕ ( V ⊗ V ), this construction results in a filtered quadratic algebra .
A graded quadratic algebra A as above admits a quadratic dual : the quadratic algebra generated by V * and with quadratic relations forming the orthogonal complement of S in V * ⊗ V * .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quadratic_algebra |
In mathematics , a quadratic equation (from Latin quadratus ' square ' ) is an equation that can be rearranged in standard form as [ 1 ] a x 2 + b x + c = 0 , {\displaystyle ax^{2}+bx+c=0\,,} where the variable x represents an unknown number, and a , b , and c represent known numbers, where a ≠ 0 . (If a = 0 and b ≠ 0 then the equation is linear , not quadratic.) The numbers a , b , and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient , the linear coefficient and the constant coefficient or free term . [ 2 ]
The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root . If all the coefficients are real numbers , there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation [ 3 ] a x 2 + b x + c = a ( x − r ) ( x − s ) = 0 {\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0} where r and s are the solutions for x .
The quadratic formula x = − b ± b 2 − 4 a c 2 a {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} expresses the solutions in terms of a , b , and c . Completing the square is one of several ways for deriving the formula.
Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC. [ 4 ] [ 5 ]
Because the quadratic equation involves only one unknown, it is called " univariate ". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation . In particular, it is a second-degree polynomial equation, since the greatest power is two.
A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots . When there is only one distinct root, it can be interpreted as two roots with the same value, called a double root . When there are no real roots, the coefficients can be considered as complex numbers with zero imaginary part , and the quadratic equation still has two complex-valued roots, complex conjugates of each-other with a non-zero imaginary part. A quadratic equation whose coefficients are arbitrary complex numbers always has two complex-valued roots which may or may not be distinct.
The solutions of a quadratic equation can be found by several alternative methods.
It may be possible to express a quadratic equation ax 2 + bx + c = 0 as a product ( px + q )( rx + s ) = 0 . In some cases, it is possible, by simple inspection, to determine values of p , q , r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0 . Solving these two linear equations provides the roots of the quadratic.
For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. [ 6 ] : 202–207 If one is given a quadratic equation in the form x 2 + bx + c = 0 , the sought factorization has the form ( x + q )( x + s ) , and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule" [ 7 ] and is related to Vieta's formulas ). As an example, x 2 + 5 x + 6 factors as ( x + 3)( x + 2) . The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.
Except for special cases such as where b = 0 or c = 0 , factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. [ 6 ] : 207
The process of completing the square makes use of the algebraic identity x 2 + 2 h x + h 2 = ( x + h ) 2 , {\displaystyle x^{2}+2hx+h^{2}=(x+h)^{2},} which represents a well-defined algorithm that can be used to solve any quadratic equation. [ 6 ] : 207 Starting with a quadratic equation in standard form, ax 2 + bx + c = 0
We illustrate use of this algorithm by solving 2 x 2 + 4 x − 4 = 0 2 x 2 + 4 x − 4 = 0 {\displaystyle 2x^{2}+4x-4=0} x 2 + 2 x − 2 = 0 {\displaystyle \ x^{2}+2x-2=0} x 2 + 2 x = 2 {\displaystyle \ x^{2}+2x=2} x 2 + 2 x + 1 = 2 + 1 {\displaystyle \ x^{2}+2x+1=2+1} ( x + 1 ) 2 = 3 {\displaystyle \left(x+1\right)^{2}=3} x + 1 = ± 3 {\displaystyle \ x+1=\pm {\sqrt {3}}} x = − 1 ± 3 {\displaystyle \ x=-1\pm {\sqrt {3}}}
The plus–minus symbol "±" indicates that both x = − 1 + 3 {\textstyle x=-1+{\sqrt {3}}} and x = − 1 − 3 {\textstyle x=-1-{\sqrt {3}}} are solutions of the quadratic equation. [ 8 ]
Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula. [ 9 ] The mathematical proof will now be briefly summarized. [ 10 ] It can easily be seen, by polynomial expansion , that the following equation is equivalent to the quadratic equation: ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 . {\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.} Taking the square root of both sides, and isolating x , gives: x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax 2 + 2 bx + c = 0 or ax 2 − 2 bx + c = 0 , [ 11 ] where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.
A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.
A lesser known quadratic formula, as used in Muller's method , provides the same roots via the equation x = 2 c − b ± b 2 − 4 a c . {\displaystyle x={\frac {2c}{-b\pm {\sqrt {b^{2}-4ac}}}}.} This can be deduced from the standard quadratic formula by Vieta's formulas , which assert that the product of the roots is c / a . It also follows from dividing the quadratic equation by x 2 {\displaystyle x^{2}} giving c x − 2 + b x − 1 + a = 0 , {\displaystyle cx^{-2}+bx^{-1}+a=0,} solving this for x − 1 , {\displaystyle x^{-1},} and then inverting.
One property of this form is that it yields one valid root when a = 0 , while the other root contains division by zero, because when a = 0 , the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0 , the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0 .
When neither a nor c is zero, the equality between the standard quadratic formula and Muller's method, 2 c − b − b 2 − 4 a c = − b + b 2 − 4 a c 2 a , {\displaystyle {\frac {2c}{-b-{\sqrt {b^{2}-4ac}}}}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\,,} can be verified by cross multiplication , and similarly for the other choice of signs.
It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a , which is always possible since a is non-zero. This produces the reduced quadratic equation : [ 12 ]
x 2 + p x + q = 0 , {\displaystyle x^{2}+px+q=0,}
where p = b / a and q = c / a . This monic polynomial equation has the same solutions as the original.
The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is x = − p 2 ± ( p 2 ) 2 − q . {\displaystyle x=-{\frac {p}{2}}\pm {\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}\,.}
In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta : [ 13 ] Δ = b 2 − 4 a c . {\displaystyle \Delta =b^{2}-4ac.} A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
The function f ( x ) = ax 2 + bx + c is a quadratic function . [ 16 ] The graph of any quadratic function has the same general shape, which is called a parabola . The location and size of the parabola, and how it opens, depend on the values of a , b , and c . If a > 0 , the parabola has a minimum point and opens upward. If a < 0 , the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex . The x -coordinate of the vertex will be located at x = − b 2 a {\displaystyle \scriptstyle x={\tfrac {-b}{2a}}} , and the y -coordinate of the vertex may be found by substituting this x -value into the function. The y -intercept is located at the point (0, c ) .
The solutions of the quadratic equation ax 2 + bx + c = 0 correspond to the roots of the function f ( x ) = ax 2 + bx + c , since they are the values of x for which f ( x ) = 0 . If a , b , and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x - coordinates of the points where the graph touches the x -axis. If the discriminant is positive, the graph touches the x -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x -axis.
The term x − r {\displaystyle x-r} is a factor of the polynomial a x 2 + b x + c {\displaystyle ax^{2}+bx+c} if and only if r is a root of the quadratic equation a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.} It follows from the quadratic formula that a x 2 + b x + c = a ( x − − b + b 2 − 4 a c 2 a ) ( x − − b − b 2 − 4 a c 2 a ) . {\displaystyle ax^{2}+bx+c=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right).} In the special case b 2 = 4 ac where the quadratic has only one distinct root ( i.e. the discriminant is zero), the quadratic polynomial can be factored as a x 2 + b x + c = a ( x + b 2 a ) 2 . {\displaystyle ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}.}
The solutions of the quadratic equation a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} may be deduced from the graph of the quadratic function f ( x ) = a x 2 + b x + c , {\displaystyle f(x)=ax^{2}+bx+c,} which is a parabola .
If the parabola intersects the x -axis in two points, there are two real roots , which are the x -coordinates of these two points (also called x -intercept).
If the parabola is tangent to the x -axis, there is a double root, which is the x -coordinate of the contact point between the graph and parabola.
If the parabola does not intersect the x -axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be. [ 17 ]
Let h and k be respectively the x -coordinate and the y -coordinate of the vertex of the parabola (that is the point with maximal or minimal y -coordinate. The quadratic function may be rewritten y = a ( x − h ) 2 + k . {\displaystyle y=a(x-h)^{2}+k.} Let d be the distance between the point of y -coordinate 2 k on the axis of the parabola, and a point on the parabola with the same y -coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is h , and their imaginary part are ± d . That is, the roots are h + i d and h − i d , {\displaystyle h+id\quad {\text{and}}\quad h-id,} or in the case of the example of the figure 5 + 3 i and 5 − 3 i . {\displaystyle 5+3i\quad {\text{and}}\quad 5-3i.}
Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis , where real numbers are approximated by floating point numbers (called "reals" in many programming languages ). In this context, the quadratic formula is not completely stable .
This occurs when the roots have different order of magnitude , or, equivalently, when b 2 and b 2 − 4 ac are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, r , can be computed as ( c / a ) / R {\displaystyle (c/a)/R} where R is the root that is bigger in magnitude. This is equivalent to using the formula
x = − 2 c b ± b 2 − 4 a c {\displaystyle x={\frac {-2c}{b\pm {\sqrt {b^{2}-4ac}}}}}
using the plus sign if b > 0 {\displaystyle b>0} and the minus sign if b < 0. {\displaystyle b<0.}
A second form of cancellation can occur between the terms b 2 and 4 ac of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots. [ 11 ] [ 18 ]
The golden ratio is found as the positive solution of the quadratic equation x 2 − x − 1 = 0. {\displaystyle x^{2}-x-1=0.}
The equations of the circle and the other conic sections — ellipses , parabolas , and hyperbolas —are quadratic equations in two variables.
Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation.
The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.
Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.
The equation given by Fuss' theorem , giving the relation among the radius of a bicentric quadrilateral 's inscribed circle , the radius of its circumscribed circle , and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral .
Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.
In physics , for motion with constant acceleration a {\displaystyle a} , the displacement or position x {\displaystyle x} of a moving body can be expressed as a quadratic function of time t {\displaystyle t} given the initial position x 0 {\displaystyle x_{0}} and initial velocity v 0 {\displaystyle v_{0}} : x = x 0 + v 0 t + 1 2 a t 2 {\textstyle x=x_{0}+v_{0}t+{\frac {1}{2}}at^{2}} .
In chemistry , the pH of a solution of weak acid can be calculated from the negative base-10 logarithm of the positive root of a quadratic equation in terms of the acidity constant and the analytical concentration of the acid.
Babylonian mathematicians , as early as 2000 BC (displayed on Old Babylonian clay tablets ) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the Third Dynasty of Ur . [ 19 ] In modern notation, the problems typically involved solving a pair of simultaneous equations of the form: x + y = p , x y = q , {\displaystyle x+y=p,\ \ xy=q,} which is equivalent to the statement that x and y are the roots of the equation: [ 20 ] : 86 z 2 + q = p z . {\displaystyle z^{2}+q=pz.}
The steps given by Babylonian scribes for solving the above rectangle problem, in terms of x and y , were as follows:
In modern notation this means calculating x = p 2 + ( p 2 ) 2 − q {\displaystyle x={\frac {p}{2}}+{\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}} , which is equivalent to the modern day quadratic formula for the larger real root (if any) x = − b + b 2 − 4 a c 2 a {\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}} with a = 1 , b = − p , and c = q .
Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus , dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. [ 21 ] Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots. [ 22 ] [ 23 ] Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art , a Chinese treatise on mathematics. [ 23 ] [ 24 ] These early geometric methods do not appear to have had a general formula. Euclid , the Greek mathematician , produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work Arithmetica , the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive. [ 25 ]
In 628 AD, Brahmagupta , an Indian mathematician , gave in his book Brāhmasphuṭasiddhānta the first explicit (although still not completely general) solution of the quadratic equation ax 2 + bx = c as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." [ 26 ] This is equivalent to x = 4 a c + b 2 − b 2 a . {\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}.} The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as linear indeterminate equations (originally of type ax / c = y ). Muhammad ibn Musa al-Khwarizmi (9th century) developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process. [ 27 ] He also described the method of completing the square and recognized that the discriminant must be positive, [ 27 ] [ 28 ] : 230 which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. [ 28 ] : 234 While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions, [ 27 ] : 191 as well as irrational numbers as solutions. [ 29 ] Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root , cube root or fourth root ) as solutions to quadratic equations or as coefficients in an equation. [ 30 ] The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations. [ 31 ]
The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation. [ 32 ] His solution was largely based on Al-Khwarizmi's work. [ 27 ] The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi . [ 33 ] By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. [ 34 ] In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today.
Vieta's formulas (named after François Viète ) are the relations x 1 + x 2 = − b a , x 1 x 2 = c a {\displaystyle x_{1}+x_{2}=-{\frac {b}{a}},\quad x_{1}x_{2}={\frac {c}{a}}} between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation ( x − x 1 ) ( x − x 2 ) = x 2 − ( x 1 + x 2 ) x + x 1 x 2 = 0 {\displaystyle \left(x-x_{1}\right)\left(x-x_{2}\right)=x^{2}-\left(x_{1}+x_{2}\right)x+x_{1}x_{2}=0} with the equation x 2 + b a x + c a = 0. {\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0.}
The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex , the vertex's x -coordinate is located at the average of the roots (or intercepts). Thus the x -coordinate of the vertex is x V = x 1 + x 2 2 = − b 2 a . {\displaystyle x_{V}={\frac {x_{1}+x_{2}}{2}}=-{\frac {b}{2a}}.} The y -coordinate can be obtained by substituting the above result into the given quadratic equation, giving y V = − b 2 4 a + c = − b 2 − 4 a c 4 a . {\displaystyle y_{V}=-{\frac {b^{2}}{4a}}+c=-{\frac {b^{2}-4ac}{4a}}.} Also, these formulas for the vertex can be deduced directly from the formula (see Completing the square ) a x 2 + b x + c = a ( x + b 2 a ) 2 − b 2 − 4 a c 4 a . {\displaystyle ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}-4ac}{4a}}.}
For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If | x 2 | << | x 1 | , then x 1 + x 2 ≈ x 1 , and we have the estimate: x 1 ≈ − b a . {\displaystyle x_{1}\approx -{\frac {b}{a}}.} The second Vieta's formula then provides: x 2 = c a x 1 ≈ − c b . {\displaystyle x_{2}={\frac {c}{ax_{1}}}\approx -{\frac {c}{b}}.} These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b ), which causes round-off error in a numerical evaluation. The figure shows the difference between [ clarification needed ] (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient b increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.
This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see Step response ).
In the days before calculators, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis , that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots. [ 35 ] Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations.
It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution . Consider the following alternate form of the quadratic equation,
where the sign of the ± symbol is chosen so that a and c may both be positive. By substituting
and then multiplying through by cos 2 ( θ ) / c , we obtain
Introducing functions of 2 θ and rearranging, we obtain
where the subscripts n and p correspond, respectively, to the use of a negative or positive sign in equation [1] . Substituting the two values of θ n or θ p found from equations [4] or [5] into [2] gives the required roots of [1] . Complex roots occur in the solution based on equation [5] if the absolute value of sin 2 θ p exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone. [ 36 ] Calculating complex roots would require using a different trigonometric form. [ 37 ]
To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy: 4.16130 x 2 + 9.15933 x − 11.4207 = 0 {\displaystyle 4.16130x^{2}+9.15933x-11.4207=0}
If the quadratic equation a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} with real coefficients has two complex roots—the case where b 2 − 4 a c < 0 , {\displaystyle b^{2}-4ac<0,} requiring a and c to have the same sign as each other—then the solutions for the roots can be expressed in polar form as [ 38 ]
x 1 , x 2 = r ( cos θ ± i sin θ ) , {\displaystyle x_{1},\,x_{2}=r(\cos \theta \pm i\sin \theta ),}
where r = c a {\displaystyle r={\sqrt {\tfrac {c}{a}}}} and θ = cos − 1 ( − b 2 a c ) . {\displaystyle \theta =\cos ^{-1}\left({\tfrac {-b}{2{\sqrt {ac}}}}\right).}
The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method . The three coefficients a , b , c are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA. [ 39 ]
The Carlyle circle , named after Thomas Carlyle , has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis . [ 40 ] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons .
The formula and its derivation remain correct if the coefficients a , b and c are complex numbers , or more generally members of any field whose characteristic is not 2 . (In a field of characteristic 2, the element 2 a is zero and it is impossible to divide by it.)
The symbol ± b 2 − 4 a c {\displaystyle \pm {\sqrt {b^{2}-4ac}}} in the formula should be understood as "either of the two elements whose square is b 2 − 4 ac , if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2 . Even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.
In a field of characteristic 2 , the quadratic formula, which relies on 2 being a unit , does not hold. Consider the monic quadratic polynomial x 2 + b x + c {\displaystyle x^{2}+bx+c} over a field of characteristic 2 . If b = 0 , then the solution reduces to extracting a square root, so the solution is x = c {\displaystyle x={\sqrt {c}}} and there is only one root since − c = − c + 2 c = c . {\displaystyle -{\sqrt {c}}=-{\sqrt {c}}+2{\sqrt {c}}={\sqrt {c}}.} In summary, x 2 + c = ( x + c ) 2 . {\displaystyle \displaystyle x^{2}+c=(x+{\sqrt {c}})^{2}.} See quadratic residue for more information about extracting square roots in finite fields.
In the case that b ≠ 0 , there are two distinct roots, but if the polynomial is irreducible , they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R ( c ) of c to be a root of the polynomial x 2 + x + c , an element of the splitting field of that polynomial. One verifies that R ( c ) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax 2 + bx + c are b a R ( a c b 2 ) {\displaystyle {\frac {b}{a}}R\left({\frac {ac}{b^{2}}}\right)} and b a ( R ( a c b 2 ) + 1 ) . {\displaystyle {\frac {b}{a}}\left(R\left({\frac {ac}{b^{2}}}\right)+1\right).}
For example, let a denote a multiplicative generator of the group of units of F 4 , the Galois field of order four (thus a and a + 1 are roots of x 2 + x + 1 over F 4 . Because ( a + 1) 2 = a , a + 1 is the unique solution of the quadratic equation x 2 + a = 0 . On the other hand, the polynomial x 2 + ax + 1 is irreducible over F 4 , but it splits over F 16 , where it has the two roots ab and ab + a , where b is a root of x 2 + x + a in F 16 .
This is a special case of Artin–Schreier theory . | https://en.wikipedia.org/wiki/Quadratic_equation |
In mathematics , a quadratic form is a polynomial with terms all of degree two (" form " is another name for a homogeneous polynomial ). For example, 4 x 2 + 2 x y − 3 y 2 {\displaystyle 4x^{2}+2xy-3y^{2}}
is a quadratic form in the variables x and y . The coefficients usually belong to a fixed field K , such as the real or complex numbers, and one speaks of a quadratic form over K . Over the reals, a quadratic form is said to be definite if it takes the value zero only when all its variables are simultaneously zero; otherwise it is isotropic .
Quadratic forms occupy a central place in various branches of mathematics, including number theory , linear algebra , group theory ( orthogonal groups ), differential geometry (the Riemannian metric , the second fundamental form ), differential topology ( intersection forms of manifolds , especially four-manifolds ), Lie theory (the Killing form ), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form − x T Σ − 1 x {\displaystyle -\mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Sigma }}^{-1}\mathbf {x} } )
Quadratic forms are not to be confused with quadratic equations , which have only one variable and may include terms of degree less than two. A quadratic form is a specific instance of the more general concept of forms .
Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary , binary , and ternary and have the following explicit form: q ( x ) = a x 2 (unary) q ( x , y ) = a x 2 + b x y + c y 2 (binary) q ( x , y , z ) = a x 2 + b x y + c y 2 + d y z + e z 2 + f x z (ternary) {\displaystyle {\begin{aligned}q(x)&=ax^{2}&&{\textrm {(unary)}}\\q(x,y)&=ax^{2}+bxy+cy^{2}&&{\textrm {(binary)}}\\q(x,y,z)&=ax^{2}+bxy+cy^{2}+dyz+ez^{2}+fxz&&{\textrm {(ternary)}}\end{aligned}}}
where a , ..., f are the coefficients . [ 1 ]
The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers , rational numbers , or integers . In linear algebra , analytic geometry , and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field . In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring , frequently the integers Z or the p -adic integers Z p . [ 2 ] Binary quadratic forms have been extensively studied in number theory , in particular, in the theory of quadratic fields , continued fractions , and modular forms . The theory of integral quadratic forms in n variables has important applications to algebraic topology .
Using homogeneous coordinates , a non-zero quadratic form in n variables defines an ( n − 2) -dimensional quadric in the ( n − 1) -dimensional projective space . This is a basic construction in projective geometry . In this way one may visualize 3-dimensional real quadratic forms as conic sections .
An example is given by the three-dimensional Euclidean space and the square of the Euclidean norm expressing the distance between a point with coordinates ( x , y , z ) and the origin: q ( x , y , z ) = d ( ( x , y , z ) , ( 0 , 0 , 0 ) ) 2 = ‖ ( x , y , z ) ‖ 2 = x 2 + y 2 + z 2 . {\displaystyle q(x,y,z)=d((x,y,z),(0,0,0))^{2}=\left\|(x,y,z)\right\|^{2}=x^{2}+y^{2}+z^{2}.}
A closely related notion with geometric overtones is a quadratic space , which is a pair ( V , q ) , with V a vector space over a field K , and q : V → K a quadratic form on V . See § Definitions below for the definition of a quadratic form on a vector space.
The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares , which determines when an integer may be expressed in the form x 2 + y 2 , where x , y are integers. This problem is related to the problem of finding Pythagorean triples , which appeared in the second millennium BCE. [ 3 ]
In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta , which includes, among many other things, a study of equations of the form x 2 − ny 2 = c . He considered what is now called Pell's equation , x 2 − ny 2 = 1 , and found a method for its solution. [ 4 ] In Europe this problem was studied by Brouncker , Euler and Lagrange .
In 1801 Gauss published Disquisitiones Arithmeticae , a major portion of which was devoted to a complete theory of binary quadratic forms over the integers . Since then, the concept has been generalized, and the connections with quadratic number fields , the modular group , and other areas of mathematics have been further elucidated.
Any n × n matrix A determines a quadratic form q A in n variables by q A ( x 1 , … , x n ) = ∑ i = 1 n ∑ j = 1 n a i j x i x j = x T A x , {\displaystyle q_{A}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{x_{i}}{x_{j}}=\mathbf {x} ^{\mathsf {T}}A\mathbf {x} ,} where A = ( a ij ) .
Consider the case of quadratic forms in three variables x , y , z . The matrix A has the form A = [ a b c d e f g h k ] . {\displaystyle A={\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\end{bmatrix}}.}
The above formula gives q A ( x , y , z ) = a x 2 + e y 2 + k z 2 + ( b + d ) x y + ( c + g ) x z + ( f + h ) y z . {\displaystyle q_{A}(x,y,z)=ax^{2}+ey^{2}+kz^{2}+(b+d)xy+(c+g)xz+(f+h)yz.}
So, two different matrices define the same quadratic form if and only if they have the same elements on the diagonal and the same values for the sums b + d , c + g and f + h . In particular, the quadratic form q A is defined by a unique symmetric matrix A = [ a b + d 2 c + g 2 b + d 2 e f + h 2 c + g 2 f + h 2 k ] . {\displaystyle A={\begin{bmatrix}a&{\frac {b+d}{2}}&{\frac {c+g}{2}}\\{\frac {b+d}{2}}&e&{\frac {f+h}{2}}\\{\frac {c+g}{2}}&{\frac {f+h}{2}}&k\end{bmatrix}}.}
This generalizes to any number of variables as follows.
Given a quadratic form q A over the real numbers, defined by the matrix A = ( a ij ) ,
the matrix B = ( a i j + a j i 2 ) = 1 2 ( A + A T ) {\displaystyle B=\left({\frac {a_{ij}+a_{ji}}{2}}\right)={\frac {1}{2}}(A+A^{\text{T}})} is symmetric , defines the same quadratic form as A , and is the unique symmetric matrix that defines q A .
So, over the real numbers (and, more generally, over a field of characteristic different from two), there is a one-to-one correspondence between quadratic forms and symmetric matrices that determine them.
A fundamental problem is the classification of real quadratic forms under a linear change of variables .
Jacobi proved that, for every real quadratic form, there is an orthogonal diagonalization ; that is, an orthogonal change of variables that puts the quadratic form in a " diagonal form " λ 1 x ~ 1 2 + λ 2 x ~ 2 2 + ⋯ + λ n x ~ n 2 , {\displaystyle \lambda _{1}{\tilde {x}}_{1}^{2}+\lambda _{2}{\tilde {x}}_{2}^{2}+\cdots +\lambda _{n}{\tilde {x}}_{n}^{2},} where the associated symmetric matrix is diagonal . Moreover, the coefficients λ 1 , λ 2 , ..., λ n are determined uniquely up to a permutation . [ 5 ]
If the change of variables is given by an invertible matrix that is not necessarily orthogonal, one can suppose that all coefficients λ i are 0, 1, or −1. Sylvester's law of inertia states that the numbers of each 0, 1, and −1 are invariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The signature of the quadratic form is the triple ( n 0 , n + , n − ) , where these components count the number of 0s, number of 1s, and the number of −1s, respectively. Sylvester 's law of inertia shows that this is a well-defined quantity attached to the quadratic form.
The case when all λ i have the same sign is especially important: in this case the quadratic form is called positive definite (all 1) or negative definite (all −1). If none of the terms are 0, then the form is called nondegenerate ; this includes positive definite, negative definite, and isotropic quadratic form (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate bilinear form . A real vector space with an indefinite nondegenerate quadratic form of index ( p , q ) (denoting p 1s and q −1s) is often denoted as R p , q particularly in the physical theory of spacetime .
The discriminant of a quadratic form , concretely the class of the determinant of a representing matrix in K / ( K × ) 2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative". Zero corresponds to degenerate, while for a non-degenerate form it is the parity of the number of negative coefficients, (−1) n − .
These results are reformulated in a different way below.
Let q be a quadratic form defined on an n -dimensional real vector space. Let A be the matrix of the quadratic form q in a given basis. This means that A is a symmetric n × n matrix such that q ( v ) = x T A x , {\displaystyle q(v)=x^{\mathsf {T}}Ax,} where x is the column vector of coordinates of 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 ( n 0 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 . [ 6 ] 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, 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.
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 ) = a 2 q ( 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 x 1 , ..., x n and A = ( a ij ) 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 = ( x 1 , ..., x n ) 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 characteristic of K be different from 2. [ 7 ] The coefficient matrix A of q may be replaced by the symmetric matrix ( A + A T )/2 with the same quadratic form, so it may be assumed from the outset that A is symmetric. Moreover, a symmetric matrix A is uniquely determined by the corresponding quadratic form. Under an equivalence C , the symmetric matrix A of φ and the symmetric matrix B of ψ are related as follows: B = C T A C . {\displaystyle B=C^{\mathsf {T}}AC.}
The associated bilinear form of a quadratic form q is defined by b q ( x , y ) = 1 2 ( q ( x + y ) − q ( x ) − q ( y ) ) = x T A y = y T A x . {\displaystyle b_{q}(x,y)={\tfrac {1}{2}}(q(x+y)-q(x)-q(y))=x^{\mathsf {T}}Ay=y^{\mathsf {T}}Ax.}
Thus, b q is a symmetric bilinear form over K with matrix A . Conversely, any symmetric bilinear form b defines a quadratic form q ( x ) = b ( x , x ) , {\displaystyle q(x)=b(x,x),} and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.
Given an n -dimensional vector space V over a field K , a quadratic form on V is a function Q : V → K that has the following property: for some basis, the function q that maps the coordinates of v ∈ V to Q ( v ) is a quadratic form. In particular, if V = K n with its standard basis , one has q ( v 1 , … , v n ) = Q ( [ v 1 , … , v n ] ) for [ v 1 , … , v n ] ∈ K n . {\displaystyle q(v_{1},\ldots ,v_{n})=Q([v_{1},\ldots ,v_{n}])\quad {\text{for}}\quad [v_{1},\ldots ,v_{n}]\in K^{n}.}
The change of basis formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in V , although the quadratic form q depends on the choice of the basis.
A finite-dimensional vector space with a quadratic form is called a quadratic space .
The map Q is a homogeneous function of degree 2, which means that it has the property that, for all a in K and v in V : Q ( a v ) = a 2 Q ( v ) . {\displaystyle Q(av)=a^{2}Q(v).}
When the characteristic of K is not 2, the bilinear map B : V × V → K over K is defined: B ( v , w ) = 1 2 ( Q ( v + w ) − Q ( v ) − Q ( w ) ) . {\displaystyle B(v,w)={\tfrac {1}{2}}(Q(v+w)-Q(v)-Q(w)).} This bilinear form B is symmetric. That is, B ( x , y ) = B ( y , x ) for all x , y in V , and it determines Q : Q ( x ) = B ( x , x ) for all x in V .
When the characteristic of K is 2, so that 2 is not a unit , it is still possible to use a quadratic form to define a symmetric bilinear form B ′( x , y ) = Q ( x + y ) − Q ( x ) − Q ( y ) . However, Q ( x ) can no longer be recovered from this B ′ in the same way, since B ′( x , x ) = 0 for all x (and is thus alternating). [ 8 ] Alternatively, there always exists a bilinear form B ″ (not in general either unique or symmetric) such that B ″( x , x ) = Q ( x ) .
The pair ( V , Q ) consisting of a finite-dimensional vector space V over K and a quadratic map Q from V to K is called a quadratic space , and B as defined here is the associated symmetric bilinear form of Q . The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, Q is also called a quadratic form.
Two n -dimensional quadratic spaces ( V , Q ) and ( V ′, Q ′) are isometric if there exists an invertible linear transformation T : V → V ′ ( isometry ) such that Q ( v ) = Q ′ ( T v ) for all v ∈ V . {\displaystyle Q(v)=Q'(Tv){\text{ for all }}v\in V.}
The isometry classes of n -dimensional quadratic spaces over K correspond to the equivalence classes of n -ary quadratic forms over K .
Let R be a commutative ring , M be an R - module , and b : M × M → R be an R -bilinear form. [ 9 ] A mapping q : M → R : v ↦ b ( v , v ) is the associated quadratic form of b , and B : M × M → R : ( u , v ) ↦ q ( u + v ) − q ( u ) − q ( v ) is the polar form of q .
A quadratic form q : M → R may be characterized in the following equivalent ways:
Two elements v and w of V are called orthogonal if B ( v , w ) = 0 . The kernel of a bilinear form B consists of the elements that are orthogonal to every element of V . Q is non-singular if the kernel of its associated bilinear form is {0} . If there exists a non-zero v in V such that Q ( v ) = 0 , the quadratic form Q is isotropic , otherwise it is definite . This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of Q to a subspace U of V is identically zero, then U is totally singular .
The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q : that is, the group of isometries of ( V , Q ) into itself.
If a quadratic space ( A , Q ) has a product so that A is an algebra over a field , and satisfies ∀ x , y ∈ A Q ( x y ) = Q ( x ) Q ( y ) , {\displaystyle \forall x,y\in A\quad Q(xy)=Q(x)Q(y),} then it is a composition algebra .
Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form q ( x ) = a 1 x 1 2 + a 2 x 2 2 + ⋯ + a n x n 2 . {\displaystyle q(x)=a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+\cdots +a_{n}x_{n}^{2}.}
Such a diagonal form is often denoted by ⟨ a 1 , ..., a n ⟩ .
Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.
Using Cartesian coordinates in three dimensions, let x = ( x , y , z ) T , and let A be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation x T A x + b T x = 1 depends on the eigenvalues of the matrix A .
If all eigenvalues of A are non-zero, then the solution set is an ellipsoid or a hyperboloid . [ citation needed ] If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is an imaginary ellipsoid (we get the equation of an ellipsoid but with imaginary radii); if some eigenvalues are positive and some are negative, then it is a hyperboloid.
If there exist one or more eigenvalues λ i = 0 , then the shape depends on the corresponding b i . If the corresponding b i ≠ 0 , then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding b i = 0 , then the dimension i degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of b . When the solution set is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.
Quadratic forms over the ring of integers are called integral quadratic forms , whereas the corresponding modules are quadratic lattices (sometimes, simply lattices ). They play an important role in number theory and topology .
An integral quadratic form has integer coefficients, such as x 2 + xy + y 2 ; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R ), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ , meaning Q ( x , y ) ∈ Z if x , y ∈ Λ .
This is the current use of the term; in the past it was sometimes used differently, as detailed below.
Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean:
This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).
In "twos in", binary quadratic forms are of the form ax 2 + 2 bxy + cy 2 , represented by the symmetric matrix ( a b b c ) {\displaystyle {\begin{pmatrix}a&b\\b&c\end{pmatrix}}} This is the convention Gauss uses in Disquisitiones Arithmeticae .
In "twos out", binary quadratic forms are of the form ax 2 + bxy + cy 2 , represented by the symmetric matrix ( a b / 2 b / 2 c ) . {\displaystyle {\begin{pmatrix}a&b/2\\b/2&c\end{pmatrix}}.}
Several points of view mean that twos out has been adopted as the standard convention. Those include:
An integral quadratic form whose image consists of all the positive integers is sometimes called universal . Lagrange's four-square theorem shows that w 2 + x 2 + y 2 + z 2 is universal. Ramanujan generalized this aw 2 + bx 2 + cy 2 + dz 2 and found 54 multisets { a , b , c , d } that can each generate all positive integers, namely,
There are also forms whose image consists of all but one of the positive integers. For example, {1, 2, 5, 5} has 15 as the exception. Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15. | https://en.wikipedia.org/wiki/Quadratic_form |
In elementary algebra , the quadratic formula is a closed-form expression describing the solutions of a quadratic equation . Other ways of solving quadratic equations, such as completing the square , yield the same solutions.
Given a general quadratic equation of the form a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} , with x {\displaystyle x} representing an unknown, and coefficients a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} representing known real or complex numbers with a ≠ 0 {\displaystyle a\neq 0} , the values of x {\displaystyle x} satisfying the equation, called the roots or zeros , can be found using the quadratic formula,
x = − b ± b 2 − 4 a c 2 a , {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},}
where the plus–minus symbol " ± {\displaystyle \pm } " indicates that the equation has two roots. [ 1 ] Written separately, these are:
x 1 = − b + b 2 − 4 a c 2 a , x 2 = − b − b 2 − 4 a c 2 a . {\displaystyle x_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}},\qquad x_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}.}
The quantity Δ = b 2 − 4 a c {\displaystyle \textstyle \Delta =b^{2}-4ac} is known as the discriminant of the quadratic equation. [ 2 ] If the coefficients a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are real numbers then when Δ > 0 {\displaystyle \Delta >0} , the equation has two distinct real roots; when Δ = 0 {\displaystyle \Delta =0} , the equation has one repeated real root; and when Δ < 0 {\displaystyle \Delta <0} , the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other.
Geometrically, the roots represent the x {\displaystyle x} values at which the graph of the quadratic function y = a x 2 + b x + c {\displaystyle \textstyle y=ax^{2}+bx+c} , a parabola , crosses the x {\displaystyle x} -axis: the graph's x {\displaystyle x} -intercepts. [ 3 ] The quadratic formula can also be used to identify the parabola's axis of symmetry . [ 4 ]
The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} . [ 5 ] [ 6 ] [ 7 ] [ 8 ] The idea is to manipulate the equation into the form ( x + k ) 2 = s {\displaystyle \textstyle (x+k)^{2}=s} for some expressions k {\displaystyle k} and s {\displaystyle s} written in terms of the coefficients; take the square root of both sides; and then isolate x {\displaystyle x} .
We start by dividing the equation by the quadratic coefficient a {\displaystyle a} , which is allowed because a {\displaystyle a} is non-zero. Afterwards, we subtract the constant term c / a {\displaystyle c/a} to isolate it on the right-hand side:
a x 2 | + b x + c = 0 x 2 + b a x + c a = 0 x 2 + b a x = − c a . {\displaystyle {\begin{aligned}ax^{2{\vphantom {|}}}+bx+c&=0\\[3mu]x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}&=0\\[3mu]x^{2}+{\frac {b}{a}}x&=-{\frac {c}{a}}.\end{aligned}}}
The left-hand side is now of the form x 2 + 2 k x {\displaystyle \textstyle x^{2}+2kx} , and we can "complete the square" by adding a constant k 2 {\displaystyle \textstyle k^{2}} to obtain a squared binomial x 2 + 2 k x + k 2 = {\displaystyle \textstyle x^{2}+2kx+k^{2}={}} ( x + k ) 2 {\displaystyle \textstyle (x+k)^{2}} . In this example we add ( b / 2 a ) 2 {\displaystyle \textstyle (b/2a)^{2}} to both sides so that the left-hand side can be factored (see the figure):
x 2 + 2 ( b 2 a ) x + ( b 2 a ) 2 = − c a + ( b 2 a ) 2 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 . {\displaystyle {\begin{aligned}x^{2}+2\left({\frac {b}{2a}}\right)x+\left({\frac {b}{2a}}\right)^{2}&=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2}\\[5mu]\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}-4ac}{4a^{2}}}.\end{aligned}}}
Because the left-hand side is now a perfect square, we can easily take the square root of both sides:
x + b 2 a = ± b 2 − 4 a c 2 a . {\displaystyle x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}.}
Finally, subtracting b / 2 a {\displaystyle b/2a} from both sides to isolate x {\displaystyle x} produces the quadratic formula:
x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
The quadratic formula can equivalently be written using various alternative expressions, for instance
x = − b 2 a ± ( b 2 a ) 2 − c a , {\displaystyle x=-{\frac {b}{2a}}\pm {\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}},}
which can be derived by first dividing a quadratic equation by 2 a {\displaystyle 2a} , resulting in 1 2 x 2 + b 2 a x + c 2 a = 0 {\displaystyle \textstyle {\tfrac {1}{2}}x^{2}+{\tfrac {b}{2a}}x+{\tfrac {c}{2a}}=0} , then substituting the new coefficients into the standard quadratic formula. Because this variant allows re-use of the intermediately calculated quantity b 2 a {\displaystyle {\tfrac {b}{2a}}} , it can slightly reduce the arithmetic involved.
A lesser known quadratic formula, first mentioned by Giulio Fagnano , [ 9 ] describes the same roots via an equation with the square root in the denominator (assuming c ≠ 0 {\displaystyle c\neq 0} ):
x = 2 c − b ∓ b 2 − 4 a c . {\displaystyle x={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac}}}}.}
Here the minus–plus symbol " ∓ {\displaystyle \mp } " indicates that the two roots of the quadratic equation, in the same order as the standard quadratic formula, are
x 1 = 2 c − b − b 2 − 4 a c , x 2 = 2 c − b + b 2 − 4 a c . {\displaystyle x_{1}={\frac {2c}{-b-{\sqrt {b^{2}-4ac}}}},\qquad x_{2}={\frac {2c}{-b+{\sqrt {b^{2}-4ac}}}}.}
This variant has been jokingly called the "citardauq" formula ("quadratic" spelled backwards). [ 10 ]
When − b {\displaystyle -b} has the opposite sign as either + b 2 − 4 a c {\displaystyle \textstyle +{\sqrt {b^{2}-4ac}}} or − b 2 − 4 a c {\displaystyle \textstyle -{\sqrt {b^{2}-4ac}}} , subtraction can cause catastrophic cancellation , resulting in poor accuracy in numerical calculations; choosing between the version of the quadratic formula with the square root in the numerator or denominator depending on the sign of b {\displaystyle b} can avoid this problem. See § Numerical calculation below.
This version of the quadratic formula is used in Muller's method for finding the roots of general functions. It can be derived from the standard formula from the identity x 1 x 2 = c / a {\displaystyle x_{1}x_{2}=c/a} , one of Vieta's formulas . Alternately, it can be derived by dividing each side of the equation a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} by x 2 {\displaystyle \textstyle x^{2}} to get c x − 2 + b x − 1 + a = 0 {\displaystyle \textstyle cx^{-2}+bx^{-1}+a=0} , applying the standard formula to find the two roots x − 1 {\displaystyle \textstyle x^{-1}\!} , and then taking the reciprocal to find the roots x {\displaystyle x} of the original equation.
Any generic method or algorithm for solving quadratic equations can be applied to an equation with symbolic coefficients and used to derive some closed-form expression equivalent to the quadratic formula. Alternative methods are sometimes simpler than completing the square, and may offer interesting insight into other areas of mathematics.
Instead of dividing by a {\displaystyle a} to isolate x 2 {\displaystyle \textstyle x^{2}\!} , it can be slightly simpler to multiply by 4 a {\displaystyle 4a} instead to produce ( 2 a x ) 2 {\displaystyle \textstyle (2ax)^{2}\!} , which allows us to complete the square without need for fractions. Then the steps of the derivation are: [ 11 ]
Applying this method to a generic quadratic equation with symbolic coefficients yields the quadratic formula:
a x 2 + b x + c = 0 4 a 2 x 2 + 4 a b x + 4 a c = 0 4 a 2 x 2 + 4 a b x + b 2 = b 2 − 4 a c ( 2 a x + b ) 2 = b 2 − 4 a c 2 a x + b = ± b 2 − 4 a c x = − b ± b 2 − 4 a c 2 a . ) {\displaystyle {\begin{aligned}ax^{2}+bx+c&=0\\[3mu]4a^{2}x^{2}+4abx+4ac&=0\\[3mu]4a^{2}x^{2}+4abx+b^{2}&=b^{2}-4ac\\[3mu](2ax+b)^{2}&=b^{2}-4ac\\[3mu]2ax+b&=\pm {\sqrt {b^{2}-4ac}}\\[5mu]x&={\dfrac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.{\vphantom {\bigg )}}\end{aligned}}}
This method for completing the square is ancient and was known to the 8th–9th century Indian mathematician Śrīdhara . [ 12 ] Compared with the modern standard method for completing the square, this alternate method avoids fractions until the last step and hence does not require a rearrangement after step 3 to obtain a common denominator in the right side. [ 11 ]
Another derivation uses a change of variables to eliminate the linear term. Then the equation takes the form u 2 = s {\displaystyle \textstyle u^{2}=s} in terms of a new variable u {\displaystyle u} and some constant expression s {\displaystyle s} , whose roots are then u = ± s {\displaystyle u=\pm {\sqrt {s}}} .
By substituting x = u − b 2 a {\displaystyle x=u-{\tfrac {b}{2a}}} into a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} , expanding the products and combining like terms, and then solving for u 2 {\displaystyle \textstyle u^{2}\!} , we have:
a ( u − b 2 a ) 2 + b ( u − b 2 a ) + c = 0 a ( u 2 − b a u + b 2 4 a 2 ) + b ( u − b 2 a ) + c = 0 a u 2 − b u + b 2 4 a + b u − b 2 2 a + c = 0 a u 2 + 4 a c − b 2 4 a = 0 u 2 = b 2 − 4 a c 4 a 2 . {\displaystyle {\begin{aligned}a\left(u-{\frac {b}{2a}}\right)^{2}+b\left(u-{\frac {b}{2a}}\right)+c&=0\\[5mu]a\left(u^{2}-{\frac {b}{a}}u+{\frac {b^{2}}{4a^{2}}}\right)+b\left(u-{\frac {b}{2a}}\right)+c&=0\\[5mu]au^{2}-bu+{\frac {b^{2}}{4a}}+bu-{\frac {b^{2}}{2a}}+c&=0\\[5mu]au^{2}+{\frac {4ac-b^{2}}{4a}}&=0\\[5mu]u^{2}&={\frac {b^{2}-4ac}{4a^{2}}}.\end{aligned}}}
Finally, after taking a square root of both sides and substituting the resulting expression for u {\displaystyle u} back into x = u − b 2 a , {\displaystyle x=u-{\tfrac {b}{2a}},} the familiar quadratic formula emerges:
x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
The following method was used by many historical mathematicians: [ 13 ]
Let the roots of the quadratic equation a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} be α {\displaystyle \alpha } and β {\displaystyle \beta } . The derivation starts from an identity for the square of a difference (valid for any two complex numbers), of which we can take the square root on both sides:
( α − β ) 2 = ( α + β ) 2 − 4 α β α − β = ± ( α + β ) 2 − 4 α β . {\displaystyle {\begin{aligned}(\alpha -\beta )^{2}&=(\alpha +\beta )^{2}-4\alpha \beta \\[3mu]\alpha -\beta &=\pm {\sqrt {(\alpha +\beta )^{2}-4\alpha \beta }}.\end{aligned}}}
Since the coefficient a ≠ 0 {\displaystyle a\neq 0} , we can divide the quadratic equation by a {\displaystyle a} to obtain a monic polynomial with the same roots. Namely,
x 2 + b a x + c a = ( x − α ) ( x − β ) = x 2 − ( α + β ) x + α β . {\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=(x-\alpha )(x-\beta )=x^{2}-(\alpha +\beta )x+\alpha \beta .}
This implies that the sum α + β = − b a {\displaystyle \alpha +\beta =-{\tfrac {b}{a}}} and the product α β = c a {\displaystyle \alpha \beta ={\tfrac {c}{a}}} . Thus the identity can be rewritten:
α − β = ± ( − b a ) 2 − 4 c a = ± b 2 − 4 a c a . {\displaystyle \alpha -\beta =\pm {\sqrt {\left(-{\frac {b}{a}}\right)^{2}-4{\frac {c}{a}}}}=\pm {\frac {\sqrt {b^{2}-4ac}}{a}}.}
Therefore,
α = 1 2 ( α + β ) + 1 2 ( α − β ) = − b 2 a ± b 2 − 4 a c 2 a , β = 1 2 ( α + β ) − 1 2 ( α − β ) = − b 2 a ∓ b 2 − 4 a c 2 a . {\displaystyle {\begin{aligned}\alpha &={\tfrac {1}{2}}(\alpha +\beta )+{\tfrac {1}{2}}(\alpha -\beta )=-{\frac {b}{2a}}\pm {\frac {\sqrt {b^{2}-4ac}}{2a}},\\[10mu]\beta &={\tfrac {1}{2}}(\alpha +\beta )-{\tfrac {1}{2}}(\alpha -\beta )=-{\frac {b}{2a}}\mp {\frac {\sqrt {b^{2}-4ac}}{2a}}.\end{aligned}}}
The two possibilities for each of α {\displaystyle \alpha } and β {\displaystyle \beta } are the same two roots in opposite order, so we can combine them into the standard quadratic equation: x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents , [ 14 ] which is an early part of Galois theory . [ 15 ] This method can be generalized to give the roots of cubic polynomials and quartic polynomials , and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group .
This approach focuses on the roots themselves rather than algebraically rearranging the original equation. Given a monic quadratic polynomial x 2 + p x + q {\displaystyle \textstyle x^{2}+px+q} assume that α {\displaystyle \alpha } and β {\displaystyle \beta } are the two roots. So the polynomial factors as
x 2 + p x + q = ( x − α ) ( x − β ) = x 2 − ( α + β ) x + α β {\displaystyle {\begin{aligned}x^{2}+px+q&=(x-\alpha )(x-\beta )\\[3mu]&=x^{2}-(\alpha +\beta )x+\alpha \beta \end{aligned}}}
which implies p = − ( α + β ) {\displaystyle p=-(\alpha +\beta )} and q = α β {\displaystyle q=\alpha \beta } .
Since multiplication and addition are both commutative , exchanging the roots α {\displaystyle \alpha } and β {\displaystyle \beta } will not change the coefficients p {\displaystyle p} and q {\displaystyle q} : one can say that p {\displaystyle p} and q {\displaystyle q} are symmetric polynomials in α {\displaystyle \alpha } and β {\displaystyle \beta } . Specifically, they are the elementary symmetric polynomials – any symmetric polynomial in α {\displaystyle \alpha } and β {\displaystyle \beta } can be expressed in terms of α + β {\displaystyle \alpha +\beta } and α β {\displaystyle \alpha \beta } instead.
The Galois theory approach to analyzing and solving polynomials is to ask whether, given coefficients of a polynomial each of which is a symmetric function in the roots, one can "break" the symmetry and thereby recover the roots. Using this approach, solving a polynomial of degree n {\displaystyle n} is related to the ways of rearranging (" permuting ") n {\displaystyle n} terms, called the symmetric group on n {\displaystyle n} letters and denoted S n {\displaystyle S_{n}} . For the quadratic polynomial, the only ways to rearrange two roots are to either leave them be or to transpose them, so solving a quadratic polynomial is simple.
To find the roots α {\displaystyle \alpha } and β {\displaystyle \beta } , consider their sum and difference:
r 1 = α + β , r 2 = α − β . {\displaystyle r_{1}=\alpha +\beta ,\quad r_{2}=\alpha -\beta .}
These are called the Lagrange resolvents of the polynomial, from which the roots can be recovered as
α = 1 2 ( r 1 + r 2 ) , β = 1 2 ( r 1 − r 2 ) . {\displaystyle \alpha ={\tfrac {1}{2}}(r_{1}+r_{2}),\quad \beta ={\tfrac {1}{2}}(r_{1}-r_{2}).}
Because r 1 = α + β {\displaystyle r_{1}=\alpha +\beta } is a symmetric function in α {\displaystyle \alpha } and β {\displaystyle \beta } , it can be expressed in terms of p {\displaystyle p} and q , {\displaystyle q,} specifically r 1 = − p {\displaystyle r_{1}=-p} as described above. However, r 2 = α − β {\displaystyle r_{2}=\alpha -\beta } is not symmetric, since exchanging α {\displaystyle \alpha } and β {\displaystyle \beta } yields the additive inverse − r 2 = β − α {\displaystyle -r_{2}=\beta -\alpha } . So r 2 {\displaystyle r_{2}} cannot be expressed in terms of the symmetric polynomials. However, its square r 2 2 = ( α − β ) 2 {\displaystyle \textstyle r_{2}^{2}=(\alpha -\beta )^{2}} is symmetric in the roots, expressible in terms of p {\displaystyle p} and q {\displaystyle q} . Specifically r 2 2 = ( α − β ) 2 = {\displaystyle \textstyle r_{2}^{2}=(\alpha -\beta )^{2}={}} ( α + β ) 2 − 4 α β = {\displaystyle \textstyle (\alpha +\beta )^{2}-4\alpha \beta ={}} p 2 − 4 q {\displaystyle \textstyle p^{2}-4q} , which implies r 2 = ± p 2 − 4 q {\displaystyle \textstyle r_{2}=\pm {\sqrt {p^{2}-4q}}} . Taking the positive root "breaks" the symmetry, resulting in
r 1 = − p , r 2 = p 2 − 4 q {\displaystyle r_{1}=-p,\qquad r_{2}={\textstyle {\sqrt {p^{2}-4q}}}}
from which the roots α {\displaystyle \alpha } and β {\displaystyle \beta } are recovered as
x = 1 2 ( r 1 ± r 2 ) = 1 2 ( − p ± p 2 − 4 q ) {\displaystyle x={\tfrac {1}{2}}(r_{1}\pm r_{2})={\tfrac {1}{2}}{\bigl (}{-p}\pm {\textstyle {\sqrt {p^{2}-4q}}}\,{\bigr )}}
which is the quadratic formula for a monic polynomial.
Substituting p = b / a {\displaystyle p=b/a} , q = c / a {\displaystyle q=c/a} yields the usual expression for an arbitrary quadratic polynomial. The resolvents can be recognized as
1 2 r 1 = − 1 2 p = − b 2 a , r 2 2 = p 2 − 4 q = b 2 − 4 a c a 2 , {\displaystyle {\tfrac {1}{2}}r_{1}=-{\tfrac {1}{2}}p=-{\frac {b}{2a}},\qquad r_{2}^{2}=p_{2}-4q={\frac {b^{2}-4ac}{a^{2}}},}
respectively the vertex and the discriminant of the monic polynomial.
A similar but more complicated method works for cubic equations , which have three resolvents and a quadratic equation (the "resolving polynomial") relating r 2 {\displaystyle r_{2}} and r 3 {\displaystyle r_{3}} , which one can solve by the quadratic equation, and similarly for a quartic equation ( degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [ 14 ] The same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and, in fact, solutions to quintic equations in general cannot be expressed using only roots.
The quadratic formula is exactly correct when performed using the idealized arithmetic of real numbers , but when approximate arithmetic is used instead, for example pen-and-paper arithmetic carried out to a fixed number of decimal places or the floating-point binary arithmetic available on computers, the limitations of the number representation can lead to substantially inaccurate results unless great care is taken in the implementation. Specific difficulties include catastrophic cancellation in computing the sum − b ± Δ {\displaystyle \textstyle -b\pm {\sqrt {\Delta }}} if b ≈ ± Δ {\displaystyle \textstyle b\approx \pm {\sqrt {\Delta }}} ; catastrophic calculation in computing the discriminant Δ = b 2 − 4 a c {\displaystyle \textstyle \Delta =b^{2}-4ac} itself in cases where b 2 ≈ 4 a c {\displaystyle \textstyle b^{2}\approx 4ac} ; degeneration of the formula when a {\displaystyle a} , b {\displaystyle b} , or c {\displaystyle c} is represented as zero or infinite; and possible overflow or underflow when multiplying or dividing extremely large or small numbers, even in cases where the roots can be accurately represented. [ 16 ] [ 17 ]
Catastrophic cancellation occurs when two numbers which are approximately equal are subtracted. While each of the numbers may independently be representable to a certain number of digits of precision, the identical leading digits of each number cancel, resulting in a difference of lower relative precision. When b ≈ Δ {\displaystyle \textstyle b\approx {\sqrt {\Delta }}} , evaluation of − b + Δ {\displaystyle \textstyle -b+{\sqrt {\Delta }}} causes catastrophic cancellation, as does the evaluation of − b − Δ {\displaystyle \textstyle -b-{\sqrt {\Delta }}} when b ≈ − Δ {\displaystyle \textstyle b\approx -{\sqrt {\Delta }}} . When using the standard quadratic formula, calculating one of the two roots always involves addition, which preserves the working precision of the intermediate calculations, while calculating the other root involves subtraction, which compromises it. Therefore, naïvely following the standard quadratic formula often yields one result with less relative precision than expected. Unfortunately, introductory algebra textbooks typically do not address this problem, even though it causes students to obtain inaccurate results in other school subjects such as introductory chemistry. [ 18 ]
For example, if trying to solve the equation x 2 − 1634 x + 2 = 0 {\displaystyle \textstyle x^{2}-1634x+2=0} using a pocket calculator, the result of the quadratic formula x = 817 ± 667 487 {\displaystyle \textstyle x=817\pm {\sqrt {667\,487}}} might be approximately calculated as: [ 19 ]
x 1 = 817 + 816.998 776 0 = 1.633 998 776 × 10 3 , x 2 = 817 − 816.998 776 0 = 1.224 × 10 − 3 . {\displaystyle {\begin{alignedat}{3}x_{1}&=817+816.998\,776\,0&&=1.633\,998\,776\times 10^{3},\\x_{2}&=817-816.998\,776\,0&&=1.224\times 10^{-3}.\end{alignedat}}}
Even though the calculator used ten decimal digits of precision for each step, calculating the difference between two approximately equal numbers has yielded a result for x 2 {\displaystyle x_{2}} with only four correct digits.
One way to recover an accurate result is to use the identity x 1 x 2 = c / a {\displaystyle x_{1}x_{2}=c/a} . In this example x 2 {\displaystyle x_{2}} can be calculated as x 2 = 2 / x 1 = {\displaystyle x_{2}=2/x_{1}={}} 1.223 991 125 × 10 − 3 {\displaystyle 1.223\,991\,125\times 10^{-3}\!} , which is correct to the full ten digits. Another more or less equivalent approach is to use the version of the quadratic formula with the square root in the denominator to calculate one of the roots (see § Square root in the denominator above).
Practical computer implementations of the solution of quadratic equations commonly choose which formula to use for each root depending on the sign of b {\displaystyle b} . [ 20 ]
These methods do not prevent possible overflow or underflow of the floating-point exponent in computing b 2 {\displaystyle \textstyle b^{2}} or 4 a c {\displaystyle 4ac} , which can lead to numerically representable roots not being computed accurately. A more robust but computationally expensive strategy is to start with the substitution x = − u sgn ( b ) | c | / | a | {\displaystyle \textstyle x=-u\operatorname {sgn}(b){\sqrt {\vert c\vert }}{\big /}\!{\sqrt {\vert a\vert }}} , turning the quadratic equation into
u 2 − 2 | b | 2 | a | | c | u + sgn ( c ) = 0 , {\displaystyle u^{2}-2{\frac {|b|}{2{\sqrt {|a|}}{\sqrt {|c|}}}}u+\operatorname {sgn}(c)=0,}
where sgn {\displaystyle \operatorname {sgn} } is the sign function . Letting d = | b | / 2 | a | | c | {\displaystyle \textstyle d=\vert b\vert {\big /}2{\sqrt {\vert a\vert }}{\sqrt {\vert c\vert }}} , this equation has the form u 2 − 2 d u ± 1 = 0 {\displaystyle \textstyle u^{2}-2du\pm 1=0} , for which one solution is u 1 = d + d 2 ∓ 1 {\displaystyle \textstyle u_{1}=d+{\sqrt {d^{2}\mp 1}}} and the other solution is u 2 = ± 1 / u 1 {\displaystyle \textstyle u_{2}=\pm 1/u_{1}} . The roots of the original equation are then x 1 = − sgn ( b ) ( | c | / | a | ) u 1 {\displaystyle \textstyle x_{1}=-\operatorname {sgn}(b){\bigl (}{\sqrt {\vert c\vert }}{\big /}\!{\sqrt {\vert a\vert }}~\!{\bigr )}u_{1}} and x 2 = − sgn ( b ) ( | c | / | a | ) u 2 {\displaystyle \textstyle x_{2}=-\operatorname {sgn}(b){\bigl (}{\sqrt {\vert c\vert }}{\big /}\!{\sqrt {\vert a\vert }}~\!{\bigr )}u_{2}} . [ 21 ] [ 22 ]
With additional complication the expense and extra rounding of the square roots can be avoided by approximating them as powers of two, while still avoiding exponent overflow for representable roots. [ 17 ]
The earliest methods for solving quadratic equations were geometric. Babylonian cuneiform tablets contain problems reducible to solving quadratic equations. [ 23 ] The Egyptian Berlin Papyrus , dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. [ 24 ]
The Greek mathematician Euclid (circa 300 BC) used geometric methods to solve quadratic equations in Book 2 of his Elements , an influential mathematical treatise [ 25 ] Rules for quadratic equations appear in the Chinese The Nine Chapters on the Mathematical Art circa 200 BC. [ 26 ] [ 27 ] In his work Arithmetica , the Greek mathematician Diophantus (circa 250 AD) solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid. [ 25 ] His solution gives only one root, even when both roots are positive. [ 28 ]
The Indian mathematician Brahmagupta included a generic method for finding one root of a quadratic equation in his treatise Brāhmasphuṭasiddhānta (circa 628 AD), written out in words in the style of that time. [ 29 ] [ 30 ] His solution of the quadratic equation a x 2 + b x = c {\displaystyle \textstyle ax^{2}+bx=c} was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." [ 31 ] In modern notation, this can be written x = ( c ⋅ 4 a + b 2 − b ) / 2 a {\displaystyle \textstyle x={\bigl (}{\sqrt {c\cdot 4a+b^{2}}}-b{\bigr )}{\big /}2a} . The Indian mathematician Śrīdhara (8th–9th century) came up with a similar algorithm for solving quadratic equations in a now-lost work on algebra quoted by Bhāskara II . [ 32 ] The modern quadratic formula is sometimes called Sridharacharya's formula in India and Bhaskara's formula in Brazil. [ 33 ]
The 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī solved quadratic equations algebraically. [ 34 ] The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. [ 35 ] In 1637 René Descartes published La Géométrie containing special cases of the quadratic formula in the form we know today. [ 36 ]
In terms of coordinate geometry, an axis-aligned parabola is a curve whose ( x , y ) {\displaystyle (x,y)} -coordinates are the graph of a second-degree polynomial, of the form y = a x 2 + b x + c {\displaystyle \textstyle y=ax^{2}+bx+c} , where a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are real-valued constant coefficients with a ≠ 0 {\displaystyle a\neq 0} .
Geometrically, the quadratic formula defines the points ( x , 0 ) {\displaystyle (x,0)} on the graph, where the parabola crosses the x {\displaystyle x} -axis. Furthermore, it can be separated into two terms,
x = − b ± b 2 − 4 a c 2 a = − b 2 a ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}=-{\frac {b}{2a}}\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}.}
The first term describes the axis of symmetry , the line x = − b 2 a {\displaystyle x=-{\tfrac {b}{2a}}} . The second term, b 2 − 4 a c / 2 a {\displaystyle \textstyle {\sqrt {b^{2}-4ac}}{\big /}2a} , gives the distance the roots are away from the axis of symmetry.
If the parabola's vertex is on the x {\displaystyle x} -axis, then the corresponding equation has a single repeated root on the line of symmetry, and this distance term is zero; algebraically, the discriminant b 2 − 4 a c = 0 {\displaystyle \textstyle b^{2}-4ac=0} .
If the discriminant is positive, then the vertex is not on the x {\displaystyle x} -axis but the parabola opens in the direction of the x {\displaystyle x} -axis, crossing it twice, so the corresponding equation has two real roots. If the discriminant is negative, then the parabola opens in the opposite direction, never crossing the x {\displaystyle x} -axis, and the equation has no real roots; in this case the two complex-valued roots will be complex conjugates whose real part is the x {\displaystyle x} value of the axis of symmetry.
If the constants a {\displaystyle a} , b {\displaystyle b} , and/or c {\displaystyle c} are not unitless then the quantities x {\displaystyle x} and b a {\displaystyle {\tfrac {b}{a}}} must have the same units, because the terms a x 2 {\displaystyle \textstyle ax^{2}} and b x {\displaystyle bx} agree on their units. By the same logic, the coefficient c {\displaystyle c} must have the same units as b 2 a {\displaystyle {\tfrac {b^{2}}{a}}} , irrespective of the units of x {\displaystyle x} . This can be a powerful tool for verifying that a quadratic expression of physical quantities has been set up correctly.
Fagnano, Giulio Carlo (1750), "Applicazione dell' algoritmo nuovo Alla resoluzione analitica dell' equazioni del secondo, del terzo, e del quarto grado" [Application of a new algorithm to the analytical resolution of equations of the second, third, and fourth degree], Produzioni matematiche del conte Giulio Carlo di Fagnano, Marchese de' Toschi, e DiSant' Ononio (in Italian), vol. 1, Pesaro: Gavelliana, Appendice seconda, eq. 6, p. 467, doi : 10.3931/e-rara-8663 | https://en.wikipedia.org/wiki/Quadratic_formula |
In mathematics , a quadratic integral is an integral of the form ∫ d x a + b x + c x 2 . {\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}.}
It can be evaluated by completing the square in the denominator .
∫ d x a + b x + c x 2 = 1 c ∫ d x ( x + b 2 c ) 2 + ( a c − b 2 4 c 2 ) . {\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{\!2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.}
Assume that the discriminant q = b 2 − 4 ac is positive. In that case, define u and A by u = x + b 2 c , {\displaystyle u=x+{\frac {b}{2c}},} and − A 2 = a c − b 2 4 c 2 = 1 4 c 2 ( 4 a c − b 2 ) . {\displaystyle -A^{2}={\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}={\frac {1}{4c^{2}}}(4ac-b^{2}).}
The quadratic integral can now be written as ∫ d x a + b x + c x 2 = 1 c ∫ d u u 2 − A 2 = 1 c ∫ d u ( u + A ) ( u − A ) . {\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {du}{u^{2}-A^{2}}}={\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}.}
The partial fraction decomposition 1 ( u + A ) ( u − A ) = 1 2 A ( 1 u − A − 1 u + A ) {\displaystyle {\frac {1}{(u+A)(u-A)}}={\frac {1}{2A}}\!\left({\frac {1}{u-A}}-{\frac {1}{u+A}}\right)} allows us to evaluate the integral: 1 c ∫ d u ( u + A ) ( u − A ) = 1 2 A c ln ( u − A u + A ) + constant . {\displaystyle {\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}={\frac {1}{2Ac}}\ln \left({\frac {u-A}{u+A}}\right)+{\text{constant}}.}
The final result for the original integral, under the assumption that q > 0, is ∫ d x a + b x + c x 2 = 1 q ln ( 2 c x + b − q 2 c x + b + q ) + constant . {\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{\sqrt {q}}}\ln \left({\frac {2cx+b-{\sqrt {q}}}{2cx+b+{\sqrt {q}}}}\right)+{\text{constant}}.}
In case the discriminant q = b 2 − 4 ac is negative, the second term in the denominator in ∫ d x a + b x + c x 2 = 1 c ∫ d x ( x + b 2 c ) 2 + ( a c − b 2 4 c 2 ) . {\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{\!2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.} is positive. Then the integral becomes 1 c ∫ d u u 2 + A 2 = 1 c A ∫ d u / A ( u / A ) 2 + 1 = 1 c A ∫ d w w 2 + 1 = 1 c A arctan ( w ) + c o n s t a n t = 1 c A arctan ( u A ) + constant = 1 c a c − b 2 4 c 2 arctan ( x + b 2 c a c − b 2 4 c 2 ) + constant = 2 4 a c − b 2 arctan ( 2 c x + b 4 a c − b 2 ) + constant . {\displaystyle {\begin{aligned}{\frac {1}{c}}\int {\frac {du}{u^{2}+A^{2}}}&={\frac {1}{cA}}\int {\frac {du/A}{(u/A)^{2}+1}}\\[9pt]&={\frac {1}{cA}}\int {\frac {dw}{w^{2}+1}}\\[9pt]&={\frac {1}{cA}}\arctan(w)+\mathrm {constant} \\[9pt]&={\frac {1}{cA}}\arctan \left({\frac {u}{A}}\right)+{\text{constant}}\\[9pt]&={\frac {1}{c{\sqrt {{\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}}}}}\arctan \left({\frac {x+{\frac {b}{2c}}}{\sqrt {{\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}}}}\right)+{\text{constant}}\\[9pt]&={\frac {2}{\sqrt {4ac-b^{2}\,}}}\arctan \left({\frac {2cx+b}{\sqrt {4ac-b^{2}}}}\right)+{\text{constant}}.\end{aligned}}} | https://en.wikipedia.org/wiki/Quadratic_integral |
In mathematics , a quadratic irrational number (also known as a quadratic irrational or quadratic surd ) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers . [ 1 ] Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator , a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers , are algebraic numbers of degree 2 , and can therefore be expressed as
for integers a , b , c , d ; with b , c and d non-zero, and with c square-free . When c is positive, we get real quadratic irrational numbers , while a negative c gives complex quadratic irrational numbers which are not real numbers . This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable ; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set . Abu Kamil was the first mathematician to introduce irrational numbers as valid solutions to quadratic equations. [ 2 ] [ 3 ]
Quadratic irrationals are used in field theory to construct field extensions of the field of rational numbers Q . Given the square-free integer c , the augmentation of Q by quadratic irrationals using √ c produces a quadratic field Q ( √ c ). For example, the inverses of elements of Q ( √ c ) are of the same form as the above algebraic numbers:
Quadratic irrationals have useful properties, especially in relation to continued fractions , where we have the result that all real quadratic irrationals, and only real quadratic irrationals, have periodic continued fraction forms. For example
The periodic continued fractions can be placed in one-to-one correspondence with the rational numbers. The correspondence is explicitly provided by Minkowski's question mark function , and an explicit construction is given in that article. It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function. Such repeating sequences correspond to periodic orbits of the dyadic transformation (for the binary digits) and the Gauss map h ( x ) = 1 / x − ⌊ 1 / x ⌋ {\displaystyle h(x)=1/x-\lfloor 1/x\rfloor } for continued fractions.
We may rewrite a quadratic irrationality as follows:
It follows that every quadratic irrational number can be written in the form
This expression is not unique.
Fix a non-square, positive integer c {\displaystyle c} congruent to 0 {\displaystyle 0} or 1 {\displaystyle 1} modulo 4 {\displaystyle 4} , and define a set S c {\displaystyle S_{c}} as
Every quadratic irrationality is in some set S c {\displaystyle S_{c}} , since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor.
A matrix
with integer entries and α δ − β γ = 1 {\displaystyle \alpha \delta -\beta \gamma =1} can be used to transform a number y {\displaystyle y} in S c {\displaystyle S_{c}} . The transformed number is
If y {\displaystyle y} is in S c {\displaystyle S_{c}} , then z {\displaystyle z} is too.
The relation between y {\displaystyle y} and z {\displaystyle z} above is an equivalence relation . (This follows, for instance, because the above transformation gives a group action of the group of integer matrices with determinant 1 on the set S c {\displaystyle S_{c}} .) Thus, S c {\displaystyle S_{c}} partitions into equivalence classes . Each equivalence class comprises a collection of quadratic irrationalities with each pair equivalent through the action of some matrix. Serret's theorem implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail. Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail.
There are finitely many equivalence classes of quadratic irrationalities in S c {\displaystyle S_{c}} . The standard proof of this involves considering the map φ {\displaystyle \varphi } from binary quadratic forms of discriminant c {\displaystyle c} to S c {\displaystyle S_{c}} given by
A computation shows that φ {\displaystyle \varphi } is a bijection that respects the matrix action on each set. The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant.
Through the bijection φ {\displaystyle \varphi } , expanding a number in S c {\displaystyle S_{c}} in a continued fraction corresponds to reducing the quadratic form. The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.
The definition of quadratic irrationals requires them to satisfy two conditions: they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation ax 2 + bx + c = 0 are
Thus quadratic irrationals are precisely those real numbers in this form that are not rational. Since b and 2 a are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational. The answer to this is that the square root of any natural number that is not a square number is irrational.
The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma .
Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic , which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae . This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator. When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore, the square of a rational non-integer is always a non-integer; by contrapositive , the square root of an integer is always either another integer, or irrational.
Euclid used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in Euclid's Elements Book X Proposition 9. [ 4 ]
The fundamental theorem of arithmetic is not actually required to prove the result, however. There are self-contained proofs by Richard Dedekind , [ 5 ] among others. The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of 2 found by Theodor Estermann in 1975. [ 6 ] [ 7 ]
If D is a non-square natural number, then there is a natural number n such that:
so in particular
If the square root of D is rational, then it can be written as the irreducible fraction p / q , so that q is the smallest possible denominator, and hence the smallest number for which q √ D is also an integer. Then:
which is thus also an integer. But 0 < ( √ D − n ) < 1 so ( √ D − n ) q < q . Hence ( √ D − n ) q is an integer smaller than q which multiplied by √ D makes an integer. This is a contradiction, because q was defined to be the smallest such number. Therefore, √ D cannot be rational. | https://en.wikipedia.org/wiki/Quadratic_irrational_number |
In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric ( conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Let P = ( P , G , ∈ ) {\displaystyle {\mathfrak {P}}=({\mathcal {P}},{\mathcal {G}},\in )} be a projective space. A quadratic set is a non-empty subset Q {\displaystyle {\mathcal {Q}}} of P {\displaystyle {\mathcal {P}}} for which the following two conditions hold:
A quadratic set Q {\displaystyle {\mathcal {Q}}} is called non-degenerate if for every point P ∈ Q {\displaystyle P\in {\mathcal {Q}}} , the set Q P {\displaystyle {\mathcal {Q}}_{P}} is a hyperplane.
A Pappian projective space is a projective space in which Pappus's hexagon theorem holds.
The following result, due to Francis Buekenhout , is an astonishing statement for finite projective spaces.
Ovals and ovoids are special quadratic sets: Let P {\displaystyle {\mathfrak {P}}} be a projective space of dimension ≥ 2 {\displaystyle \geq 2} . A non-degenerate quadratic set O {\displaystyle {\mathcal {O}}} that does not contain lines is called ovoid (or oval in plane case).
The following equivalent definition of an oval/ovoid are more common:
Definition: (oval) A non-empty point set o {\displaystyle {\mathfrak {o}}} of a projective plane is called oval if the following properties are fulfilled:
A line g {\displaystyle g} is a exterior or tangent or secant line of the
oval if | g ∩ o | = 0 {\displaystyle |g\cap {\mathfrak {o}}|=0} or | g ∩ o | = 1 {\displaystyle |g\cap {\mathfrak {o}}|=1} or | g ∩ o | = 2 {\displaystyle |g\cap {\mathfrak {o}}|=2} respectively.
For finite planes the following theorem provides a more simple definition.
Theorem: (oval in finite plane) Let be P {\displaystyle {\mathfrak {P}}} a projective plane of order n {\displaystyle n} .
A set o {\displaystyle {\mathfrak {o}}} of points is an oval if | o | = n + 1 {\displaystyle |{\mathfrak {o}}|=n+1} and if no three points
of o {\displaystyle {\mathfrak {o}}} are collinear.
According to this theorem of Beniamino Segre , for Pappian projective planes of odd order the ovals are just conics:
Theorem: Let be P {\displaystyle {\mathfrak {P}}} a Pappian projective plane of odd order.
Any oval in P {\displaystyle {\mathfrak {P}}} is an oval conic (non-degenerate quadric ).
Definition: (ovoid) A non-empty point set O {\displaystyle {\mathcal {O}}} of a projective space is called ovoid if the following properties are fulfilled:
Example:
For finite projective spaces of dimension n {\displaystyle n} over a field K {\displaystyle K} we have: Theorem:
Counterexamples (Tits–Suzuki ovoid) show that i.g. statement b) of the theorem above is not true for char K = 2 {\displaystyle \operatorname {char} K=2} : | https://en.wikipedia.org/wiki/Quadratic_set |
In mathematics , quadrature is a historic term for the computation of areas and is thus used for computation of integrals .
The word is derived from the Latin quadratus meaning "square". The reason is that, for Ancient Greek mathematicians , the computation of an area consisted of constructing a square of the same area. In this sense, the modern term is squaring . For example, the quadrature of the circle , (or squaring the circle) is a famous old problem that has been shown, in the 19th century, to be impossible with the methods available to the Ancient Greeks,
Integral calculus , introduced in the 17th century, is a general method for computation of areas. Quadrature came to refer to the computation of any integral; such a computation is presently called more often "integral" or "integration". However, the computation of solutions of differential equations and differential systems is also called integration , and quadrature remains useful for distinguish integrals from solutions of differential equations, in contexts where both problems are considered. This is the case in numerical analysis ; see numerical quadrature . Also, reduction to quadratures and solving by quadratures means expressing solutions of differential equations in terms of integrals.
The remainder of this article is devoted to the original meaning of quadrature, namely, computation of areas.
Greek mathematicians understood the determination of an area of a figure as the process of geometrically constructing a square having the same area ( squaring ), thus the name quadrature for this process. The Greek geometers were not always successful (see squaring the circle ), but they did carry out quadratures of some figures whose sides were not simply line segments, such as the lune of Hippocrates and the parabola . By a certain Greek tradition, these constructions had to be performed using only a compass and straightedge , though not all Greek mathematicians adhered to this dictum.
For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side x = a b {\displaystyle x={\sqrt {ab}}} (the geometric mean of a and b ). For this purpose it is possible to use the following: if one draws the circle with diameter made from joining line segments of lengths a and b , then the height ( BH in the diagram) of the line segment drawn perpendicular to the diameter, from the point of their connection to the point where it crosses the circle, equals the geometric mean of a and b . A similar geometrical construction solves the problems of quadrature of a parallelogram and of a triangle.
Problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible. [ 1 ] [ 2 ] Nevertheless, for some figures a quadrature can be performed. The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity.
For the proofs of these results, Archimedes used the method of exhaustion attributed to Eudoxus . [ 3 ]
In medieval Europe, quadrature meant the calculation of area by any method. Most often the method of indivisibles was used; it was less rigorous than the geometric constructions of the Greeks, but it was simpler and more powerful. With its help, Galileo Galilei and Gilles de Roberval found the area of a cycloid arch, Grégoire de Saint-Vincent investigated the area under a hyperbola ( Opus Geometricum , 1647), [ 3 ] : 491 and Alphonse Antonio de Sarasa , de Saint-Vincent's pupil and commentator, noted the relation of this area to logarithms . [ 3 ] : 492 [ 4 ]
John Wallis algebrised this method; he wrote in his Arithmetica Infinitorum (1656) some series which are equivalent to what is now called the definite integral , and he calculated their values. Isaac Barrow and James Gregory made further progress: quadratures for some algebraic curves and spirals . Christiaan Huygens successfully performed a quadrature of the surface area of some solids of revolution .
The quadrature of the hyperbola by Gregoire de Saint-Vincent and A. A. de Sarasa provided a new function , the natural logarithm , of critical importance. With the invention of integral calculus came a universal method for area calculation. In response, the term quadrature has become traditional, and instead the modern phrase finding the area is more commonly used for what is technically the computation of a univariate definite integral . | https://en.wikipedia.org/wiki/Quadrature_(geometry) |
Quadray coordinates , also known as caltrop , tetray or Chakovian coordinates (named for David Chako [ 1 ] ), were developed by Darrel Jarmusch (in 1981 [ 2 ] ) and others, as another take on simplicial coordinates, a coordinate system using a simplex or tetrahedron as its basis polyhedron. [ 3 ]
The four basis (but not necessarily unit) vectors stem from the center of a regular tetrahedron and go to its four corners. Their coordinate addresses are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) respectively. These may be positively scaled without rotation (e.g. negation) and linearly combined to span conventional XYZ space, with at least one of the four coordinates unneeded (set to zero).
A typical application might set the edges of the basis tetrahedron as unit. The tetrahedron itself may also be defined as the unit of volume (see below).
The four quadrays may be linearly combined to provide integer coordinates for the inverse tetrahedron (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0), and for the cube, octahedron, rhombic dodecahedron and cuboctahedron of volumes 3, 4, 6 and 20 respectively, given the starting tetrahedron of unit volume.
For example, given A, B, C, D as (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) respectively, the vertices of an octahedron with the same edge length and volume four would be A + B, A + C, A + D, B + C, B + D, C + D or all eight permutations of {1,1,0,0}.
The 12 permutations of {2,1,1,0} define the vertices of the volume 20 cuboctahedron centered at (0,0,0,0). These vectors point from any given sphere to its 12 surrounding neighbors in the cubic close packing (CCP), equivalently the IVM (isotropic vector matrix) in Synergetics. Therefore CCP ball centers all have non-negative integer coordinates.
dual tetrahedron labels E,F,G,H omitted to reduce clutter
If one now calls this volume "4D" as in "four-dimensional" or "four-directional" we have primed the pump for an understanding of R. Buckminster Fuller's "4D geometry," or Synergetics .
In this American transcendentalist philosophy, the regular tetrahedron of edges one, as defined by four inter-tangent uni-radius balls, is taken as unit of volume.
A set of familiar convex polyhedra, termed "the concentric hierarchy" is nested around it, per the above table, such that the cube has volume 3, the octahedron volume 4, rhombic dodecahedron volume 6, and cuboctahedron volume 20. | https://en.wikipedia.org/wiki/Quadray_coordinates |
Divine Proportions: Rational Trigonometry to Universal Geometry is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry , called rational trigonometry . The book advocates replacing the usual basic quantities of trigonometry, Euclidean distance and angle measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa). The author claims his approach holds some advantages, such as avoiding the need for irrational numbers .
The book was "essentially self-published" [ 1 ] by Wildberger through his publishing company Wild Egg. The formulas and theorems in the book are regarded as correct mathematics but the claims about practical or pedagogical superiority are primarily promoted by Wildberger himself and have received mixed reviews.
The main idea of Divine Proportions is to replace distances by the squared Euclidean distance , which Wildberger calls the quadrance , and to replace angle measures by the squares of their sines, which Wildberger calls the spread between two lines. Divine Proportions defines both of these concepts directly from the Cartesian coordinates of points that determine a line segment or a pair of crossing lines. Defined in this way, they are rational functions of those coordinates, and can be calculated directly without the need to take the square roots or inverse trigonometric functions required when computing distances or angle measures. [ 1 ]
For Wildberger, a finitist , this replacement has the purported advantage of avoiding the concepts of limits and actual infinity used in defining the real numbers , which Wildberger claims to be unfounded. [ 2 ] [ 1 ] It also allows analogous concepts to be extended directly from the rational numbers to other number systems such as finite fields using the same formulas for quadrance and spread. [ 1 ] Additionally, this method avoids the ambiguity of the two supplementary angles formed by a pair of lines, as both angles have the same spread. This system is claimed to be more intuitive, and to extend more easily from two to three dimensions. [ 3 ] However, in exchange for these benefits, one loses the additivity of distances and angles: for instance, if a line segment is divided in two, its length is the sum of the lengths of the two pieces, but combining the quadrances of the pieces is more complicated and requires square roots. [ 1 ]
Divine Proportions is divided into four parts. Part I presents an overview of the use of quadrance and spread to replace distance and angle, and makes the argument for their advantages. Part II formalizes the claims made in part I, and proves them rigorously. [ 1 ] Rather than defining lines as infinite sets of points, they are defined by their homogeneous coordinates , which may be used in formulas for testing the incidence of points and lines. Like the sine, the cosine and tangent are replaced with rational equivalents, called the "cross" and "twist", and Divine Proportions develops various analogues of trigonometric identities involving these quantities, [ 3 ] including versions of the Pythagorean theorem , law of sines and law of cosines . [ 4 ]
Part III develops the geometry of triangles and conic sections using the tools developed in the two previous parts. [ 1 ] Well known results such as Heron's formula for calculating the area of a triangle from its side lengths, or the inscribed angle theorem in the form that the angles subtended by a chord of a circle from other points on the circle are equal, are reformulated in terms of quadrance and spread, and thereby generalized to arbitrary fields of numbers. [ 3 ] [ 5 ] Finally, Part IV considers practical applications in physics and surveying, and develops extensions to higher-dimensional Euclidean space and to polar coordinates . [ 1 ]
Divine Proportions does not assume much in the way of mathematical background in its readers, but its many long formulas, frequent consideration of finite fields, and (after part I) emphasis on mathematical rigour are likely to be obstacles to a popular mathematics audience. Instead, it is mainly written for mathematics teachers and researchers. However, it may also be readable by mathematics students, and contains exercises making it possible to use as the basis for a mathematics course. [ 1 ] [ 6 ]
The feature of the book that was most positively received by reviewers was its work extending results in distance and angle geometry to finite fields. Reviewer Laura Wiswell found this work impressive, and was charmed by the result that the smallest finite field containing a regular pentagon is F 19 {\displaystyle \mathbb {F} _{19}} . [ 1 ] Michael Henle calls the extension of triangle and conic section geometry to finite fields, in part III of the book, "an elegant theory of great generality", [ 4 ] and William Barker also writes approvingly of this aspect of the book, calling it "particularly novel" and possibly opening up new research directions. [ 6 ]
Wiswell raises the question of how many of the detailed results presented without attribution in this work are actually novel. [ 1 ] In this light, Michael Henle notes that the use of squared Euclidean distance "has often been found convenient elsewhere"; [ 4 ] for instance it is used in distance geometry , least squares statistics, and convex optimization . James Franklin points out that for spaces of three or more dimensions, modelled conventionally using linear algebra , the use of spread by Divine Proportions is not very different from standard methods involving dot products in place of trigonometric functions. [ 5 ]
An advantage of Wildberger's methods noted by Henle is that, because they involve only simple algebra, the proofs are both easy to follow and easy for a computer to verify. However, he suggests that the book's claims of greater simplicity in its overall theory rest on a false comparison in which quadrance and spread are weighed not against the corresponding classical concepts of distances, angles, and sines, but the much wider set of tools from classical trigonometry. He also points out that, to a student with a scientific calculator, formulas that avoid square roots and trigonometric functions are a non-issue, [ 4 ] and Barker adds that the new formulas often involve a greater number of individual calculation steps. [ 6 ] Although multiple reviewers felt that a reduction in the amount of time needed to teach students trigonometry would be very welcome, [ 3 ] [ 5 ] [ 7 ] Paul Campbell is skeptical that these methods would actually speed learning. [ 7 ] Gerry Leversha keeps an open mind, writing that "It will be interesting to see some of the textbooks aimed at school pupils [that Wildberger] has promised to produce, and ... controlled experiments involving student guinea pigs." [ 3 ] However, these textbooks and experiments have not been published.
Wiswell is unconvinced by the claim that conventional geometry has foundational flaws that these methods avoid. [ 1 ] While agreeing with Wiswell, Barker points out that there may be other mathematicians who share Wildberger's philosophical suspicions of the infinite, and that this work should be of great interest to them. [ 6 ]
A final issue raised by multiple reviewers is inertia: supposing for the sake of argument that these methods are better, are they sufficiently better to make worthwhile the large individual effort of re-learning geometry and trigonometry in these terms, and the institutional effort of re-working the school curriculum to use them in place of classical geometry and trigonometry? Henle, Barker, and Leversha conclude that the book has not made its case for this, [ 3 ] [ 4 ] [ 6 ] but Sandra Arlinghaus sees this work as an opportunity for fields such as her mathematical geography "that have relatively little invested in traditional institutional rigidity" to demonstrate the promise of such a replacement. [ 8 ] | https://en.wikipedia.org/wiki/Quadrea |
A quadruple bond is a type of chemical bond between two atoms involving eight electrons . This bond is an extension of the more familiar types of covalent bonds : double bonds and triple bonds . [ 1 ] Stable quadruple bonds are most common among the transition metals in the middle of the d-block , such as rhenium , tungsten , technetium , molybdenum and chromium . Typically the ligands that support quadruple bonds are π-donors , not π-acceptors . Quadruple bonds are rare as compared to double bonds and triple bonds , but hundreds of compounds with such bonds have been prepared. [ 2 ]
Chromium(II) acetate , Cr 2 ( μ -O 2 CCH 3 ) 4 (H 2 O) 2 , was the first chemical compound containing a quadruple bond to be synthesized. It was described in 1844 by E. Peligot , although its distinctive bonding was not recognized for more than a century. [ 3 ]
The first crystallographic study of a compound with a quadruple bond was provided by Soviet chemists for salts of Re 2 Cl 2− 8 . [ 4 ] The very short Re–Re distance was noted. This short distance (and the salt's diamagnetism ) indicated Re–Re bonding. These researchers, however, misformulated the anion as a derivative of Re(II), i.e., Re 2 Cl 4− 8 .
Soon thereafter, F. Albert Cotton and Charles B. Harris reported the crystal structure of potassium octachlorodirhenate or K 2 [Re 2 Cl 8 ]·2H 2 O. [ 5 ] This structural analysis indicated that the previous characterization was mistaken. Cotton and Harris formulated a molecular orbital rationale for the bonding that explicitly indicated a quadruple bond. [ 3 ] The rhenium–rhenium bond length in this compound is only 224 pm . In molecular orbital theory , the bonding is described as σ 2 π 4 δ 2 with one sigma bond , two pi bonds and one delta bond .
The [Re 2 Cl 8 ] 2− ion adopts an eclipsed conformation as shown at left. The delta bonding orbital is then formed by overlap of the d orbitals on each rhenium atom, which are perpendicular to the Re–Re axis and lie in between the Re–Cl bonds. The d orbitals directed along the Re–Cl bonds are stabilized by interaction with chloride ligand orbitals and do not contribute to Re–Re bonding. [ 6 ] In contrast, the [Os 2 Cl 8 ] 2− ion with two more electrons (σ 2 π 4 δ 2 δ* 2 ) has an Os–Os triple bond and a staggered geometry . [ 6 ]
Many other compounds with quadruple bonds between transition metal atoms have been described, often by Cotton and his coworkers. Isoelectronic with the dirhenium compound is the salt K 4 [Mo 2 Cl 8 ] ( potassium octachlorodimolybdate ). [ 7 ] An example of a ditungsten compound with a quadruple bond is ditungsten tetra(hpp) .
Quadruple bonds between atoms of main-group elements are unknown. For the diatomic carbon (C 2 ) molecule as an example, molecular orbital theory shows that there are two sets of paired electrons in the sigma system (one bonding, one antibonding), and two sets of paired electrons in a degenerate π-bonding set of orbitals. This adds up to a bond order of 2, meaning that there exists a double bond between the two carbon atoms. The molecular orbital diagram of diatomic carbon would show that there are two pi bonds and no sigma bonds. A 2012 paper by S. Shaik et al. suggests that a quadruple bond exists in dicarbon, [ 8 ] but this is disputed. [ 9 ] | https://en.wikipedia.org/wiki/Quadruple_bond |
In general relativity , the quadrupole formula describes the gravitational waves that are emitted from a system of masses in terms of the (mass) quadrupole moment . The formula reads
where h ¯ i j {\displaystyle {\bar {h}}_{ij}} is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave. G {\displaystyle G} is the gravitational constant, c {\displaystyle c} the speed of light in vacuum, and I i j {\displaystyle I_{ij}} is the mass quadrupole moment. [ 1 ]
It is useful to express the gravitational wave strain in the transverse traceless gauge, by replacing the mass quadrupole moment I i j {\displaystyle I_{ij}} with the transverse traceless projection I i j T T {\displaystyle I_{ij}^{TT}} , which is defined as:
where n {\displaystyle \mathbf {n} } is a unit vector in the direction of the observer, r n ≡ r ⋅ n {\displaystyle r_{n}\equiv \mathbf {r} \cdot \mathbf {n} } , and r 2 ≡ r ⋅ r {\displaystyle r^{2}\equiv \mathbf {r} \cdot \mathbf {r} } . [ 2 ]
The total energy carried away by gravitational waves can be expressed as:
where I i j T {\displaystyle I_{ij}^{T}} is the traceless mass quadrupole moment, which is given by:
The formula was first obtained by Albert Einstein in 1918. After a long history of debate on its physical correctness, observations of energy loss due to gravitational radiation in the Hulse–Taylor binary discovered in 1974 confirmed the result, with agreement up to 0.2 percent (by 2005). [ 3 ] | https://en.wikipedia.org/wiki/Quadrupole_formula |
In experimental physics , a quadrupole ion trap or paul trap is a type of ion trap that uses dynamic electric fields to trap charged particles . They are also called radio frequency (RF) traps or Paul traps in honor of Wolfgang Paul , who invented the device [ 1 ] [ 2 ] and shared the Nobel Prize in Physics in 1989 for this work. [ 3 ] It is used as a component of a mass spectrometer or a trapped ion quantum computer .
A charged particle, such as an atomic or molecular ion , feels a force from an electric field . It is not possible to create a static configuration of electric fields that traps the charged particle in all three directions (this restriction is known as Earnshaw's theorem ). It is possible, however, to create an average confining force in all three directions by use of electric fields that change in time. To do so, the confining and anti-confining directions are switched at a rate faster than it takes the particle to escape the trap. The traps are also called "radio frequency" traps because the switching rate is often at a radio frequency .
The quadrupole is the simplest electric field geometry used in such traps, though more complicated geometries are possible for specialized devices. The electric fields are generated from electric potentials on metal electrodes. A pure quadrupole is created from hyperbolic electrodes, though cylindrical electrodes are often used for ease of fabrication. Microfabricated ion traps exist where the electrodes lie in a plane with the trapping region above the plane. [ 4 ] There are two main classes of traps, depending on whether the oscillating field provides confinement in three or two dimensions. In the two-dimension case (a so-called "linear RF trap"), confinement in the third direction is provided by static electric fields.
The 3D trap itself generally consists of two hyperbolic metal electrodes with their foci facing each other and a hyperbolic ring electrode halfway between the other two electrodes. The ions are trapped in the space between these three electrodes by AC (oscillating) and DC (static) electric fields . The AC radio frequency voltage oscillates between the two hyperbolic metal end cap electrodes if ion excitation is desired; the driving AC voltage is applied to the ring electrode. The ions are first pulled up and down axially while being pushed in radially. The ions are then pulled out radially and pushed in axially (from the top and bottom). In this way the ions move in a complex motion that generally involves the cloud of ions being long and narrow and then short and wide, back and forth, oscillating between the two states. Since the mid-1980s most 3D traps (Paul traps) have used ~1 mTorr of helium . The use of damping gas and the mass-selective instability mode developed by Stafford et al. led to the first commercial 3D ion traps. [ 5 ]
The quadrupole ion trap has two main configurations: the three-dimensional form described above and the linear form made of 4 parallel electrodes. A simplified rectilinear configuration is also used. [ 6 ] The advantage of the linear design is its greater storage capacity (in particular of Doppler-cooled ions) and its simplicity, but this leaves a particular constraint on its modeling. The Paul trap is designed to create a saddle-shaped field to trap a charged ion, but with a quadrupole, this saddle-shaped electric field cannot be rotated about an ion in the centre. It can only 'flap' the field up and down. For this reason, the motions of a single ion in the trap are described by Mathieu equations , which can only be solved numerically by computer simulations.
The intuitive explanation and lowest order approximation is the same as strong focusing in accelerator physics . Since the field affects the acceleration, the position lags behind (to lowest order by half a period). So the particles are at defocused positions when the field is focusing and vice versa. Being farther from center, they experience a stronger field when the field is focusing than when it is defocusing.
Ions in a quadrupole field experience restoring forces that drive them back toward the center of the trap. The motion of the ions in the field is described by solutions to the Mathieu equation . [ 7 ] When written for ion motion in a trap, the equation is
where u {\displaystyle u} represents the x, y and z coordinates, ξ {\displaystyle \xi } is a dimensionless variable given by ξ = Ω t / 2 {\displaystyle \xi =\Omega t/2} , and a u {\displaystyle a_{u}\,} and q u {\displaystyle q_{u}} are dimensionless trapping parameters. The parameter Ω {\displaystyle \Omega } is the radial frequency of the potential applied to the ring electrode. By using the chain rule , it can be shown that
Substituting Equation 2 into the Mathieu Equation 1 yields
Multiplying by m and rearranging terms shows us that
By Newton's laws of motion , the above equation represents the force on the ion. This equation can be exactly solved using the Floquet theorem or the standard techniques of multiple scale analysis . [ 8 ] The particle dynamics and time averaged density of charged particles in a Paul trap can also be obtained by the concept of ponderomotive force .
The forces in each dimension are not coupled, thus the force acting on an ion in, for example, the x dimension is
Here, ϕ {\displaystyle \phi } is the quadrupolar potential, given by
where ϕ 0 {\displaystyle \phi _{0}} is the applied electric potential and λ {\displaystyle \lambda } , σ {\displaystyle \sigma } , and γ {\displaystyle \gamma } are weighting factors, and r 0 {\displaystyle r_{0}} is a size parameter constant. In order to satisfy Laplace's equation , ∇ 2 ϕ 0 = 0 {\displaystyle \nabla ^{2}\phi _{0}=0} , it can be shown that
For an ion trap, λ = σ = 1 {\displaystyle \lambda =\sigma =1} and γ = − 2 {\displaystyle \gamma =-2} and for a quadrupole mass filter , λ = − σ = 1 {\displaystyle \lambda =-\sigma =1} and γ = 0 {\displaystyle \gamma =0} .
Transforming equation 6 into a cylindrical coordinate system with x = r cos θ {\displaystyle x=r\cos \theta } , y = r sin θ {\displaystyle y=r\sin \theta } , and z = z {\displaystyle z=z} and applying the Pythagorean trigonometric identity sin 2 θ + cos 2 θ = 1 {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1} gives
The applied electric potential is a combination of RF and DC given by
where Ω = 2 π ν {\displaystyle \Omega =2\pi \nu } and ν {\displaystyle \nu } is the applied frequency in hertz .
Substituting Equation 8 into Equation 6 with λ = 1 {\displaystyle \lambda =1} gives
Substituting equation 9 into equation 5 leads to
Comparing terms on the right hand side of equation 1 and equation 10 leads to
and
Further q x = q y {\displaystyle q_{x}=q_{y}\,} ,
and
The trapping of ions can be understood in terms of stability regions in q u {\displaystyle q_{u}} and a u {\displaystyle a_{u}} space. The boundaries of the shaded regions in the figure are the boundaries of stability in the two directions (also known as boundaries of bands). The domain of overlap of the two regions is the trapping domain. For calculation of these boundaries and similar diagrams as above see Müller-Kirsten. [ 9 ]
The linear ion trap uses a set of quadrupole rods to confine ions radially and a static electrical potential on-end electrodes to confine the ions axially. [ 11 ] The linear form of the trap can be used as a selective mass filter, or as an actual trap by creating a potential well for the ions along the axis of the electrodes. [ 12 ] Advantages of the linear trap design are increased ion storage capacity, faster scan times, and simplicity of construction (although quadrupole rod alignment is critical, adding a quality control constraint to their production. This constraint is additionally present in the machining requirements of the 3D trap). [ 13 ]
The cylindrical ion trap (CIT) emerged as a derivative of the quadrupole ion trap with simpler geometric structure in which the electrodes are arranged in a cylindrical shape rather than the traditional hyperbolic or linear configuration. [ 14 ]
The cylindrical ion trap consists of a central cylindrical electrode (ring electrode) and two end-cap electrodes. By applying a combination of static ( DC ) and oscillating ( RF ) voltages to these electrodes, a three-dimensional quadrupole field is generated. The ions are trapped in the center of this field due to the restoring forces created by the electric fields, which confine the ions along the axis and radial directions. [ 15 ]
Ion traps with a cylindrical rather than a hyperbolic ring electrode [ 16 ] [ 17 ] [ 18 ] [ 19 ] [ 20 ] have been developed and microfabricated in arrays to develop miniature mass spectrometers for chemical detection in medical diagnosis and other fields. However, the reduction in ion storage volumes remains a problem in small ion traps. [ 14 ]
Quadrupole traps can also be "unfolded" to create the same effect using a set of planar electrodes. [ 21 ] This trap geometry can be made using standard micro-fabrication techniques, including the top metal layer in a standard CMOS microelectronics process, [ 22 ] and is a key technology for scaling trapped ion quantum computers to useful numbers of qubits.
A combined radio frequency trap is a combination of a Paul ion trap and a Penning trap . [ 23 ] One of the main bottlenecks of a quadrupole ion trap is that it can confine only single-charged species or multiple species with similar masses. But in certain applications like antihydrogen production it is important to confine two species of charged particles of widely varying masses. To achieve this objective, a uniform magnetic field is added in the axial direction of the quadrupole ion trap.
The digital ion trap (DIT) is a quadrupole ion trap (linear or 3D) that differs from conventional traps by the driving waveform. A DIT is driven by digital signals, typically rectangular waveforms [ 24 ] [ 25 ] that are generated by switching rapidly between discrete voltage levels. Major advantages of the DIT are its versatility [ 26 ] and virtually unlimited mass range. The digital ion trap has been developed mainly as a mass analyzer. | https://en.wikipedia.org/wiki/Quadrupole_ion_trap |
QualNet is a commercial network simulation tool owned by Keysight Technologies (formerly Scalable Network Technologies). [ 1 ]
QualNet uses parallel discrete event simulation technology to model various layers of the network stack including the physical layer of wireless radios. QualNet simulations may be customized to include terrain data , antenna patterns , experimental or specialized network protocols (ex mesh networking ), and mobility (nodes move during the simulation). Alternatives to QualNet include NS-2 and OPNET . QualNet was used since the early 2000s to research and develop wireless communication technologies. [ improper synthesis? ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] [ 13 ]
This simulation software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/QualNet |
The Qualcomm Emergency Download mode , commonly known as Qualcomm EDL mode and officially known as Qualcomm HS-USB QD-Loader 9008 [ 1 ] is a feature implemented in the boot ROM of a system on a chip by Qualcomm which can be used to recover bricked smartphones . [ 2 ] [ 3 ] On Google's Pixel 3 , the feature was accidentally shown to users after the phone was bricked. [ 4 ]
For a device to support EDL it must be using Qualcomm hardware. The Snapdragon family is very widely used.
The Android Debug Bridge can be utilized to get access to EDL mode, with the command adb reboot edl . [ 2 ]
The Qualcomm Product Support Tool (QPST) is normally used internally by service center executives for low-level firmware flashing to revive Android devices from a hard-brick or to fix persistent software issues. To flash the firmware, the tool communicates with supported devices via EDL. [ 5 ] [ unreliable source? ] The QPST has not been officially released by Qualcomm. [ citation needed ]
Qualcomm Download (QDL) is a tool to communicate with Qualcomm System On a Chip bootroms to install or execute code. The source code is maintained by Bjorn Andersson also known as andersson. [ 6 ] [ non-primary source needed ]
Qualcomm implemented motherboards always include a test point. These can vary between phone models. Generally, test points are a pair of contacts, which can be some way apart. EDL can be accessed by opening the back of the phone, finding the location of the test point, which depends on the model, and using a pair of metal tweezers to short the connectors and boot the phone into EDL. Further software tools are needed for actions in EDL mode.
Qualcomm implemented a feature in motherboards with the presence of EDL, where they can be booted to EDL via an EDL Deep Flash Cable. This specific cable has the general appearance of a button present in the cable. The button can be represented as a switch, to be able to make the phone boot into EDL mode. With the use of the cable, in most devices and cases, it will not be necessary to use the test points. The cable also works on hard-bricked devices to boot them into EDL mode. This cable works by having a button present between D+ and GND.
This mobile technology related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Qualcomm_EDL_mode |
The Toq (pronounced “tok” as in "tick-tock") is a smartwatch developed by Qualcomm released as a proof of concept to OEMs and was released in limited quantities in December 2013. The Toq was first unveiled at Qualcomm's annual Uplinq event on September 4, 2013 in San Diego . It syncs with Android 4.0+ smartphones , allowing users to scan through texts, emails, phone calls, and other notifications. It features a Mirasol display , which like E Ink e-reader screens, can be easily viewed in direct sunlight. Unlike most ereaders, it can display colors and can refresh fast enough for watching videos, it also includes speech recognition technology from Nuance to allow users to dictate replies to text messages. [ 1 ] The Toq has a backlight for when there is no outside light source. [ 2 ] | https://en.wikipedia.org/wiki/Qualcomm_Toq |
The Snapdragon Qualcomm VR820 is a virtual reality reference platform [ 1 ] that was released in Q4 2016, with the first commercial devices based on the platform available shortly thereafter. It is based on the Qualcomm Snapdragon 820 processor, released on November 10, 2015.
Source: [ 2 ]
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Qualcomm_VR_820 |
In the European Union, the Qualified Person Responsible For Pharmacovigilance ( QPPV ) is an individual, usually an employee of a pharmaceutical company, who is personally responsible for the safety of the human pharmaceutical products marketed by that company in the EU. This function was established in 2004 by article 23 of regulation (EC) No 726/2004. The article establishes that the holder of a marketing authorization for a drug for human use must have a QPPV. When a company submits an application for permission to bring a medicinal product onto the market, the company submits a description of its system for monitoring the safety of the product in actual use (a pharmacovigilance system) and proof that the services of a QPPV are in place. [ 1 ]
"The holder of an authorisation for a medicinal product for human use granted in accordance with the provisions of this Regulation shall have permanently and continuously at his disposal an appropriately qualified person responsible for pharmacovigilance."
Per Article 23, the QPPV is responsible for:
The Good Pharmacovigilance Practices (GVP) provide further guidance regarding these responsibilities of the QPPV (Module I, Section I.C.1.3: Role of the qualified person responsible for pharmacovigilance in the EU. Per GVP, the QPPV is responsible for the establishment and maintenance of the marketing authorisation holder's pharmacovigilance system and therefore shall have sufficient authority to influence the performance of the quality system and the pharmacovigilance activities and to promote, maintain and improve compliance with the legal requirements. The QPPV is specifically responsible for:
The QPPV must reside in the EU, and should be permanently and continuously at the disposal of the MAH. Each company (i.e. Applicant/Marketing Authorisation Holder or group of Marketing Authorisation Holders using a common pharmacovigilance system) should appoint one QPPV responsible for overall pharmacovigilance for all medicinal products for which the company holds marketing authorisations within the EU .
Detailed information on the role and responsibilities of the QPPV, and guidance for a Marketing Authorisation Holder on how to adequately support the QPPV are specified in Guideline on good pharmacovigilance practices (GVP; Module I – Pharmacovigilance systems and their quality systems)." [ 2 ] At a minimum the QPPV should be appropriately qualified , with documented experience in all aspects of pharmacovigilance in order to fulfil the responsibilities and tasks of the post. If the QPPV is not medically qualified, access to a medically qualified person should be available.
In addition, there are national regulations in some EU member states that also require a nominated individual in that country who has specific legal obligations as a QPPV at a national level. | https://en.wikipedia.org/wiki/Qualified_Person_Responsible_For_Pharmacovigilance |
Classical qualitative inorganic analysis is a method of analytical chemistry which seeks to find the elemental composition of inorganic compounds . It is mainly focused on detecting ions in an aqueous solution , therefore materials in other forms may need to be brought to this state before using standard methods. The solution is then treated with various reagents to test for reactions characteristic of certain ions, which may cause color change, precipitation and other visible changes. [ 1 ] [ 2 ]
Qualitative inorganic analysis is that branch or method of analytical chemistry which seeks to establish the elemental composition of inorganic compounds through various reagents.
According to their properties, cations are usually classified into six groups. [ 1 ] Each group has a common reagent which can be used to separate them from the solution . To obtain meaningful results, the separation must be done in the sequence specified below, as some ions of an earlier group may also react with the reagent of a later group, causing ambiguity as to which ions are present. This happens because cationic analysis is based on the solubility products of the ions. As the cation gains its optimum concentration needed for precipitation it precipitates and hence allowing us to detect it. The division and precise details of separating into groups vary slightly from one source to another; given below is one of the commonly used schemes.
The 1st analytical group of cations consists of ions which form insoluble chlorides . As such, the group reagent to separate them is hydrochloric acid , usually used at a concentration of 1–2 M . Concentrated HCl must not be used, because it forms a soluble complex ([PbCl 4 ] 2− ) with Pb 2+ . Consequently, the Pb 2+ ion would go undetected.
The most important cations in the 1st group are Ag + , Hg 2+ 2 , and Pb 2+ . The chlorides of these elements cannot be distinguished from each other by their colour - they are all white solid compounds. PbCl 2 is soluble in hot water, and can therefore be differentiated easily. Ammonia is used as a reagent to distinguish between the other two. While AgCl dissolves in ammonia (due to the formation of the complex ion [Ag(NH 3 ) 2 ] + ), Hg 2 Cl 2 gives a black precipitate consisting of a mixture of chloro-mercuric amide and elemental mercury. Furthermore, AgCl is reduced to silver under light, which gives samples a violet colour.
The silver ammonia complex can react with bismuth ions and iodide to generate orange or brown Ag 2 BiI 5 precipitate. [ 3 ]
PbCl 2 is far more soluble than the chlorides of the other two ions, especially in hot water. Therefore, HCl in concentrations which completely precipitate Hg 2+ 2 and Ag + may not be sufficient to do the same to Pb 2+ . Higher concentrations of Cl − cannot be used for the before mentioned reasons. Thus, a filtrate obtained after first group analysis of Pb 2+ contains an appreciable concentration of this cation, enough to give the test of the second group, viz. formation of an insoluble sulfide. For this reason, Pb 2+ is usually also included in the 2nd analytical group.
A signature reaction of lead ions involve the formation of a yellow lead chromate precipitate upon treatment with chromate ions. This precipitate doesn't dissolve in ammonia (unlike Cu(II) and Ag(I)) or acetic acid (unlike Cu(II) and Hg(II)). [ 3 ]
This group can be determined by adding the salt in water and then adding dilute hydrochloric acid. A white precipitate is formed, to which ammonia is then added. If the precipitate is insoluble, then Pb 2+ is present; if the precipitate is soluble, then Ag + is present, and if the white precipitate turns black, then Hg 2+ 2 is present.
Hg 2+ 2 ions, after oxidation in the presence of chloride ions to HgCl 4 2- , can form a characteristic orange-red precipitate of Cu 2 HgI 4 with the addition of Cu 2+ and I − . [ 3 ]
Confirmation test for Pb 2+ :
Confirmation test for Ag + :
Confirmation test for Hg 2+ 2 :
The 2nd analytical group of cations consists of ions which form acid-insoluble sulfides . Cations in the 2nd group include: Cd 2+ , Bi 3+ , Cu 2+ , As 3+ , As 5+ , Sb 3+ , Sb 5+ , Sn 2+ , Sn 4+ and Hg 2+ . Pb 2+ is usually also included here in addition to the first group. Although these methods refer to solutions that contain sulfide (S 2− ), these solutions actually only contain H 2 S and bisulfide (HS − ). Sulfide (S 2− ) does not exist in appreciable concentrations in water.
The reagent used can be any substance that gives S 2− ions in such solutions; most commonly used are hydrogen sulfide (at 0.2-0.3 M), thioacetamide (at 0.3-0.6 M), addition of hydrogen sulfide can often prove to be a lumbersome process and therefore sodium sulfide can also serve the purpose. The test with the sulfide ion must be conducted in the presence of dilute HCl. Its purpose is to keep the sulfide ion concentration at a required minimum, so as to allow the precipitation of 2nd group cations alone. If dilute acid is not used, the early precipitation of 4th group cations (if present in solution) may occur, thus leading to misleading results. Acids beside HCl are rarely used. Sulfuric acid may lead to the precipitation of the 5th group cations, whereas nitric acid oxidises the sulfide ion in the reagent, forming colloidal sulfur.
The precipitates of these cations are almost indistinguishable, except for CdS , which is yellow. All the precipitates, except for HgS , are soluble in dilute nitric acid. HgS is soluble only in aqua regia , which can be used to separate it from the rest. The action of ammonia is also useful in differentiating the cations. CuS dissolves in ammonia forming an intense blue solution, whereas CdS dissolves forming a colourless solution. The sulfides of As 3+ , As 5+ , Sb 3+ , Sb 5+ , Sn 2+ , Sn 4+ are soluble in yellow ammonium sulfide , where they form polysulfide complexes.
This group is determined by adding the salt in water and then adding dilute hydrochloric acid (to make the medium acidic) followed by hydrogen sulfide gas. Usually it is done by passing hydrogen sulfide over the test tube for detection of 1st group cations. If it forms a reddish-brown or black precipitate then Bi 3+ , Cu 2+ , Hg 2+ or Pb 2+ is present. Otherwise, if it forms a yellow precipitate, then Cd 2+ or Sn 4+ is present; or if it forms a brown precipitate, then Sn 2+ must be present; or if a red orange precipitate is formed, then Sb 3+ is present.
Confirmation test for copper:
Confirmation test for bismuth:
Confirmation test for mercury:
The 3rd analytical group of cations includes ions which form hydroxides that are insoluble even at low concentrations.
Cations in the 3rd group are, among others: Fe 2+ , Fe 3+ , Al 3+ , and Cr 3+ .
The group is determined by making a solution of the salt in water and adding ammonium chloride and ammonium hydroxide. Ammonium chloride is added to ensure low concentration of hydroxide ions.
The formation of a reddish-brown precipitate indicates Fe 3+ ; a gelatinous white precipitate indicates Al 3+ ; and a green precipitate indicates Cr 3+ or Fe 2+ . These last two are distinguished by adding sodium hydroxide in excess to the green precipitate. If the precipitate dissolves, Cr 3+ is indicated; otherwise, Fe 2+ is present.
The 4th analytical group of cations includes ions that precipitate as sulfides at pH 9. The reagent used is ammonium sulfide or Na 2 S 0.1 M added to the ammonia/ammonium chloride solution used to detect group 3 cations.
It includes: Zn 2+ , Ni 2+ , Co 2+ , and Mn 2+ . Zinc will form a white precipitate, nickel and cobalt a black precipitate and manganese a brick/flesh colored precipitate. Dimethylglyoxime can be used to confirm nickel presence, while ammonium thiocyanate in ether will turn blue in the presence of cobalt. This group is sometimes denoted as IIIB since groups III and IV are tested for at the same time, with the addition of sulfide being the only difference.
This includes ions which form sulfides that are insoluble at high concentrations. The reagents used are H 2 S in the presence of NH 4 OH. NH 4 OH is used to increase the concentration of the sulfide ion, by the common ion effect - hydroxide ions from NH 4 OH combine with H + ions from H 2 S, which shifts the equilibrium in favor of the ionized form:
They contain Zn 2+ , Mn 2+ , Ni 2+ and Co 2+
Ions in 5th analytical group of cations form carbonates that are insoluble in water. The reagent usually used is (NH 4 ) 2 CO 3 (at around 0.2 M), with a neutral or slightly basic pH. All the cations in the previous groups are separated beforehand, since many of them also form insoluble carbonates.
The most important ions in the 5th group are Ba 2+ , Ca 2+ , and Sr 2+ . After separation, the easiest way to distinguish between these ions is by testing flame colour: barium gives a yellow-green flame, calcium gives brick red, and strontium, crimson red.
Cations which are left after carefully separating previous groups are considered to be in the sixth analytical group. The most important ones are Mg 2+ , Li + , Na + and K + . All the ions are distinguished by flame color: lithium gives a red flame, sodium gives bright yellow (even in trace amounts), potassium gives violet, and magnesium, colorless (although magnesium metal burns with a bright white flame). Magnesium can also be distinguished from other cations in this group by adding sodium hydroxide to drive the pH to 11 or higher, which selectively precipitates Mg(OH) 2 .
The 1st group of anions consist of CO 2− 3 , HCO − 3 , CH 3 COO − , S 2− , SO 2− 3 , S 2 O 2− 3 and NO − 2 . The reagent for Group 1 anions is dilute hydrochloric acid (HCl) or dilute sulfuric acid (H 2 SO 4 ).
The 2nd group of anions consist of Cl − , Br − , I − , NO − 3 and C 2 O 2− 4 . The group reagent for Group 2 anion is concentrated sulfuric acid (H 2 SO 4 ).
After addition of the acid, chlorides, bromides and iodides will form precipitates with silver nitrate . The precipitates are white, pale yellow, and yellow, respectively. The silver halides formed are completely soluble, partially soluble, or not soluble at all, respectively, in aqueous ammonia solution.
Chlorides are confirmed by the chromyl chloride test . When the salt is heated with K 2 Cr 2 O 7 and concentrated H 2 SO 4 , red vapours of chromyl chloride (CrO 2 Cl 2 ) are produced. Passing this gas through a solution of NaOH produces a yellow solution of Na 2 CrO 4 . The acidified solution of Na 2 CrO 4 gives a yellow precipitate with the addition of (CH 3 COO) 2 Pb .
Bromides and iodides are confirmed by the layer test . A sodium carbonate extract is made from the solution containing bromide or iodide, and CHCl 3 or CS 2 is added to the solution, which separates into two layers: an orange colour in the CHCl 3 or CS 2 layer indicates the presence of Br − , and a violet colour indicates the presence of I − .
Nitrates give brown fumes with concentrated H 2 SO 4 due to formation of NO 2 . This is intensified upon adding copper turnings. Nitrate ion is confirmed by adding an aqueous solution of the salt to FeSO 4 and pouring concentrated H 2 SO 4 slowly along the sides of the test tube, which produces a brown ring around the walls of the tube, at the junction of the two liquids caused by the formation of Fe(NO) 2+ . [ 4 ]
Upon treatment with concentrated sulfuric acid, oxalates yield colourless CO 2 and CO gases. These gases burn with a bluish flame and turn lime water milky. Oxalates also decolourise KMnO 4 and give a white precipitate with CaCl 2 .
The 3rd group of anions consist of SO 2− 4 , PO 3− 4 and BO 3− 3 . They react neither with concentrated nor diluted H 2 SO 4 .
Qualitative inorganic analysis is now used only as a pedagogical tool. Modern techniques such as atomic absorption spectroscopy and ICP-MS are able to quickly detect the presence and concentrations of elements using a very small amount of sample.
The sodium carbonate test (not to be confused with sodium carbonate extract test) is used to distinguish between some common metal ions, which are precipitated as their respective carbonates. The test can distinguish between copper (Cu), iron (Fe), and calcium (Ca), zinc (Zn) or lead (Pb). Sodium carbonate solution is added to the salt of the metal. A blue precipitate indicates Cu 2+ ion. A dirty green precipitate indicates Fe 2+ ion. A yellow-brown precipitate indicates Fe 3+ ion. A white precipitate indicates Ca 2+ , Zn 2+ , or Pb 2+ ion. The compounds formed are, respectively, basic copper carbonate , iron(II) carbonate , iron(III) oxide , calcium carbonate , zinc carbonate , and lead(II) carbonate . This test is used to precipitate the ion present as almost all carbonates are insoluble. While this test is useful for telling these cations apart, it fails if other ions are present, because most metal carbonates are insoluble and will precipitate. In addition, calcium, zinc, and lead ions all produce white precipitates with carbonate, making it difficult to distinguish between them. Instead of sodium carbonate, sodium hydroxide may be added, this gives nearly the same colours, except that lead and zinc hydroxides are soluble in excess alkali, and can hence be distinguished from calcium. See qualitative inorganic analysis for the complete sequence of tests used for qualitative cation analysis. | https://en.wikipedia.org/wiki/Qualitative_inorganic_analysis |
Qualitative properties are properties that are observed and can generally not be measured with a numerical result, unlike quantitative properties , which have numerical characteristics.
Qualitative properties are properties that are observed and can generally not be measured with a numerical result. [ 1 ] They are contrasted to quantitative properties which have numerical characteristics.
Although measuring something in qualitative terms is difficult, most people can (and will) make a judgement about a behaviour on the basis of how they feel treated. This indicates that qualitative properties are closely related to emotional impressions.
A test method can result in qualitative data about something. This can be a categorical result or a binary classification (e.g., pass/fail, go/no go , conform /non-conform). It can sometimes be an engineering judgement.
The data that all share a qualitative property form a nominal category . A variable which codes for the presence or absence of such a property is called a binary categorical variable , or equivalently a dummy variable . | https://en.wikipedia.org/wiki/Qualitative_property |
Qualitative Reasoning (QR) is an area of research within Artificial Intelligence (AI) that automates reasoning about continuous aspects of the physical world, such as space, time, and quantity, for the purpose of problem solving and planning using qualitative rather than quantitative information. [ 1 ] Precise numerical values or quantities are avoided, and qualitative values are used instead (e.g., high, low, zero, rising, falling, etc.). [ 2 ]
Qualitative reasoning creates non-numerical descriptions of physical systems and their behavior, preserving important behavioral properties and qualitative distinctions. [ 3 ] The goal of qualitative reasoning research is to develop representation and reasoning methods that enable computer programs to reason about the behavior of physical systems, without precise quantitative information. An example is observing pouring rain and the steadily rising water level of a river, which is sufficient information to take action against possible flooding without knowing the exact water level, the rate of change, or the time the river might flood. [ 4 ]
The principles used are motivated by human cognition .
The principles of qualitative reasoning include: [ 5 ]
The techniques which have been developed for qualitative reasoning permit the simulation of quantitative systems which are subject to multiple constraints in the form of inequalities as well as equalities. It can permit the simulation of certain important systems, such as ecosystems, which might otherwise be too complex to model. Qualitative reasoning provides a method for modeling with quantitative inequalities in addition to qualities.
Successful application areas include process control , system verification, explanation, [ 2 ] autonomous spacecraft support, simulation and explanation of the behavior of structures, [ 6 ] failure analysis and on-board diagnosis of vehicle systems, automated generation of control software for photocopiers, conceptual knowledge capture in ecology, and intelligent aids for human learning. [ 3 ] | https://en.wikipedia.org/wiki/Qualitative_reasoning |
In mathematics , the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov . There are relatively few differential equations that can be solved explicitly, but using tools from analysis and topology , one can "solve" them in the qualitative sense, obtaining information about their properties. [ 1 ]
It was used by Benjamin Kuipers in the book Qualitative reasoning: modeling and simulation with incomplete knowledge to demonstrate how the theory of PDEs can be applied even in situations where only qualitative knowledge is available. | https://en.wikipedia.org/wiki/Qualitative_theory_of_differential_equations |
Quality, cost, delivery ( QCD ), sometimes expanded to quality , cost , delivery , morale, safety ( QCDMS ), [ 1 ] is a management approach originally developed by the British automotive industry . [ 2 ] QCD assess different components of the production process and provides feedback in the form of facts and figures that help managers make logical decisions. By using the gathered data, it is easier for organizations to prioritize their future goals . [ 3 ] QCD helps break down processes to organize and prioritize efforts before they grow overwhelming. [ 4 ]
QCD is a "three-dimensional" approach. If there is a problem with even one dimension, the others will inevitably suffer as well. One dimension cannot be sacrificed for the sake of the other two. [ 5 ]
Quality is the ability of a product or service to meet and exceed customer expectations. It is the result of the efficiency of the entire production process formed of people, material, and machinery. Customer requirements determine the quality scope.
Quality is a competitive advantage ; poor quality often results in bad business. The U.S. business organizations in the 1970s focused more on cost and productivity . That approach led to Japanese businesses capturing a major share of the U.S. market. [ 6 ] It was not until the late 1970s and the beginning of the 1980s that the quality factor drastically shifted and became a strategic approach, created by Harvard professor David Garvin. [ 7 ] This approach focuses on preventing mistakes and puts a great emphasis on customer satisfaction. [ 8 ]
David A. Garvin lists eight dimensions of quality : [ 9 ] [ 10 ]
The quality of a product depends almost entirely on the quality of its raw material. Suppliers and manufacturers must work together to eliminate defects and achieve higher quality. Small and medium-sized enterprises (SMEs) should discuss with their suppliers how quality improvements can affect the overall performance of the supply chain. Quality assurance can reduce testing , scrapping, reworks, and production costs. [ 12 ]
The biggest costs in most businesses are the four basic types of manufacturing costs : [ 5 ] [ 15 ]
In addition, there are business costs that stay the same, regardless of the production output. Business costs include:
Businesses desire to reduce costs to increase their operating profit and bottom line. Cost reduction strategies include:
Logistics are an essential part in providing good customer service on time. [ 19 ] [ 20 ] Logistics customer service can be separated into three elements:
QCD offers a method of measuring both simple and complicated business processes . It also represents a basis for comparing businesses: for example, a business measuring a supplier's delivery performance may compare its findings with the business's own performance. [ 21 ]
The "quality, cost, delivery, and flexibility" (QCDF) approach, includes flexibility as the capacity to adapt to changes or modifications in the input quality, output quality, product specifications, and delivery schedules.
There are seven measures used to increase profitability. [ 22 ]
Not getting things right the first time [ 23 ] means wasted resources, effort and time. This all leads to excessive costs for the company and poor-quality, high-priced products for the customer. NRFT measures the quality of a product and is expressed in “number of defective parts per million”. The number of defective products is divided by the total quantity of finished products. This figure is then multiplied by 10^6 to get the number of defective parts per million. [ 3 ]
NRFT can be measured internally (defective parts identified within the production process) or externally (defective parts identified outside the production process (e.g. by the supplier or the customer). [ 24 ]
DSA analyses how well a supplier delivers what the customer wants and when they want it. The goal is to achieve 100% on-time delivery without any special deliveries or overtime payments, which only increase the delivery cost. DSA measures the actual delivery performance against the planned delivery schedule. [ 3 ] Failed deliveries include:
PP is measured by the time it takes (in staff hours) to produce a good-quality product. Obtaining high PP is only possible when:
The ST ratio shows how quickly a company turns raw materials into finished, ready-to-be-sold products. The quicker the better. A low ST means that the money is tied up in stock, and the company has fewer funds to invest in other parts of its business. [ 3 ]
The OEE shows how well a company uses its equipment and staff.
OEE is calculated on the base of three elements:
VAPP shows how well people are used to turn raw materials into finished goods. In order to calculate VAPP, three things need to be taken into account:
FSU measures the sales revenue generated by a square meter of factory floor space. [ 25 ] Usually to achieve higher FSU the floor space has to be reduced. That means eliminating inventory and reducing the necessary space to a minimum. [ 24 ] | https://en.wikipedia.org/wiki/Quality,_cost,_delivery |
In response theory , the quality of an excited system is related to the number of excitation frequencies to which it can respond. In the case of a homogeneous, isotropic system, the quality is proportional to the FWHM .
This sense of the phrase is the precursor of the usage of the word in music theory . In music theory, quality is the number of harmonics of a fundamental frequency of an instrument (the higher the quality, the richer the sound ).
This physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quality_(physics) |
The rapid development in the multidisciplinary field of tissue engineering has resulted in a variety of new and innovative medicinal products, often carrying living cells, intended to repair, regenerate or replace damaged human tissue. Tissue engineered medicinal products (TEMPs) vary in terms of the type and origin of cells and the product’s complexity. As all medicinal products , the safety and efficacy of TEMPs must be consistent throughout the manufacturing process. Quality control and assurance are of paramount importance and products are constantly assessed throughout the manufacturing process to ensure their safety, efficacy, consistency and reproducibility between batches. [ 1 ] The European Medicines Agency (EMA) is responsible for the development, assessment and supervision of medicines in the EU. The appointed committees are involved in referral procedures concerning safety or the balance of benefit/risk of a medicinal product. In addition, the committees organize inspections with regards to the conditions under which medicinal products are being manufactured. For example, the compliance with good manufacturing practice (GMP) , good clinical practice (GCP) , good laboratory practice (GLP) and pharmacovigilance (PhV) . [ 2 ]
When quality control of TEMPs is considered, a risk assessment needs to be conducted. A risk is defined as a "potentially unfavourable effect that can be attributed to the clinical use of advanced therapy medicinal products (ATMPs) and is of concern to the patient and/or to other populations (e.g. caregivers and off-spring)". [ 3 ] Some risks include immunogenicity , disease transmission, tumor formation, treatment failure, undesirable tissue formation, and inadvertent germ transduction. A risk factor is defined as a "qualitative or quantitative characteristic that contributes to a specific risk following handling and/or administration of an ATMP". [ 3 ] The integration of all available information on risks and risk factors is called risk profiling. Due to the fact that every TEMP is different, the risks associated with each one of them vary and, subsequently, the procedures that must be implemented to ensure its quality are also unique to the product. Once the risks associated with the TEMP are identified, the appropriate tests must be developed and validated accordingly. Thus, there is no standard set of tests for the quality control of TEMPs. The EMA has released a set of regulatory guidelines on the topics to be considered by companies involved in the development and marketing of medicines for use in the European Union. [ 3 ] [ 4 ] [ 5 ] [ 6 ] These guidelines have to be followed in order for the marketing authorization of a product to be issued. Fictitious examples of risk analysis for further elucidation of the process are provided in the EMA guidelines. [ 3 ]
Careful and detailed documentation concerning the characteristics of the starting materials (e.g. history of the cell line derivation and cell banking ) and manufacturing process steps (e.g. procurement of tissue or cells and manipulation) must be maintained. The cellular part of every cell-based medicinal product must be characterized in terms of identity, purity, potency, viability and suitability for the intended use. The non-cellular constituents must be also characterized with regards to their intended function in the final product. For example, scaffolds or membranes that are used to support the cells must be identified and characterized in terms of porosity , density, microscopic structure and particular size. The same requirement for characterization applies for biologically active molecules, such as growth factors or cytokines . [ 5 ]
Proper quality control involves the release testing of the final product through updated and validated methods. The release specifications of the product must be selected on the basis of the parameters defined during the characterization studies and the appropriate release tests must be performed. In case a release test cannot be performed on the final product but only on previous stages of the manufacturing, exceptions can be made after proper justification. However, in these cases adequate quality control has to rise from the manufacturing process. Specifications about the stability of the product, the presence or not of genetically modified cells , structural components and whether it is a combination product must also be defined. [ 5 ] | https://en.wikipedia.org/wiki/Quality_control_in_tissue_engineering |
A quality control system (QCS) refers to a system used to measure and control the quality of moving sheet processes on-line as in the paper produced by a paper machine. Generally, a control system is concerned with measurement and control of one or multiple properties in time in a single dimension. A QCS is designed to continuously measure and control the material properties of the moving sheet in two dimensions: in the machine direction (MD) and the cross-machine direction (CD). The ultimate goal is maintaining a good and homogeneous quality and meeting users' economic goals. A basic quality measurement system generally includes basis weight and moisture profile measurements and in addition average basis weight of the paper web and moisture control related to these variables. Caliper is also one of the basic measurements. [ 1 ] [ 2 ]
Other commonly used continuous measurements include: ash content, color, brightness, smoothness and gloss, coat weight, formation, porosity, fiber orientation, and surface properties (topography). [ 3 ]
QCS is used in paper machines , board machines, tissue machines, pulp drying machines, and other plastic or metal film processes
In modern systems QCS applications can be embedded to distributed control systems .
Sensors measuring the paper quality (online meters) are attached to a sensor platform that move across the web guided by the scanner beam. A typical crossing time for a sensor platform is 10–30 s (an 8 m web, 60 cm/s). The sensor platform scans across the paper web and continuously measures paper characteristics from edge to edge. It can also be directed and stopped to a specific, fixed point on the web to measure the machine-direction (MD) variation at a single point.
The QCS scanner beam is an essential part of a QCS system. Wide machines and accurate profile calculations require beam stability and accuracy of mechanical movement. As high accuracy in demanding and variable conditions is required, the sensitive sensors must be securely fastened. The most important goal is maintaining the exact respective position of the upper and lower measurement platforms in relation to their distance from each other, in MD and in CD . This is achieved through a robust construction and by reducing the effects of temperature and other environmental effects and through a moving mechanism with minimized backlash of the measurement platform.
Usually the scanner beam also contains all the cables and the air, cooling liquid and protection gas pipes. The base of the scanner beam contains elements that dampen vertical vibration.
A basic quality measurement system generally includes basis weight and moisture profile measurements and in addition average basis weight of the paper web and moisture control related to these variables. Caliper is also one of the basic measurements. [ 4 ]
Other commonly used continuous measurements include: ash content, color, brightness, smoothness and gloss, coat weight, formation, porosity, fiber orientation, and surface properties (topography).
Online sensors are set on the scanner beam to scan across the web. Typical crossing time of the web in new systems is 10–30 s (8m web, 60 cm/s). If the web speed is 1200 m/min and web width 8.5 m, the web moves 280 m during a scan, and the sensor moves the same distance diagonally across the web. The measurements are taken on the diagonal line and act as basis for profile (machine and cross-direction) and variation calculations. This value is subject to integration depending on the machine speed. If the measurement signal sampling frequency is in size range 2000/s, then the smallest measurement element is about 0.2 cm in cross-direction. The measurement data is integrated to eliminate a small-scale formation variation from the measurement result.
The measurement value is averaged so that each sensor gives one measurement value per one data box which is typically from 5 mm to one centimeter of web width. For a 1 meter wide web, for instance, 100 - 200 measurement values are taken. These measurement values from a single scan (profile points) are called 'raw profiles'. In modern quality control systems, the width of these data boxes can be changed, and accurate profiles can be formed using several thousands profile data boxes.
Typically, the sensor's output is the instantaneous value, the profile average value and the complete profile.
The requirements for an ideal paper machine online sensor include the following:
the sensor is calibrated to a natural constant during the measurement;
the sensor and the related electronics include fault diagnostics;
digital processing of the signal is possible from the start without destroying the possibility of analyzing large frequency components;
the sensor system does not disturb the production;
the measurements are performed real-time and can be adjusted without delays;
the measurement concerns the entire production, not just small sample values.
It must be possible to distinguish between the machine-directional and cross-directional deviation and the residual deviation as the control system handles these three deviations separately and in different ways. The earlier systems calculated a long-term average profile to filter the profile. As several quality profiles can be adjusted automatically it is important to get the right profile data with high resolution quickly to the control system. This is especially important during changes, after breaks and during grade changes. In advanced systems, algorithms are used to calculate the profile data. | https://en.wikipedia.org/wiki/Quality_control_system_for_paper,_board_and_tissue_machines |
Quality engineering is the discipline of engineering concerned with the principles and practice of product and service quality assurance and control . [ 1 ] In software development, it is the management, development, operation and maintenance of IT systems and enterprise architectures with high quality standard. [ 2 ] [ 3 ] [ 4 ]
Quality engineering is the discipline of engineering that creates and implements strategies for quality assurance in product development and production as well as software development. [ 5 ]
Quality Engineers focus on optimizing product quality which W. Edwards Deming defined as:
Quality engineering body of knowledge includes: [ 6 ]
Auditor : Quality engineers may be responsible for auditing their own companies or their suppliers for compliance to international quality standards such as ISO9000 and AS9100 . They may also be independent auditors under an auditing body. [ 7 ]
Process quality : Quality engineers may be tasked with value stream mapping and statistical process control to determine if a process is likely to produce a defective product. They may create inspection plans and criteria to ensure defective parts are detected prior to completion. [ 8 ]
Supplier quality : Quality engineers may be responsible for auditing suppliers or performing root cause and corrective action at their facility or overseeing such activity to prevent the delivery of defective products.
IT services are increasingly interlinked in workflows across platform boundaries, device and organisational boundaries, for example in cyber-physical systems, business-to-business workflows or when using cloud services. In such contexts, quality engineering facilitates the necessary all-embracing consideration of quality attributes.
In such contexts an "end-to-end" view of quality from management to operation is vital. Quality engineering integrates methods and tools from enterprise architecture -management, Software product management , IT service management , software engineering and systems engineering , and from software quality management and information security management . This means that quality engineering goes beyond the classic disciplines of software engineering, information security management or software product management since it integrates management issues (such as business and IT strategy, risk management, business process views, knowledge and information management, operative performance management), design considerations (including the software development process , requirements analysis , software testing ) and operative considerations (such as configuration, monitoring, IT service management ). In many of the fields where it is used, quality engineering is closely linked to compliance with legal and business requirements, contractual obligations and standards. As far as quality attributes are concerned, reliability, security and safety of IT services play a predominant role.
In quality engineering, quality objectives are implemented in a collaborative process. This process requires the interaction of largely independent actors whose knowledge is based on different sources of information.
Quality objectives describe basic requirements for software quality . In quality engineering they often address the quality attributes of availability, security, safety, reliability and performance. With the help of quality models like ISO/IEC 25000 and methods like the Goal Question Metric approach it is possible to attribute metrics to quality objectives. This allows measuring the degree of attainment of quality objectives. This is a key component of the quality engineering process and, at the same time, is a prerequisite for its continuous monitoring and control. To ensure effective and efficient measuring of quality objectives the integration of core numbers, which were identified manually (e.g. by expert estimates or reviews), and automatically identified metrics (e.g. by statistical analysis of source codes or automated regression tests) as a basis for decision-making is favourable. [ 9 ]
Composite indicators are increasingly used in quality engineering to summarize various software quality metrics into a single score. The Quality Engineering Score (QE Score) is one such example, combining multiple quality dimensions into a continuously updated indicator to support monitoring and decision-making. The approach is publicly documented and has been presented at professional conferences such as the French Software Testing Days. [ 10 ] [ 11 ]
The end-to-end quality management approach to quality engineering requires numerous actors with different responsibilities and tasks, different expertise and involvement in the organisation.
Different roles involved in quality engineering:
Typically, these roles are distributed over geographic and organisational boundaries. Therefore, appropriate measures need to be taken to coordinate the heterogeneous tasks of the various roles in quality engineering and to consolidate and synchronize the data and information necessary to fulfill the tasks, and make them available to each actor in an appropriate form.
Knowledge management plays an important part in quality engineering. [ 12 ] The quality engineering knowledge base comprises manifold structured and unstructured data , ranging from code repositories via requirements specifications, standards, test reports and enterprise architecture models to system configurations and runtime logs. Software and system models play an important role in mapping this knowledge. The data of the quality engineering knowledge base are generated, processed and made available both manually as well as tool-based in a geographically, organisationally and technically distributed context. Of prime importance is the focus on quality assurance tasks, early recognition of risks, and appropriate support for the collaboration of actors.
This results in the following requirements for a quality engineering knowledge base:
The quality engineering process comprises all tasks carried out manually and in a (semi-)automated way to identify, fulfil and measure any quality features in a chosen context. The process is a highly collaborative one in the sense that it requires interaction of actors, widely acting independently from each other.
The quality engineering process has to integrate any existing sub-processes that may comprise highly structured processes such as IT service management and processes with limited structure such as agile software development . Another important aspect is change-driven procedure, where change events, such as changed requirements are dealt with in the local context of information and actors affected by such change. A pre-requisite for this is methods and tools, which support change propagation and change handling.
The objective of an efficient quality engineering process is the coordination of automated and manual quality assurance tasks. Code review or elicitation of quality objectives are examples of manual tasks, while regression tests and the collection of code metrics are examples for automatically performed tasks. The quality engineering process (or its sub-processes) can be supported by tools such as ticketing systems or security management tools.
Associations | https://en.wikipedia.org/wiki/Quality_engineering |
The quality intellectual property metric (QIP) is an international standard, developed by Virtual Socket Interface Alliance ( VSIA ) [ 1 ] for measuring Intellectual Property (IP) or Silicon intellectual property (SIP) quality and examining the practices used to design, integrate and support the SIP. SIP hardening is required to facilitate the reuse of IP in integrated circuit design .
Many computer processors use a system-on-a-chip (SoC) design, which is intended to include all of a device's functions on a single chip. As a result, these chips need to include numerous technical standards that the device will use. One solution to designing such a chip is the reuse of high quality IP. Reusing IP from others means that the chip designer does not need to redesign these elements. IP quality is the key to successful SoC designs, but it is one of the SoC’s most challenging problems.
Hong Kong Science and Technology Parks Corporation ( HKSTP ) and Hong Kong University of Science and Technology ( HKUST ) started to develop a SIP verification and quality measures framework in 2005, based on QIP metric. The objective is to develop a technical framework for SIP quality measures and evaluation based on QIP. Third-party SIP evaluation service is provided by HKSTP , so that IP integrators can know the quality of their desired SIP cores. [ 3 ]
Integrated circuit (IC) designers have developed their own IC design , and such design can be reused by other IC designers. Other IC designers can reduce their risk of IC design , because such parts of design (IP cores) are well established and known to work well. But other IC designers do not know the quality of such IP cores provided. The original IP providers cannot provide the IP cores to other IC designers for evaluation, as it could allow competitors to steal their technologies. The solution is to have a third-party review the quality and value of the IP.
HKSTP is the third party between them and can act as a judge body, to evaluate the quality of IP cores of various IP providers, based on the public standard QIP metric . Other IP designers can know and choose their IP provider, based on the information provided by the third party, HKSTP . The IP providers can also get the evaluation report and improve the quality of their IP cores.
There are soft SIP and hard SIP verification and quality measures. | https://en.wikipedia.org/wiki/Quality_intellectual_property_metric |
Quality of measurements made in chemistry and other areas is an important issue in today's world as measurements influence quality of life , cross-border trade and commerce . In this respect, EN ISO 17025 is the main standard used by testing and calibration laboratories as to appropriately tackle quality management related issues. While chapter four of the standard deals with management requirements, chapter five describes requirements for technical competence. Management related issues can be found in other standards as well e.g. ISO 9000 , however the technical requirements are specific for calibration and testing laboratories.
As for technical competence, the following topics are especially important:
As it is important to have globally harmonised approach to the above-mentioned topics, various training and teaching material have been developed at the European level, as to support implementation e.g. by EURACHEM and by TrainMiC. | https://en.wikipedia.org/wiki/Quality_of_analytical_results |
Quality of experience ( QoE ) is a measure of the delight or annoyance of a customer's experiences with a service (e.g., web browsing, phone call, TV broadcast). [ 1 ] QoE focuses on the entire service experience; it is a holistic concept, similar to the field of user experience , but with its roots in telecommunication. [ 2 ] QoE is an emerging multidisciplinary field based on social psychology , cognitive science , economics , and engineering science , focused on understanding overall human quality requirements.
In 2013, within the context of the COST Action QUALINET , QoE has been defined as: [ 1 ]
The degree of delight or annoyance of the user of an application or service. It results from the fulfillment of his or her expectations with respect to the utility and / or enjoyment of the application or service in the light of the user’s personality and current state.
This definition has been adopted in 2016 by the International Telecommunication Union in Recommendation ITU-T P.10/G.100. [ 3 ] Before, various definitions of QoE had existed in the domain, with the above-mentioned definition now finding wide acceptance in the community.
QoE has historically emerged from Quality of Service (QoS), which attempts to objectively measure service parameters (such as packet loss rates or average throughput ). QoS measurement is most of the time not related to a customer, but to the media or network itself. QoE however is a purely subjective measure from the user's perspective of the overall quality of the service provided, by capturing people's aesthetic and hedonic needs. [ 4 ]
QoE looks at a vendor's or purveyor's offering from the standpoint of the customer or end user, and asks, "What mix of goods, services, and support, do you think will provide you with the perception that the total product is providing you with the experience you desired and/or expected?" It then asks, "Is this what the vendor/purveyor has actually provided?" If not, "What changes need to be made to enhance your total experience?" In short, QoE provides an assessment of human expectations, feelings, perceptions, cognition and satisfaction with respect to a particular product, service or application. [ 5 ]
QoE is a blueprint of all human subjective and objective quality needs and experiences arising from the interaction of a person with technology and with business entities in a particular context. [ 4 ] Although QoE is perceived as subjective, it is an important measure that counts for customers of a service. Being able to measure it in a controlled manner helps operators understand what may be wrong with their services and how to improve them.
QoE aims at taking into consideration every factor that contributes to a user's perceived quality of a system or service. This includes system, human and contextual factors. [ 6 ] The following so-called "influence factors" have been identified and classified by Reiter et al.: [ 6 ]
Studies in the field of QoE have typically focused on system factors, primarily due to its origin in the QoS and network engineering domains. Through the use of dedicated test laboratories, the context is often sought to be kept constant.
QoE is strongly related to but different from the field of User Experience (UX), which also focuses on users' experiences with services. Historically, QoE has emerged from telecommunication research, while UX has its roots in Human–Computer Interaction . [ 2 ] Both fields can be considered multi-disciplinary. In contrast to UX, the goal of improving QoE for users was more strongly motivated by economic needs. [ 7 ]
Wechsung and De Moor identify the following key differences between the fields: [ 2 ]
Historical lack of theoretical frameworks
Theoretic background in hedonic psychology
Empirical– positivist research
Interpretative and constructivist research
As a measure of the end-to-end performance at the service level from the user's perspective, QoE is an important metric for the design of systems and engineering processes. This is particularly relevant for video services because – due to their high traffic demands –, bad network performance may highly affect the user's experience. [ 8 ] [ 9 ] So, when designing systems, the expected output, i.e. the expected QoE, is often taken into account – also as a system output metric and optimization goal.
To measure this level of QoE, human ratings can be used. The mean opinion score (MOS) is a widely used measure for assessing the quality of media signals. It is a limited form of QoE measurement, relating to a specific media type, in a controlled environment and without explicitly taking into account user expectations. The MOS as an indicator of experienced quality has been used for audio and speech communication, as well as for the assessment of quality of Internet video, television and other multimedia signals, [ 10 ] and web browsing. [ 11 ] Due to inherent limitations in measuring QoE in a single scalar value, the usefulness of the MOS is often debated. [ 12 ]
Subjective quality evaluation requires a lot of human resources, establishing it as a time-consuming process. Objective evaluation methods can provide quality results faster, but require dedicated computing resources. Since such instrumental video quality algorithms are often developed based on a limited set of subjective data, their QoE prediction accuracy may be low when compared to human ratings.
QoE metrics are often measured at the end devices and can conceptually be seen as the remaining quality after the distortion introduced during the preparation of the content and the delivery through the network, until it reaches the decoder at the end device. There are several elements in the media preparation and delivery chain, and some of them may introduce distortion. This causes degradation of the content, and several elements in this chain can be considered as "QoE-relevant" for the offered services. The causes of degradation are applicable for any multimedia service, that is, not exclusive to video or speech. Typical degradations occur at the encoding system ( compression degradation ), transport network, access network (e.g., packet loss or packet delay ), home network (e.g. WiFi performance) and end device (e.g. decoding performance).
Several QoE-centric network management and bandwidth management solutions have been proposed, which aim to improve the QoE delivered to the end-users. [ 13 ] [ 14 ] [ 15 ] [ 16 ]
When managing a network, QoE fairness may be taken into account in order to keep the users sufficiently satisfied (i.e., high QoE) in a fair manner. From a QoE perspective, network resources and multimedia services should be managed in order to guarantee specific QoE levels instead of classical QoS parameters, which are unable to reflect the actual delivered QoE. A pure QoE-centric management is challenged by the nature of the Internet itself, as the Internet protocols and architecture were not originally designed to support today's complex and high demanding multimedia services.
As an example for an implementation of QoE management, network nodes can become QoE-aware by estimating the status of the multimedia service as perceived by the end-users. [ 17 ] This information can then be used to improve the delivery of the multimedia service over the network and proactively improve the users' QoE. [ 18 ] This can be achieved, for example, via traffic shaping . QoE management gives the service provider and network operator the capability to minimize storage and network resources by allocating only the resources that are sufficient to maintain a specific level of user satisfaction.
As it may involve limiting resources for some users or services in order to increase the overall network performance and QoE, the practice of QoE management requires that net neutrality regulations are considered. [ 19 ] | https://en.wikipedia.org/wiki/Quality_of_experience |
Quantal neurotransmitter release is the process by which neurons communicate by releasing neurotransmitters in discrete, measurable units known as quanta . Each quantum represents the contents of a single synaptic vesicle , which fuses with the presynaptic membrane to release neurotransmitters into the synaptic cleft . This process is tightly regulated by calcium ion signaling and specialized SNARE protein complexes that enable vesicle docking and fusion. Following release, synaptic vesicles are recycled through multiple pathways to maintain synaptic function. Disruptions in this mechanism are linked to neurological disorders such as autism spectrum disorder , Alzheimer's disease , and myasthenia gravis .
Neurotransmitters are released into the synapse in small packages called quanta , which are stored inside structures called synaptic vesicles. One quantum generates a miniature end plate potential (MEPP) which is the smallest amount of stimulation that one neuron can send to another neuron. [ 1 ] Quantal release is the mechanism by which most traditional endogenous neurotransmitters are transmitted throughout the body. The aggregate sum of many MEPPs is an end plate potential (EPP). A normal end plate potential usually causes the postsynaptic neuron to reach its threshold of excitation and elicit an action potential . [ 1 ] Electrical synapses do not use quantal neurotransmitter release and instead use gap junctions between neurons to send current flows between neurons. The goal of any synapse is to produce either an excitatory postsynaptic potential (EPSP) or an inhibitory postsynaptic potential (IPSP), which generate or repress the expression, respectively, of an action potential in the postsynaptic neuron. It is estimated that an action potential will trigger the release of approximately 20% of an axon terminal 's neurotransmitter load. [ 2 ]
Neurotransmitters are synthesized in the axon terminal where they are stored in vesicles. These neurotransmitter-filled vesicles are the quanta that will be released into the synapse. Quantal vesicles release their contents into the synapse by binding to the presynaptic membrane and combining their phospholipid bilayers . Individual quanta may randomly diffuse into the synapse and cause a subsequent MEPP. Spontaneous release happens randomly, without being triggered by a signal or action potential.
Spontaneous neurotransmitter release occurs randomly, independent of Ca²⁺ influx, while evoked release is action potential-dependent and triggered by calcium channel activation. The differential regulation of these two forms of release contributes to synaptic plasticity and fine-tuning of neuronal communication. [ 3 ]
Calcium ion signaling to the axon terminal is the usual signal for presynaptic release of neurotransmitters. Calcium ion diffusion into the presynaptic membrane signals the axon terminal to release quanta to generate either an IPSP or EPSP in the postsynaptic membrane. Different neurotransmitters cause different effects on the postsynaptic neuron, either exciting or inhibiting it. Action potentials that transmit down to the axon terminal will depolarize the terminal's membrane and cause a conformational change in the membrane's calcium ion channels. These calcium channels will adopt an "open" configuration that will allow only calcium ions to enter the axon terminal. The influx of calcium ions will further depolarize the interior of the axon terminal and will signal the quanta in the axon terminal to bind to the presynaptic membrane. [ 1 ] Once bound, the vesicles will fuse into the membrane and the neurotransmitters will be released into the membrane by exocytosis .
When an action potential reaches the axon terminal, it causes calcium ions to flow into the neuron through voltage-gated calcium channels. These calcium ions bind to a protein called synaptotagmin , which acts as a calcium sensor. Once bound to calcium, synaptotagmin interacts with the SNARE complex , a group of proteins that includes synaptobrevin , syntaxin , and SNAP-25 . Together, these proteins pull the synaptic vesicle close to the presynaptic membrane and promote membrane fusion. This fusion releases the vesicle’s neurotransmitter content into the synaptic cleft in a process known as exocytosis . The SNARE complex ensures that neurotransmitter release is rapid and tightly controlled, allowing neurons to communicate with high precision. [ 4 ]
Neurotransmitter release occurs in two forms: spontaneous release and evoked release. Spontaneous release happens without any stimulation from an action potential . Instead, single vesicles randomly fuse with the presynaptic membrane, likely due to baseline activity in the nerve terminal . This form of release plays a role in maintaining synaptic structure and modulating baseline neuronal activity. In contrast, evoked release is triggered by an action potential. When the action potential reaches the axon terminal, it opens voltage-gated calcium channels , allowing calcium ions to enter. The sudden increase in intracellular calcium concentration activates the SNARE complex and leads to rapid, synchronized fusion of multiple vesicles. Evoked release is responsible for fast synaptic transmission and is essential for most forms of information transfer between neurons. [ 5 ]
While the basic calcium-triggered mechanism for evoked release is well understood, the precise signaling hierarchy among calcium channels and receptors in the presynaptic membrane remains under investigation. Research suggests that different calcium channel types vary in their excitability and efficiency, with certain channels being preferentially activated to regulate the strength and timing of quantal release. [ 6 ]
Estimating the time course of quantal release—how quickly and reliably vesicles fuse after stimulation—has been a valuable tool for studying synaptic function. While this approach is not equally effective across all types of synapses, it has provided insights into how the kinetics of release can vary depending on presynaptic architecture and receptor distribution. [ 7 ]
Synaptic vesicles are organized into functionally distinct pools that regulate neurotransmitter availability during synaptic activity:
These vesicle pools work together to ensure reliable neurotransmission. The Readily Releasable Pool (RRP) supports fast, immediate communication by providing vesicles that are already docked at the membrane and ready to fuse. During sustained or intense neural activity, vesicles from the Reserve Pool are recruited to maintain neurotransmitter output. After vesicles release their contents, they enter the Recycling Pool , where they are reprocessed and prepared for future use. This dynamic system allows the synapse to quickly adapt to changing activity levels and maintain efficiency. Disruption in vesicle pool regulation can impair synaptic strength and contribute to neurological dysfunction. [ 8 ]
As described above, the synaptic vesicle will remain fused to the presynaptic membrane after its neurotransmitter contents have been released into the synapse. If vesicles weren’t recycled, the axon terminal membrane would keep expanding and disrupt normal synaptic function. The axon terminal compensates for this problem by reuptaking the vesicle by endocytosis and reusing its components to form new synaptic vesicles. [ 1 ] The exact mechanism and signaling cascade which triggers synaptic vesicle recycling is still unknown.
No one method of synaptic vesicle recycling seems to hold true in all scenarios, which suggests the existence of multiple pathways for synaptic vesicle recycling. Multiple proteins have been linked with synaptic vesicle reuptake and then subsequently been linked to different synaptic vesicle recycling pathways. Clathrin-mediated endocytosis (CME) and activity-dependent bulk endocytosis (ADBE) are the two most predominant forms of synaptic vesicle recycling, with ADBE being more active during periods of high neuronal activity and CME being active for long periods of time after neuronal activity has ceased. [ 9 ]
Synaptic vesicles are retrieved through two primary pathways:
The interaction between dynamin and syndapin, as described by Clayton et al. (2009), plays a key role in regulating both CME and ADBE.
This regulation is activity-dependent: during low-frequency stimulation, CME ensures precise vesicle recycling, while during intense neuronal activity, ADBE provides a rapid, large-scale retrieval system to prevent membrane buildup. The balance between these two pathways allows neurons to adapt to different levels of synaptic demand and maintain transmission fidelity across a wide range of firing conditions. [ 10 ]
Disruptions in quantal neurotransmitter release are implicated in several neurological disorders:[6]
Understanding these processes at the molecular level helps in the development of therapeutic interventions targeting synaptic dysfunction.
Disruptions in quantal neurotransmitter release are implicated in several neurological and neurodevelopmental disorders. In autism spectrum disorder (ASD), altered synaptic vesicle cycling and imbalances in spontaneous vs. evoked neurotransmitter release have been linked to abnormal neuronal connectivity and impaired social behavior . Studies in model organisms suggest that synaptic dysfunction may underlie both the cognitive symptoms and the frequent co-occurrence of sleep disturbances observed in ASD patients. [ 11 ]
In neurodegenerative diseases such as Alzheimer’s and Parkinson’s , evidence shows that impaired vesicle trafficking and neurotransmitter release contribute to synaptic failure and progressive neuronal loss. Early disruption in synaptic function is now recognized as a hallmark of these diseases, often preceding large-scale cell death .
Additionally, in myasthenia gravis , a disorder of the neuromuscular junction, reduced quantal content and impaired synaptic transmission lead to characteristic muscle weakness and fatigue. Understanding the molecular underpinnings of quantal release has helped guide therapeutic strategies targeting synaptic efficiency and receptor sensitivity . [ 11 ] | https://en.wikipedia.org/wiki/Quantal_neurotransmitter_release |
Quantasomes are particles found in the thylakoid membrane of chloroplasts in which photosynthesis takes place. They are embedded in a paracrystalline array on the surface of thylakoid discs in chloroplasts. They are composed of lipids and proteins that include various photosynthetic pigments and redox carriers. For this reason they are considered to be photosynthetic units. They occur in 2 sizes: the smaller quantasome is thought to represent the site of photosystem I, the larger to represent the site of photosystem II. [ clarification needed ]
[ 1 ] [ 2 ] [ 3 ] [ 4 ]
This photosynthesis article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantasome |
Quantemol Ltd is based in University College London initiated by Professor Jonathan Tennyson FRS and Dr. Daniel Brown in 2004. The company initially developed a unique software tool, Quantemol-N , which provides full accessibility to the highly sophisticated UK molecular R-matrix codes, used to model electron polyatomic molecule interactions. Since then Quantemol has widened to further types of simulation, with plasmas and industrial plasma tools, in Quantemol-VT in 2013 and launched in 2016 a sustainable database Quantemol-DB , representing the chemical and radiative transport properties of a wide range of plasmas.
The Quantemol-N software system has been developed to simplify use of UK R-matrix codes . It provides an interface for non specialists to perform ab initio electron-molecule scattering calculations. Quantemol-N calculates a variety of observables for electron molecule collisions including:
Quantemol-N is capable of tackling a variety of problems;
A study on the key benchmark molecule; water, gave results more accurate than obtainable experimentally ( Faure et al. 2004 ).
Experimentally, there are problems measuring large cross sections at low angles; this applies to any molecule with a large dipole moment. Being a simulation, this is not a problem for Quantemol-N.
Quantemol-Electron Collisions is a python-based software enabling calculations of electron-molecule scattering cross sections using a suite of up to date R-matrix codes (UKRMol+) and other methods such as Binary Encounter Bethe (BEB) model, BEf- scaling and dissociative electron attachment cross-section estimation. It was launched in 2019 and its major differences from Quantemol-N are the use of UKRMol+ instead of UKRMol and utilising Molpro software for molecular target setups. These changes resulted in higher accuracy of calculations and improved usability as molecular geometry optimisation/generation and symmetry identification is performed by Molpro.
Quantemol-EC calculates a variety of observables for electron molecule collisions including:
In the same way as Quantemol-N, Quantemol-EC can be used for closed-shell and open-shell molecules, radicals, neutral and positively charged species.
For resonance fits:
For calculating electron attachment:
For calculating Binary Encounter Bethe (BEB) model:
For calculating BE-f Scaling:
Quantemol-Virtual Tool is an expert software system for the simulation of industrial plasma processing tools. Q-VT builds upon the comprehensively validated Hybrid Plasma Equipment Model (HPEM) codes developed by renowned plasma physicist Professor Mark Kushner for simulating non-equilibrium low pressure (up to 1 Torr) plasma processes. Q-VT includes an intuitive user interface, data visualisation and analysis capabilities, and convenient job/batch management.
Applications include:
What Q-VT can model:
Benefits of Q-VT
The Quantemol database ( QDB or Quantemol-DB ) is a database of plasma processes developed by Quantemol Ltd at University College London in 2016. The database contains chemistry data for plasma chemistry modelling with pre-assembled and validated chemistry sets, and is updated by Quantemol and contributing users. A peer-reviewed article detailing the database and service was published in 2017. [ 1 ] One of the most challenging aspects in plasma modelling is insufficient chemistry data. The purpose of QDB is to provide a forum for collaborative effort between academia and industrial research to access, compare and improve the understanding of plasma chemistry sets influencing plasma behaviour.
The principles established for the validation of chemistry sets are that:
This methodology is specifically applied to atomic and molecular calculations using the principles established in the publication "Uncertainty Estimates of Theoretical Atomic and Molecular Data", which was produced for the International Atomic Energy Agency and focused on "data that are most important for high-temperature plasma modeling" with the "ultimate goal to develop guidelines for self-validation of computational theory for A+M [Atomic and Molecular] processes".
It is recognised that while the validation of chemistry sets directly may still be uncertain, the validation of data produced by models using this data will often be more easily obtained.
QDB users are invited to validate chemistry sets either directly or by validating the results of models which use these chemistry sets as inputs. Validation of the chemistry sets provided in the database will be based on the foundations of Uncertainty Quantification for calculations of complex systems. [ 2 ]
For chemistry simulation, the scaling law based on the parameter study is a common methodology for this validation. [ 3 ] For higher dimensional simulation, the behaviour of the species and the surface will be used for comparison. [ 4 ]
Referencing is provided for users downloading chemistry sets, to ensure that relevant citations to chemistry set and validating experiments are included and can be used for publications.
Rate coefficients of each reaction are included in the validated chemistry set for a similar range of temperature and pressure.
The main validation method for individual reactions is compared with alternative theoretical calculations/estimations and experimental measurements. For unknown reactions different calculation methods are used:
The Application Programming Interface (API) is a set of protocols and tools for linking the database with plasma modelling software Quantemol-VT. An API specifies how software components should interact and APIs are used when chemistries can be accessible in the graphical user interface (GUI) of the plasma modelling software.
The database has a library of sticking coefficients for atomic oxygen, atomic fluorine, fluorocarbons, and silane radicals. For surface mechanisms such as specific etches, the database provides a set of individual reactions with their associated probabilities. For energy-dependent reactions, the formula and the value of the used parameters are provided.
This application helps to gather data which is already in Quantemol-DB related to feedstock gases of the plasma and assemble a new chemistry set and preferred format for downloading or running a Global Model or Boltzmann Solver.
The online global model calculates the reactor averaged particle densities and the electron temperature for a given set of process parameters in plasma.
The model is solving equations:
Particle density balance for heavy species
Charge neutrality
Electron energy density balance
The output includes volume average densities of species and electron temperature .
Calculations can be set up for both pre-assembled and self-generated chemistry sets using the Dynamic Chemistry app.
Detailed documentation can be found here .
The Boltzmann Solver is based on the formalism described in S. D. Rockwood, " Elastic and Inelastic Cross Sections for Electron-Hg Scattering from Hg Transport Data", Physical Review A 8, 2348-2358 (1973) and it was extended to a non-uniform energy grid.
The solver calculates EEDFs, effective electron temperature, and rate coefficients for electron collisions in the chemistry set for a gas temperature of choice, suitable for discharges with non- Maxwellian distributions .
Calculations can be set up for both pre-assembled and self-generated chemistry sets using the Dynamic Chemistry app. | https://en.wikipedia.org/wiki/Quantemol-DB |
An interferon-gamma release assay ( IGRA ) is a diagnostic tool for indicating a latent tuberculosis infection (LTBI). IGRAs are surrogate markers of Mycobacterium tuberculosis infection and indicate a cellular immune response to M. tuberculosis if the latter is present.
IGRAs cannot distinguish between latent infection and active tuberculosis (TB) disease, and should not be used as a sole method for diagnosis of active TB, which is a microbiological diagnosis. A positive IGRA result may not necessarily indicate TB infection, but can also be caused by infection with non-tuberculous mycobacteria. A negative IGRA does not rule out active TB disease; a number of studies have shown that up to a quarter of patients with active TB have negative IGRA results.
Because IGRAs are not affected by Bacille Calmette–Guérin (BCG) vaccination status, IGRAs are useful for evaluation of LTBI in BCG-vaccinated individuals, particularly in settings where BCG vaccination is administered after infancy or multiple (booster) BCG vaccinations are given. In contrast, the specificity of tuberculin skin test (TST) varies depending on timing of BCG and whether repeated (booster) vaccinations are given.
Quantiferon , also known as QFT, is the registered trademark of an interferon gamma release assay (IGRA) for tuberculosis diagnosis manufactured by QIAGEN . The QFT-GIT assay is an ELISA -based, whole-blood test that uses peptides from three TB antigens ( ESAT-6 , CFP-10 , and TB7.7 ) in an in-tube format. The result is reported as quantification of IFN-gamma in international units (IU) per mL. An individual is considered positive for M. tuberculosis infection if the IFN-gamma response to TB antigens is above the test cut-off (after subtracting the background IFN-gamma response in the negative control).
Since IGRAs are more costly and technically complex to do than the Mantoux test , in their 2011 policy statement, the WHO did not recommend replacing the Mantoux test by IGRAs as a public health intervention in low- and middle-income countries. [ 1 ]
Quantiferon-TB Gold In-Tube (QFT-GIT) , the third generation test, has replaced Quantiferon-TB (QFT) and Quantiferon-Gold, which are no longer marketed.
According to the U.S. Centers for Disease Control , [ 2 ] in 2001, the Quantiferon-TB test (QFT) was approved by the Food and Drug Administration (FDA) as an aid for detecting latent Mycobacterium tuberculosis infection. This test is an in vitro diagnostic aid that measures a component of cell-mediated immune reactivity to M. tuberculosis . The test is based on the quantification of interferon-gamma (IFN-γ) released from sensitized lymphocytes in whole blood incubated overnight with purified protein derivative (PPD) from M. tuberculosis and control antigens.
Tuberculin skin testing (TST) has been used for years as an aid in diagnosing latent tuberculosis infection (LTBI) and includes measurement of the delayed type hypersensitivity response 48–72 hours after intradermal injection of PPD. TST and QFT do not measure the same components of the immunologic response and are not interchangeable. Assessment of the accuracy of these tests is limited by lack of a standard for confirming LTBI. [ citation needed ]
As a diagnostic test, QFT:
Compared with TST, QFT results are less subject to reader bias and error. In a CDC-sponsored multicenter trial, QFT and TST results were moderately concordant (overall kappa value = 0.60). The level of concordance was adversely affected by prior bacille Calmette-Guérin (BCG) vaccination , immune reactivity to nontuberculous mycobacteria (NTM), and a prior positive TST. [ 3 ] In addition to the multicenter study, two other published studies have demonstrated moderate concordance between TST and QFT. [ 4 ] [ 5 ] However, one of the five sites involved in the CDC study reported less agreement. [ 6 ] Although there have been studies confirming the increased future risk of active TB in individuals with positive TST, the same was not true for those with a positive IGRA result. A recently published study [ 7 ] demonstrated that a positive IGRA result is predictive of future active TB risk. Moreover, IGRA was at least as sensitive and was more specific compared to traditional TST. In this study of immunocompetent recently exposed close contacts of active TB cases, the progression rate to active disease among untreated QFT positive individuals was significantly greater than for untreated TST positives (14.6% versus 2.3%). Although the numbers were small, all of the close contacts who went on to develop active TB were QFT positive, but only 83% were TST positive. [ citation needed ]
As noted above, prior BCG vaccination can produce false positive TST results. In a study of military personnel returning from missions, about one-half of the positive TSTs were falsely positive. [ 8 ] In a more recent study of military returning from missions, Franken et al. [ 9 ] reported evidence suggesting false positive TST results are common and that QFT testing could guide more targeted treatment and alleviate unnecessary anti-tuberculous treatment.
The FDA's cutpoint for a positive result was established at >0.34 International Units/millilitre (IU/ml), though this has proven functionally problematic in low prevalence areas, such as among US and Canadian healthcare workers. In areas of low risk and low prevalence, the positive predictive value of any test is diminished. In the case of serially screened North American healthcare workers, QFT results just above this cutpoint produce false-positive test results that upon repeat testing revert to negative, [ 10 ] where tuberculosis screening is often mandated on an annual basis. [ 11 ] Research at Stanford University and the Veterans Administration has reported the use of a retesting (or borderline) zone below 1.1 IU/ml mitigates 76% of the false-positives, or reversions. [ 12 ] [ 13 ]
Limitations of QFT include the need to draw blood and process it within 16 hours after collection and limited laboratory and clinical experience with the assay. There is need for further study of the utility of QFT in predicting the progression to active tuberculosis, particularly in children and immunocompromised hosts. [ 12 ]
To its disadvantage, QFT can yield false positive results with Mycobacterium szulgai , Mycobacterium kansasii , and Mycobacterium marinum . [ 14 ]
The Quantiferon-TB Gold test (QFT-G) is a whole-blood test for use as an aid in diagnosing Mycobacterium tuberculosis infection, including latent tuberculosis infection (LTBI) and tuberculosis (TB) disease. [ 15 ] This test was approved by the U.S. Food and Drug Administration (FDA) in 2005.
Blood samples are mixed with antigens (substances that can produce an immune response) and controls. For QFT-G, the antigens include mixtures of synthetic peptides representing two M. tuberculosis proteins, ESAT-6 and CFP-10 . After incubation of the blood with antigens for 16 to 24 hours, the amount of interferon-gamma (IFN-gamma) is measured.
If the patient is infected with M. tuberculosis , their white blood cells will release IFN-gamma in response to contact with the TB antigens . The QFT-G results are based on the amount of IFN-gamma that is released in response to the antigens.
Clinical evaluation and additional tests (such as a chest radiograph , sputum smear, and culture) are needed to differentiate between a diagnosis of latent TB or active TB.
Advantages of the test are:
Disadvantages and limitations of the test are:
On 10/10/2007 the US FDA [ 19 ] gave approval for the Quantiferon TB Gold In Tube to be marketed in the US
The FDA states:
Approval for a modification of the Quantiferon-TB gold to an in-tube collection system that consists of three blood collection tubes, nil, tb antigen, and mitogen. The device, as modified, will be marketed under the trade name quantiferon-tb gold in-tube and is indicated for use as an in vitro diagnostic test using a peptide cocktail simulating esat-6, cfp-10 and tb 7.7(p4) proteins to stimulate cells in heparinized whole blood drawn directly into specialized blood collection tubes. Detection of interferon-y by enzyme-linked immunosorbent assay (elisa) is used to identify in vitro responses to these peptide antigens that are associated with mycobacterium tuberculosis infection.
According to the FDA approved package insert [ 20 ] Quantiferon TB Gold In Tube has a consistent specificity of >99% in low risk individuals and a sensitivity as high as 92% in individuals with active disease, depending on setting and extent of disease. The specificity in two studies of a few hundred people is 96-98% in a health immunised population.
The package insert also advises that the kit provides three collection tubes which have had antigens dried onto their walls and that these tubes must be transferred to an incubator within 16 hours of blood collection.
On 25 June 2010, the US Centers for Disease Control and Prevention (CDC) updated the tuberculosis (TB) testing guidelines providing guidance to US public health officials, clinicians, and laboratory workers regarding screening for and diagnosis of TB infection. The updated guidelines provide new direction for TB control in the US. [ 11 ]
Previously, Quantiferon-TB Gold was able to be used in any situation in which the Tuberculin Skin Test (TST) was used, without preference. The 2010 guidelines establish a new benchmark because they recommend IGRAs as the preferred TB testing method in many patients, including those who are BCG vaccinated or are unlikely to return for TST reading. [ citation needed ]
In January 2008 the CDC advised - via their TB Notes Newsletter [ 21 ] - TB controllers and others of a link [ 22 ] to a list of laboratories in the US and Canada offering to perform the Quantiferon Gold test.
The California Tuberculosis Controllers Association have also provided a list of public health laboratories [ 23 ] in California that are testing with Quantiferon. | https://en.wikipedia.org/wiki/Quantiferon |
In mathematics and empirical science , quantification (or quantitation ) is the act of counting and measuring that maps human sense observations and experiences into quantities . Quantification in this sense is fundamental to the scientific method .
Some measure of the undisputed general importance of quantification in the natural sciences can be gleaned from the following comments:
This meaning of quantification comes under the heading of pragmatics . [ clarification needed ]
In some instances in the natural sciences a seemingly intangible concept may be quantified by creating a scale—for example, a pain scale in medical research, or a discomfort scale at the intersection of meteorology and human physiology such as the heat index measuring the combined perceived effect of heat and humidity , or the wind chill factor measuring the combined perceived effects of cold and wind.
In the social sciences , quantification is an integral part of economics and psychology . Both disciplines gather data – economics by empirical observation and psychology by experimentation – and both use statistical techniques such as regression analysis to draw conclusions from it.
In some instances a seemingly intangible property may be quantified by asking subjects to rate something on a scale —for example, a happiness scale or a quality-of-life scale —or by the construction of a scale by the researcher, as with the index of economic freedom . In other cases, an unobservable variable may be quantified by replacing it with a proxy variable with which it is highly correlated—for example, per capita gross domestic product is often used as a proxy for standard of living or quality of life .
Frequently in the use of regression, the presence or absence of a trait is quantified by employing a dummy variable , which takes on the value 1 in the presence of the trait or the value 0 in the absence of the trait.
Quantitative linguistics is an area of linguistics that relies on quantification. For example, [ 7 ] indices of grammaticalization of morphemes , such as phonological shortness, dependence on surroundings, and fusion with the verb, have been developed and found to be significantly correlated across languages with stage of evolution of function of the morpheme.
The ease of quantification is one of the features used to distinguish hard and soft sciences from each other. Scientists often consider hard sciences to be more scientific or rigorous, but this is disputed by social scientists who maintain that appropriate rigor includes the qualitative evaluation of the broader contexts of qualitative data. In some social sciences such as sociology , quantitative data are difficult to obtain, either because laboratory conditions are not present or because the issues involved are conceptual but not directly quantifiable. Thus in these cases qualitative methods are preferred. [ citation needed ] | https://en.wikipedia.org/wiki/Quantification_(science) |
In logic , a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula . For instance, the universal quantifier ∀ {\displaystyle \forall } in the first-order formula ∀ x P ( x ) {\displaystyle \forall xP(x)} expresses that everything in the domain satisfies the property denoted by P {\displaystyle P} . On the other hand, the existential quantifier ∃ {\displaystyle \exists } in the formula ∃ x P ( x ) {\displaystyle \exists xP(x)} expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.
The most commonly used quantifiers are ∀ {\displaystyle \forall } and ∃ {\displaystyle \exists } . These quantifiers are standardly defined as duals ; in classical logic : each can be defined in terms of the other using negation . They can also be used to define more complex quantifiers, as in the formula ¬ ∃ x P ( x ) {\displaystyle \neg \exists xP(x)} which expresses that nothing has the property P {\displaystyle P} . Other quantifiers are only definable within second-order logic or higher-order logics . Quantifiers have been generalized beginning with the work of Andrzej Mostowski and Per Lindström .
In a first-order logic statement, quantifications in the same type (either universal quantifications or existential quantifications) can be exchanged without changing the meaning of the statement, while the exchange of quantifications in different types changes the meaning. As an example, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications.
First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all". However, many natural language quantifiers can only be analyzed in terms of generalized quantifiers .
For a finite domain of discourse D = { a 1 , . . . a n } {\displaystyle D=\{a_{1},...a_{n}\}} , the universally quantified formula ∀ x ∈ D P ( x ) {\displaystyle \forall x\in D\;P(x)} is equivalent to the logical conjunction P ( a 1 ) ∧ . . . ∧ P ( a n ) {\displaystyle P(a_{1})\land ...\land P(a_{n})} .
Dually, the existentially quantified formula ∃ x ∈ D P ( x ) {\displaystyle \exists x\in D\;P(x)} is equivalent to the logical disjunction P ( a 1 ) ∨ . . . ∨ P ( a n ) {\displaystyle P(a_{1})\lor ...\lor P(a_{n})} .
For example, if B = { 0 , 1 } {\displaystyle B=\{0,1\}} is the set of binary digits , the formula ∀ x ∈ B x = x 2 {\displaystyle \forall x\in B\;x=x^{2}} abbreviates 0 = 0 2 ∧ 1 = 1 2 {\displaystyle 0=0^{2}\land 1=1^{2}} , which evaluates to true .
Consider the following statement ( using dot notation for multiplication ):
This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages , this is immediately a problem, since syntax rules are expected to generate finite statements.
The example above is fortunate in that there is a procedure to generate all the conjuncts. However, if the same assertion were to be made about real numbers instead of natural numbers , there would be no way to enumerate all the conjuncts, since real numbers cannot be enumerated . [ dubious – discuss ] Succinct equivalent formulations, which avoid these problems, uses universal quantification :
A similar analysis applies to the disjunction ,
which can be rephrased using existential quantification :
It is possible to devise abstract algebras whose models include formal languages with quantification, but progress has been slow [ clarification needed ] and interest in such algebra has been limited. Three approaches have been devised to date:
The two most common quantifiers are the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is " ∀ ", a rotated letter " A ", which stands for "for all" or "all". The corresponding symbol for the existential quantifier is " ∃ ", a rotated letter " E ", which stands for "there exists" or "exists". [ 1 ] [ 2 ]
An example of translating a quantified statement in a natural language such as English would be as follows. Given the statement, "Each of Peter's friends either likes to dance or likes to go to the beach (or both)", key aspects can be identified and rewritten using symbols including quantifiers. So, let X be the set of all Peter's friends, P ( x ) the predicate " x likes to dance", and Q ( x ) the predicate " x likes to go to the beach". Then the above sentence can be written in formal notation as ∀ x ∈ X , ( P ( x ) ∨ Q ( x ) ) {\displaystyle \forall {x}{\in }X,(P(x)\lor Q(x))} , which is read, "for every x that is a member of X , P applies to x or Q applies to x ".
Some other quantified expressions are constructed as follows,
for a formula P . These two expressions (using the definitions above) are read as "there exists a friend of Peter who likes to dance" and "all friends of Peter like to dance", respectively. Variant notations include, for set X and set members x :
All of these variations also apply to universal quantification.
Other variations for the universal quantifier are
Some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified; for a given mathematical theory, this can be done in several ways:
One can use any variable as a quantified variable in place of any other, under certain restrictions in which variable capture does not occur. Even if the notation uses typed variables, variables of that type may be used.
Informally or in natural language, the "∀ x " or "∃ x " might appear after or in the middle of P ( x ). Formally, however, the phrase that introduces the dummy variable is placed in front.
Mathematical formulas mix symbolic expressions for quantifiers with natural language quantifiers such as,
Keywords for uniqueness quantification include:
Further, x may be replaced by a pronoun . For example,
The order of quantifiers is critical to meaning, as is illustrated by the following two propositions:
This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the order of quantifiers is reversed is different:
This is clearly false; it asserts that there is a single natural number s that is the square of every natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables.
A less trivial example from mathematical analysis regards the concepts of uniform and pointwise continuity, whose definitions differ only by an exchange in the positions of two quantifiers. A function f from R to R is called
In the former case, the particular value chosen for δ can be a function of both ε and x , the variables that precede it.
In the latter case, δ can be a function only of ε (i.e., it has to be chosen independent of x ). For example, f ( x ) = x 2 satisfies pointwise, but not uniform continuity (its slope is unbound). In contrast, interchanging the two initial universal quantifiers in the definition of pointwise continuity does not change the meaning.
As a general rule, swapping two adjacent universal quantifiers with the same scope (or swapping two adjacent existential quantifiers with the same scope) doesn't change the meaning of the formula (see Example here ), but swapping an existential quantifier and an adjacent universal quantifier may change its meaning.
The maximum depth of nesting of quantifiers in a formula is called its " quantifier rank ".
If D is a domain of x and P ( x ) is a predicate dependent on object variable x , then the universal proposition can be expressed as
∀ x ∈ D P ( x ) . {\displaystyle \forall x\!\in \!D\;P(x).}
This notation is known as restricted or relativized or bounded quantification . Equivalently one can write,
∀ x ( x ∈ D → P ( x ) ) . {\displaystyle \forall x\;(x\!\in \!D\to P(x)).}
The existential proposition can be expressed with bounded quantification as
∃ x ∈ D P ( x ) , {\displaystyle \exists x\!\in \!D\;P(x),}
or equivalently
∃ x ( x ∈ D ∧ P ( x ) ) . {\displaystyle \exists x\;(x\!\in \!\!D\land P(x)).}
Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:
¬ ( ∀ x ∈ D P ( x ) ) ≡ ∃ x ∈ D ¬ P ( x ) , {\displaystyle \neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),}
which shows that to disprove a "for all x " proposition, one needs no more than to find an x for which the predicate is false. Similarly,
¬ ( ∃ x ∈ D P ( x ) ) ≡ ∀ x ∈ D ¬ P ( x ) , {\displaystyle \neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),}
to disprove a "there exists an x " proposition, one needs to show that the predicate is false for all x .
In classical logic , every formula is logically equivalent to a formula in prenex normal form , that is, a string of quantifiers and bound variables followed by a quantifier-free formula.
Quantifier elimination is a concept of simplification used in mathematical logic , model theory , and theoretical computer science . Informally, a quantified statement " ∃ x {\displaystyle \exists x} such that ..." can be viewed as a question "When is there an x {\displaystyle x} such that ...?", and the statement without quantifiers can be viewed as the answer to that question. [ 8 ]
One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest.
Every quantification involves one specific variable and a domain of discourse or range of quantification of that variable. The range of quantification specifies the set of values that the variable takes. In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, say, asserting that a predicate holds for some natural number or for some real number . Expository conventions often reserve some variable names such as " n " for natural numbers, and " x " for real numbers, although relying exclusively on naming conventions cannot work in general, since ranges of variables can change in the course of a mathematical argument.
A universally quantified formula over an empty range (like ∀ x ∈ ∅ x ≠ x {\displaystyle \forall x\!\in \!\varnothing \;x\neq x} ) is always vacuously true . Conversely, an existentially quantified formula over an empty range (like ∃ x ∈ ∅ x = x {\displaystyle \exists x\!\in \!\varnothing \;x=x} ) is always false.
A more natural way to restrict the domain of discourse uses guarded quantification . For example, the guarded quantification
means
In some mathematical theories , a single domain of discourse fixed in advance is assumed. For example, in Zermelo–Fraenkel set theory , variables range over all sets. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above, to express
in Zermelo–Fraenkel set theory, one would write
where N is the set of all natural numbers.
Mathematical semantics is the application of mathematics to study the meaning of expressions in a formal language. It has three elements: a mathematical specification of a class of objects via syntax , a mathematical specification of various semantic domains and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones. This article only addresses the issue of how quantifier elements are interpreted.
The syntax of a formula can be given by a syntax tree. A quantifier has a scope , and an occurrence of a variable x is free if it is not within the scope of a quantification for that variable. Thus in
∀ x ( ∃ y B ( x , y ) ) ∨ C ( y , x ) {\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)}
the occurrence of both x and y in C ( y , x ) is free, while the occurrence of x and y in B ( y , x ) is bound (i.e. non-free).
An interpretation for first-order predicate calculus assumes as given a domain of individuals X . A formula A whose free variables are x 1 , ..., x n is interpreted as a Boolean -valued function F ( v 1 , ..., v n ) of n arguments, where each argument ranges over the domain X . Boolean-valued means that the function assumes one of the values T (interpreted as truth) or F (interpreted as falsehood). The interpretation of the formula
∀ x n A ( x 1 , … , x n ) {\displaystyle \forall x_{n}A(x_{1},\ldots ,x_{n})}
is the function G of n -1 arguments such that G ( v 1 , ..., v n -1 ) = T if and only if F ( v 1 , ..., v n -1 , w ) = T for every w in X . If F ( v 1 , ..., v n -1 , w ) = F for at least one value of w , then G ( v 1 , ..., v n -1 ) = F . Similarly the interpretation of the formula
∃ x n A ( x 1 , … , x n ) {\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})}
is the function H of n -1 arguments such that H ( v 1 , ..., v n -1 ) = T if and only if F ( v 1 , ..., v n -1 , w ) = T for at least one w and H ( v 1 , ..., v n -1 ) = F otherwise.
The semantics for uniqueness quantification requires first-order predicate calculus with equality. This means there is given a distinguished two-placed predicate "="; the semantics is also modified accordingly so that "=" is always interpreted as the two-place equality relation on X . The interpretation of
∃ ! x n A ( x 1 , … , x n ) {\displaystyle \exists !x_{n}A(x_{1},\ldots ,x_{n})}
then is the function of n -1 arguments, which is the logical and of the interpretations of
∃ x n A ( x 1 , … , x n ) ∀ y , z ( A ( x 1 , … , x n − 1 , y ) ∧ A ( x 1 , … , x n − 1 , z ) ⟹ y = z ) . {\displaystyle {\begin{aligned}\exists x_{n}&A(x_{1},\ldots ,x_{n})\\\forall y,z&{\big (}A(x_{1},\ldots ,x_{n-1},y)\wedge A(x_{1},\ldots ,x_{n-1},z)\implies y=z{\big )}.\end{aligned}}}
Each kind of quantification defines a corresponding closure operator on the set of formulas, by adding, for each free variable x , a quantifier to bind x . [ 9 ] For example, the existential closure of the open formula n >2 ∧ x n + y n = z n is the closed formula ∃ n ∃ x ∃ y ∃ z ( n >2 ∧ x n + y n = z n ); the latter formula, when interpreted over the positive integers, is known to be false by Fermat's Last Theorem . As another example, equational axioms, like x + y = y + x , are usually meant to denote their universal closure , like ∀ x ∀ y ( x + y = y + x ) to express commutativity .
None of the quantifiers previously discussed apply to a quantification such as
One possible interpretation mechanism can be obtained as follows: Suppose that in addition to a semantic domain X , we have given a probability measure P defined on X and cutoff numbers 0 < a ≤ b ≤ 1. If A is a formula with free variables x 1 ,..., x n whose interpretation is
the function F of variables v 1 ,..., v n then the interpretation of
∃ m a n y x n A ( x 1 , … , x n − 1 , x n ) {\displaystyle \exists ^{\mathrm {many} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}
is the function of v 1 ,..., v n -1 which is T if and only if
P { w : F ( v 1 , … , v n − 1 , w ) = T } ≥ b {\displaystyle \operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\geq b}
and F otherwise. Similarly, the interpretation of
∃ f e w x n A ( x 1 , … , x n − 1 , x n ) {\displaystyle \exists ^{\mathrm {few} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})}
is the function of v 1 ,..., v n -1 which is F if and only if
0 < P { w : F ( v 1 , … , v n − 1 , w ) = T } ≤ a {\displaystyle 0<\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\leq a}
and T otherwise.
A few other quantifiers have been proposed over time. In particular, the solution quantifier, [ 10 ] : 28 noted § ( section sign ) and read "those". For example,
[ § n ∈ N n 2 ≤ 4 ] = { 0 , 1 , 2 } {\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}}
is read "those n in N such that n 2 ≤ 4 are in {0,1,2}." The same construct is expressible in set-builder notation as
{ n ∈ N : n 2 ≤ 4 } = { 0 , 1 , 2 } . {\displaystyle \{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.}
Contrary to the other quantifiers, § yields a set rather than a formula. [ 11 ]
Some other quantifiers sometimes used in mathematics include:
Term logic , also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Term logic treated All , Some and No in the 4th century BC, in an account also touching on the alethic modalities .
In 1827, George Bentham published his Outline of a New System of Logic: With a Critical Examination of Dr. Whately's Elements of Logic , describing the principle of the quantifier, but the book was not widely circulated. [ 12 ]
William Hamilton claimed to have coined the terms "quantify" and "quantification", most likely in his Edinburgh lectures c. 1840. Augustus De Morgan confirmed this in 1847, but modern usage began with De Morgan in 1862 where he makes statements such as "We are to take in both all and some-not-all as quantifiers". [ 13 ]
Gottlob Frege , in his 1879 Begriffsschrift , was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates . He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀ x ~, or contraposition . Frege's treatment of quantification went largely unremarked until Bertrand Russell 's 1903 Principles of Mathematics .
In work that culminated in Peirce (1885), Charles Sanders Peirce and his student Oscar Howard Mitchell independently invented universal and existential quantifiers, and bound variables . Peirce and Mitchell wrote Π x and Σ x where we now write ∀ x and ∃ x . Peirce's notation can be found in the writings of Ernst Schröder , Leopold Loewenheim , Thoralf Skolem , and Polish logicians into the 1950s. Most notably, it is the notation of Kurt Gödel 's landmark 1930 paper on the completeness of first-order logic , and 1931 paper on the incompleteness of Peano arithmetic . Per Martin-Löf adopted a similar notation for dependent products and sums in his intuitionistic type theory , which are conceptually related to quantification.
Peirce's approach to quantification also influenced William Ernest Johnson and Giuseppe Peano , who invented yet another notation, namely ( x ) for the universal quantification of x and (in 1897) ∃ x for the existential quantification of x . Hence for decades, the canonical notation in philosophy and mathematical logic was ( x ) P to express "all individuals in the domain of discourse have the property P ", and "(∃ x ) P " for "there exists at least one individual in the domain of discourse having the property P ". Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell , Quine , and Alonzo Church . In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s.
Around 1895, Peirce began developing his existential graphs , whose variables can be seen as tacitly quantified. Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations.) Peirce's graphical logic has attracted some attention in recent years by those researching heterogeneous reasoning and diagrammatic inference . | https://en.wikipedia.org/wiki/Quantifier_(logic) |
Quantifier elimination is a concept of simplification used in mathematical logic , model theory , and theoretical computer science . Informally, a quantified statement " ∃ x {\displaystyle \exists x} such that ..." can be viewed as a question "When is there an x {\displaystyle x} such that ...?", and the statement without quantifiers can be viewed as the answer to that question. [ 1 ]
One way of classifying formulas is by the amount of quantification . Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest.
A theory has quantifier elimination if for every formula α {\displaystyle \alpha } , there exists another formula α Q F {\displaystyle \alpha _{QF}} without quantifiers that is equivalent to it ( modulo this theory).
An example from mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative: [ 1 ]
∃ x ∈ R . ( a ≠ 0 ∧ a x 2 + b x + c = 0 ) ⟺ a ≠ 0 ∧ b 2 − 4 a c ≥ 0 {\displaystyle \exists x\in \mathbb {R} .(a\neq 0\wedge ax^{2}+bx+c=0)\ \ \Longleftrightarrow \ \ a\neq 0\wedge b^{2}-4ac\geq 0}
Here the sentence on the left-hand side involves a quantifier ∃ x ∈ R {\displaystyle \exists x\in \mathbb {R} } , whereas the equivalent sentence on the right does not.
Examples of theories that have been shown decidable using quantifier elimination are Presburger arithmetic , [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] algebraically closed fields , real closed fields , [ 7 ] [ 8 ] atomless Boolean algebras , term algebras , dense linear orders , [ 7 ] abelian groups , [ 9 ] random graphs , as well as many of their combinations such as Boolean algebra with Presburger arithmetic, and term algebras with queues .
Quantifier eliminator for the theory of the real numbers as an ordered additive group is Fourier–Motzkin elimination ; for the theory of the field of real numbers it is the Tarski–Seidenberg theorem . [ 7 ]
Quantifier elimination can also be used to show that "combining" decidable theories leads to new decidable theories (see Feferman–Vaught theorem ).
If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining α Q F {\displaystyle \alpha _{QF}} for each α {\displaystyle \alpha } ? If there is such a method we call it a quantifier elimination algorithm . If there is such an algorithm, then decidability for the theory reduces to deciding the truth of the quantifier-free sentences . Quantifier-free sentences have no variables, so their validity in a given theory can often be computed, which enables the use of quantifier elimination algorithms to decide validity of sentences.
Various model-theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.
Every first-order theory with quantifier elimination is model complete . Conversely, a model-complete theory, whose theory of universal consequences has the amalgamation property , has quantifier elimination. [ 10 ]
The models of the theory of the universal consequences of a theory T {\displaystyle T} are precisely the substructures of the models of T {\displaystyle T} . [ 10 ] The theory of linear orders does not have quantifier elimination. However the theory of its universal consequences has the amalgamation property.
To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals , that is, show that each formula of the form:
∃ x . ⋀ i = 1 n L i {\displaystyle \exists x.\bigwedge _{i=1}^{n}L_{i}}
where each L i {\displaystyle L_{i}} is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of literals, then if F {\displaystyle F} is a quantifier-free formula, we can write it in disjunctive normal form
⋁ j = 1 m ⋀ i = 1 n L i j , {\displaystyle \bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij},}
and use the fact that
∃ x . ⋁ j = 1 m ⋀ i = 1 n L i j {\displaystyle \exists x.\bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij}}
is equivalent to
⋁ j = 1 m ∃ x . ⋀ i = 1 n L i j . {\displaystyle \bigvee _{j=1}^{m}\exists x.\bigwedge _{i=1}^{n}L_{ij}.}
Finally, to eliminate a universal quantifier
∀ x . F {\displaystyle \forall x.F}
where F {\displaystyle F} is quantifier-free, we transform ¬ F {\displaystyle \lnot F} into disjunctive normal form, and use the fact that ∀ x . F {\displaystyle \forall x.F} is equivalent to ¬ ∃ x . ¬ F . {\displaystyle \lnot \exists x.\lnot F.}
In early model theory, quantifier elimination was used to demonstrate that various theories possess properties like decidability and completeness . A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas. This technique can be used to show that Presburger arithmetic is decidable.
Theories could be decidable yet not admit quantifier elimination. Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the language of its valid formulas is countable , it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula). [ citation needed ]
Example: Nullstellensatz for algebraically closed fields and for differentially closed fields . [ clarification needed ] | https://en.wikipedia.org/wiki/Quantifier_elimination |
In mathematical logic , the quantifier rank of a formula is the depth of nesting of its quantifiers . It plays an essential role in model theory .
The quantifier rank is a property of the formula itself (i.e. the expression in a language). Thus two logically equivalent formulae can have different quantifier ranks, when they express the same thing in different ways.
Let φ {\displaystyle \varphi } be a first-order formula. The quantifier rank of φ {\displaystyle \varphi } , written qr ( φ ) {\displaystyle \operatorname {qr} (\varphi )} , is defined as:
Remarks
For fixed-point logic , with a least fixed-point operator LFP {\displaystyle \operatorname {LFP} } : qr ( [ LFP ϕ ] y ) = 1 + qr ( ϕ ) {\displaystyle \operatorname {qr} ([\operatorname {LFP} _{\phi }]y)=1+\operatorname {qr} (\phi )} .
This mathematical logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantifier_rank |
In analytical chemistry , quantitative analysis is the determination of the absolute or relative abundance (often expressed as a concentration ) of one, several or all particular substance(s) present in a sample. [ 1 ] It relates to the determination of percentage of constituents in any given sample. [ 2 ]
Once the presence of certain substances in a sample is known, the study of their absolute or relative abundance could help in determining specific properties. Knowing the composition of a sample is very important, and several ways have been developed to make it possible, like gravimetric [ 3 ] and volumetric analysis . Gravimetric analysis yields more accurate data about the composition of a sample than volumetric analysis but also takes more time to perform in the laboratory. Volumetric analysis, on the other hand, doesn't take that much time and can produce satisfactory results. Volumetric analysis can be simply a titration based in a neutralization reaction but it can also be a precipitation or a complex forming reaction as well as a titration based in a redox reaction. However, each method in quantitative analysis has a general specification, in neutralization reactions, for example, the reaction that occurs is between an acid and a base, which yields a salt and water, hence the name neutralization. In the precipitation reactions the standard solution is in the most cases silver nitrate which is used as a reagent to react with the ions present in the sample and to form a highly insoluble precipitate . Precipitation methods are often called simply as argentometry . In the two other methods the situation is the same. Complex forming titration is a reaction that occurs between metal ions and a standard solution that is in the most cases EDTA (Ethylene Diamine Tetra Acetic acid). In the redox titration that reaction is carried out between an oxidizing agent and a reduction agent . There are some more methods like Liebig method / Duma's method / Kjeldahl's method and Carius method for estimation of organic compounds.
For example, quantitative analysis performed by mass spectrometry on biological samples can determine, by the relative abundance ratio of specific proteins , indications of certain diseases, like cancer.
The general expression Qualitative Analysis [...] refers to analyses in which substances are identified or classified on the basis of their chemical or physical properties, such as chemical reactivity, solubility, molecular weight, melting point, radioactivity properties (emission, absorption), mass spectra, nuclear half-life, etc. Quantitative Analysis refers to analyses in which the amount or concentration of an analyte may be determined (estimated) and expressed as a numerical value in appropriate units. Qualitative Analysis may take place with Quantitative Analysis, but Quantitative Analysis requires the identification (qualification) of the analyte for which numerical estimates are given.
The term "quantitative analysis" is often used in comparison (or contrast) with "qualitative analysis", which seeks information about the identity or form of substance present. For instance, a chemist might be given an unknown solid sample. They will use "qualitative" techniques (perhaps NMR or IR spectroscopy ) to identify the compounds present, and then quantitative techniques to determine the amount of each compound in the sample. Careful procedures for recognizing the presence of different metal ions have been developed, although they have largely been replaced by modern instruments; these are collectively known as qualitative inorganic analysis . Similar tests for identifying organic compounds (by testing for different functional groups ) are also known.
Many techniques can be used for either qualitative or quantitative measurements. For instance, suppose an indicator solution changes color in the presence of a metal ion. It could be used as a qualitative test: does the indicator solution change color when a drop of sample is added? It could also be used as a quantitative test, by studying the color of the indicator solution with different concentrations of the metal ion. (This would probably be done using ultraviolet-visible spectroscopy .) | https://en.wikipedia.org/wiki/Quantitative_analysis_(chemistry) |
Quantitative biology is an umbrella term encompassing the use of mathematical, statistical or computational techniques to study life and living organisms . The central theme and goal of quantitative biology is the creation of predictive models based on fundamental principles governing living systems . [ 1 ] [ 2 ]
The subfields of biology that employ quantitative approaches include:
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Quantitative_biology |
Quantitative comparative linguistics is the use of quantitative analysis as applied to comparative linguistics . Examples include the statistical fields of lexicostatistics and glottochronology , and the borrowing of phylogenetics from biology.
Statistical methods have been used for the purpose of quantitative analysis in comparative linguistics for more than a century. During the 1950s, the Swadesh list emerged: a standardised set of lexical concepts found in most languages, as words or phrases, that allow two or more languages to be compared and contrasted empirically.
Probably the first published quantitative historical linguistics study was by Sapir in 1916, [ 1 ] while Kroeber and Chretien in 1937 [ 2 ] investigated nine Indo-European (IE) languages using 74 morphological and phonological features (extended in 1939 by the inclusion of Hittite). Ross [ 3 ] in 1950 carried out an investigation into the theoretical basis for such studies. Swadesh, using word lists, developed lexicostatistics and glottochronology in a series of papers [ 4 ] published in the early 1950s but these methods were widely criticised [ 5 ] though some of the criticisms were seen as unjustified by other scholars. Embleton published a book on "Statistics in Historical Linguistics" in 1986 which reviewed previous work and extended the glottochronological method. Dyen, Kruskal and Black carried out a study of the lexicostatistical method on a large IE database in 1992. [ 6 ]
During the 1990s, there was renewed interest in the topic, based on the application of methods of computational phylogenetics and cladistics . Such projects often involved collaboration by linguistic scholars, and colleagues with expertise in information science and/or biological anthropology . These projects often sought to arrive at an optimal phylogenetic tree (or network), to represent a hypothesis about the evolutionary ancestry and perhaps its language contacts. Pioneers in these methods included the founders of CPHL: computational phylogenetics in historical linguistics (CPHL project): Donald Ringe , Tandy Warnow , Luay Nakhleh and Steven N. Evans .
In the mid-1990s a group at Pennsylvania University computerised the comparative method and used a different IE database with 20 ancient languages. [ 7 ] In the biological field several software programs were then developed which could have application to historical linguistics. In particular a group at the University of Auckland developed a method that gave controversially old dates for IE languages. [ 8 ] A conference on "Time-depth in Historical Linguistics" was held in August 1999 at which many applications of quantitative methods were discussed. [ 9 ] Subsequently many papers have been published on studies of various language groups as well as comparisons of the methods.
Greater media attention was generated in 2003 after the publication by anthropologists Russell Gray and Quentin Atkinson of a short study on Indo-European languages in Nature . Gray and Atkinson attempted to quantify, in a probabilistic sense, the age and relatedness of modern Indo-European languages and, sometimes, the preceding proto-languages.
The proceedings of an influential 2004 conference, Phylogenetic Methods and the Prehistory of Languages were published in 2006, edited by Peter Forster and Colin Renfrew .
Computational phylogenetic analyses have been performed for:
The standard method for assessing language relationships has been the comparative method . However this has a number of limitations. Not all linguistic material is suitable as input and there are issues of the linguistic levels on which the method operates. The reconstructed languages are idealized and different scholars can produce different results. Language family trees are often used in conjunction with the method and "borrowings" must be excluded from the data, which is difficult when borrowing is within a family. It is often claimed that the method is limited in the time depth over which it can operate. The method is difficult to apply and there is no independent test. [ 28 ] Thus alternative methods have been sought that have a formalised method, quantify the relationships and can be tested.
A goal of comparative historical linguistics is to identify instances of genetic relatedness amongst languages. [ 29 ] The steps in quantitative analysis are (i) to devise a procedure based on theoretical grounds, on a particular model or on past experience, etc. (ii) to verify the procedure by applying it to some data where there exists a large body of linguistic opinion for comparison (this may lead to a revision of the procedure of stage (i) or at the extreme of its total abandonment) (iii) to apply the procedure to data where linguistic opinions have not yet been produced, have not yet been firmly established or perhaps are even in conflict. [ 30 ]
Applying phylogenetic methods to languages is a multi-stage process: (a) the encoding stage - getting from real languages to some expression of the relationships between them in the form of numerical or state data, so that those data can then be used as input to phylogenetic methods (b) the representation stage - applying phylogenetic methods to extract from those numerical and/or state data a signal that is converted into some useful form of representation, usually two dimensional graphical ones such as trees or networks, which synthesise and "collapse" what are often highly complex multi dimensional relationships in the signal (c) the interpretation stage - assessing those tree and network representations to extract from them what they actually mean for real languages and their relationships through time. [ 31 ]
An output of a quantitative historical linguistic analysis is normally a tree or a network diagram. This allows summary visualisation of the output data but is not the complete result. A tree is a connected acyclic graph, consisting of a set of vertices (also known as "nodes") and a set of edges ("branches") each of which connects a pair of vertices. [ 32 ] An internal node represents a linguistic ancestor in a phylogenic tree or network. Each language is represented by a path, the paths showing the different states as it evolves. There is only one path between every pair of vertices. Unrooted trees plot the relationship between the input data without assumptions regarding their descent. A rooted tree explicitly identifies a common ancestor, often by specifying a direction of evolution or by including an "outgroup" that is known to be only distantly related to the set of languages being classified. Most trees are binary, that is a parent has two children. A tree can always be produced even though it is not always appropriate. A different sort of tree is that only based on language similarities / differences. In this case the internal nodes of the graph do not represent ancestors but are introduced to represent the conflict between the different splits ("bipartitions") in the data analysis. The "phenetic distance" is the sum of the weights (often represented as lengths) along the path between languages. Sometimes an additional assumption is made that these internal nodes do represent ancestors.
When languages converge, usually with word adoption ("borrowing"), a network model is more appropriate. There will be additional edges to reflect the dual parentage of a language. These edges will be bidirectional if both languages borrow from one another. A tree is thus a simple network, however there are many other types of network. A phylogentic network is one where the taxa are represented by nodes and their evolutionary relationships are represented by branches. [ 33 ] Another type is that based on splits, and is a combinatorial generalisation of the split tree. A given set of splits can have more than one representation thus internal nodes may not be ancestors and are only an "implicit" representation of evolutionary history as distinct from the "explicit" representation of phylogenetic networks. In a splits network the phrenetic distance is that of the shortest path between two languages. A further type is the reticular network which shows incompatibilities (due to for example to contact) as reticulations and its internal nodes do represent ancestors. A network may also be constructed by adding contact edges to a tree. The last main type is the consensus network formed from trees. These trees may be as a result of bootstrap analysis or samples from a posterior distribution.
Change happens continually to languages, but not usually at a constant rate, [ 34 ] with its cumulative effect producing splits into dialects, languages and language families. It is generally thought that morphology changes slowest and phonology the quickest. As change happens, less and less evidence of the original language remains. Finally there could be loss of any evidence of relatedness. Changes of one type may not affect other types, for example sound changes do not affect cognacy. Unlike biology, it cannot be assumed that languages all have a common origin and establishing relatedness is necessary. In modelling it is often assumed for simplicity that the characters change independently but this may not be the case. Besides borrowing, there can also be semantic shifts and polymorphism.
Analysis can be carried out on the "characters" of languages or on the "distances" of the languages. In the former case the input to a language classification generally takes the form of a data matrix where the rows correspond to the various languages being analysed and the columns correspond to different features or characters by which each language may be described. These features are of two types cognates or typological data. Characters can take one or more forms (homoplasy) and can be lexical, morphological or phonological. Cognates are morphemes (lexical or grammatical) or larger constructions. Typological characters can come from any part of the grammar or lexicon. If there are gaps in the data these have to be coded.
In addition to the original database of (unscreened) data, in many studies subsets are formed for particular purposes (screened data).
In lexicostatistics the features are the meanings of words, or rather semantic slots. Thus the matrix entries are a series of glosses. As originally devised by Swadesh the single most common word for a slot was to be chosen, which can be difficult and subjective because of semantic shift. Later methods may allow more than one meaning to be incorporated.
Some methods allow constraints to be placed on language contact geography (isolation by distance) and on sub-group split times.
Swadesh originally published a 200 word list but later refined it into a 100 word one. [ 35 ] A commonly used IE database is that by Dyen, Kruskal and Black which contains data for 95 languages, though the original is known to contain a few errors. Besides the raw data it also contains cognacy judgements. This is available online. [ 36 ] The database of Ringe, Warnow and Taylor has information on 24 IE languages, with 22 phonological characters, 15 morphological characters and 333 lexical characters. Gray and Atkinson used a database of 87 languages with 2449 lexical items, based on the Dyen set with the addition of three ancient languages. They incorporated the cognacy judgements of a number of scholars. Other databases have been drawn up for African, Australian and Andean language families, amongst others.
Coding of the data may be in binary form or in multistate form. The former is often used but does result in a bias. It has been claimed that there is a constant scale factor between the two coding methods, and that allowance can be made for this. However, another study suggests that the topology may change [ 37 ]
The word slots are chosen to be as culture- and borrowing- free as possible. The original Swadesh lists are most commonly used but many others have been devised for particular purposes. Often these are shorter than Swadesh's preferred 100 item list. Kessler has written a book on "The Significance of Word Lists [ 38 ] while McMahon and McMahon carried out studies on the effects of reconstructability and retentiveness. [ 28 ] The effect of increasing the number of slots has been studied and a law of diminishing returns found, with about 80 being found satisfactory. [ 39 ] However some studies have used less than half this number.
Generally each cognate set is represented as a different character but differences between words can also be measured as a distance measurement by sound changes. Distances may also be measured letter by letter.
Traditionally these have been seen as more important than lexical ones and so some studies have put additional weighting on this type of character. Such features were included in the Ringe, Warnow and Taylor IE database for example. However other studies have omitted them.
Examples of these features include glottalised constants, tone systems, accusative alignment in nouns, dual number, case number correspondence, object-verb order, and first person singular pronouns. These will be listed in the WALS database, though this is only sparsely populated for many languages yet. [ 40 ]
Some analysis methods incorporate a statistical model of language evolution and use the properties of the model to estimate the evolution history. Statistical models are also used for simulation of data for testing purposes. A stochastic process can be used to describe how a set of characters evolves within a language. The probability with which a character will change can depend on the branch but not all characters evolve together, nor is the rate identical on all branches. It is often assumed that each character evolves independently but this is not always the case. Within a model borrowing and parallel development (homoplasy) may also be modelled, as well as polymorphisms.
Chance resemblances produce a level of noise against which the required signal of relatedness has to be found. A study was carried out by Ringe [ 41 ] into the effects of chance on the mass comparison method. This showed that chance resemblances were critical to the technique and that Greenberg's conclusions could not be justified, though the mathematical procedure used by Ringe was later criticised.
With small databases sampling errors can be important.
In some cases with a large database and exhaustive search of all possible trees or networks is not feasible because of running time limitations. Thus there is a chance that the optimum solution is not found by heuristic solution-space search methods.
Loanwords can severely affect the topology of a tree so efforts are made to exclude borrowings. However, undetected ones sometimes still exist. McMahon and McMahon [ 42 ] showed that around 5% borrowing can affect the topology while 10% has significant effects. In networks borrowing produces reticulations. Minett and Wang [ 43 ] examined ways of detecting borrowing automatically.
Dating of language splits can be determined if it is known how the characters evolve along each branch of a tree. The simplest assumption is that all characters evolve at a single constant rate with time and that this is independent of the tree branch. This was the assumption made in glottochronology. However, studies soon showed that there was variation between languages, some probably due to the presence of unrecognised borrowing. [ 44 ] A better approach is to allow rate variation, and the gamma distribution is usually used because of its mathematical convenience. Studies have also been carried out that show that the character replacement rate depends on the frequency of use. [ 45 ] Widespread borrowing can bias divergence time estimates by making languages seem more similar and hence younger. However, this also makes the ancestor's branch length longer so that the root is unaffected. [ 46 ]
This aspect is the most controversial part of quantitative comparative linguistics.
There is a need to understand how a language classification method works in order to determine its assumptions and limitations. It may only be valid under certain conditions or be suitable for small databases. The methods differ in their data requirements, their complexity and running time. The methods also differ in their optimisation criteria.
These two methods are similar but the maximum parsimony method's objective is to find the tree (or network) in which the minimum number of evolutionary changes occurs. In some implementations the characters can be given weights and then the objective is to minimise the total weighted sum of the changes. The analysis produces unrooted trees unless an outgroup is used or directed characters. Heuristics are used to find the best tree but optimisation is not guaranteed. The method is often implemented using the programs PAUP or TNT.
Maximum compatibility also uses characters, with the objective of finding the tree on which the maximum number of characters evolve without homoplasy. Again the characters can be weighted and when this occurs the objective is to maximise the sum of the weights of compatible characters. It also produces unrooted trees unless additional information is incorporated. There are no readily available heuristics available that are accurate with large databases. This method has only been used by Ringe's group. [ 47 ]
In these two methods there are often several trees found with the same score so the usual practice is to find a consensus tree via an algorithm. A majority consensus has bipartitions in more than half of the input trees while a greedy consensus adds bipartitions to the majority tree. The strict consensus tree is the least resolved and contains those splits that are in every tree.
Bootstrapping (a statistical resampling strategy) is used to provide branch support values. The technique randomly picks characters from the input data matrix and then the same analysis is used. The support value is the fraction of the runs with that bipartition in the observed tree. However, bootstrapping is very time consuming.
Both of these methods use explicit evolution models. The maximum likelihood method optimises the probability of producing the
observed data, while Bayesian analysis estimates the probability of each tree and so produces a probability distribution. A random walk is made through the "model-tree space". Both take an indeterminate time to run, and stopping may be arbitrary so a decision is a problem. However, both produce support information for each branch.
The assumptions of these methods are overt and are verifiable. The complexity of the model can be increased if required. The model parameters are estimated directly from the input data so assumptions about evolutionary rate are avoided.
This method produces an explicit phylogenic network having an underlying tree with additional contact edges. Characters can be borrowed but evolve without homoplasy. To produce such networks, a graph-theoretic algorithm [ 48 ] has been used.
The input lexical data is coded in binary form, with one character for each state of the original multi-state character. The method allows homoplasy and constraints on split times. A likelihood-based analysis method is used, with evolution expressed as a rate matrix. Cognate gain and loss is modelled with a gamma distribution to allow rate variation and with rate smoothing. Because of the vast number of possible trees with many languages, Bayesian inference is used to search for the optimal tree. A Markov Chain Monte Carlo algorithm [ 49 ] generates a sample of trees as an approximation to the posterior probability distribution. A summary of this distribution can be provided as a greedy consensus tree or network with support values. The method also provides date estimates.
The method is accurate when the original characters are binary, and evolve identically and independently of each other under a rates-across-sites model with gamma distributed rates; the dates are accurate when the rate of change is constant. Understanding the performance of the method when the original characters are multi-state is more complicated, since the binary encoding produces characters that are not independent, while the method assumes independence.
This method [ 50 ] is an outgrowth of Gray and Atkinson's. Rather than having two parameters for a character, this method uses three. The birth rate, death rate of a cognate are specified and its borrowing rate. The birth rate is a Poisson random variable with a single birth of a cognate class but separate deaths of branches are allowed (Dollo parsimony). The method does not allow homoplasy but allows polymorphism and constraints. Its major problem is that it cannot handle missing data (this issue has since been resolved by Ryder and Nicholls. [ 51 ] Statistical techniques are used to fit the model to the data. Prior information may be incorporated and an MCMC research is made of possible reconstructions. The method has been applied to Gray and Nichol's database and seems to give similar results.
These use a triangular matrix of pairwise language comparisons. The input character matrix is used to compute the distance matrix either using the Hamming distance or the Levenshtein distance . The former measures the proportion of matching characters while the latter allows costs of the various possible transforms to be included. These methods are fast compared with wholly character based ones. However, these methods do result in information loss.
The "Unweighted Pairwise Group Method with Arithmetic-mean" ( UPGMA ) is a clustering technique which operates by repeatedly joining the two languages that have the smallest distance between them. It operates accurately with clock-like evolution but otherwise it can be in error. This is the method used in Swadesh's original lexicostatistics.
This is a technique for dividing data into natural groups. [ 52 ] The data could be characters but is more usually distance measures. The character counts or distances are used to generate the splits and to compute weights (branch lengths) for the splits. The weighted splits are then represented in a tree or network based on minimising the number of changes between each pair of taxa. There are fast algorithms for generating the collection of splits. The weights are determined from the taxon to taxon distances. Split decomposition is effective when the number of taxa is small or when the signal is not too complicated.
This method operates on distance data, computes a transformation of the input matrix and then computes the minimum distance of the pairs of languages. [ 53 ] It operates correctly even if the languages do not evolve with a lexical clock. A weighted version of the method may also be used. The method produces an output tree. It is claimed to be the closest method to manual techniques for tree construction.
It uses a similar algorithm to neighbor joining. [ 54 ] Unlike Split Decomposition it does not fuse nodes immediately but waits until a node has been paired a second time. The tree nodes are then replaced by two and the distance matrix reduced. It can handle large and complicated data sets. However, the output is a phenogram rather than a phylogram. This is the most popular network method.
This was an early network method that has been used for some language analysis. It was originally developed for genetic sequences with more than one possible origin. [ 55 ] Network collapses the alternative trees into a single network. Where there are multiple histories a reticulation (a box shape) is drawn. It generates a list of characters incompatible with a tree.
This uses a declarative knowledge representation formalism and the methods of Answer Set Programming. [ 56 ] One such solver is CMODELS which can be used for small problems but larger ones require heuristics. Preprocessing is used to determine the informative characters. CMODELS transforms them into a propositional theory that uses a SAT solver to compute the models of this theory.
Fitch and Kitch are maximum likelihood based programs in PHYLIP that allow a tree to be rearranged after each addition, unlike NJ. Kitch differs from Fitch in assuming a constant rate of change throughout the tree while Fitch allows for different rates down each branch. [ 57 ]
Holm introduced a method in 2000 to deal with some known problems of lexicostatistical analysis. These are the "symplesiomorphy trap", where shared archaisms are difficult to distinguish from shared innovations, and the "proportionality "trap" when later changes can obscure early ones. Later he introduced a refined method, called SLD, to take account of the variable word distribution across languages. [ 58 ] The method does not assume aconstant rate of change.
A number of fast converging analysis methods have been developed for use with large databases (>200 languages). One of these is the Disk Covering Method (DCM). [ 59 ] This has been combined with existing methods to give improved performance. A paper on the DCM-NJ+MP method is given by the same authors in "The performance of Phylogenetic Methods on Trees of Bounded Diameter", [ full citation needed ] where it is compared with the NJ method.
These models compare the letters of words rather than their phonetics. Dunn et al. [ 60 ] studied 125 typological characters across 16 Austronesian and 15 Papuan languages. They compared their results to an MP tree and one constructed by traditional analysis. Significant differences were found. Similarly Wichmann and Saunders [ 61 ] used 96 characters to study 63 American languages.
A method that has been suggested for initial inspection of a set of languages to see if they are related was mass comparison . However, this has been severely criticised and fell into disuse. Recently Kessler has resurrected a computerised version of the method but using rigorous hypothesis testing. [ 62 ] The aim is to make use of similarities across more than two languages at a time. In another paper [ 63 ] various criteria for comparing word lists are evaluated. It was found that the IE and Uralic families could be reconstructed but there was no evidence for a joint super-family.
This method uses stable lexical fields, such as stance verbs, to try to establish long-distance relationships. [ 64 ] Account is taken of convergence and semantic shifts to search for ancient cognates. A model is outlined and the results of a pilot study are presented.
The Automated Similarity Judgment Program (ASJP) is similar to lexicostatistics , but the judgement of similarities is done by a computer program following a consistent set of rules. [ 65 ] Trees are generated using standard phylogenetic methods. ASJP uses 7 vowel symbols and 34 consonant symbols. There are also various modifiers. Two words are judged similar if at least two consecutive consonants in the respective words are identical while vowels are also taken into account. The proportion of words with the same meaning judged to be similar for a pair of languages is the Lexical Similarity Percentage (LSP). The Phonological Similarity Percentage (PSP) is also calculated. PSP is then subtracted from the LSP yielding the Subtracted Similarity Percentage (SSP) and the ASJP distance is 100-SSP. Currently there are data on over 4,500 languages and dialects in the ASJP database [ 66 ] from which a tree of the world's languages was generated. [ 67 ]
This measures the orthographical distance between words to avoid the subjectivity of cognacy judgements. [ 68 ] It determines the minimum number of operations needed to transform one word into another, normalised by the length of the longer word. A tree is constructed from the distance data by the UPGMA technique.
Heggarty has proposed a means of providing a measure of the degrees of difference between cognates, rather than just yes/no answers. [ 69 ] This is based on examining many (>30) features of the phonetics of the glosses in comparison with the protolanguage. This could require a large amount of work but Heggarty claims that only a representative sample of sounds is necessary. He also examined the rate of change of the phonetics and found a large rate variation, so that it was unsuitable for glottochronology. A similar evaluation of the phonetics had earlier been carried out by Grimes and Agard for Romance languages, but this used only six points of comparison. [ 70 ]
Standard mathematical techniques are available for measuring the similarity/difference of two trees. For consensus trees the Consistency Index (CI) is a measure of homoplasy. For one character it is the ratio of the minimimum conceivable number of steps on any one tree (= 1 for binary trees) divided by the number of reconstructed steps on the tree. The CI of a tree is the sum of the character CIs divided by the number of characters. [ 71 ] It represents the proportion of patterns correctly assigned.
The Retention Index (RI) measures the amount of similarity in a character. It is the ratio (g - s) / (g - m) where g is the greatest number of steps of a character on any tree, m is the minimum number of steps on any tree, and s is the minimum steps on a particular tree. There is also a Rescaled CI which is the product of the CI and RI.
For binary trees the standard way of comparing their topology is to use the Robinson-Foulds metric . [ 72 ] This distance is the average of the number of false positives and false negatives in terms of branch occurrence. R-F rates above 10% are considered poor matches. For other sorts of trees and for networks there is yet no standard method of comparison.
Lists of incompatible characters are produced by some tree producing methods. These can be extremely helpful in analysing the output. Where heuristic methods are used repeatability is an issue. However, standard mathematical techniques are used to overcome this problem.
In order to evaluate the methods a well understood family of languages is chosen, with a reliable dataset. This family is often the IE one but others have been used. After applying the methods to be compared to the database, the resulting trees are compared with the reference tree determined by traditional linguistic methods. The aim is to have no conflicts in topology, for example no missing sub-groups, and compatible dates. The families suggested for this analysis by Nichols and Warnow [ 73 ] are Germanic, Romance, Slavic, Common Turkic, Chinese, and Mixe Zoque as well as older groups such as Oceanic and IE.
Although the use of real languages does add realism and provides real problems, the above method of validation suffers from the fact that the true evolution of the languages is unknown. By generating a set of data from a simulated evolution correct tree is known. However it will be a simplified version of reality. Thus both evaluation techniques should be used.
To assess the robustness of a solution it is desirable to vary the input data and constraints, and observe the output. Each variable is changed slightly in turn. This analysis has been carried out in a number of cases and the methods found to be robust, for example by Atkinson and Gray. [ 74 ]
During the early 1990s, linguist Donald Ringe , with computer scientists Luay Nakhleh and Tandy Warnow , statistician Steven N. Evans and others, began collaborating on research in quantitative comparative linguistic projects. They later founded the CHPL project , the goals of which include: "producing and maintaining real linguistic datasets, in particular of Indo-European languages", "formulating statistical models that capture the evolution of historical linguistic data", "designing simulation tools and accuracy measures for generating synthetic data for studying the performance of reconstruction methods", and "developing and implementing statistically-based as well as combinatorial methods for reconstructing language phylogenies, including phylogenetic networks". [ 75 ]
A comparison of coding methods was carried out by Rexova et al. (2003). [ 76 ] They created a reduced data set from the Dyen database but with the addition of Hittite. They produced a standard multistate matrix where the 141 character states corresponds to individual cognate classes, allowing polymorphism. They also joined some cognate classes, to reduce subjectivity and polymorphic states were not allowed. Lastly they produced a binary matrix where each class of words was treated as a separate character. The matrices were analysed by PAUP. It was found that using the binary matrix produced changes near the root of the tree.
McMahon and McMahon (2003) used three PHYLIP programs (NJ, Fitch and Kitch) on the DKB dataset. [ 77 ] They found that the results produced were very similar. Bootstrapping was used to test the robustness of any part of the tree. Later they used subsets of the data to assess its retentiveness and reconstructability. [ 42 ] The outputs showed topological differences which were attributed to borrowing. They then also used Network, Split Decomposition, Neighbor-net and SplitsTree on several data sets. Significant differences were found between the latter two methods. Neighbor-net was considered optimal for discerning language contact.
In 2005, Nakhleh, Warnow, Ringe and Evans carried out a comparison of six analysis methods using an Indo-European database. [ 78 ] The methods compared were UPGMA, NJ MP, MC, WMC and GA. The PAUP software package was used for UPGMA, NJ, and MC as well as computing the majority consensus trees. The RWT database was used but 40 characters were removed due to evidence of polymorphism. Then a screened database was produced excluding all characters that clearly exhibited parallel development, so eliminating 38 features. The trees were evaluated on the basis of the number of incompatible characters and on agreement with established sub-grouping results. They found that UPGMA was clearly worst but there was not a lot of difference between the other methods. The results depended on the data set used. It was found that weighting the characters was important, which requires linguistic judgement.
Saunders (2005) [ 79 ] compared NJ, MP, GA and Neighbor-Net on a combination of lexical and typological data. He recommended use of the GA method but Nichols and Warnow have some concerns about the study methodology. [ 80 ]
Cysouw et al. (2006) [ 81 ] compared Holm's original method with NJ, Fitch, MP and SD. They found Holm's method to be less accurate than the others.
In 2013, François Barbancon, Warnow, Evans, Ringe and Nakleh (2013) studied various tree reconstruction methods using simulated data. [ 82 ] Their simulated data varied in the number of contact edges, the degree of homoplasy, the deviation from a lexical clock, and the deviation from the rates-across-sites assumption. It was found that the accuracy of the unweighted methods (MP, NJ, UPGMA, and GA) were consistent in all the conditions studied, with MP being the best. The accuracy of the two weighted methods (WMC and WMP) depended on the appropriateness of the weighting scheme. With low homoplasy the weighted methods generally produced the more accurate results but inappropriate weighting could make these worse than MP or GA under moderate or high homoplasy levels.
Choice of an appropriate model is critical for the production of good phylogenetic analyses. Both underparameterised or overly restrictive models may produce aberrant behaviour when their underlying assumptions are violated, while overly complex or overparameterised models require long run times and their parameters may be overfit. [ 83 ] The most common method of model selection is the "Likelihood Ratio Test" which produces an estimate of the fit between the model and the data, but as an alternative the Akaike Information Criterion or the Bayesian Information Criterion can be used. Model selection computer programs are available. | https://en.wikipedia.org/wiki/Quantitative_comparative_linguistics |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.