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In mathematics, a weighted Voronoi diagram in n dimensions is a generalization of a Voronoi diagram . The Voronoi cells in a weighted Voronoi diagram are defined in terms of a distance function. The distance function may specify the usual Euclidean distance , or may be some other, special distance function. In weighted Voronoi diagrams, each site has a weight that influences the distance computation. The idea is that larger weights indicate more important sites, and such sites will get bigger Voronoi cells.
In a multiplicatively weighted Voronoi diagram , the distance between a point and a site is divided by the (positive) weight of the site. [ 1 ] In the plane under the ordinary Euclidean distance , the multiplicatively weighted Voronoi diagram is also called circular Dirichlet tessellation [ 2 ] [ 3 ] and its edges are circular arcs and straight line segments. A Voronoi cell may be non-convex, disconnected and may have holes. This diagram arises, e.g., as a model of crystal growth , where crystals from different points may grow with different speed. Since crystals may grow in empty space only and are continuous objects, a natural variation is the crystal Voronoi diagram , in which the cells are defined somewhat differently.
In an additively weighted Voronoi diagram , weights are subtracted from the distances. In the plane under the ordinary Euclidean distance this diagram is also known as the hyperbolic Dirichlet tessellation and its edges are arcs of hyperbolas and straight line segments. [ 1 ]
The power diagram is defined when weights are subtracted from the squared Euclidean distance. It can also be defined using the power distance defined from a set of circles. [ 4 ]
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https://en.wikipedia.org/wiki/Weighted_Voronoi_diagram
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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average ), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.
If all the weights are equal, then the weighted mean is the same as the arithmetic mean . While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox .
Given two school classes — one with 20 students, one with 30 students — and test grades in each class as follows:
The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students): x ¯ = 4300 50 = 86. {\displaystyle {\bar {x}}={\frac {4300}{50}}=86.}
Or, this can be accomplished by weighting the class means by the number of students in each class. The larger class is given more "weight":
Thus, the weighted mean makes it possible to find the mean average student grade without knowing each student's score. Only the class means and the number of students in each class are needed.
Since only the relative weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such a linear combination is called a convex combination .
Using the previous example, we would get the following weights:
Then, apply the weights like this:
Formally, the weighted mean of a non-empty finite tuple of data ( x 1 , x 2 , … , x n ) {\displaystyle \left(x_{1},x_{2},\dots ,x_{n}\right)} ,
with corresponding non-negative weights ( w 1 , w 2 , … , w n ) {\displaystyle \left(w_{1},w_{2},\dots ,w_{n}\right)} is
which expands to:
Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights may not be negative in order for the equation to work [ a ] . Some may be zero, but not all of them (since division by zero is not allowed).
The formulas are simplified when the weights are normalized such that they sum up to 1, i.e., ∑ i = 1 n w i ′ = 1 {\textstyle \sum \limits _{i=1}^{n}{w_{i}'}=1} .
For such normalized weights, the weighted mean is equivalently:
One can always normalize the weights by making the following transformation on the original weights:
The ordinary mean 1 n ∑ i = 1 n x i {\textstyle {\frac {1}{n}}\sum \limits _{i=1}^{n}{x_{i}}} is a special case of the weighted mean where all data have equal weights.
If the data elements are independent and identically distributed random variables with variance σ 2 {\displaystyle \sigma ^{2}} , the standard error of the weighted mean , σ x ¯ {\displaystyle \sigma _{\bar {x}}} , can be shown via uncertainty propagation to be:
For the weighted mean of a list of data for which each element x i {\displaystyle x_{i}} potentially comes from a different probability distribution with known variance σ i 2 {\displaystyle \sigma _{i}^{2}} , all having the same mean, one possible choice for the weights is given by the reciprocal of variance:
The weighted mean in this case is:
and the standard error of the weighted mean (with inverse-variance weights) is:
Note this reduces to σ x ¯ 2 = σ 0 2 / n {\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}/n} when all σ i = σ 0 {\displaystyle \sigma _{i}=\sigma _{0}} .
It is a special case of the general formula in previous section,
The equations above can be combined to obtain:
The significance of this choice is that this weighted mean is the maximum likelihood estimator of the mean of the probability distributions under the assumption that they are independent and normally distributed with the same mean.
The weighted sample mean, x ¯ {\displaystyle {\bar {x}}} , is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one).
If the observations have expected values E ( x i ) = μ i , {\displaystyle E(x_{i})={\mu _{i}},} then the weighted sample mean has expectation E ( x ¯ ) = ∑ i = 1 n w i ′ μ i . {\displaystyle E({\bar {x}})=\sum _{i=1}^{n}{w_{i}'\mu _{i}}.} In particular, if the means are equal, μ i = μ {\displaystyle \mu _{i}=\mu } , then the expectation of the weighted sample mean will be that value, E ( x ¯ ) = μ . {\displaystyle E({\bar {x}})=\mu .}
When treating the weights as constants, and having a sample of n observations from uncorrelated random variables , all with the same variance and expectation (as is the case for i.i.d random variables), then the variance of the weighted mean can be estimated as the multiplication of the unweighted variance by Kish's design effect (see proof ):
With σ ^ y 2 = ∑ i = 1 n ( y i − y ¯ ) 2 n − 1 {\displaystyle {\hat {\sigma }}_{y}^{2}={\frac {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}{n-1}}} , w ¯ = ∑ i = 1 n w i n {\displaystyle {\bar {w}}={\frac {\sum _{i=1}^{n}w_{i}}{n}}} , and w 2 ¯ = ∑ i = 1 n w i 2 n {\displaystyle {\overline {w^{2}}}={\frac {\sum _{i=1}^{n}w_{i}^{2}}{n}}}
However, this estimation is rather limited due to the strong assumption about the y observations. This has led to the development of alternative, more general, estimators.
From a model based perspective, we are interested in estimating the variance of the weighted mean when the different y i {\displaystyle y_{i}} are not i.i.d random variables. An alternative perspective for this problem is that of some arbitrary sampling design of the data in which units are selected with unequal probabilities (with replacement). [ 1 ] : 306
In Survey methodology , the population mean, of some quantity of interest y , is calculated by taking an estimation of the total of y over all elements in the population ( Y or sometimes T ) and dividing it by the population size – either known ( N {\displaystyle N} ) or estimated ( N ^ {\displaystyle {\hat {N}}} ). In this context, each value of y is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The survey sampling procedure yields a series of Bernoulli indicator values ( I i {\displaystyle I_{i}} ) that get 1 if some observation i is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.: Poisson sampling ). The probability of some element to be chosen, given a sample, is denoted as P ( I i = 1 ∣ Some sample of size n ) = π i {\displaystyle P(I_{i}=1\mid {\text{Some sample of size }}n)=\pi _{i}} , and the one-draw probability of selection is P ( I i = 1 | one sample draw ) = p i ≈ π i n {\displaystyle P(I_{i}=1|{\text{one sample draw}})=p_{i}\approx {\frac {\pi _{i}}{n}}} (If N is very large and each p i {\displaystyle p_{i}} is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities. [ 2 ] : 42, 43, 51 I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as cluster sampling design).
Since each element ( y i {\displaystyle y_{i}} ) is fixed, and the randomness comes from it being included in the sample or not ( I i {\displaystyle I_{i}} ), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term: y i ′ = y i I i {\displaystyle y'_{i}=y_{i}I_{i}} . With the following expectancy: E [ y i ′ ] = y i E [ I i ] = y i π i {\displaystyle E[y'_{i}]=y_{i}E[I_{i}]=y_{i}\pi _{i}} ; and variance: V [ y i ′ ] = y i 2 V [ I i ] = y i 2 π i ( 1 − π i ) {\displaystyle V[y'_{i}]=y_{i}^{2}V[I_{i}]=y_{i}^{2}\pi _{i}(1-\pi _{i})} .
When each element of the sample is inflated by the inverse of its selection probability, it is termed the π {\displaystyle \pi } -expanded y values, i.e.: y ˇ i = y i π i {\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}} . A related quantity is p {\displaystyle p} -expanded y values: y i p i = n y ˇ i {\displaystyle {\frac {y_{i}}{p_{i}}}=n{\check {y}}_{i}} . [ 2 ] : 42, 43, 51, 52 As above, we can add a tick mark if multiplying by the indicator function. I.e.: y ˇ i ′ = I i y ˇ i = I i y i π i {\displaystyle {\check {y}}'_{i}=I_{i}{\check {y}}_{i}={\frac {I_{i}y_{i}}{\pi _{i}}}}
In this design based perspective, the weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the selection probability (i.e.: the inflation factor). I.e.: w i = 1 π i ≈ 1 n × p i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}\approx {\frac {1}{n\times p_{i}}}} .
If the population size N is known we can estimate the population mean using Y ¯ ^ known N = Y ^ p w r N ≈ ∑ i = 1 n w i y i ′ N {\displaystyle {\hat {\bar {Y}}}_{{\text{known }}N}={\frac {{\hat {Y}}_{pwr}}{N}}\approx {\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{N}}} .
If the sampling design is one that results in a fixed sample size n (such as in pps sampling ), then the variance of this estimator is:
The general formula can be developed like this:
The population total is denoted as Y = ∑ i = 1 N y i {\displaystyle Y=\sum _{i=1}^{N}y_{i}} and it may be estimated by the (unbiased) Horvitz–Thompson estimator , also called the π {\displaystyle \pi } -estimator. This estimator can be itself estimated using the pwr -estimator (i.e.: p {\displaystyle p} -expanded with replacement estimator, or "probability with replacement" estimator). With the above notation, it is: Y ^ p w r = 1 n ∑ i = 1 n y i ′ p i = ∑ i = 1 n y i ′ n p i ≈ ∑ i = 1 n y i ′ π i = ∑ i = 1 n w i y i ′ {\displaystyle {\hat {Y}}_{pwr}={\frac {1}{n}}\sum _{i=1}^{n}{\frac {y'_{i}}{p_{i}}}=\sum _{i=1}^{n}{\frac {y'_{i}}{np_{i}}}\approx \sum _{i=1}^{n}{\frac {y'_{i}}{\pi _{i}}}=\sum _{i=1}^{n}w_{i}y'_{i}} . [ 2 ] : 51
The estimated variance of the pwr -estimator is given by: [ 2 ] : 52 Var ( Y ^ p w r ) = n n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 {\displaystyle \operatorname {Var} ({\hat {Y}}_{pwr})={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}} where w y ¯ = ∑ i = 1 n w i y i n {\displaystyle {\overline {wy}}=\sum _{i=1}^{n}{\frac {w_{i}y_{i}}{n}}} .
The above formula was taken from Sarndal et al. (1992) (also presented in Cochran 1977), but was written differently. [ 2 ] : 52 [ 1 ] : 307 (11.35) The left side is how the variance was written and the right side is how we've developed the weighted version:
Var ( Y ^ pwr ) = 1 n 1 n − 1 ∑ i = 1 n ( y i p i − Y ^ p w r ) 2 = 1 n 1 n − 1 ∑ i = 1 n ( n n y i p i − n n ∑ i = 1 n w i y i ) 2 = 1 n 1 n − 1 ∑ i = 1 n ( n y i π i − n ∑ i = 1 n w i y i n ) 2 = n 2 n 1 n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 = n n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 {\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{\text{pwr}})&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {y_{i}}{p_{i}}}-{\hat {Y}}_{pwr}\right)^{2}\\&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {n}{n}}{\frac {y_{i}}{p_{i}}}-{\frac {n}{n}}\sum _{i=1}^{n}w_{i}y_{i}\right)^{2}={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(n{\frac {y_{i}}{\pi _{i}}}-n{\frac {\sum _{i=1}^{n}w_{i}y_{i}}{n}}\right)^{2}\\&={\frac {n^{2}}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\\&={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\end{aligned}}}
And we got to the formula from above.
An alternative term, for when the sampling has a random sample size (as in Poisson sampling ), is presented in Sarndal et al. (1992) as: [ 2 ] : 182
Var ( Y ¯ ^ pwr (known N ) ) = 1 N 2 ∑ i = 1 n ∑ j = 1 n ( Δ ˇ i j y ˇ i y ˇ j ) {\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)}
With y ˇ i = y i π i {\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}} . Also, C ( I i , I j ) = π i j − π i π j = Δ i j {\displaystyle C(I_{i},I_{j})=\pi _{ij}-\pi _{i}\pi _{j}=\Delta _{ij}} where π i j {\displaystyle \pi _{ij}} is the probability of selecting both i and j. [ 2 ] : 36 And Δ ˇ i j = 1 − π i π j π i j {\displaystyle {\check {\Delta }}_{ij}=1-{\frac {\pi _{i}\pi _{j}}{\pi _{ij}}}} , and for i=j: Δ ˇ i i = 1 − π i π i π i = 1 − π i {\displaystyle {\check {\Delta }}_{ii}=1-{\frac {\pi _{i}\pi _{i}}{\pi _{i}}}=1-\pi _{i}} . [ 2 ] : 43
If the selection probability are uncorrelated (i.e.: ∀ i ≠ j : C ( I i , I j ) = 0 {\displaystyle \forall i\neq j:C(I_{i},I_{j})=0} ), and when assuming the probability of each element is very small, then:
We assume that ( 1 − π i ) ≈ 1 {\displaystyle (1-\pi _{i})\approx 1} and that Var ( Y ^ pwr (known N ) ) = 1 N 2 ∑ i = 1 n ∑ j = 1 n ( Δ ˇ i j y ˇ i y ˇ j ) = 1 N 2 ∑ i = 1 n ( Δ ˇ i i y ˇ i y ˇ i ) = 1 N 2 ∑ i = 1 n ( ( 1 − π i ) y i π i y i π i ) = 1 N 2 ∑ i = 1 n ( w i y i ) 2 {\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{{\text{pwr (known }}N{\text{)}}})&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left({\check {\Delta }}_{ii}{\check {y}}_{i}{\check {y}}_{i}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left((1-\pi _{i}){\frac {y_{i}}{\pi _{i}}}{\frac {y_{i}}{\pi _{i}}}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left(w_{i}y_{i}\right)^{2}\end{aligned}}}
The previous section dealt with estimating the population mean as a ratio of an estimated population total ( Y ^ {\displaystyle {\hat {Y}}} ) with a known population size ( N {\displaystyle N} ), and the variance was estimated in that context. Another common case is that the population size itself ( N {\displaystyle N} ) is unknown and is estimated using the sample (i.e.: N ^ {\displaystyle {\hat {N}}} ). The estimation of N {\displaystyle N} can be described as the sum of weights. So when w i = 1 π i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}} we get N ^ = ∑ i = 1 n w i I i = ∑ i = 1 n I i π i = ∑ i = 1 n 1 ˇ i ′ {\displaystyle {\hat {N}}=\sum _{i=1}^{n}w_{i}I_{i}=\sum _{i=1}^{n}{\frac {I_{i}}{\pi _{i}}}=\sum _{i=1}^{n}{\check {1}}'_{i}} . With the above notation, the parameter we care about is the ratio of the sums of y i {\displaystyle y_{i}} s, and 1s. I.e.: R = Y ¯ = ∑ i = 1 N y i π i ∑ i = 1 N 1 π i = ∑ i = 1 N y ˇ i ∑ i = 1 N 1 ˇ i = ∑ i = 1 N w i y i ∑ i = 1 N w i {\displaystyle R={\bar {Y}}={\frac {\sum _{i=1}^{N}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}_{i}}{\sum _{i=1}^{N}{\check {1}}_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y_{i}}{\sum _{i=1}^{N}w_{i}}}} . We can estimate it using our sample with: R ^ = Y ¯ ^ = ∑ i = 1 N I i y i π i ∑ i = 1 N I i 1 π i = ∑ i = 1 N y ˇ i ′ ∑ i = 1 N 1 ˇ i ′ = ∑ i = 1 N w i y i ′ ∑ i = 1 N w i 1 i ′ = ∑ i = 1 n w i y i ′ ∑ i = 1 n w i 1 i ′ = y ¯ w {\displaystyle {\hat {R}}={\hat {\bar {Y}}}={\frac {\sum _{i=1}^{N}I_{i}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}I_{i}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}'_{i}}{\sum _{i=1}^{N}{\check {1}}'_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y'_{i}}{\sum _{i=1}^{N}w_{i}1'_{i}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}={\bar {y}}_{w}} . As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write: y ¯ w = ∑ i = 1 n w i y i ∑ i = 1 n w i {\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y_{i}}{\sum _{i=1}^{n}w_{i}}}} . This will be the estimand for specific values of y and w, but the statistical properties comes when including the indicator variable y ¯ w = ∑ i = 1 n w i y i ′ ∑ i = 1 n w i 1 i ′ {\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}} . [ 2 ] : 162, 163, 176
This is called a Ratio estimator and it is approximately unbiased for R . [ 2 ] : 182
In this case, the variability of the ratio depends on the variability of the random variables both in the numerator and the denominator - as well as their correlation. Since there is no closed analytical form to compute this variance, various methods are used for approximate estimation. Primarily Taylor series first-order linearization, asymptotics, and bootstrap/jackknife. [ 2 ] : 172 The Taylor linearization method could lead to under-estimation of the variance for small sample sizes in general, but that depends on the complexity of the statistic. For the weighted mean, the approximate variance is supposed to be relatively accurate even for medium sample sizes. [ 2 ] : 176 For when the sampling has a random sample size (as in Poisson sampling ), it is as follows: [ 2 ] : 182
If π i ≈ p i n {\displaystyle \pi _{i}\approx p_{i}n} , then either using w i = 1 π i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}} or w i = 1 p i {\displaystyle w_{i}={\frac {1}{p_{i}}}} would give the same estimator, since multiplying w i {\displaystyle w_{i}} by some factor would lead to the same estimator. It also means that if we scale the sum of weights to be equal to a known-from-before population size N , the variance calculation would look the same. When all weights are equal to one another, this formula is reduced to the standard unbiased variance estimator.
The Taylor linearization states that for a general ratio estimator of two sums ( R ^ = Y ^ Z ^ {\displaystyle {\hat {R}}={\frac {\hat {Y}}{\hat {Z}}}} ), they can be expanded around the true value R, and give: [ 2 ] : 178
R ^ = Y ^ Z ^ = ∑ i = 1 n w i y i ′ ∑ i = 1 n w i z i ′ ≈ R + 1 Z ∑ i = 1 n ( y i ′ π i − R z i ′ π i ) {\displaystyle {\hat {R}}={\frac {\hat {Y}}{\hat {Z}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}z'_{i}}}\approx R+{\frac {1}{Z}}\sum _{i=1}^{n}\left({\frac {y'_{i}}{\pi _{i}}}-R{\frac {z'_{i}}{\pi _{i}}}\right)}
And the variance can be approximated by: [ 2 ] : 178, 179
V ( R ^ ) ^ = 1 Z ^ 2 ∑ i = 1 n ∑ j = 1 n ( Δ ˇ i j y i − R ^ z i π i y j − R ^ z j π j ) = 1 Z ^ 2 [ V ( Y ^ ) ^ + R ^ V ( Z ^ ) ^ − 2 R ^ C ^ ( Y ^ , Z ^ ) ] {\displaystyle {\widehat {V({\hat {R}})}}={\frac {1}{{\hat {Z}}^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\frac {y_{i}-{\hat {R}}z_{i}}{\pi _{i}}}{\frac {y_{j}-{\hat {R}}z_{j}}{\pi _{j}}}\right)={\frac {1}{{\hat {Z}}^{2}}}\left[{\widehat {V({\hat {Y}})}}+{\hat {R}}{\widehat {V({\hat {Z}})}}-2{\hat {R}}{\hat {C}}({\hat {Y}},{\hat {Z}})\right]} .
The term C ^ ( Y ^ , Z ^ ) {\displaystyle {\hat {C}}({\hat {Y}},{\hat {Z}})} is the estimated covariance between the estimated sum of Y and estimated sum of Z. Since this is the covariance of two sums of random variables , it would include many combinations of covariances that will depend on the indicator variables. If the selection probability are uncorrelated (i.e.: ∀ i ≠ j : Δ i j = C ( I i , I j ) = 0 {\displaystyle \forall i\neq j:\Delta _{ij}=C(I_{i},I_{j})=0} ), this term would still include a summation of n covariances for each element i between y i ′ = I i y i {\displaystyle y'_{i}=I_{i}y_{i}} and z i ′ = I i z i {\displaystyle z'_{i}=I_{i}z_{i}} . This helps illustrate that this formula incorporates the effect of correlation between y and z on the variance of the ratio estimators.
When defining z i = 1 {\displaystyle z_{i}=1} the above becomes: [ 2 ] : 182
V ( R ^ ) ^ = V ( y ¯ w ) ^ = 1 N ^ 2 ∑ i = 1 n ∑ j = 1 n ( Δ ˇ i j y i − y ¯ w π i y j − y ¯ w π j ) . {\displaystyle {\widehat {V({\hat {R}})}}={\widehat {V({\bar {y}}_{w})}}={\frac {1}{{\hat {N}}^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\frac {y_{i}-{\bar {y}}_{w}}{\pi _{i}}}{\frac {y_{j}-{\bar {y}}_{w}}{\pi _{j}}}\right).}
If the selection probability are uncorrelated (i.e.: ∀ i ≠ j : Δ i j = C ( I i , I j ) = 0 {\displaystyle \forall i\neq j:\Delta _{ij}=C(I_{i},I_{j})=0} ), and when assuming the probability of each element is very small (i.e.: ( 1 − π i ) ≈ 1 {\displaystyle (1-\pi _{i})\approx 1} ), then the above reduced to the following: V ( y ¯ w ) ^ = 1 N ^ 2 ∑ i = 1 n ( ( 1 − π i ) y i − y ¯ w π i ) 2 = 1 ( ∑ i = 1 n w i ) 2 ∑ i = 1 n w i 2 ( y i − y ¯ w ) 2 . {\displaystyle {\widehat {V({\bar {y}}_{w})}}={\frac {1}{{\hat {N}}^{2}}}\sum _{i=1}^{n}\left((1-\pi _{i}){\frac {y_{i}-{\bar {y}}_{w}}{\pi _{i}}}\right)^{2}={\frac {1}{(\sum _{i=1}^{n}w_{i})^{2}}}\sum _{i=1}^{n}w_{i}^{2}(y_{i}-{\bar {y}}_{w})^{2}.}
A similar re-creation of the proof (up to some mistakes at the end) was provided by Thomas Lumley in crossvalidated. [ 3 ]
We have (at least) two versions of variance for the weighted mean: one with known and one with unknown population size estimation. There is no uniformly better approach, but the literature presents several arguments to prefer using the population estimation version (even when the population size is known). [ 2 ] : 188 For example: if all y values are constant, the estimator with unknown population size will give the correct result, while the one with known population size will have some variability. Also, when the sample size itself is random (e.g.: in Poisson sampling ), the version with unknown population mean is considered more stable. Lastly, if the proportion of sampling is negatively correlated with the values (i.e.: smaller chance to sample an observation that is large), then the un-known population size version slightly compensates for that.
For the trivial case in which all the weights are equal to 1, the above formula is just like the regular formula for the variance of the mean (but notice that it uses the maximum likelihood estimator for the variance instead of the unbiased variance. I.e.: dividing it by n instead of (n-1)).
It has been shown, by Gatz et al. (1995), that in comparison to bootstrapping methods, the following (variance estimation of ratio-mean using Taylor series linearization) is a reasonable estimation for the square of the standard error of the mean (when used in the context of measuring chemical constituents): [ 4 ] : 1186
where w ¯ = ∑ w i n {\displaystyle {\bar {w}}={\frac {\sum w_{i}}{n}}} . Further simplification leads to
Gatz et al. mention that the above formulation was published by Endlich et al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size, [ 5 ] based on the formulation published by Cochran (1977), as an approximation to the ratio mean. However, Endlich et al. didn't seem to publish this derivation in their paper (even though they mention they used it), and Cochran's book includes a slightly different formulation. [ 1 ] : 155 Still, it's almost identical to the formulations described in previous sections.
Because there is no closed analytical form for the variance of the weighted mean, it was proposed in the literature to rely on replication methods such as the Jackknife and Bootstrapping . [ 1 ] : 321
For uncorrelated observations with variances σ i 2 {\displaystyle \sigma _{i}^{2}} , the variance of the weighted sample mean is [ citation needed ]
whose square root σ x ¯ {\displaystyle \sigma _{\bar {x}}} can be called the standard error of the weighted mean (general case) . [ citation needed ]
Consequently, if all the observations have equal variance, σ i 2 = σ 0 2 {\displaystyle \sigma _{i}^{2}=\sigma _{0}^{2}} , the weighted sample mean will have variance
where 1 / n ≤ ∑ i = 1 n w i ′ 2 ≤ 1 {\textstyle 1/n\leq \sum _{i=1}^{n}{w_{i}'^{2}}\leq 1} . The variance attains its maximum value, σ 0 2 {\displaystyle \sigma _{0}^{2}} , when all weights except one are zero. Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have σ x ¯ = σ 0 / n {\textstyle \sigma _{\bar {x}}=\sigma _{0}/{\sqrt {n}}} , i.e., it degenerates into the standard error of the mean , squared.
Because one can always transform non-normalized weights to normalized weights, all formulas in this section can be adapted to non-normalized weights by replacing all w i ′ = w i ∑ i = 1 n w i {\displaystyle w_{i}'={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}} .
Typically when a mean is calculated it is important to know the variance and standard deviation about that mean. When a weighted mean μ ∗ {\displaystyle \mu ^{*}} is used, the variance of the weighted sample is different from the variance of the unweighted sample.
The biased weighted sample variance σ ^ w 2 {\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}} is defined similarly to the normal biased sample variance σ ^ 2 {\displaystyle {\hat {\sigma }}^{2}} :
where ∑ i = 1 N w i = 1 {\displaystyle \sum _{i=1}^{N}w_{i}=1} for normalized weights. If the weights are frequency weights (and thus are random variables), it can be shown [ citation needed ] that σ ^ w 2 {\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}} is the maximum likelihood estimator of σ 2 {\displaystyle \sigma ^{2}} for iid Gaussian observations.
For small samples, it is customary to use an unbiased estimator for the population variance. In normal unweighted samples, the N in the denominator (corresponding to the sample size) is changed to N − 1 (see Bessel's correction ). In the weighted setting, there are actually two different unbiased estimators, one for the case of frequency weights and another for the case of reliability weights .
If the weights are frequency weights (where a weight equals the number of occurrences), then the unbiased estimator is:
This effectively applies Bessel's correction for frequency weights. For example, if values { 2 , 2 , 4 , 5 , 5 , 5 } {\displaystyle \{2,2,4,5,5,5\}} are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample { 2 , 4 , 5 } {\displaystyle \{2,4,5\}} with corresponding weights { 2 , 1 , 3 } {\displaystyle \{2,1,3\}} , and we get the same result either way.
If the frequency weights { w i } {\displaystyle \{w_{i}\}} are normalized to 1, then the correct expression after Bessel's correction becomes
where the total number of samples is ∑ i = 1 N w i {\displaystyle \sum _{i=1}^{N}w_{i}} (not N {\displaystyle N} ). In any case, the information on total number of samples is necessary in order to obtain an unbiased correction, even if w i {\displaystyle w_{i}} has a different meaning other than frequency weight.
The estimator can be unbiased only if the weights are not standardized nor normalized , these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction).
If the weights are instead reliability weights (non-random values reflecting the sample's relative trustworthiness, often derived from sample variance), we can determine a correction factor to yield an unbiased estimator. Assuming each random variable is sampled from the same distribution with mean μ {\displaystyle \mu } and actual variance σ actual 2 {\displaystyle \sigma _{\text{actual}}^{2}} , taking expectations we have,
where V 1 = ∑ i = 1 N w i {\displaystyle V_{1}=\sum _{i=1}^{N}w_{i}} and V 2 = ∑ i = 1 N w i 2 {\displaystyle V_{2}=\sum _{i=1}^{N}w_{i}^{2}} . Therefore, the bias in our estimator is ( 1 − V 2 V 1 2 ) {\displaystyle \left(1-{\frac {V_{2}}{V_{1}^{2}}}\right)} , analogous to the ( N − 1 N ) {\displaystyle \left({\frac {N-1}{N}}\right)} bias in the unweighted estimator (also notice that V 1 2 / V 2 = N e f f {\displaystyle \ V_{1}^{2}/V_{2}=N_{eff}} is the effective sample size ). This means that to unbias our estimator we need to pre-divide by 1 − ( V 2 / V 1 2 ) {\displaystyle 1-\left(V_{2}/V_{1}^{2}\right)} , ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution. The final unbiased estimate of sample variance is:
where E [ s w 2 ] = σ actual 2 {\displaystyle \operatorname {E} [s_{\mathrm {w} }^{2}]=\sigma _{\text{actual}}^{2}} . The degrees of freedom of this weighted, unbiased sample variance vary accordingly from N − 1 down to 0. The standard deviation is simply the square root of the variance above.
As a side note, other approaches have been described to compute the weighted sample variance. [ 7 ]
In a weighted sample, each row vector x i {\displaystyle \mathbf {x} _{i}} (each set of single observations on each of the K random variables) is assigned a weight w i ≥ 0 {\displaystyle w_{i}\geq 0} .
Then the weighted mean vector μ ∗ {\displaystyle \mathbf {\mu ^{*}} } is given by
And the weighted covariance matrix is given by: [ 8 ]
Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights.
If the weights are frequency weights , the unbiased weighted estimate of the covariance matrix C {\displaystyle \textstyle \mathbf {C} } , with Bessel's correction, is given by: [ 8 ]
This estimator can be unbiased only if the weights are not standardized nor normalized , these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction).
In the case of reliability weights , the weights are normalized :
(If they are not, divide the weights by their sum to normalize prior to calculating V 1 {\displaystyle V_{1}} :
Then the weighted mean vector μ ∗ {\displaystyle \mathbf {\mu ^{*}} } can be simplified to
and the unbiased weighted estimate of the covariance matrix C {\displaystyle \mathbf {C} } is: [ 9 ]
The reasoning here is the same as in the previous section.
Since we are assuming the weights are normalized, then V 1 = 1 {\displaystyle V_{1}=1} and this reduces to:
If all weights are the same, i.e. w i / V 1 = 1 / N {\displaystyle w_{i}/V_{1}=1/N} , then the weighted mean and covariance reduce to the unweighted sample mean and covariance above.
The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood estimate. We simply replace the variance σ 2 {\displaystyle \sigma ^{2}} by the covariance matrix C {\displaystyle \mathbf {C} } and the arithmetic inverse by the matrix inverse (both denoted in the same way, via superscripts); the weight matrix then reads: [ 10 ]
W i = C i − 1 . {\displaystyle \mathbf {W} _{i}=\mathbf {C} _{i}^{-1}.}
The weighted mean in this case is: x ¯ = C x ¯ ( ∑ i = 1 n W i x i ) , {\displaystyle {\bar {\mathbf {x} }}=\mathbf {C} _{\bar {\mathbf {x} }}\left(\sum _{i=1}^{n}\mathbf {W} _{i}\mathbf {x} _{i}\right),} (where the order of the matrix–vector product is not commutative ), in terms of the covariance of the weighted mean: C x ¯ = ( ∑ i = 1 n W i ) − 1 , {\displaystyle \mathbf {C} _{\bar {\mathbf {x} }}=\left(\sum _{i=1}^{n}\mathbf {W} _{i}\right)^{-1},}
For example, consider the weighted mean of the point [1 0] with high variance in the second component and [0 1] with high variance in the first component. Then
then the weighted mean is:
which makes sense: the [1 0] estimate is "compliant" in the second component and the [0 1] estimate is compliant in the first component, so the weighted mean is nearly [1 1].
In the general case, suppose that X = [ x 1 , … , x n ] T {\displaystyle \mathbf {X} =[x_{1},\dots ,x_{n}]^{T}} , C {\displaystyle \mathbf {C} } is the covariance matrix relating the quantities x i {\displaystyle x_{i}} , x ¯ {\displaystyle {\bar {x}}} is the common mean to be estimated, and J {\displaystyle \mathbf {J} } is a design matrix equal to a vector of ones [ 1 , … , 1 ] T {\displaystyle [1,\dots ,1]^{T}} (of length n {\displaystyle n} ). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by:
and
where:
Consider the time series of an independent variable x {\displaystyle x} and a dependent variable y {\displaystyle y} , with n {\displaystyle n} observations sampled at discrete times t i {\displaystyle t_{i}} . In many common situations, the value of y {\displaystyle y} at time t i {\displaystyle t_{i}} depends not only on x i {\displaystyle x_{i}} but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean z {\displaystyle z} for a window size m {\displaystyle m} .
In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction 0 < Δ < 1 {\displaystyle 0<\Delta <1} at each time step. Setting w = 1 − Δ {\displaystyle w=1-\Delta } we can define m {\displaystyle m} normalized weights by
where V 1 {\displaystyle V_{1}} is the sum of the unnormalized weights. In this case V 1 {\displaystyle V_{1}} is simply
approaching V 1 = 1 / ( 1 − w ) {\displaystyle V_{1}=1/(1-w)} for large values of m {\displaystyle m} .
The damping constant w {\displaystyle w} must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then the following properties of exponentially decreasing weights are useful in making a suitable choice: at step ( 1 − w ) − 1 {\displaystyle (1-w)^{-1}} , the weight approximately equals e − 1 ( 1 − w ) = 0.39 ( 1 − w ) {\displaystyle {e^{-1}}(1-w)=0.39(1-w)} , the tail area the value e − 1 {\displaystyle e^{-1}} , the head area 1 − e − 1 = 0.61 {\displaystyle {1-e^{-1}}=0.61} . The tail area at step n {\displaystyle n} is ≤ e − n ( 1 − w ) {\displaystyle \leq {e^{-n(1-w)}}} . Where primarily the closest n {\displaystyle n} observations matter and the effect of the remaining observations can be ignored safely, then choose w {\displaystyle w} such that the tail area is sufficiently small.
The concept of weighted average can be extended to functions. [ 11 ] Weighted averages of functions play an important role in the systems of weighted differential and integral calculus. [ 12 ]
Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that χ 2 {\displaystyle \chi ^{2}} is too large. The correction that must be made is
where χ ν 2 {\displaystyle \chi _{\nu }^{2}} is the reduced chi-squared :
The square root σ ^ x ¯ {\displaystyle {\hat {\sigma }}_{\bar {x}}} can be called the standard error of the weighted mean (variance weights, scale corrected) .
When all data variances are equal, σ i = σ 0 {\displaystyle \sigma _{i}=\sigma _{0}} , they cancel out in the weighted mean variance, σ x ¯ 2 {\displaystyle \sigma _{\bar {x}}^{2}} , which again reduces to the standard error of the mean (squared), σ x ¯ 2 = σ 2 / n {\displaystyle \sigma _{\bar {x}}^{2}=\sigma ^{2}/n} , formulated in terms of the sample standard deviation (squared),
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https://en.wikipedia.org/wiki/Weighted_arithmetic_mean
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Weighted correlation network analysis , also known as weighted gene co-expression network analysis (WGCNA), is a widely used data mining method especially for studying biological networks based on pairwise correlations between variables. While it can be applied to most high-dimensional data sets, it has been most widely used in genomic applications. It allows one to define modules (clusters), intramodular hubs, and network nodes with regard to module membership, to study the relationships between co-expression modules, and to compare the network topology of different networks (differential network analysis). WGCNA can be used as a data reduction technique (related to oblique factor analysis ), as a clustering method (fuzzy clustering), as a feature selection method (e.g. as gene screening method), as a framework for integrating complementary (genomic) data (based on weighted correlations between quantitative variables), and as a data exploratory technique. [ 1 ] Although WGCNA incorporates traditional data exploratory techniques, its intuitive network language and analysis framework transcend any standard analysis technique. Since it uses network methodology and is well suited for integrating complementary genomic data sets, it can be interpreted as systems biologic or systems genetic data analysis method. By selecting intramodular hubs in consensus modules, WGCNA also gives rise to network based meta analysis techniques. [ 2 ]
The WGCNA method was developed by Steve Horvath , a professor of human genetics at the David Geffen School of Medicine at UCLA and of biostatistics at the UCLA Fielding School of Public Health and his colleagues at UCLA, and (former) lab members (in particular Peter Langfelder, Bin Zhang, Jun Dong). Much of the work arose from collaborations with applied researchers. In particular, weighted correlation networks were developed in joint discussions with cancer researchers Paul Mischel , Stanley F. Nelson, and neuroscientists Daniel H. Geschwind , Michael C. Oldham, according to the acknowledgement section in. [ 1 ]
A weighted correlation network can be interpreted as special case of a weighted network , dependency network or correlation network. Weighted correlation network analysis can be attractive for the following reasons:
First, one defines a gene co-expression similarity measure which is used to define the network. We denote the gene co-expression similarity measure of a pair of genes i and j by s i j {\displaystyle s_{ij}} . Many co-expression studies use the absolute value of the correlation as an unsigned co-expression similarity measure,
s i j u n s i g n e d = | c o r ( x i , x j ) | {\displaystyle s_{ij}^{unsigned}=|cor(x_{i},x_{j})|}
where gene expression profiles x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} consist of the expression of genes i and j across multiple samples. However, using the absolute value of the correlation may obfuscate biologically relevant information, since no distinction is made between gene repression and activation. In contrast, in signed networks the similarity between genes reflects the sign of the correlation of their expression profiles. Varied transformation (or scaling) approaches can be considered if a signed co-expression measure between gene expression profiles x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} is needed. For example, one can (linearly) scale the correlations to be within the [ 0 , 1 ] {\displaystyle [0,1]} range by performing a simple transformation of the correlations as follows:
s i j s i g n e d = 0.5 + 0.5 c o r ( x i , x j ) {\displaystyle s_{ij}^{signed}=0.5+0.5cor(x_{i},x_{j})}
As the unsigned measure s i j u n s i g n e d {\displaystyle s_{ij}^{unsigned}} , the signed similarity s i j s i g n e d {\displaystyle s_{ij}^{signed}} takes on a value between 0 and 1. Note that the unsigned similarity between two oppositely expressed genes ( c o r ( x i , x j ) = − 1 {\displaystyle cor(x_{i},x_{j})=-1} ) equals 1 while it equals 0 for the signed similarity. Similarly, while the unsigned co-expression measure of two genes with zero correlation remains zero, the signed similarity equals 0.5.
Next, an adjacency matrix (network), A = [ a i j ] {\displaystyle A=[a_{ij}]} , is used to quantify how strongly genes are connected to one another. A {\displaystyle A} is defined by thresholding the co-expression similarity matrix S = [ s i j ] {\displaystyle S=[s_{ij}]} . 'Hard' thresholding (dichotomizing) the similarity measure S {\displaystyle S} results in an unweighted gene co-expression network. Specifically an unweighted network adjacency is defined to be 1 if s i j > τ {\displaystyle s_{ij}>\tau } and 0 otherwise.
Because hard thresholding encodes gene connections in a binary fashion, it can be sensitive to the choice of the threshold and result in the loss of co-expression information. [ 3 ] The continuous nature of the co-expression information can be preserved by employing soft thresholding, which results in a weighted network. Specifically, WGCNA uses the following power function assess their connection strength:
a i j = ( s i j ) β {\textstyle a_{ij}=(s_{ij})^{\beta }} ,
where the power β {\displaystyle \beta } is the soft thresholding parameter. The default values β = 6 {\displaystyle \beta =6} and β = 12 {\displaystyle \beta =12} are used for unsigned and signed networks, respectively. Alternatively, β {\displaystyle \beta } can be chosen using the scale-free topology criterion which amounts to choosing the smallest value of β {\displaystyle \beta } such that approximate scale free topology is reached. [ 3 ]
Since l o g ( a i j ) = β l o g ( s i j ) {\displaystyle log(a_{ij})=\beta log(s_{ij})} , the weighted network adjacency is linearly related to the co-expression similarity on a logarithmic scale. Note that a high power β {\displaystyle \beta } transforms high similarities into high adjacencies, while pushing low similarities towards 0. Since this soft-thresholding procedure applied to a pairwise correlation matrix leads to weighted adjacency matrix, the ensuing analysis is referred to as weighted gene co-expression network analysis.
A major step in the module centric analysis is to cluster genes into network modules using a network proximity measure. Roughly speaking, a pair of genes has a high proximity if it is closely interconnected. By convention, the maximal proximity between two genes is 1 and the minimum proximity is 0. Typically, WGCNA uses the topological overlap measure (TOM) as proximity. [ 9 ] [ 10 ] which can also be defined for weighted networks. [ 3 ] The TOM combines the adjacency of two genes and the connection strengths these two genes share with other "third party" genes. The TOM is a highly robust measure of network interconnectedness (proximity). This proximity is used as input of average linkage hierarchical clustering. Modules are defined as branches of the resulting cluster tree using the dynamic branch cutting approach. [ 11 ] Next the genes inside a given module are summarized with the module eigengene , which can be considered as the best summary of the standardized module expression data. [ 4 ] The module eigengene of a given module is defined as the first principal component of the standardized expression profiles. Eigengenes define robust biomarkers, [ 12 ] and can be used as features in complex machine learning models such as Bayesian networks . [ 13 ] To find modules that relate to a clinical trait of interest, module eigengenes are correlated with the clinical trait of interest, which gives rise to an eigengene significance measure. Eigengenes can be used as features in more complex predictive models including decision trees and Bayesian networks. [ 12 ] One can also construct co-expression networks between module eigengenes (eigengene networks), i.e. networks whose nodes are modules. [ 14 ] To identify intramodular hub genes inside a given module, one can use two types of connectivity measures. The first, referred to as k M E i = c o r ( x i , M E ) {\displaystyle kME_{i}=cor(x_{i},ME)} , is defined based on correlating each gene with the respective module eigengene. The second, referred to as kIN, is defined as a sum of adjacencies with respect to the module genes. In practice, these two measures are equivalent. [ 4 ] To test whether a module is preserved in another data set, one can use various network statistics, e.g. Z s u m m a r y {\displaystyle Zsummary} . [ 6 ]
WGCNA has been widely used for analyzing gene expression data (i.e. transcriptional data), e.g. to find intramodular hub genes. [ 2 ] [ 15 ] Such as, WGCNA study reveals novel transcription factors are associated with Bisphenol A (BPA) dose-response. [ 16 ]
It is often used as data reduction step in systems genetic applications where modules are represented by "module eigengenes" e.g. [ 17 ] [ 18 ] Module eigengenes can be used to correlate modules with clinical traits. Eigengene networks are coexpression networks between module eigengenes (i.e. networks whose nodes are modules) .
WGCNA is widely used in neuroscientific applications, e.g. [ 19 ] [ 20 ] and for analyzing genomic data including microarray data, [ 21 ] single cell RNA-Seq data [ 22 ] [ 23 ] DNA methylation data, [ 24 ] miRNA data, peptide counts [ 25 ] and microbiota data (16S rRNA gene sequencing). [ 26 ] Other applications include brain imaging data, e.g. functional MRI data. [ 27 ]
The WGCNA R software package [ 28 ] provides functions for carrying out all aspects of weighted network analysis (module construction, hub gene selection, module preservation statistics, differential network analysis, network statistics). The WGCNA package is available from the Comprehensive R Archive Network (CRAN), the standard repository for
R add-on packages.
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https://en.wikipedia.org/wiki/Weighted_correlation_network_analysis
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In statistics , the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean .
Given a sample x = ( x 1 , x 2 … , x n ) {\displaystyle x=(x_{1},x_{2}\dots ,x_{n})} and weights w = ( w 1 , w 2 , … , w n ) {\displaystyle w=(w_{1},w_{2},\dots ,w_{n})} , it is calculated as: [ 1 ]
The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values.
If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean. [ 1 ]
This statistics -related article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Weighted_geometric_mean
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Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell has exactly the same number of nearest, next nearest, nearest of next nearest etc. neighbors and hence they are called regular lattice. Often physicists and mathematicians study phenomena which require disordered lattice where each cell do not have exactly the same number of neighbors rather the number of neighbors can vary wildly. For instance, if one wants to study the spread of disease, viruses, rumors etc. then the last thing one would look for is the square lattice. In such cases a disordered lattice is necessary. One way of constructing a disordered lattice is by doing the following.
Starting with a square, say of unit area, and dividing randomly at each step only one block, after
picking it preferentially with respect to ares, into four smaller blocks creates weighted planar stochastic lattice (WPSL) . Essentially it is a disordered planar lattice as its block size and their coordination number are random.
In applied mathematics, a weighted planar stochastic lattice (WPSL) is a structure that has properties in common with those of lattices and those of graphs . In general, space-filling planar cellular structures can be useful in a wide variety of seemingly disparate physical and biological systems.
Examples include grain in polycrystalline structures, cell texture and tissues in biology, acicular texture in martensite growth, tessellated pavement on ocean shores, soap froths
and agricultural land division according to ownership etc. [ 1 ] [ 2 ] [ 3 ] The question of how these structures appear and the understanding of their topological and geometrical properties have always been an interesting proposition among scientists in general and physicists in particular.
Several models prescribe how to generate cellular structures. Often these
structures can mimic directly the structures found in nature and they are able to capture the essential properties that we find in natural structures.
In general, cellular structures appear through random tessellation , tiling, or subdivision of a plane into contiguous and non-overlapping cells. For instance, Voronoi diagram and Apollonian packing are formed by partitioning or tiling of a plane into contiguous and non-overlapping convex polygons and disks respectively. [ 2 ] [ 4 ]
Regular planar lattices like square lattices, triangular lattices, honeycomb lattices, etc., are the simplest example of the cellular structure in which every cell has exactly the same size and the same coordination number. The planar Voronoi diagram , on the other hand, has neither a fixed cell size nor a fixed coordination number. Its coordination number distribution is rather Poissonian in nature. [ 5 ] That is, the distribution is peaked about the mean where it is almost impossible to find cells which have significantly higher or fewer coordination number than the mean. Recently, Hassan et al proposed a lattice, namely the weighted planar stochastic lattice. For instance, unlike a network or a graph, it has properties of lattices as its sites are spatially embedded. On the other hand, unlike lattices, its dual (obtained by considering the center of each block of the lattice as a node and the common border between blocks as links) display the property of networks as its degree distribution follows a power law . Besides, unlike regular lattices, the sizes of its cells are not equal; rather, the distribution of the area size of its blocks obeys dynamic scaling , [ 6 ] whose coordination number distribution follows a power-law. [ 7 ] [ 8 ]
The construction process of the WPSL can be described as follows. It starts with a square of unit area which we regard as an initiator. The generator then divides the initiator, in the first step, randomly with uniform probability into four smaller blocks. In the second step and thereafter, the generator is applied to only one of the blocks. The question is: How do we pick that block when there is more than one block? The most generic choice would be to pick preferentially according to their areas so that the higher the area the higher the probability to be picked. For instance, in step one, the generator divides the initiator randomly into four smaller blocks. Let us label their areas starting from the top left corner and moving clockwise as a 1 , a 2 , a 3 {\displaystyle a_{1},a_{2},a_{3}} and a 4 {\displaystyle a_{4}} . But of course the way we label is totally arbitrary and will bear no consequence to the final results of any observable quantities. Note that a i {\displaystyle a_{i}} is the area of the i {\displaystyle i} th block which can be well regarded as the probability of picking the i {\displaystyle i} th block. These probabilities are naturally normalized ∑ i a i = 1 {\displaystyle \sum _{i}a_{i}=1} since we choose the area of the initiator equal to one. In step two, we pick one of the four blocks preferentially with respect to their areas. Consider that we pick the block 3 {\displaystyle 3} and apply the generator onto it to divide it randomly into four smaller blocks. Thus the label 3 {\displaystyle 3} is now redundant and hence we recycle it to label the top left corner while the rest of three new blocks are labelled a 5 , a 6 {\displaystyle a_{5},a_{6}} and a 7 {\displaystyle a_{7}} in a clockwise fashion. In general, in the j {\displaystyle j} th step, we pick one out of 3 j − 2 {\displaystyle 3j-2} blocks preferentially with respect to area and divide randomly into four blocks. The detailed algorithm can be found in Dayeen and Hassan [ 6 ] and Hassan, Hassan, and Pavel. [ 8 ]
This process of lattice generation can also be described as follows. Consider that the substrate is a square of unit area and at each time step a seed is nucleated from which two orthogonal partitioning lines parallel to the sides of the substrate are grown until intercepted by existing lines. It results in partitioning the square into ever smaller mutually exclusive rectangular blocks. Note that the higher the area of a block, the higher is the probability that the seed will be nucleated in it to divide that into four smaller blocks since seeds are sown at random on the substrate. It can also describes kinetics of fragmentation of two-dimensional objects. [ 9 ] [ 10 ]
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https://en.wikipedia.org/wiki/Weighted_planar_stochastic_lattice
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The weighted product model ( WPM ) is a popular multi-criteria decision analysis (MCDA) / multi-criteria decision making (MCDM) method. It is similar to the weighted sum model (WSM) in that it produces a simple score, but has the very important advantage of overcoming the issue of 'adding apples and pears' i.e. adding together quantities measured in different units. While there are various ways of normalizing the data beforehand, the results of the weighted sum model differ according to which normalization is chosen. The weighted product approach does not require any normalization because it uses multiplication instead of addition to aggregate the data.
As with all MCDA / MCDM methods, given is a finite set of decision alternatives described in terms of a number of decision criteria. Each decision alternative is compared with the others by multiplying a number of ratios, one for each decision criterion. Each ratio is raised to the power equivalent to the relative weight of the corresponding criterion.
Suppose that a given MCDA problem is defined on m alternatives and n decision criteria. Furthermore, let us assume that all the criteria are benefit criteria. That is, the higher the values are, the better it is. Next suppose that w j denotes the relative weight of importance of the criterion C j and a ij is the performance value of alternative A i when it is evaluated in terms of criterion C j . Then, if one wishes to compare the two alternatives A K and A L (where m ≥ K , L ≥ 1) then, the following product has to be calculated: [ 1 ]
If the ratio P ( A K / A L ) is greater than or equal to the value 1, then it indicates that alternative A K is more desirable than alternative A L (in the maximization case). If we are interested in determining the best alternative, then the best alternative is the one that is better than or at least equal to all other alternatives.
The WPM is often called dimensionless analysis because its mathematical structure eliminates any units of measure. [ 1 ] [ 2 ]
Therefore, the WPM can be used in single- and multi-dimensional MCDA / MCDM problems. That is, on decision problems where the alternatives are described in terms that use different units of measurement. An advantage of this method is that instead of the actual values it can use relative ones.
The following is a simple numerical example which illustrates how the calculations for this method can be carried out. As data we use the same numerical values as in the numerical example described for the weighted sum model . These numerical data are repeated next for easier reference.
This simple decision problem is based on three alternatives denoted as A 1 , A 2 , and A 3 each described in terms of four criteria C 1 , C 2 , C 3 and C 4 . Next, let the numerical data for this problem be as in the following decision matrix:
The above table specifies that the relative weight of the first criterion is 0.20, the relative weight for the second criterion is 0.15 and so on. Similarly, the value of the first alternative (i.e., A 1 ) in terms of the first criterion is equal to 25, the value of the same alternative in terms of the second criterion is equal to 20 and so on. However, now the restriction to express all criteria in terms of the same measurement unit is not needed. That is, the numbers under each criterion may be expressed in different units.
When the WPM is applied on the previous data, then the following values are derived:
Similarly, we also get:
Therefore, the best alternative is A 1 , since it is superior to all the other alternatives. Furthermore, the following ranking of all three alternatives is as follows: A 1 > A 2 > A 3 (where the symbol ">" stands for "better than").
An alternative approach with the WPM method is for the decision maker to use only products without the previous ratios. [ 1 ] [ 2 ] That is, to use the following variant of main formula given earlier:
In the previous expression the term P ( A K ) denotes the total performance value (i.e., not a relative one) of alternative A K when all the criteria are considered simultaneously under the WPM model. Then, when the previous data are used, exactly the same ranking is derived. Some interesting properties of this method are discussed in the 2000 book by Triantaphyllou on MCDA / MCDM . [ 1 ]
An illustrative application is provided by Watters and Tofallis. [ 3 ]
The choice of values for the weights is usually difficult. The simple default of equal weighting is sometimes used. Scoring methods such as WSM and WPM may be used for rankings (universities, countries, consumer products etc.), and the weights will determine the order in which these entities are placed. There is often much argument about the appropriateness of the chosen weights, and whether they are biased or display favouritism. One approach for overcoming this issue is to automatically generate the weights from the data. [ 4 ] This has the advantage of avoiding personal input and so is more objective. The so-called Automatic Democratic Method for weight generation has two key steps:
(1) For each alternative, identify the weights which will maximize its score, subject to the condition that these weights do not lead to any of the alternatives exceeding a score of 100%.
(2) Fit an equation to these optimal scores using regression so that the regression equation predicts these scores as closely as possible using the criteria data as explanatory variables. The regression coefficients then provide the final weights.
Some of the first references to this method are due to Bridgman [ 5 ] and Miller and Starr. [ 6 ] The tutorial article by Tofallis describes its advantages over the weighted sum approach. [ 7 ]
More details on this method are given in the MCDM book by Triantaphyllou. [ 1 ]
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https://en.wikipedia.org/wiki/Weighted_product_model
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Weighted random early detection ( WRED ) is a queueing discipline for a network scheduler suited for congestion avoidance . [ 1 ] It is an extension to random early detection (RED) where a single queue may have several different sets of queue thresholds. Each threshold set is associated to a particular traffic class .
For example, a queue may have lower thresholds for lower priority packet. A queue buildup will cause the lower priority packets to be dropped, hence protecting the higher priority packets in the same queue. In this way quality of service prioritization is made possible for important packets from a pool of packets using the same buffer. [ 2 ]
It is more likely that standard traffic will be dropped instead of higher prioritized traffic.
On Cisco switches WRED is restricted to
Non-IP traffic will be dropped more often than TCP/IP traffic because it is treated with the lowest possible precedence.
WRED proceeds in this order when a packet arrives:
The average queue size depends on the previous average as well as the current size of the queue. The calculation formula is given below:
a v g = o ∗ ( 1 − 2 − n ) + c ∗ ( 2 − n ) {\displaystyle avg=o*(1-2^{-n})+c*(2^{-n})\,\!}
where n {\displaystyle n} is the user-configurable exponential weight factor, o {\displaystyle o} is the old average and c {\displaystyle c} is the current queue size. The previous average is more important for high values of n {\displaystyle n} . Peaks and lows in queue size are smoothed by a high value. For low values of n {\displaystyle n} , the average queue size is close to the current queue size.
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https://en.wikipedia.org/wiki/Weighted_random_early_detection
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In decision theory , the weighted sum model ( WSM ), [ 1 ] [ 2 ] also called weighted linear combination ( WLC ) [ 3 ] or simple additive weighting ( SAW ), [ 4 ] is the best known and simplest multi-criteria decision analysis (MCDA) / multi-criteria decision making method for evaluating a number of alternatives in terms of a number of decision criteria.
In general, suppose that a given MCDA problem is defined on m alternatives and n decision criteria. Furthermore, let us assume that all the criteria are benefit criteria, that is, the higher the values are, the better it is. Next suppose that w j denotes the relative weight of importance of the criterion C j and a ij is the performance value of alternative A i when it is evaluated in terms of criterion C j . Then, the total (i.e., when all the criteria are considered simultaneously) importance of alternative A i , denoted as A i WSM-score , is defined as follows:
For the maximization case, the best alternative is the one that yields the maximum total performance value. [ 2 ] [ clarification needed ]
It is very important to state here that it is applicable only when all the data are expressed in exactly the same unit. If this is not the case, then the final result is equivalent to "adding apples and oranges."
For a simple numerical example suppose that a decision problem of this type is defined on three alternative choices A 1 , A 2 , A 3 each described in terms of four criteria C 1 , C 2 , C 3 and C 4 . Furthermore, let the numerical data for this problem be as in the following decision matrix:
For instance, the relative weight of the first criterion is equal to 0.20, the relative weight for the second criterion is 0.15 and so on. Similarly, the value of the first alternative (i.e., A 1 ) in terms of the first criterion is equal to 25, the value of the same alternative in terms of the second criterion is equal to 20 and so on.
When the previous formula is applied on these numerical data the WSM scores for the three alternatives are:
Similarly, one gets:
Thus, the best choice (in the maximization case) is either alternative A 2 or A 3 (because they both have the maximum WSM score which is equal to 22.00). These numerical results imply the following ranking of these three alternatives: A 2 = A 3 > A 1 (where the symbol ">" stands for "greater than").
The choice of values for the weights is rarely easy. The simple default of equal weighting is sometimes used when all criteria are measured in the same units. Scoring methods may be used for rankings (universities, countries, consumer products etc.), and the weights will determine the order in which these entities are placed. There is often much argument about the appropriateness of the chosen weights, and whether they are biased or display favouritism. One approach for overcoming this issue is to automatically generate the weights from the data. [ 5 ] This has the advantage of avoiding personal input and so is more objective. The so-called Automatic Democratic Method for weight generation has two key steps:
(1) For each alternative, identify the weights which will maximize its score, subject to the condition that these weights do not lead to any of the alternatives exceeding a score of 100%.
(2) Fit an equation to these optimal scores using regression so that the regression equation predicts these scores as closely as possible using the criteria data as explanatory variables. The regression coefficients then provide the final weights.
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https://en.wikipedia.org/wiki/Weighted_sum_model
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The process of frequency weighting involves emphasizing the contribution of particular aspects of a phenomenon (or of a set of data ) over others to an outcome or result; thereby highlighting those aspects in comparison to others in the analysis . That is, rather than each variable in the data set contributing equally to the final result, some of the data is adjusted to make a greater contribution than others. This is analogous to the practice of adding (extra) weight to one side of a pair of scales in order to favour either the buyer or seller.
While weighting may be applied to a set of data, such as epidemiological data, it is more commonly applied to measurements of light, heat, sound, gamma radiation , and in fact any stimulus that is spread over a spectrum of frequencies.
In the measurement of loudness , for example, a weighting filter is commonly used to emphasise frequencies around 3 to 6 kHz where the human ear is most sensitive, while attenuating very high and very low frequencies to which the ear is insensitive. A commonly used weighting is the A-weighting curve, which results in units of dBA sound pressure level. Because the frequency response of human hearing varies with loudness, the A-weighting curve is correct only at a level of 40- phon and other curves known as B- , C- and D-weighting are also used, the latter being particularly intended for the measurement of aircraft noise.
In broadcasting and audio equipment measurements 468-weighting is the preferred weighting to use because it was specifically devised to allow subjectively valid measurements on noise, rather than pure tones. It is often not realised that equal loudness curves, and hence A-weighting, really apply only to tones, as tests with noise bands show increased sensitivity in the 5 to 7 kHz region on noise compared to tones.
Other weighting curves are used in rumble measurement and flutter measurement to properly assess subjective effect.
In each field of measurement, special units are used to indicate a weighted measurement as opposed to a basic physical measurement of energy level. For sound, the unit is the phon (1 kHz equivalent level).
In the fields of acoustics and audio engineering, it is common to use a standard curve referred to as A-weighting , one of a set that are said to be derived from equal-loudness contours .
Auditory frequency weighting functions for marine mammals were introduced by Southall et al. (2007). [ 1 ]
In the measurement of gamma rays or other ionising radiation, a radiation monitor or dosimeter will commonly use a filter to attenuate those energy levels or wavelengths that cause the least damage to the human body but letting through those that do the most damage, so any source of radiation may be measured in terms of its true danger rather than just its strength. The resulting unit is the sievert or microsievert.
Another use of weighting is in television, in which the red, green and blue components of the signal are weighted according to their perceived brightness. This ensures compatibility with black and white receivers and also benefits noise performance and allows separation into meaningful luminance and chrominance signals for transmission.
Skin damage due to sun exposure is very wavelength dependent over the UV range 295 to 325 nm, with power at the shorter wavelength causing around 30 times as much damage as the longer one. In the calculation of UV Index , a weighting curve is used which is known as the McKinlay-Diffey Erythema action spectrum. [1] Archived 2010-06-13 at the Wayback Machine
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https://en.wikipedia.org/wiki/Weighting
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A weighting curve is a graph of a set of factors, that are used to 'weight' measured values of a variable according to their importance in relation to some outcome. An important example is frequency weighting in sound level measurement where a specific set of weighting curves known as A- , B- , C- , and D-weighting as defined in IEC 61672 [ 1 ] are used. Unweighted measurements of sound pressure do not correspond to perceived loudness because the human ear is less sensitive at low and high frequencies, with the effect more pronounced at lower sound levels. The four curves are applied to the measured sound level, for example by the use of a weighting filter in a sound level meter, to arrive at readings of loudness in phons or in decibels (dB) above the threshold of hearing (see A-weighting ).
Although A-weighting with a slow RMS detector, as commonly used in sound level meters [ 1 ] is frequently used when measuring noise in audio circuits, a different weighting curve, ITU-R 468 noise weighting uses a psophometric weighting curve and a quasi-peak detector. [ 2 ] This method, formerly known as CCIR weighting , is preferred by the telecommunications industry, broadcasters, and some equipment manufacturers as it reflects more accurately the audibility of pops and short bursts of random noise as opposed to pure tones. Psophometric weighting is used in telephony and telecommunications where narrow-band circuits are common.
Hearing weighting curves are also used for sound in water. [ 3 ]
Acoustics is by no means the only subject which finds use for weighting curves however, and they are widely used in deriving measures of effect for sun exposure, gamma radiation exposure, and many other things. In the measurement of gamma rays or other ionising radiation , a radiation monitor or dosimeter will commonly use a filter to attenuate those energy levels or wavelengths that cause the least damage to the human body, while letting through those that do the most damage, so that any source of radiation may be measured in terms of its true danger rather than just its "strength". The sievert is a unit of weighted radiation dose for ionising radiation , which supersedes the older weighted unit the rem ( roentgen equivalent man ).
Weighting is also applied to the measurement of sunlight when assessing the risk of skin damage through sunburn , since different wavelengths have different biological effects. Common examples are the SPF of sunscreen, and the ultraviolet index .
Another use of weighting is in television, where the red, green, and blue components of the signal are weighted according to their perceived brightness. This ensures compatibility with black-and-white receivers, and also benefits noise performance and allows separation into meaningful luminance and chrominance signals for transmission.
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https://en.wikipedia.org/wiki/Weighting_curve
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A weighting filter is used to emphasize or suppress some aspects of a phenomenon compared to others, for measurement or other purposes.
In each field of audio measurement, special units are used to indicate a weighted measurement as opposed to a basic physical measurement of energy level. For sound, the unit is the phon (1 kHz equivalent level).
Sound has three basic components, the wavelength , frequency , and speed . In sound measurement, we measure the loudness of the sound in decibels (dB). Decibels are logarithmic with 0 dB as the reference. [ 1 ] There are also a range of frequencies that sounds can have. Frequency is the number of times a sine wave repeats itself in a second. [ 2 ] Normal auditory systems can usually hear between 20 and 20,000 Hz. [ 2 ] When we measure sound, the measurement instrument takes the incoming auditory signal and analyzes it for these different features. Weighting filters in these instruments then filter out certain frequencies and decibel levels depending on the filter. A weighted filters are most similar to natural human hearing. This allows the sound level meter to determine what decibel level the incoming sound would likely be for a normal hearing human's auditory system.
In the measurement of loudness, for example, an A-weighting filter is commonly used to emphasize frequencies around 3–6 kHz where the human ear is most sensitive, while attenuating very high and very low frequencies to which the ear is insensitive. The aim is to ensure that measured loudness corresponds well with subjectively perceived loudness.
A-weighting is only really valid for relatively quiet sounds and for pure tones as it is based on the 40-phon Fletcher–Munson equal-loudness contour . [ 3 ] The B and C curves were intended for louder sounds (though they are less used) while the D curve is used in assessing loud aircraft noise ( IEC 537 ). B curves filter out more medium loudness levels when compared to an A curves. [ 3 ] This curve is rarely ever used in the assessment or monitoring of noise levels anymore. [ 4 ] C curves differ from both A and B in the fact that they filter less of the lower and higher frequencies. [ 3 ] The filter is a much flatter shape and is used in sound measurement in especially loud and noisy environments. [ 3 ] A weighted curves follow a 40 phon curve while C weighted follows a 100 phon curve. [ 4 ] The three curves differ not in their measurement of exposure levels, but in the frequencies measured. A weighted curves allow more frequencies equal to or less than 500 Hz through, which is most representative of the human ear. [ 4 ]
There are a variety of reasons for measuring sound. This includes following regulations to protect worker's hearing , following noise ordinances , in telecommunications , and many more. At the basis of sound measurement is the idea of breaking down an incoming signal based on its different properties. Every incoming sinusoidal wave of sound has a frequency and amplitude. Using this information, a sound level can be deduced from the root-sums-of-squares of the amplitudes of all the incoming auditory information. [ 4 ] Whether using a sound level meter or a noise dosimeter , the processing is somewhat similar. With a calibrated sound level meter, the incoming sounds are going to be picked up by the microphone and then measured by the internal electronic circuits. [ 5 ] The sound measurement that the device outputs can be filtered through an A, B, or C weighting curve. The curve used will have slight effects on the resulting decibel level.
In the field of telecommunications , weighting filters are widely used in the measurement of electrical noise on telephone circuits, and in the assessment of noise as perceived through the acoustic response of different types of instrument (handset). Other noise-weighting curves have existed, e.g. DIN standards. The term psophometric weighting , though referring in principle to any weighting curve intended for noise measurement, is often used to refer to a particular weighting curve, used in telephony for narrow-bandwidth voiceband speech circuits.
A-weighted decibels are abbreviated dB(A) or dBA. When acoustic ( calibrated microphone) measurements are being referred to, then the units used will be dB SPL ( sound pressure level ) referenced to 20 micropascals = 0 dB SPL. [ 6 ] [ nb 1 ]
The A-weighting curve has been widely adopted for environmental noise measurement, and is standard in many sound level meters (see ITU-R 468 weighting for a further explanation).
A-weighting is also in common use for assessing potential hearing damage caused by loud noise, though this seems to be based on the widespread availability of sound level meters incorporating A-Weighting rather than on any good experimental evidence to suggest that such use is valid. The distance of the measuring microphone from a sound source is often "forgotten", when SPL measurements are quoted, making the data useless. In the case of environmental or aircraft noise , distance need not be quoted as it is the level at the point of measurement that is needed, but when measuring refrigerators and similar appliances the distance should be stated; where not stated it is usually one metre (1 m). An extra complication here is the effect of a reverberant room, and so noise measurement on appliances should state "at 1 m in an open field" or "at 1 m in anechoic chamber ". Measurements made outdoors will approximate well to anechoic conditions. [ citation needed ]
A-weighted SPL measurements of noise level are increasingly to be found on sales literature for domestic appliances such as refrigerators and freezers, and computer fans. Although the threshold of hearing is typically around 0 dB SPL, this is in fact very quiet indeed, and appliances are more likely to have noise levels of 30 to 40 dB SPL.
Human sensitivity to noise in the region of 6 kHz became particularly apparent in the late 1960s with the introduction of compact cassette recorders and Dolby-B noise reduction . A-weighted noise measurements were found to give misleading results because they did not give sufficient prominence to the 6 kHz region where the noise reduction was having greatest effect, and sometimes one piece of equipment would even measure worse than another and yet sound better, because of differing spectral content.
ITU-R 468 noise weighting was therefore developed to more accurately reflect the subjective loudness of all types of noise, as opposed to tones. This curve, which came out of work done by the BBC Research Department, and was standardised by the CCIR and later adopted by many other standards bodies ( IEC , BSI /) and, as of 2006 [update] , is maintained by the ITU. Noise measurements using this weighting typically also use a quasi-peak detector law rather than slow averaging. This also helps to quantify the audibility of bursty noise, ticks and pops that might go undetected with a slow rms measurement.
ITU-R 468 noise weighting with quasi-peak detection is widely used in Europe, [ 7 ] especially in telecommunications, and in broadcasting particularly after it was adopted by the Dolby corporation who realised its superior validity for their purposes. Its advantages over A-weighting seem to be less well appreciated in the US and in consumer electronics, where the use of A-weighting predominates—probably because A-weighting produces a 9 to 12 dB "better" specification, see specsmanship . [ citation needed ] [ neutrality is disputed ] It is commonly used by broadcasters in Britain, Europe, and former countries of the British Empire such as Australia and South Africa.
Though the noise level of 16-bit audio systems (such as CD players) is commonly quoted (on the basis of calculations that take no account of subjective effect) as −96 dB relative to FS (full scale), the best 468-weighted results are in the region of −68 dB relative to Alignment Level (commonly defined as 18 dB below FS) i.e. −86 dB relative to FS.
The use of weighting curves is in no way to be regarded as 'cheating', provided that the proper curve is used. Nothing of relevance is being 'hidden', and even when, for example, hum is present at 50 or 100 Hz at a level above the quoted (weighted) noise floor this is of no importance because our ears are very insensitive to low frequencies at low levels, so it will not be heard. A-weighting is often used to compare and qualify ADCs , for instance, because it more accurately represents the way noise shaping hides dither noise in the ultrasonic range.
In the measurement of gamma rays or other ionising radiation , a radiation monitor or dosimeter will commonly use a filter to attenuate those energy levels or wavelengths that cause the least damage to the human body, while letting through those that do the most damage, so that any source of radiation may be measured in terms of its true danger rather than just its 'strength'. The sievert is a unit of weighted radiation dose for ionising radiation , which supersedes the older unit the REM ( roentgen equivalent man).
Weighting is also applied to the measurement of sunlight when assessing the risk of skin damage through sunburn , since different wavelengths have different biological effects. Common examples are the SPF of sunscreen, and the UV index .
Another use of weighting is in television, where the red, green and blue components of the signal are weighted according to their perceived brightness. This ensures compatibility with black and white receivers, and also benefits noise performance and allows separation into meaningful luminance and chrominance signals for transmission.
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https://en.wikipedia.org/wiki/Weighting_filter
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In mathematics , the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number τ ( G ) {\displaystyle \tau (G)} of a simply connected simple algebraic group defined over a number field is 1. In this case, simply connected means "not having a proper algebraic covering" in the algebraic group theory sense, which is not always the topologists' meaning .
Weil ( 1959 ) calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. The first observation does not hold for all groups: Ono (1963) found examples where the Tamagawa numbers are not integers. The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.
Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups . K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups . Kottwitz (1988) proved it for all groups satisfying the Hasse principle , which at the time was known for all groups without E 8 factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E 8 case (see strong approximation in algebraic groups ), thus completing the proof of Weil's conjecture. In 2011, Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over function fields over finite fields , [ 1 ] formally published in Gaitsgory & Lurie (2019) , and a future proof using a version of the Grothendieck- Lefschetz trace formula will be published in a second volume.
Ono (1965) used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups.
For spin groups , the conjecture implies the known Smith–Minkowski–Siegel mass formula . [ 1 ]
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https://en.wikipedia.org/wiki/Weil's_conjecture_on_Tamagawa_numbers
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In mathematics , the Weil algebra of a Lie algebra g , introduced by Cartan ( 1951 ) based on unpublished work of André Weil , is a differential graded algebra given by the Koszul algebra Λ( g *)⊗ S ( g *) of its dual g *.
This algebra -related article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Weil_algebra
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In mathematics , the Weil conjectures were highly influential proposals by André Weil ( 1949 ). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory .
The conjectures concern the generating functions (known as local zeta functions ) derived from counting points on algebraic varieties over finite fields . A variety V over a finite field with q elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers N k of points over the extension field with q k elements.
Weil conjectured that such zeta functions for smooth varieties are rational functions , satisfy a certain functional equation , and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function , a kind of generating function for prime integers, which obeys a functional equation and (conjecturally) has its zeros restricted by the Riemann hypothesis . The rationality was proved by Bernard Dwork ( 1960 ), the functional equation by Alexander Grothendieck ( 1965 ), and the analogue of the Riemann hypothesis by Pierre Deligne ( 1974 ).
The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae ( Mazur 1974 ), concerned with roots of unity and Gaussian periods . In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons; and assumes that p is a prime number congruent to 1 modulo 3. Then there is a cyclic cubic field inside the cyclotomic field of p th roots of unity, and a normal integral basis of periods for the integers of this field (an instance of the Hilbert–Speiser theorem ). Gauss constructs the order-3 periods, corresponding to the cyclic group ( Z / p Z ) × of non-zero residues modulo p under multiplication and its unique subgroup of index three. Gauss lets R {\displaystyle {\mathfrak {R}}} , R ′ {\displaystyle {\mathfrak {R}}'} , and R ″ {\displaystyle {\mathfrak {R}}''} be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to exp(2 πi / p ) , he notes that these periods have a multiplication table that is accessible to calculation. Products are linear combinations of the periods, and he determines the coefficients. He sets, for example, ( R R ) {\displaystyle ({\mathfrak {R}}{\mathfrak {R}})} equal to the number of elements of Z / p Z which are in R {\displaystyle {\mathfrak {R}}} and which, after being increased by one, are also in R {\displaystyle {\mathfrak {R}}} . He proves that this number and related ones are the coefficients of the products of the periods. To see the relation of these sets to the Weil conjectures, notice that if α and α + 1 are both in R {\displaystyle {\mathfrak {R}}} , then there exist x and y in Z / p Z such that x 3 = α and y 3 = α + 1 ; consequently, x 3 + 1 = y 3 . Therefore ( R R ) {\displaystyle ({\mathfrak {R}}{\mathfrak {R}})} is related to the number of solutions to x 3 + 1 = y 3 in the finite field Z / p Z . The other coefficients have similar interpretations. Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves , and as a byproduct he proves the analog of the Riemann hypothesis.
The Weil conjectures in the special case of algebraic curves were conjectured by Emil Artin ( 1924 ). The case of curves over finite fields was proved by Weil, finishing the project started by Hasse's theorem on elliptic curves over finite fields. Their interest was obvious enough from within number theory : they implied upper bounds for exponential sums , a basic concern in analytic number theory ( Moreno 2001 ).
What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology . Given that finite fields are discrete in nature, and topology speaks only about the continuous , the detailed formulation of Weil (based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers , the Lefschetz fixed-point theorem and so on.
The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry . This took two decades (it was a central aim of the work and school of Alexander Grothendieck ) building up on initial suggestions from Serre . The rationality part of the conjectures was proved first by Bernard Dwork ( 1960 ), using p -adic methods. Grothendieck (1965) and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology , a new cohomology theory developed by Grothendieck and Michael Artin for attacking the Weil conjectures, as outlined in Grothendieck (1960) .
Of the four conjectures, the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of Serre (1960) of an analogue of the Weil conjectures for Kähler manifolds , Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles ( Kleiman 1968 ). However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem , which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by Deligne ( 1974 ), using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.
Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.
Suppose that X is a non-singular n -dimensional projective algebraic variety over the field F q with q elements. The zeta function ζ ( X , s ) of X is by definition
where N m is the number of points of X defined over the degree m extension F q m of F q .
The Weil conjectures state:
The simplest example (other than a point) is to take X to be the projective line. The number of points of X over a field with q m elements is just N m = q m + 1 (where the " + 1 " comes from the " point at infinity "). The zeta function is just
It is easy to check all parts of the Weil conjectures directly. For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1.
It is not much harder to do n -dimensional projective space. The number of points of X over a field with q m elements is just N m = 1 + q m + q 2 m + ⋯ + q nm . The zeta function is just
It is again easy to check all parts of the Weil conjectures directly. ( Complex projective space gives the relevant Betti numbers, which nearly determine the answer.)
The number of points on the projective line and projective space are so easy to calculate because they can be written as disjoint unions of a finite number of copies of affine spaces. It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians and flag varieties, which have the same "paving" property.
These give the first non-trivial cases of the Weil conjectures (proved by Hasse). If E is an elliptic curve over a finite field with q elements, then the number of points of E defined over the field with q m elements is 1 − α m − β m + q m , where α and β are complex conjugates with absolute value √ q .
The zeta function is
The Betti numbers are given by the torus , 1,2,1, and the numerator is a quadratic.
As an example, consider the hyperelliptic curve [ 1 ]
which is of genus g = 2 {\displaystyle g=2} and dimension n = 1 {\displaystyle n=1} . At first viewed as a curve C / Q {\displaystyle C/\mathbb {Q} } defined over the rational numbers Q {\displaystyle \mathbb {Q} } , this curve has good reduction at all primes 5 ≠ q ∈ P {\displaystyle 5\neq q\in \mathbb {P} } . So, after reduction modulo q ≠ 5 {\displaystyle q\neq 5} , one obtains a hyperelliptic curve C / F q : y 2 + h ( x ) y = f ( x ) {\displaystyle C/{\bf {F}}_{q}:y^{2}+h(x)y=f(x)} of genus 2, with h ( x ) = 1 , f ( x ) = x 5 ∈ F q [ x ] {\displaystyle h(x)=1,f(x)=x^{5}\in {\bf {F}}_{q}[x]} . Taking q = 41 {\displaystyle q=41} as an example, the Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} , i = 0 , 1 , 2 , {\displaystyle i=0,1,2,} and the zeta function of C / F 41 {\displaystyle C/{\bf {F}}_{41}} assume the form
The values c 1 = − 9 {\displaystyle c_{1}=-9} and c 2 = 71 {\displaystyle c_{2}=71} can be determined by counting the numbers of solutions ( x , y ) {\displaystyle (x,y)} of y 2 + y = x 5 {\displaystyle y^{2}+y=x^{5}} over F 41 {\displaystyle {\bf {F}}_{41}} and F 41 2 {\displaystyle {\bf {F}}_{41^{2}}} , respectively, and adding 1 to each of these two numbers to allow for the point at infinity ∞ {\displaystyle \infty } . This counting yields N 1 = 33 {\displaystyle N_{1}=33} and N 2 = 1743 {\displaystyle N_{2}=1743} . It follows: [ 2 ]
The zeros of P 1 ( T ) {\displaystyle P_{1}(T)} are z 1 := 0.12305 + − 1 ⋅ 0.09617 {\displaystyle z_{1}:=0.12305+{\sqrt {-1}}\cdot 0.09617} and z 2 := − 0.01329 + − 1 ⋅ 0.15560 {\displaystyle z_{2}:=-0.01329+{\sqrt {-1}}\cdot 0.15560} (the decimal expansions of these real and imaginary parts are cut off after the fifth decimal place) together with their complex conjugates z 3 := z ¯ 1 {\displaystyle z_{3}:={\bar {z}}_{1}} and z 4 := z ¯ 2 {\displaystyle z_{4}:={\bar {z}}_{2}} . So, in the factorisation P 1 ( T ) = ∏ j = 1 4 ( 1 − α 1 , j T ) {\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{1,j}T)} , we have α 1 , j = 1 / z j {\displaystyle \alpha _{1,j}=1/z_{j}} . As stated in the third part (Riemann hypothesis) of the Weil conjectures, | α 1 , j | = 41 {\displaystyle |\alpha _{1,j}|={\sqrt {41}}} for j = 1 , 2 , 3 , 4 {\displaystyle j=1,2,3,4} .
The non-singular, projective, complex manifold that belongs to C / Q {\displaystyle C/\mathbb {Q} } has the Betti numbers B 0 = 1 , B 1 = 2 g = 4 , B 2 = 1 {\displaystyle B_{0}=1,B_{1}=2g=4,B_{2}=1} . [ 3 ] As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} , for all primes q ≠ 5 {\displaystyle q\neq 5} : d e g ( P i ) = B i , i = 0 , 1 , 2 {\displaystyle {\rm {deg}}(P_{i})=B_{i},\,i=0,1,2} .
An Abelian surface is a two-dimensional Abelian variety . This is, they are projective varieties that also have the structure of a group , in a way that is compatible with the group composition and taking inverses. Elliptic curves represent one -dimensional Abelian varieties. As an example of an Abelian surface defined over a finite field, consider the Jacobian variety X := Jac ( C / F 41 ) {\displaystyle X:={\text{Jac}}(C/{\bf {F}}_{41})} of the genus 2 curve [ 4 ]
which was introduced in the section on hyperelliptic curves. The dimension of X {\displaystyle X} equals the genus of C {\displaystyle C} , so n = 2 {\displaystyle n=2} . There are algebraic integers α 1 , … , α 4 {\displaystyle \alpha _{1},\ldots ,\alpha _{4}} such that [ 5 ]
The zeta-function of X {\displaystyle X} is given by
where q = 41 {\displaystyle q=41} , T = q − s = d e f exp ( − s ⋅ log ( 41 ) ) {\displaystyle T=q^{-s}\,{\stackrel {\rm {def}}{=}}\,{\text{exp}}(-s\cdot {\text{log}}(41))} , and s {\displaystyle s} represents the complex variable of the zeta-function. The Weil polynomials P i ( T ) {\displaystyle P_{i}(T)} have the following specific form ( Kahn 2020 ):
for i = 0 , 1 , … , 4 {\displaystyle i=0,1,\ldots ,4} , and
is the same for the curve C {\displaystyle C} (see section above) and its Jacobian variety X {\displaystyle X} . This is, the inverse roots of P i ( T ) {\displaystyle P_{i}(T)} are the products α j 1 ⋅ … ⋅ α j i {\displaystyle \alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}} that consist of i {\displaystyle i} many, different inverse roots of P 1 ( T ) {\displaystyle P_{1}(T)} . Hence, all coefficients of the polynomials P i ( T ) {\displaystyle P_{i}(T)} can be expressed as polynomial functions of the parameters c 1 = − 9 {\displaystyle c_{1}=-9} , c 2 = 71 {\displaystyle c_{2}=71} and q = 41 {\displaystyle q=41} appearing in P 1 ( T ) = 1 + c 1 T + c 2 T 2 + q c 1 T 3 + q 2 T 4 . {\displaystyle P_{1}(T)=1+c_{1}T+c_{2}T^{2}+qc_{1}T^{3}+q^{2}T^{4}.} Calculating these polynomial functions for the coefficients of the P i ( T ) {\displaystyle P_{i}(T)} shows that
Polynomial P 1 {\displaystyle P_{1}} allows for calculating the numbers of elements of the Jacobian variety Jac ( C ) {\displaystyle {\text{Jac}}(C)} over the finite field F 41 {\displaystyle {\bf {F}}_{41}} and its field extension F 41 2 {\displaystyle {\bf {F}}_{41^{2}}} : [ 6 ] [ 7 ]
The inverses α i , j {\displaystyle \alpha _{i,j}} of the zeros of P i ( T ) {\displaystyle P_{i}(T)} do have the expected absolute value of 41 i / 2 {\displaystyle 41^{i/2}} (Riemann hypothesis). Moreover, the maps α i , j ⟼ 41 2 / α i , j , {\displaystyle \alpha _{i,j}\longmapsto 41^{2}/\alpha _{i,j},} j = 1 , … , deg P i , {\displaystyle j=1,\ldots ,\deg P_{i},} correlate the inverses of the zeros of P i ( T ) {\displaystyle P_{i}(T)} and the inverses of the zeros of P 4 − i ( T ) {\displaystyle P_{4-i}(T)} . A non-singular, complex, projective, algebraic variety Y {\displaystyle Y} with good reduction at the prime 41 to X = Jac ( C / F 41 ) {\displaystyle X={\text{Jac}}(C/{\bf {F}}_{41})} must have Betti numbers B 0 = B 4 = 1 , B 1 = B 3 = 4 , B 2 = 6 {\displaystyle B_{0}=B_{4}=1,B_{1}=B_{3}=4,B_{2}=6} , since these are the degrees of the polynomials P i ( T ) . {\displaystyle P_{i}(T).} The Euler characteristic E {\displaystyle E} of X {\displaystyle X} is given by the alternating sum of these degrees/Betti numbers: E = 1 − 4 + 6 − 4 + 1 = 0 {\displaystyle E=1-4+6-4+1=0} .
By taking the logarithm of
it follows that
Aside from the values M 1 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} already known, you can read off from this Taylor series all other numbers M m {\displaystyle M_{m}} , m ∈ N {\displaystyle m\in \mathbb {N} } , of F 41 m {\displaystyle {\bf {F}}_{41^{m}}} -rational elements of the Jacobian variety, defined over F 41 {\displaystyle {\bf {F}}_{41}} , of the curve C / F 41 {\displaystyle C/{\bf {F}}_{41}} : for instance, M 3 = 4755796375 = 5 3 ⋅ 11 ⋅ 61 ⋅ 56701 {\displaystyle M_{3}=4755796375=5^{3}\cdot 11\cdot 61\cdot 56701} and M 4 = 7984359145125 = 3 4 ⋅ 5 3 ⋅ 11 ⋅ 2131 ⋅ 33641 {\displaystyle M_{4}=7984359145125=3^{4}\cdot 5^{3}\cdot 11\cdot 2131\cdot 33641} . In doing so, m 1 | m 2 {\displaystyle m_{1}|m_{2}} always implies M m 1 | M m 2 {\displaystyle M_{m_{1}}|M_{m_{2}}} since then, Jac ( C / F 41 m 1 ) {\displaystyle {\text{Jac}}(C/{\bf {F}}_{41^{m_{1}}})} is a subgroup of Jac ( C / F 41 m 2 ) {\displaystyle {\text{Jac}}(C/{\bf {F}}_{41^{m_{2}}})} .
Weil suggested that the conjectures would follow from the existence of a suitable " Weil cohomology theory " for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties. His idea was that if F is the Frobenius automorphism over the finite field, then the number of points of the variety X over the field of order q m is the number of fixed points of F m (acting on all points of the variety X defined over the algebraic closure). In algebraic topology the number of fixed points of an automorphism can be worked out using the Lefschetz fixed-point theorem , given as an alternating sum of traces on the cohomology groups . So if there were similar cohomology groups for varieties over finite fields, then the zeta function could be expressed in terms of them.
The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic p . The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2-dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve. However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals. The same argument eliminates the possibility of the coefficient field being the reals or the p -adic numbers, because the quaternion algebra is still a division algebra over these fields. However it does not eliminate the possibility that the coefficient field is the field of ℓ -adic numbers for some prime ℓ ≠ p , because over these fields the division algebra splits and becomes a matrix algebra, which can act on a 2-dimensional vector space. Grothendieck and Michael Artin managed to construct suitable cohomology theories over the field of ℓ -adic numbers for each prime ℓ ≠ p , called ℓ -adic cohomology .
By the end of 1964 Grothendieck together with Artin and Jean-Louis Verdier (and the earlier 1960 work by Dwork) proved the Weil conjectures apart from the most difficult third conjecture above (the "Riemann hypothesis" conjecture) (Grothendieck 1965). The general theorems about étale cohomology allowed Grothendieck to prove an analogue of the Lefschetz fixed-point formula for the ℓ -adic cohomology theory, and by applying it to the Frobenius automorphism F he was able to prove the conjectured formula for the zeta function:
where each polynomial P i is the determinant of I − TF on the ℓ -adic cohomology group H i .
The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from Poincaré duality for ℓ -adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between ℓ -adic and ordinary cohomology for complex varieties.
More generally, Grothendieck proved a similar formula for the zeta function (or "generalized L-function") of a sheaf F 0 :
as a product over cohomology groups:
The special case of the constant sheaf gives the usual zeta function.
Verdier (1974) , Serre (1975) , Katz (1976) and Freitag & Kiehl (1988) gave expository accounts of the first proof of Deligne (1974) . Much of the background in ℓ -adic cohomology is described in ( Deligne 1977 ).
Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps:
The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks. This is done by studying the zeta functions of the even powers E k of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups. The crucial idea of considering even k powers of E was inspired by the paper Rankin ( 1939 ), who used a similar idea with k = 2 for bounding the Ramanujan tau function . Langlands (1970 , section 8) pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture , and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.
The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.
Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem . Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallée Poussin , used by Deligne to show that various L -series do not have zeros with real part 1.
A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N ( x ) β /2 , and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β .
Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then R i f ! takes mixed sheaves of weight ≤ β to mixed sheaves of weight ≤ β + i .
The original Weil conjectures follow by taking f to be a morphism from a smooth projective variety to a point and considering the constant sheaf Q ℓ on the variety. This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound.
In general R i f ! does not take pure sheaves to pure sheaves. However it does when a suitable form of Poincaré duality holds, for example if f is smooth and proper, or if one works with perverse sheaves rather than sheaves as in Beilinson, Bernstein & Deligne (1982) .
Inspired by the work of Witten (1982) on Morse theory , Laumon (1987) found another proof, using Deligne's ℓ -adic Fourier transform , which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Kiehl & Weissauer (2001) used Laumon's proof as the basis for their exposition of Deligne's theorem. Katz (2001) gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Kedlaya (2006) gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology .
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https://en.wikipedia.org/wiki/Weil_conjectures
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In arithmetic geometry , the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A , defined over K . John Tate ( 1958 ) named it for François Châtelet ( 1946 ) who introduced it for elliptic curves , and André Weil ( 1955 ), who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties , in particular for elliptic curves, because of its connection with infinite descent .
It can be defined directly from Galois cohomology , as H 1 ( G K , A ) {\displaystyle H^{1}(G_{K},A)} , where G K {\displaystyle G_{K}} is the absolute Galois group of K . It is of particular interest for local fields and global fields , such as algebraic number fields . For K a finite field , Friedrich Karl Schmidt ( 1931 ) proved that the Weil–Châtelet group is trivial for elliptic curves, and Serge Lang ( 1956 ) proved that it is trivial for any connected algebraic group.
The Tate–Shafarevich group of an abelian variety A defined over a number field K consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of K .
The Selmer group , named after Ernst S. Selmer , of A with respect to an isogeny f : A → B {\displaystyle f\colon A\to B} of abelian varieties is a related group which can be defined in terms of Galois cohomology as
where A v [ f ] denotes the f - torsion of A v and κ v {\displaystyle \kappa _{v}} is the local Kummer map
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https://en.wikipedia.org/wiki/Weil–Châtelet_group
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The weak mixing angle or Weinberg angle [ 2 ] is a parameter in the Weinberg–Salam theory (by Steven Weinberg and Abdus Salam ) of the electroweak interaction , part of the Standard Model of particle physics, and is usually denoted as θ W . It is the angle by which spontaneous symmetry breaking rotates the original W 0 and B 0 vector boson plane, producing as a result the Z 0 boson, and the photon . [ 3 ] Its measured value is slightly below 30°, but also varies, very slightly increasing, depending on how high the relative momentum of the particles involved in the interaction is that the angle is used for. [ 4 ]
The algebraic formula for the combination of the W 0 and B 0 vector bosons (i.e. 'mixing') that simultaneously produces the massive Z 0 boson and the massless photon ( γ ) is expressed by the formula
( γ Z 0 ) = ( cos θ w sin θ w − sin θ w cos θ w ) ( B 0 W 0 ) . {\displaystyle {\begin{pmatrix}\gamma ~\\{\textsf {Z}}^{0}\end{pmatrix}}={\begin{pmatrix}\quad \cos \theta _{\textsf {w}}&\sin \theta _{\textsf {w}}\\-\sin \theta _{\textsf {w}}&\cos \theta _{\textsf {w}}\end{pmatrix}}{\begin{pmatrix}{\textsf {B}}^{0}\\{\textsf {W}}^{0}\end{pmatrix}}.} [ 3 ]
The weak mixing angle also gives the relationship between the masses of the W and Z bosons (denoted as m W and m Z ),
m Z = m W cos θ w . {\displaystyle m_{\textsf {Z}}={\frac {m_{\textsf {W}}}{\,\cos \theta _{\textsf {w}}}}\,.}
The angle can be expressed in terms of the SU(2) L and U(1) Y couplings ( weak isospin g and weak hypercharge g ′ , respectively),
cos θ w = g g 2 + g ′ 2 {\displaystyle \cos \theta _{\textsf {w}}={\frac {\quad g~}{\ {\sqrt {g^{2}+g'^{\ 2}~}}\ }}\quad } and sin θ w = g ′ g 2 + g ′ 2 . {\displaystyle \quad \sin \theta _{\textsf {w}}={\frac {\quad g'~}{\ {\sqrt {g^{2}+g'^{\ 2}~}}\ }}~.}
The electric charge is then expressible in terms of it, e = g sin θ w = g ′ cos θ w (refer to the figure).
Because the value of the mixing angle is currently determined empirically, in the absence of any superseding theoretical derivation it is mathematically defined as
cos θ w = m W m Z . {\displaystyle \cos \theta _{\textsf {w}}={\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}~.} [ 5 ]
The value of θ w varies as a function of the momentum transfer , ∆ q , at which it is measured. This variation, or ' running ', is a key prediction of the electroweak theory. The most precise measurements have been carried out in electron–positron collider experiments at a value of ∆ q = 91.2 GeV /c , corresponding to the mass of the Z 0 boson, m Z .
In practice, the quantity sin 2 θ w is more frequently used. The 2004 best estimate of sin 2 θ w , at ∆ q = 91.2 GeV/ c , in the MS scheme is 0.231 20 ± 0.000 15 , which is an average over measurements made in different processes, at different detectors. Atomic parity violation experiments yield values for sin 2 θ w at smaller values of ∆ q , below 0.01 GeV/ c , but with much lower precision. In 2005 results were published from a study of parity violation in Møller scattering in which a value of sin 2 θ w = 0.2397 ± 0.0013 was obtained at ∆ q = 0.16 GeV/ c , establishing experimentally the so-called 'running' of the weak mixing angle. These values correspond to a Weinberg angle varying between 28.7° and 29.3° ≈ 30° . LHCb measured in 7 and 8 TeV proton–proton collisions an effective angle of sin 2 θ eff w = 0.23142 , [ 6 ] though the value of ∆ q for this measurement is determined by the partonic collision energy, which is close to the Z boson mass.
CODATA 2022 [ 4 ] gives the value
sin 2 θ w = 1 − ( m W m Z ) 2 = 0.22305 ( 23 ) . {\displaystyle \sin ^{2}\theta _{\textsf {w}}=1-\left({\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}\right)^{2}=0.22305(23)~.} [ b ]
The massless photon ( γ ) couples to the unbroken electric charge, Q = T 3 + 1 / 2 Y w , while the Z 0 boson couples to the broken charge T 3 − Q sin 2 θ w .
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https://en.wikipedia.org/wiki/Weinberg_angle
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In computer security , a weird machine is a computational artifact where additional code execution can happen outside the original specification of the program. [ 1 ] It is closely related to the concept of weird instructions , which are the building blocks of an exploit based on crafted input data. [ 2 ]
The concept of weird machine is a theoretical framework to understand the existence of exploits for security vulnerabilities. Exploits exist empirically, but were not studied from a theoretical perspective prior to the emergence of the framework of weird machines.
From a theoretical perspective, the emergence of weird machines becomes clear when one considers software as a way to restrict the number of reachable states and state transitions of a computer: The general-purpose CPU is, through software, specialized to simulate a finite-state machine (with potentially very large state space). Many states the CPU could be in are excluded, and certain state transitions are ruled out - for example those that violate the software's security requirements. When the system is somehow moved into a state that "makes no sense" when viewed from the perspective of the intended finite-state machine (through memory corruption, hardware failure, or other programming mistakes), the software will keep transforming the broken state into new broken states, triggered by further user input. A new computational device arises: The weird machine which can reach different states of the CPU than the programmer never anticipated, and which does so in reaction to the attacker controlled inputs.
While expected, valid input activates the normal, intended functionality in a computer program , input that was unexpected by the program developer may activate unintended functionality. The weird machine consists of this unintended functionality that can be programmed with selected inputs in an exploit .
In a classical attack taking advantage of a stack buffer overflow , the input given to a vulnerable program is crafted and delivered so that it itself becomes executed as program code . However, if the data areas of the program memory have been protected so that they cannot be executed directly like this, the input may instead take the form of pointers into pieces of existing program code that then become executed in an unexpected order to generate the functionality of the exploit. These snippets of code that are used by the exploit are referred to as gadgets in the context of return-oriented programming .
Through interpretation of data as code, weird machine functionality that is by definition outside the original program specification can be reached also by proof-carrying code (PCC), which has been formally proven to function in a certain specific way. [ 3 ] This disparity is essentially caused by a disconnect between formal abstract modelling of a computer program and its real-world instance, which can be influenced by events that are not captured in the original abstraction, such as memory errors or power outages.
Weird machine behaviors are observed even in hardware. For instance, it has been shown that one can do computation with only MOV instructions in x86. [ 4 ]
Two central categories of mitigation to the problems caused by weird machine functionality include input validation within the software and protecting against problems arising from the platform on which the program runs, such as memory errors. Input validation aims to limit the scope and forms of unexpected inputs e.g. through whitelists of allowed inputs, so that the software program itself would not end up in an unexpected state by interpreting the data internally. Equally importantly, secure programming practices such as protecting against buffer overflows make it less likely that input data becomes interpreted in unintended ways by lower layers, such as the hardware on which the program is executed.
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https://en.wikipedia.org/wiki/Weird_machine
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The Weismann barrier , proposed by August Weismann , is the strict distinction between the "immortal" germ cell lineages producing gametes and "disposable" somatic cells in animals (but not plants), in contrast to Charles Darwin 's proposed pangenesis mechanism for inheritance. [ 1 ] [ 2 ]
In more precise terminology, hereditary information is copied only from germline cells to somatic cells . This means that new information from somatic mutation is not passed on to the germline. This barrier concept implies that somatic mutations are not inherited. [ 3 ] [ 4 ]
Weismann set out the concept in his 1892 book ″Das Keimplasma: eine Theorie der Vererbung″ (German for The Germ Plasm : a theory of inheritance ). [ 5 ] The use of this theory, commonly in the context of the germ plasm theory of the late 19th century, before the development of better-based and more sophisticated concepts of genetics in the early 20th century, is sometimes referred to as Weismannism . [ 6 ] Some authors distinguish Weismannist development (either preformistic or epigenetic ) that in which there is a distinct germline, from somatic embryogenesis . [ 7 ] This type of development is correlated with the evolution of death of the somatic line.
The Weismann barrier was of great importance in its day and among other influences it effectively banished certain Lamarckian concepts: in particular, it would make Lamarckian inheritance from changes to the body (the soma) difficult or impossible. [ 8 ] It remains important, but has however required qualification in the light of modern understanding of horizontal gene transfer and some other genetic and histological developments. [ 9 ]
The Russian biologist and historian Zhores A. Medvedev , reviewing Weismann's theory a century later, considered that the accuracy of genome replicative and other synthetic systems alone could not explain the "immortal" germ cell lineages proposed by Weismann. Rather Medvedev thought that known features of the biochemistry and genetics of sexual reproduction indicated the presence of unique information maintenance and restoration processes at the different stages of gametogenesis . In particular, Medvedev considered that the most important opportunities for information maintenance of germ cells are created by recombination during meiosis and DNA repair ; he saw these as processes within the germ cells that were capable of restoring the integrity of DNA and chromosomes from the types of damage that caused irreversible ageing in somatic cells . [ 10 ]
Basal animals such as sponges ( Porifera ) and corals ( Anthozoa ) contain multipotent stem cell lineages, that give rise to both somatic and reproductive cells. The Weismann barrier appears to be of a more recent evolutionary origin among animals. [ 11 ]
In plants, genetic changes in somatic lines can and do result in genetic changes in the germ lines, because the germ cells are produced by somatic cell lineages (vegetative meristems ), which may be old enough (many years) to have accumulated multiple mutations since seed germination, some of them subject to natural selection. [ 12 ] It is noteworthy in this context that, generally speaking, adult, reproducing plants tend to produce many more offspring in number than animal organisms.
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https://en.wikipedia.org/wiki/Weismann_barrier
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The Weissenberg number ( Wi ) is a dimensionless number used in the study of viscoelastic flows. It is named after Karl Weissenberg . The dimensionless number compares the elastic forces to the viscous forces. It can be variously defined, but it is usually given by the relation of stress relaxation time of the fluid and a specific process time. For instance, in simple steady shear, the Weissenberg number, often abbreviated as Wi or We, is defined as the shear rate γ ˙ {\displaystyle {\dot {\gamma }}} times the relaxation time λ {\displaystyle \lambda } . Using the Maxwell model and the Oldroyd-B model , the elastic forces can be written as the first Normal force (N 1 ).
Since this number is obtained from scaling the evolution of the stress, it contains choices for the shear or elongation rate, and the length-scale. Therefore the exact definition of all non dimensional numbers should be given as well as the number itself.
While Wi is similar to the Deborah number and is often confused with it in technical literature, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flows with a constant stretch history, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released.
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https://en.wikipedia.org/wiki/Weissenberg_number
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The Weissman score is a performance metric for lossless compression applications. It was developed by Tsachy Weissman , a professor at Stanford University , and Vinith Misra, a graduate student, at the request of producers for HBO's television series Silicon Valley , a television show about a fictional tech start-up working on a data compression algorithm . [ 1 ] [ 2 ] [ 3 ] [ 4 ] It compares both required time and compression ratio of measured applications, with those of a de facto standard according to the data type .
The formula is the following; where r is the compression ratio , T is the time required to compress, the overlined ones are the same metrics for a standard compressor, and alpha is a scaling constant. [ 1 ] W = α r r ¯ log T ¯ log T {\displaystyle W=\alpha {r \over {\overline {r}}}{\log {\overline {T}} \over \log {T}}}
The Weissman score has been used by Daniel Reiter Horn and Mehant Baid of Dropbox to explain real-world work on lossless compression. According to the authors it "favors compression speed over ratio in most cases." [ 5 ]
This example shows the score for the data of the Hutter Prize , [ 6 ] using the paq8f as a standard and 1 as the scaling constant.
Although the value is relative to the standards against which it is compared, the unit used to measure the times changes the score (see examples 1 and 2). This is a consequence of the requirement that the argument of the logarithmic function must be dimensionless . The multiplier also can't have a numeric value of 1 or less, because the logarithm of 1 is 0 (examples 3 and 4), and the logarithm of any value less than 1 is negative (examples 5 and 6); that would result in scores of value 0 (even with changes), undefined, or negative (even if better than positive).
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https://en.wikipedia.org/wiki/Weissman_score
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The Weisz–Prater criterion is a method used to estimate the influence of pore diffusion on reaction rates in heterogeneous catalytic reactions. [ 1 ] If the criterion is satisfied, pore diffusion limitations are negligible. The criterion is N W − P = R R p 2 C s D e f f ≤ 3 β {\displaystyle N_{W-P}={\dfrac {{\mathfrak {R}}R_{p}^{2}}{C_{s}D_{eff}}}\leq 3\beta } Where R {\displaystyle {\mathfrak {R}}} is the reaction rate per volume of catalyst, R p {\displaystyle R_{p}} is the catalyst particle radius, C s {\displaystyle C_{s}} is the reactant concentration at the particle surface, and D e f f {\displaystyle D_{eff}} is the effective diffusivity . Diffusion is usually in the Knudsen regime when average pore radius is less than 100 nm. For a given effectiveness factor, η {\displaystyle \eta } , and reaction order , n, the quantity β {\displaystyle \beta } is defined by the equation: η = 3 R p 3 ∫ 0 R p [ 1 − β ( 1 − r / R p ) n ] r 2 d r {\displaystyle \eta ={\dfrac {3}{R_{p}^{3}}}\int _{0}^{R_{p}}[1-\beta (1-r/R_{p})^{n}]r^{2}\ dr} for small values of beta this can be approximated using the binomial theorem : η = 1 − n β 4 {\displaystyle \eta =1-{\dfrac {n\beta }{4}}} Assuming η = 0.95 {\displaystyle \eta =0.95} with a reaction order n = 2 {\displaystyle n=2} gives value of β {\displaystyle \beta } equal to 0.1. Therefore, for many conditions, if N W − P ≤ 0.3 {\displaystyle N_{W-P}\leq 0.3} then pore diffusion limitations can be excluded. [ 2 ]
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https://en.wikipedia.org/wiki/Weisz–Prater_criterion
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Weitao Yang ( Chinese : 杨伟涛 ; pinyin : Yáng Wěitāo ; born March 31, 1961) is a Chinese-born American chemist who is the Philip Handler Professor of Chemistry at Duke University . His main contributions to chemistry include density functional theory development, and its applications to chemistry.
Yang was born in Chaozhou , Guangdong , China, on March 31, 1961. He entered Peking University in 1978 as part of the first generation of college students after the Cultural Revolution . He received his BS in chemistry in 1982, after which he studied theoretical chemistry as a PhD student with Robert G. Parr at the University of North Carolina, Chapel Hill. He completed his PhD degree in 1986 and worked as a postdoc with Parr (1986–1987) and William H. Miller (1988–1989). Yang currently works in the department of chemistry at Duke University since 1990, as assistant professor, associate professor, professor, and Philip Handler Professor of Chemistry.
Yang's main contributions to theoretical chemistry range from fundamental theory to applications of density functional theory. He (with Parr) developed the concepts of the Fukui function , [ 1 ] hardness, and softness [ 2 ] in density functional theory. He also justified the theoretical ground of potential functional (as in Optimized-Effective-Potential methods) and fractional-number-of-electron approaches.
The Lee–Yang–Parr (shortened to LYP) [ 3 ] correlation functional by Yang and his coworkers is extensively used in chemistry and was the second most cited article in chemistry from 1999 to 2006. Yang developed the Divide and Conquer algorithm [ 4 ] for linear-scaling density functional theory. In application, Yang also developed methods in QM/MM simulation for large chemistry and biology systems. His book with Robert G. Parr, Density-Functional Theory of Atoms and Molecules , is considered to be the basic textbook in the field of density functional theory .
He is married to Helen Wen Yang and they have two children. [ 6 ]
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https://en.wikipedia.org/wiki/Weitao_Yang
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In mathematics , Weitzenböck's inequality , named after Roland Weitzenböck , states that for a triangle of side lengths a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , and area Δ {\displaystyle \Delta } , the following inequality holds:
Equality occurs if and only if the triangle is equilateral. Pedoe's inequality is a generalization of Weitzenböck's inequality. The Hadwiger–Finsler inequality is a strengthened version of Weitzenböck's inequality.
Rewriting the inequality above allows for a more concrete geometric interpretation, which in turn provides an immediate proof. [ 1 ]
Now the summands on the left side are the areas of equilateral triangles erected over the sides of the original triangle and hence the inequation states that the sum of areas of the equilateral triangles is always greater than or equal to threefold the area of the original triangle.
This can now be shown by replicating area of the triangle three times within the equilateral triangles. To achieve that the Fermat point is used to partition the triangle into three obtuse subtriangles with a 120 ∘ {\displaystyle 120^{\circ }} angle and each of those subtriangles is replicated three times within the equilateral triangle next to it. This only works if every angle of the triangle is smaller than 120 ∘ {\displaystyle 120^{\circ }} , since otherwise the Fermat point is not located in the interior of the triangle and becomes a vertex instead. However if one angle is greater or equal to 120 ∘ {\displaystyle 120^{\circ }} it is possible to replicate the whole triangle three times within the largest equilateral triangle, so the sum of areas of all equilateral triangles stays greater than the threefold area of the triangle anyhow.
The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. Even so, the result is not too difficult to derive using Heron's formula for the area of a triangle:
It can be shown that the area of the inner Napoleon's triangle , which must be nonnegative, is [ 2 ]
so the expression in parentheses must be greater than or equal to 0.
This method assumes no knowledge of inequalities except that all squares are nonnegative.
and the result follows immediately by taking the positive square root of both sides. From the first inequality we can also see that equality occurs only when a = b = c {\displaystyle a=b=c} and the triangle is equilateral.
This proof assumes knowledge of the AM–GM inequality .
As we have used the arithmetic-geometric mean inequality, equality only occurs when a = b = c {\displaystyle a=b=c} and the triangle is equilateral.
Write x = cot A , c = cot A + cot B > 0 {\displaystyle x=\cot A,c=\cot A+\cot B>0} so the sum S = cot A + cot B + cot C = c + 1 − x ( c − x ) c {\displaystyle S=\cot A+\cot B+\cot C=c+{\frac {1-x(c-x)}{c}}} and c S = c 2 − x c + x 2 + 1 = ( x − c 2 ) 2 + ( c 3 2 − 1 ) 2 + c 3 ≥ c 3 {\displaystyle cS=c^{2}-xc+x^{2}+1=\left(x-{\frac {c}{2}}\right)^{2}+\left({\frac {c{\sqrt {3}}}{2}}-1\right)^{2}+c{\sqrt {3}}\geq c{\sqrt {3}}} i.e. S ≥ 3 {\displaystyle S\geq {\sqrt {3}}} . But cot A = b 2 + c 2 − a 2 4 Δ {\displaystyle \cot A={\frac {b^{2}+c^{2}-a^{2}}{4\Delta }}} , so S = a 2 + b 2 + c 2 4 Δ {\displaystyle S={\frac {a^{2}+b^{2}+c^{2}}{4\Delta }}} .
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https://en.wikipedia.org/wiki/Weitzenböck's_inequality
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In mathematics , in particular in differential geometry , mathematical physics , and representation theory , a Weitzenböck identity , named after Roland Weitzenböck , expresses a relationship between two second-order elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck formulae are implemented for G -invariant self-adjoint operators between vector bundles associated to some principal G -bundle , although the precise conditions under which such a formula exists are difficult to formulate. This article focuses on three examples of Weitzenböck identities: from Riemannian geometry , spin geometry , and complex analysis .
In Riemannian geometry there are two notions of the Laplacian on differential forms over an oriented compact Riemannian manifold M . The first definition uses the divergence operator δ defined as the formal adjoint of the de Rham operator d : ∫ M ⟨ α , δ β ⟩ := ∫ M ⟨ d α , β ⟩ {\displaystyle \int _{M}\langle \alpha ,\delta \beta \rangle :=\int _{M}\langle d\alpha ,\beta \rangle } where α is any p -form and β is any ( p + 1 )-form, and ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the metric induced on the bundle of ( p + 1 )-forms. The usual form Laplacian is then given by Δ = d δ + δ d . {\displaystyle \Delta =d\delta +\delta d.}
On the other hand, the Levi-Civita connection supplies a differential operator ∇ : Ω p M → Ω 1 M ⊗ Ω p M , {\displaystyle \nabla :\Omega ^{p}M\rightarrow \Omega ^{1}M\otimes \Omega ^{p}M,} where Ω p M is the bundle of p -forms. The Bochner Laplacian is given by Δ ′ = ∇ ∗ ∇ {\displaystyle \Delta '=\nabla ^{*}\nabla } where ∇ ∗ {\displaystyle \nabla ^{*}} is the adjoint of ∇ {\displaystyle \nabla } . This is also known as the connection or rough Laplacian.
The Weitzenböck formula then asserts that Δ ′ − Δ = A {\displaystyle \Delta '-\Delta =A} where A is a linear operator of order zero involving only the curvature.
The precise form of A is given, up to an overall sign depending on curvature conventions, by A = 1 2 ⟨ R ( θ , θ ) # , # ⟩ + Ric ( θ , # ) , {\displaystyle A={\frac {1}{2}}\langle R(\theta ,\theta )\#,\#\rangle +\operatorname {Ric} (\theta ,\#),} where
If M is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = ð 2 on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator ∇ : S M → T ∗ M ⊗ S M . {\displaystyle \nabla :SM\rightarrow T^{*}M\otimes SM.} As in the case of Riemannian manifolds, let Δ ′ = ∇ ∗ ∇ {\displaystyle \Delta '=\nabla ^{*}\nabla } . This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenböck formula yields: Δ ′ − Δ = − 1 4 S c {\displaystyle \Delta '-\Delta =-{\frac {1}{4}}Sc} where Sc is the scalar curvature. This result is also known as the Lichnerowicz formula .
If M is a compact Kähler manifold , there is a Weitzenböck formula relating the ∂ ¯ {\displaystyle {\bar {\partial }}} -Laplacian (see Dolbeault complex ) and the Euclidean Laplacian on ( p , q )-forms . Specifically, let Δ = ∂ ¯ ∗ ∂ ¯ + ∂ ¯ ∂ ¯ ∗ , {\displaystyle \Delta ={\bar {\partial }}^{*}{\bar {\partial }}+{\bar {\partial }}{\bar {\partial }}^{*},} and Δ ′ = − ∑ k ∇ k ∇ k ¯ {\displaystyle \Delta '=-\sum _{k}\nabla _{k}\nabla _{\bar {k}}} in a unitary frame at each point.
According to the Weitzenböck formula, if α ∈ Ω ( p , q ) M {\displaystyle \alpha \in \Omega ^{(p,q)}M} , then Δ ′ α − Δ α = A ( α ) {\displaystyle \Delta ^{\prime }\alpha -\Delta \alpha =A(\alpha )} where A {\displaystyle A} is an operator of order zero involving the curvature. Specifically, if α = α i 1 i 2 … i p j ¯ 1 j ¯ 2 … j ¯ q {\displaystyle \alpha =\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {j}}_{q}}} in a unitary frame, then A ( α ) = − ∑ k , j s Ric j ¯ α k ¯ α i 1 i 2 … i p j ¯ 1 j ¯ 2 … k ¯ … j ¯ q {\displaystyle A(\alpha )=-\sum _{k,j_{s}}\operatorname {Ric} _{{\bar {j}}_{\alpha }}^{\bar {k}}\alpha _{i_{1}i_{2}\dots i_{p}{\bar {j}}_{1}{\bar {j}}_{2}\dots {\bar {k}}\dots {\bar {j}}_{q}}} with k in the s -th place.
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Welan gum is an exopolysaccharide used as a rheology modifier in industrial applications such as cement manufacturing. [ 1 ] It is produced by fermentation of sugar by bacteria of the genus Alcaligenes . The molecule consists of repeating tetrasaccharide units with single branches of L-mannose or L-rhamnose . [ 2 ] In solution, the gum retains viscosity at elevated temperature, and is stable in a wide pH range, in the presence of calcium ion, and with high concentration of glycols. [ 3 ] [ 4 ] [ 5 ]
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In mathematics , Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space . The bounds are important tools in the design and analysis of certain methods in telecommunication engineering, particularly in coding theory . The bounds were originally published in a 1974 paper by L. R. Welch . [ 1 ]
If { x 1 , … , x m } {\displaystyle \{x_{1},\ldots ,x_{m}\}} are unit vectors in C n {\displaystyle \mathbb {C} ^{n}} , define c max = max i ≠ j | ⟨ x i , x j ⟩ | {\displaystyle c_{\max }=\max _{i\neq j}|\langle x_{i},x_{j}\rangle |} , where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the usual inner product on C n {\displaystyle \mathbb {C} ^{n}} . Then the following inequalities hold for k = 1 , 2 , … {\displaystyle k=1,2,\dots } : ( c max ) 2 k ≥ 1 m − 1 [ m ( n + k − 1 k ) − 1 ] . {\displaystyle (c_{\max })^{2k}\geq {\frac {1}{m-1}}\left[{\frac {m}{\binom {n+k-1}{k}}}-1\right].} Welch bounds are also sometimes stated in terms of the averaged squared overlap between the set of vectors. In this case, one has the inequality [ 2 ] [ 3 ] [ 4 ] 1 m 2 ∑ i , j = 1 m | ⟨ x i , x j ⟩ | 2 k ≥ 1 ( n + k − 1 k ) . {\displaystyle {\frac {1}{m^{2}}}\sum _{i,j=1}^{m}|\langle x_{i},x_{j}\rangle |^{2k}\geq {\frac {1}{\binom {n+k-1}{k}}}.}
If m ≤ n {\displaystyle m\leq n} , then the vectors { x i } {\displaystyle \{x_{i}\}} can form an orthonormal set in C n {\displaystyle \mathbb {C} ^{n}} . In this case, c max = 0 {\displaystyle c_{\max }=0} and the bounds are vacuous. Consequently, interpretation of the bounds is only meaningful if m > n {\displaystyle m>n} . This will be assumed throughout the remainder of this article.
The "first Welch bound," corresponding to k = 1 {\displaystyle k=1} , is by far the most commonly used in applications. Its proof proceeds in two steps, each of which depends on a more basic mathematical inequality. The first step invokes the Cauchy–Schwarz inequality and begins by considering the m × m {\displaystyle m\times m} Gram matrix G {\displaystyle G} of the vectors { x i } {\displaystyle \{x_{i}\}} ; i.e.,
The trace of G {\displaystyle G} is equal to the sum of its eigenvalues. Because the rank of G {\displaystyle G} is at most n {\displaystyle n} , and it is a positive semidefinite matrix, G {\displaystyle G} has at most n {\displaystyle n} positive eigenvalues with its remaining eigenvalues all equal to zero. Writing the non-zero eigenvalues of G {\displaystyle G} as λ 1 , … , λ r {\displaystyle \lambda _{1},\ldots ,\lambda _{r}} with r ≤ n {\displaystyle r\leq n} and applying the Cauchy-Schwarz inequality to the inner product of an r {\displaystyle r} -vector of ones with a vector whose components are these eigenvalues yields
The square of the Frobenius norm (Hilbert–Schmidt norm) of G {\displaystyle G} satisfies
Taking this together with the preceding inequality gives
Because each x i {\displaystyle x_{i}} has unit length, the elements on the main diagonal of G {\displaystyle G} are ones, and hence its trace is T r G = m {\displaystyle \mathrm {Tr} \;G=m} . So,
or
The second part of the proof uses an inequality encompassing the simple observation that the average of a set of non-negative numbers can be no greater than the largest number in the set. In mathematical notation, if a ℓ ≥ 0 {\displaystyle a_{\ell }\geq 0} for ℓ = 1 , … , L {\displaystyle \ell =1,\ldots ,L} , then
The previous expression has m ( m − 1 ) {\displaystyle m(m-1)} non-negative terms in the sum, the largest of which is c max 2 {\displaystyle c_{\max }^{2}} . So,
or
which is precisely the inequality given by Welch in the case that k = 1 {\displaystyle k=1} .
In certain telecommunications applications, it is desirable to construct sets of vectors that meet the Welch bounds with equality. Several techniques have been introduced to obtain so-called Welch Bound Equality (WBE) sets of vectors for the k = 1 {\displaystyle k=1} bound.
The proof given above shows that two separate mathematical inequalities are incorporated into the Welch bound when k = 1 {\displaystyle k=1} . The Cauchy–Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix G {\displaystyle G} are equal, which happens precisely when the vectors { x 1 , … , x m } {\displaystyle \{x_{1},\ldots ,x_{m}\}} constitute a tight frame for C n {\displaystyle \mathbb {C} ^{n}} .
The other inequality in the proof is satisfied with equality if and only if | ⟨ x i , x j ⟩ | {\displaystyle |\langle x_{i},x_{j}\rangle |} is the same for every choice of i ≠ j {\displaystyle i\neq j} . In this case, the vectors are equiangular . So this Welch bound is met with equality if and only if the set of vectors { x i } {\displaystyle \{x_{i}\}} is an equiangular tight frame in C n {\displaystyle \mathbb {C} ^{n}} .
Similarly, the Welch bounds stated in terms of average squared overlap, are saturated for all k ≤ t {\displaystyle k\leq t} if and only if the set of vectors is a t {\displaystyle t} -design in the complex projective space C P n − 1 {\displaystyle \mathbb {CP} ^{n-1}} . [ 4 ]
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https://en.wikipedia.org/wiki/Welch_bounds
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In statistics and uncertainty analysis , the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom of a linear combination of independent sample variances , also known as the pooled degrees of freedom , [ 1 ] [ 2 ] corresponding to the pooled variance .
For n sample variances s i 2 ( i = 1, ..., n ) , each respectively having ν i degrees of freedom, often one computes the linear combination.
where k i {\displaystyle k_{i}} is a real positive number, typically k i = 1 ν i + 1 {\displaystyle k_{i}={\frac {1}{\nu _{i}+1}}} . In general, the probability distribution of χ' cannot be expressed analytically. However, its distribution can be approximated by another chi-squared distribution , whose effective degrees of freedom are given by the Welch–Satterthwaite equation
There is no assumption that the underlying population variances σ i 2 are equal. This is known as the Behrens–Fisher problem .
The result can be used to perform approximate statistical inference tests. The simplest application of this equation is in performing Welch's t -test .
An improved equation was derived to reduce underestimating the effective degrees of freedom if the pooled sample variances have small degrees of freedom. Examples are jackknife and imputation-based variance estimates. [ 3 ]
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Weld-On is a division of IPS Corporation, a manufacturer of solvent cements , primers , and cleaners for PVC , CPVC , and ABS plastic piping systems. Weld-On products are commonly used for joining plastic pipes and fittings . Weld-On also manufactures specialty products from repair adhesives for leaking pipes, pipe thread sealants / joint compounds, to test plugs for pipeline pressure testing. Their products are most commonly utilized in the irrigation , industrial, pool & spa , electrical conduit , and plumbing industries.
Headquartered in California, Weld-On has operations throughout the United States, as well as in China, and a worldwide network of sales representatives and distributors.
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The weld access hole or rat hole is a structural engineering technique in which a part of the web of an I-beam or T-beam is cut out at the end or ends of the beam. The hole in the web allows a welder to weld the flange to another part of the structure with a continuous weld the full width on both top and bottom sides of the flange. Without the weld access hole, the middle of the flange would be blocked by the web and inaccessible for welding.
The hole also minimizes the induction of thermal stresses with a combination of partially releasing the welded section, avoiding welding the T section where the flange joins the web and improving cooling conditions. [ 1 ] [ 2 ]
The configuration adopted for web access holes affects how the beam joint will bend when under stress. [ 3 ]
Welds may be classified as either Complete Joint Penetration (CJP) or Partial Joint Penetration (PJP). CJP welds extend completely through the thickness of components joined. A CJP weld transmits the full load-carrying capacity of the structural components joined, [ 4 ] and is important for seismic safety.
Complete penetration usually requires welding on both sides of a joint. Weld access holes in the web of a beam make it possible to weld both sides of a flange, making the flange joint stronger, at the expense of the web joint.
The strength of a flange joint is important, because the flange resists bending moment of a beam.
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A weld neck flange (also known as a high-hub flange [ 1 ] and tapered hub flange [ 2 ] ) is a type of flange . There are two designs. The regular type is used with pipes. The long type is unsuitable for pipes and is used in process plant. [ 3 ] A weld neck flange consists of a circular fitting with a protruding rim around the circumference. Generally machined from a forging , these flanges are typically butt welded to a pipe. The rim has a series of drilled holes that permit the flange to be affixed to another flange with bolts. [ 4 ]
Such flanges are suitable for use in hostile environments that have extremes of temperature, pressure or other sources of stress. [ 4 ] The resilience of this type of flange is achieved by sharing the environmental stress with the pipe with which it is welded. [ 1 ] This type of flange has been used successfully at pressures up to 5,000 psi . [ 2 ]
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Advanced thermoplastic composites (ACM) have a high strength fibres held together by a thermoplastic matrix. Advanced thermoplastic composites are becoming more widely used in the aerospace, marine, automotive and energy industry. This is due to the decreasing cost and superior strength to weight ratios, over metallic parts. Advance thermoplastic composite have excellent damage tolerance, corrosion resistant, high fracture toughness , high impact resistance, good fatigue resistance, low storage cost, and infinite shelf life. [ 1 ] Thermoplastic composites also have the ability to be formed and reformed, repaired and fusion welded.
Fusion bonding is a category of techniques for welding thermoplastic composites. It requires the melting of the joint interface, which decreases the viscosity of the polymer and allows for intermolecular diffusion. These polymer chains then diffuse across the joint interface and become entangled, giving the joint its strength.
There are many welding techniques that can be used to fusion bond thermoplastic composites. These different techniques can be broken down into three classifications for their ways of generating heat; frictional heating, external heating and electromagnetic heating. Some of these techniques can be very limited and only used for specific joints and geometries.
Friction welding is best used for parts that are small and flat. The welding equipment is often expensive, but produces high-quality welds.
Two flat parts are brought together under pressure with one fixed in place and the other vibrating back-and-forth parallel to the joint. Frictional heat is then generated till the polymers are softened or melted. Once the desired temperature is met, the vibration motion stops, the polymer solidifies and a weld joint is made. The two most important welding parameters that affect the mechanical performance are welding pressure and time. Developing parameters for different advance thermoplastic composite can be challenging because the high elastic modulus of the material will have a higher heat generation, requiring less weld time. The pressure can affect the fiber orientation which also greatly impact the mechanical performance. Lap shear joints tend to have the best mechanical performance from the higher volume fraction of fibers at the weld interface. Overall linear vibration welding can achieve high production rates with excellent strength, but is limited to the joint geometries that are flat.
Spin welding is not a very common welding technique for advanced thermoplastic composites because this can only be done with parts that have a circular geometry. This is done by one part remaining stationary while the other is continuously rotated with pressure applied to the weld interface. Rotational velocity will vary throughout different radii of the Interface. This will result in a temperature gradient as a function of the radius, resulting in different shrinkage for the fibers causing high residual stresses. The orientation of the fibers will also contribute to high residual stress and reduction in strength.
Ultrasonics welding is one of the most commonly used technique for welding advanced thermoplastic composites. This is due for its ability to maintain high weld strength, hermetic sealing, and high production rates. This welding technique operates at high vibrational frequencies (10–70 kHz) [ 2 ] and low amplitude. The direction of vibration is perpendicular to the joint surface, but can also be parallel to the joint for hermetic application. Heat is generated from the surface and intermolecular friction due to the vibrational. On the surface of the joint there are small asperities called energy directors, where the vibrational energy concentrates and induces melting. Design of the energy director and optimized parameters can be critical to improve the quality of the weld to reducing any fiber disruption during welding. Energy directors that are triangular or semi-circle often achieve the highest strength. [ 2 ] With optimize welding parameters and joint design weld strength, up to 80% of the base material can be retained for advanced thermoplastic composites. [ 3 ] However, welding can cause damage to the fibers, which will result in premature failure. Ultrasonic welding of advanced thermoplastic composites is used for making automotive parts, medical devices and battery housing.
Thermal welding can produce good weld quality although extra precautions need to be taken to prevent high residual stress, warping, and decohesion. Other thermal welding techniques are not commonly used due their high heat input, which can damage the composite.
Laser welding of advanced thermoplastic composites is a process by which the LASER (Light Amplification of Simulated Emission of electromagnetic Radiation), a highly focused coherent beam of light melts the composite tin various ways. Taking advantage of joint design and material properties, lasers can be applied either directly or indirectly to create the welded joint. There are processing methods that take advantage of material structure/properties to create the weld joint. Welding variables affect weld quality in both positive and negative ways depending on how they are manipulated.
When a laser beam impinges on a material, it excites electrons in the outer most shell of the atom. The return of those electrons to the relaxed state induces thermal heating through conversion to vibrational states which propagate to the surrounding material. [ 4 ]
This method involves using infrared radiation to heat the surfaces the composites to be welded and then clamping until and holding the parts together. [ 5 ]
This method involves laser melting a polymer post and pressing a die into the molten post to create a rivet-like button to joint materials like metals. [ 5 ] This process can be used to join metallic joints to composite structures.
This method utilizes one laser transparent (LT) and one laser absorbing (LA) material. Typically, the components are layered as a sandwich with the laser beam passing through the LT layer and irradiating the surface of the LA. This creates a melt layer at the interface of two components leading to a weld. [ 4 ]
To understand how the properties of a composite affect is weldability, the effects of the individual constituents (fiber, matrix, additives, etc.) need to be understood. The effect of each will be noted separately and then the combined effects will be discussed.
A laser beam can interact in one of three ways when it contacts the polymer matrix. It can be absorbed, transmitted, or reflected. The amount of absorption determines the amount of energy available for welding. The reflectivity is affected by the index of refraction according to this relation: R = ( n − m ) 2 ( n + m ) 2 {\displaystyle R={\frac {(n-m)^{2}}{(n+m)^{2}}}} , where n is the index of refraction of the polymer and m is the index of refraction of air. [ 5 ]
Absorption can be affected by the following structural characteristics of the polymer to be discussed below: crystallinity, chemical bonding, and concentration of additives.
Increased crystallinity tends to cause lower laser beam transmission because of scattering caused by changes in the index of refraction encountered when going from one phase to the next or because of changing crystallographic orientation. [ 5 ] Increased crystallinity can cause the transmission to increase monotonically as a function of polymer thickness. The relationship follows the Lambert-Bouuger's Law: I t = I 0 e − ( α t ) {\displaystyle I_{t}=I_{0}e^{-(\alpha t)}} , where I t {\displaystyle I_{t}} is the intensity of the laser beam at a given depth or thickness, t. I 0 {\displaystyle I_{0}} is the intensity of laser beam at its source. α {\displaystyle \alpha } is the absorption constant of the polymer. [ 5 ] By the same token, amorphous polymers lack this trend with thickness. [ 5 ]
Polymers absorb EMR (Electro Magnetic Radiation) in a specific wavelength of light depending on what functional groups are present on the polymer. For instance, bending of the C-H bond on the − C H 2 − {\displaystyle -CH_{2}-} at 6800 nm. [ 5 ] Many polymers have vibrational modes at wavelengths greater than 1100 nm, so to apply methods such as TTIr, laser sources must produce photons at wavelengths shorter than that. Therefore, Nd:YAG lasers (1064 nm) and diode lasers (800-950 nm) can pass through the LT until they impinge on the intended modified polymer or additive that results in absorption, whereas CO 2 {\displaystyle {\ce {CO_2}}} lasers (10,640 nm) will be absorbed too easily as it passes through the LT. [ 5 ]
Reinforcements such as fibers or short particles. Reinforcing fibers can be added to increase the strength of a composite.
Some reinforcements like carbon fibers have high thermal conductivity and can dissipate the heat of welding, thus requiring more energy input than with other reinforcement materials such as glass. Glass reinforcements can cause scattering of the beam. [ 6 ]
The orientation of the continuous fibers can affect the width of welds being made. When the welding direction is parallel to the orientation of the fibers, the weld width is usually narrower due to heat being channeled through the fibers to the front and the rear of the weld. [ 6 ]
Increased volume fraction of reinforcements such as glass can scatter the laser beam, thus allowing less to be transmitted to the weld joint. When this happens, the amount of energy necessary to fuse the joint may increase. The increase if not done carefully can cause damage to the transparent part of a TTIr weld joint. [ 6 ]
Some additives can be intentionally added to absorb laser energy. This technique is especially useful in concentrating the weld joint to the mated surfaces of two materials that are relatively transparent to the laser beam. For example, carbon black increases absorption of the laser beam. There can be some unintended consequences of using these absorbing additives. Increasing the concentration of carbon black in a polymer can decrease the depth of heating and increase the peak temperature at the weld joint. Surface damage can occur if the concentration of carbon black becomes excessive. [ 5 ]
Some additives such as the highly selective materials used in the Clearweld process are applied only to the mating surfaces between the plastics to be joined. Some of the chemicals such as cyanines only absorb in a narrow wavelength band centered around 785 nm. [ 4 ] This methodology initially was applied only to plastics, but has recently been applied to composites such as carbon fiber reinforced PEEK. [ 7 ]
Other additives called clarifiers can do the opposite of carbon black by increasing laser beam transmission by reducing crystallinity in polymers. [ 5 ]
Despite the fact that both pigments and dyes can both add color to a polymer, they behave differently. A dye is soluble in a polymer, whereas a pigment is not.
During TTIr, although it takes more energy per unit length to achieve fusion with QS than with CW, QS offers the advantage of achieving higher weld strength and weldability of low transmissive materials such as continuous glass fiber thermoplastics. [ 6 ] Greater strength is imparted because full fusion is achieved without damaging the surface of the transparent material.
Electromagnetic welding is capable of welding complex parts with also the possibility of reopening welds for replacement or repair. To achieve good welds the design of the coil and implant is important for uniform heating.
Implant resistance welding can be a low cost solution for welding parts that are flat or with curved surfaces. The heating element used is often a metal mesh or carbon strips, which provides uniform heating. However, advanced thermoplastic composites that contain conductive fibers can’t be used due to unwanted power leakages.
Induction welding uses a implant or susceptor that is placed at the weld interface and embedded with conductive material such as metal or carbon fibers. An induction coil is then place near the weld joint, which induces a current in embedded in the material used to generate heat. When welding carbon fiber, carbon and graphite fiber mats with higher electrical resistance are used to concentrate the heat at the weld interface. This has the ability to weld complex geometry structures with great weld strength.
The heat generated during welding thermoplastic composite, induces residual stresses in the joint. These stresses can greatly reduce the strength and performance of the part. Upon cooling from welding the matrix and fibers will have different coefficients of thermal expansion, which introduces the residual stress. Things such as heat input, cooling rates, volume fraction of the fibers, and matrix material will influence the residual stress. Another important factor to consider is the orientation of the fibers. During the molten state of welding, fibers can reorient themselves in a manner that reduces weld strength.
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https://en.wikipedia.org/wiki/Welding_of_advanced_thermoplastic_composites
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Robot welding is the use of mechanized programmable tools ( robots ), which completely automate a welding process by both performing the weld and handling the part. Processes such as gas metal arc welding , while often automated, are not necessarily equivalent to robot welding, since a human operator sometimes prepares the materials to be welded. Robot welding is commonly used for resistance spot welding and arc welding in high production applications, such as the automotive industry .
Robot welding is a relatively new application of robotics , even though robots were first introduced into U.S. industry during the 1960s. The use of robots in welding did not take off until the 1980s, when the automotive industry began using robots extensively for spot welding . Since then, both the number of robots used in industry and the number of their applications has grown greatly. In 2005, more than 120,000 robots were in use in North American industry, about half of them for welding. [ 1 ] Growth is primarily limited by high equipment costs, and the resulting restriction to high-production applications.
Robot arc welding has begun growing quickly just recently, [ when? ] and already it commands about 20 percent of industrial robot applications. The major components of arc welding robots are the manipulator or the mechanical unit and the controller, which acts as the robot's "brain". The manipulator is what makes the robot move, and the design of these systems can be categorized into several common types, such as SCARA and cartesian coordinate robot , which use different coordinate systems to direct the arms of the machine.
The robot may weld a pre-programmed position, be guided by machine vision , or by a combination of the two methods. [ 2 ] However, the many benefits of robotic welding have proven to make it a technology that helps many original equipment manufacturers increase accuracy, repeat-ability, and throughput [ 3 ] One welding robot can do the work of several human welders. [ 4 ] [ 5 ] For example, in arc welding, which produces hot sparks and smoke, a human welder can keep his torch on the work for roughly thirty percent of the time; for robots, the percentage is about 90. [ 6 ]
The technology of signature image processing has been developed since the late 1990s for analyzing electrical data in real time collected from automated, robotic welding, thus enabling the optimization of welds.
Advantages of robot welding include: [ 7 ] [ 8 ] [ 9 ] [ 5 ] [ 10 ]
Disadvantages of robot welding include: [ 11 ] [ 12 ] [ 13 ]
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The Weldon process is a process developed in 1866 by Walter Weldon for recovering manganese dioxide for re-use in chlorine manufacture. Commercial operations started at the Gamble works in St. Helens in 1869. The process is described in considerable detail in the book, The Alkali Industry, by J.R. Partington,D.Sc.
The common method to manufacture chlorine at the time, was to react manganese dioxide (and related oxides) with hydrochloric acid to give chlorine:
Weldon's contribution was to develop a process to recycle the manganese. The waste manganese(II) chloride solution is treated with lime, steam and oxygen, producing calcium manganite(IV) :
The resulting calcium manganite can be reacted with HCl as in related processes:
The manganese(II) chloride can be recycled, while the calcium chloride is a waste byproduct. [ citation needed ]
The Weldon process was first replaced by the Deacon process and later by the Chloralkali process .
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Welfare biology is a proposed cross-disciplinary field of research to study the positive and negative well-being of sentient individuals in relation to their environment. Yew-Kwang Ng first advanced the field in 1995. Since then, its establishment has been advocated for by a number of writers, including philosophers, who have argued for the importance of creating the research field, particularly in relation to wild animal suffering . Some researchers have put forward examples of existing research that welfare biology could draw upon and suggested specific applications for the research's findings.
Welfare biology was first proposed by the welfare economist Yew-Kwang Ng , in his 1995 paper "Towards welfare biology: Evolutionary economics of animal consciousness and suffering". In the paper, Ng defines welfare biology as the "study of living things and their environment with respect to their welfare (defined as net happiness, or enjoyment minus suffering)." He also distinguishes between "affective" and "non-affective" sentients, affective sentients being individuals with the capacity for perceiving the external world and experiencing pleasure or pain, while non-affective sentients have the capacity for perception, with no corresponding experience; Ng argues that because the latter experience no pleasure or suffering, "[t]heir welfare is necessarily zero, just like nonsentients". He concludes, based on his modelling of evolutionary dynamics, that suffering dominates enjoyment in nature. [ 1 ]
Matthew Clarke and Ng, in 2006, used Ng's welfare biology framework to analyse the costs, benefits and welfare implications of the culling of kangaroos—classified as affective sentients—in Puckapunyal , Australia. They concluded that while their discussion "may give some support to the culling of kangaroos or other animals in certain circumstances, a more preventive measure may be superior to the resort to culling". [ 2 ] In the same year, Thomas Eichner and Rüdiger Pethi analyzed Ng's model, raising concern regarding a lack of appropriate determinants of the welfare of organisms because of the infancy of welfare biology. [ 3 ]
In 2016, Ng argued that welfare biology could help resolve the paradox within animal welfare science , first raised by Marian Dawkins , on the difficulty of studying animal feelings, by answering "difficult questions regarding animal welfare"; in the paper, Ng also offered various practical proposals for improving the welfare of captive animals. [ 4 ] Todd K. Shackelford and Sayma H. Chowdhury, in response to Ng, argued that rather than focusing on improving the welfare of captive animals, that it would be better to not breed them in the first place, as this would "eliminate their suffering altogether". [ 5 ]
Ng published an update to his original 1995 paper, with Zach Groff, in 2019, which found an error in his original model, leading to a negation of the original conclusion and a revised conception of the extent of suffering in nature as less pessimistic. [ 6 ]
Researchers in environmental economics have drawn attention to Ng's claim in his original paper that the "time is ripe for the recognition of welfare biology as a valid field of scientific study", yet after 25 years, welfare biology as a field of research has yet to take off. [ 7 ]
Some researchers have emphasised the importance of life history theory to welfare biology, as they argue certain traits of life history may predispose certain individuals to worse welfare outcomes and that this has a strong relationship with habitat fragmentation sensitivity. [ 8 ] It has also been suggested that while welfare biology, as a field in its infancy, lacks sufficient empirical studies on the welfare of wild animals, it can make up for this through the use of existing demographic data, currently used to inform biodiversity conservation , to inform future research efforts. [ 9 ] Reviewing the welfare implications of fire on wild animals has been cited as an example of using knowledge drawn from existing ecology studies to establish the field of welfare biology and identify future directions of research. [ 10 ] The application of welfare biology to rewilding projects has additionally been a subject of investigation, with "collaboration between local people, conservationists, authorities and policymakers" suggested as a means of establishing welfare biology as a discipline. [ 11 ]
In the 2019 book, The Routledge Handbook of Animal Ethics , moral philosophers Catia Faria and Oscar Horta contribute a chapter on welfare biology. They propose that welfare biology could evolve from animal welfare science, extending its scope beyond the study of animals under human control. Additionally, Faria and Horta suggest that welfare biology could originate from ecology , emphasizing the impact of environments on the well-being of sentient beings. They express concern over what they perceive as the minimization of animal well-being, attributed to prevalent speciesist and environmentalist beliefs among life scientists and the general public. This, they argue, could impede the advancement of welfare biology. Faria and Horta conclude that the potential benefits of developing welfare biology are significant, given the widespread suffering of animals in the wild, challenging the idealized views of nature. [ 12 ]
Animal Ethics and Wild Animal Initiative are two organizations working on promoting the establishment of welfare biology as a field of research. [ 13 ]
Catia Faria and Oscar Horta have proposed urban welfare ecology as a subdiscipline of welfare biology, which would study the well-being of animals living in urban, suburban and industrial ecosystems. They suggest that much research has already been carried out on animals in these areas, but with the intention of eliminating their negative impact on humans, or to conserve animals of particular species. Faria and Horta argue that such knowledge can be applied to help mitigate the harms that these animals experience and that such environments are perfect for intervention experiments because such ecosystems are already greatly under human control and that the findings could be applied to improve the assessments of the well-being of animals in other ecosystems. [ 12 ]
Some writers in the field of animal ethics have argued that there are compelling moral reasons to reduce the suffering of sentient individuals and that following this line of reasoning, humans should undertake interventions to reduce the suffering of wild animals; [ 14 ] [ 15 ] they claim that because ecosystems are not sentient, that they consequently lack the capacity to care about biodiversity , while arguing that sentient animals do have interests in their welfare. [ 16 ] As a result, they argue that there are strong justifications for ecologists to shift their resources currently used for conservation biology, to welfare biology. [ 16 ] [ 17 ] It has also been asserted that if one is to accept an obligation to undertake systematic and large-scale efforts to help wild animals, that this would first require several important questions to be answered and that large-scale actions should only be carried out after a long phase of successful small-scale trials. [ 18 ]
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In mathematics , a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined , ill defined or ambiguous . [ 1 ] A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if f {\displaystyle f} takes real numbers as input, and if f ( 0.5 ) {\displaystyle f(0.5)} does not equal f ( 1 / 2 ) {\displaystyle f(1/2)} then f {\displaystyle f} is not well defined (and thus not a function). [ 2 ] The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory.
A function that is not well defined is not the same as a function that is undefined . For example, if f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} , then even though f ( 0 ) {\displaystyle f(0)} is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of f {\displaystyle f} .
Let A 0 , A 1 {\displaystyle A_{0},A_{1}} be sets, let A = A 0 ∪ A 1 {\displaystyle A=A_{0}\cup A_{1}} and "define" f : A → { 0 , 1 } {\displaystyle f:A\rightarrow \{0,1\}} as f ( a ) = 0 {\displaystyle f(a)=0} if a ∈ A 0 {\displaystyle a\in A_{0}} and f ( a ) = 1 {\displaystyle f(a)=1} if a ∈ A 1 {\displaystyle a\in A_{1}} .
Then f {\displaystyle f} is well defined if A 0 ∩ A 1 = ∅ {\displaystyle A_{0}\cap A_{1}=\emptyset \!} . For example, if A 0 := { 2 , 4 } {\displaystyle A_{0}:=\{2,4\}} and A 1 := { 3 , 5 } {\displaystyle A_{1}:=\{3,5\}} , then f ( a ) {\displaystyle f(a)} would be well defined and equal to mod ( a , 2 ) {\displaystyle \operatorname {mod} (a,2)} .
However, if A 0 ∩ A 1 ≠ ∅ {\displaystyle A_{0}\cap A_{1}\neq \emptyset } , then f {\displaystyle f} would not be well defined because f ( a ) {\displaystyle f(a)} is "ambiguous" for a ∈ A 0 ∩ A 1 {\displaystyle a\in A_{0}\cap A_{1}} . For example, if A 0 := { 2 } {\displaystyle A_{0}:=\{2\}} and A 1 := { 2 } {\displaystyle A_{1}:=\{2\}} , then f ( 2 ) {\displaystyle f(2)} would have to be both 0 and 1, which makes it ambiguous. As a result, the latter f {\displaystyle f} is not well defined and thus not a function.
In order to avoid the quotation marks around "define" in the previous simple example, the "definition" of f {\displaystyle f} could be broken down into two logical steps:
While the definition in step 1 is formulated with the freedom of any definition and is certainly effective (without the need to classify it as "well defined"), the assertion in step 2 has to be proven. That is, f {\displaystyle f} is a function if and only if A 0 ∩ A 1 = ∅ {\displaystyle A_{0}\cap A_{1}=\emptyset } , in which case f {\displaystyle f} – as a function – is well defined.
On the other hand, if A 0 ∩ A 1 ≠ ∅ {\displaystyle A_{0}\cap A_{1}\neq \emptyset } , then for an a ∈ A 0 ∩ A 1 {\displaystyle a\in A_{0}\cap A_{1}} , we would have that ( a , 0 ) ∈ f {\displaystyle (a,0)\in f} and ( a , 1 ) ∈ f {\displaystyle (a,1)\in f} , which makes the binary relation f {\displaystyle f} not functional (as defined in Binary relation#Special types of binary relations ) and thus not well defined as a function. Colloquially, the "function" f {\displaystyle f} is also called ambiguous at point a {\displaystyle a} (although there is per definitionem never an "ambiguous function"), and the original "definition" is pointless. Despite these subtle logical problems, it is quite common to use the term definition (without apostrophes) for "definitions" of this kind, for three reasons:
Questions regarding the well-definedness of a function often arise when the defining equation of a function refers not only to the arguments themselves, but also to elements of the arguments, serving as representatives . This is sometimes unavoidable when the arguments are cosets and when the equation refers to coset representatives. The result of a function application must then not depend on the choice of representative.
For example, consider the following function:
where n ∈ Z , m ∈ { 4 , 8 } {\displaystyle n\in \mathbb {Z} ,m\in \{4,8\}} and Z / m Z {\displaystyle \mathbb {Z} /m\mathbb {Z} } are the integers modulo m and n ¯ m {\displaystyle {\overline {n}}_{m}} denotes the congruence class of n mod m .
N.B.: n ¯ 4 {\displaystyle {\overline {n}}_{4}} is a reference to the element n ∈ n ¯ 8 {\displaystyle n\in {\overline {n}}_{8}} , and n ¯ 8 {\displaystyle {\overline {n}}_{8}} is the argument of f {\displaystyle f} .
The function f {\displaystyle f} is well defined, because:
As a counter example, the converse definition:
does not lead to a well-defined function, since e.g. 1 ¯ 4 {\displaystyle {\overline {1}}_{4}} equals 5 ¯ 4 {\displaystyle {\overline {5}}_{4}} in Z / 4 Z {\displaystyle \mathbb {Z} /4\mathbb {Z} } , but the first would be mapped by g {\displaystyle g} to 1 ¯ 8 {\displaystyle {\overline {1}}_{8}} , while the second would be mapped to 5 ¯ 8 {\displaystyle {\overline {5}}_{8}} , and 1 ¯ 8 {\displaystyle {\overline {1}}_{8}} and 5 ¯ 8 {\displaystyle {\overline {5}}_{8}} are unequal in Z / 8 Z {\displaystyle \mathbb {Z} /8\mathbb {Z} } .
In particular, the term well-defined is used with respect to (binary) operations on cosets. In this case, one can view the operation as a function of two variables, and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some n can be defined naturally in terms of integer addition.
The fact that this is well-defined follows from the fact that we can write any representative of [ a ] {\displaystyle [a]} as a + k n {\displaystyle a+kn} , where k {\displaystyle k} is an integer. Therefore,
similar holds for any representative of [ b ] {\displaystyle [b]} , thereby making [ a + b ] {\displaystyle [a+b]} the same, irrespective of the choice of representative.
For real numbers, the product a × b × c {\displaystyle a\times b\times c} is unambiguous because ( a × b ) × c = a × ( b × c ) {\displaystyle (a\times b)\times c=a\times (b\times c)} ; hence the notation is said to be well defined . [ 1 ] This property, also known as associativity of multiplication, guarantees the result does not depend on the sequence of multiplications; therefore, a specification of the sequence can be omitted. The subtraction operation is non-associative; despite that, there is a convention that a − b − c {\displaystyle a-b-c} is shorthand for ( a − b ) − c {\displaystyle (a-b)-c} , thus it is considered "well-defined". On the other hand, Division is non-associative, and in the case of a / b / c {\displaystyle a/b/c} , parenthesization conventions are not well established; therefore, this expression is often considered ill-defined.
Unlike with functions, notational ambiguities can be overcome by means of additional definitions (e.g., rules of precedence , associativity of the operator). For example, in the programming language C , the operator - for subtraction is left-to-right-associative , which means that a-b-c is defined as (a-b)-c , and the operator = for assignment is right-to-left-associative , which means that a=b=c is defined as a=(b=c) . [ 3 ] In the programming language APL there is only one rule: from right to left – but parentheses first.
A solution to a partial differential equation is said to be well-defined if it is continuously determined by boundary conditions as those boundary conditions are changed. [ 1 ]
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In mathematical logic , propositional logic and predicate logic , a well-formed formula , abbreviated WFF or wff , often simply formula , is a finite sequence of symbols from a given alphabet that is part of a formal language . [ 1 ]
The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff". [ 12 ]
A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation . Two key uses of formulas are in propositional logic and predicate logic.
A key use of formulas is in propositional logic and predicate logic such as first-order logic . In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven.
Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. This distinction between the vague notion of "property" and the inductively-defined notion of well-formed formula has roots in Weyl's 1910 paper "Uber die Definitionen der mathematischen Grundbegriffe". [ 13 ] Thus the same formula may be written more than once, and a formula might in principle be so long that it cannot be written at all within the physical universe.
Formulas themselves are syntactic objects. They are given meanings by interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as a formula.
The formulas of propositional calculus , also called propositional formulas , [ 14 ] are expressions such as ( A ∧ ( B ∨ C ) ) {\displaystyle (A\land (B\lor C))} . Their definition begins with the arbitrary choice of a set V of propositional variables . The alphabet consists of the letters in V along with the symbols for the propositional connectives and parentheses "(" and ")", all of which are assumed to not be in V . The formulas will be certain expressions (that is, strings of symbols) over this alphabet.
The formulas are inductively defined as follows:
This definition can also be written as a formal grammar in Backus–Naur form , provided the set of variables is finite:
Using this grammar, the sequence of symbols
is a formula, because it is grammatically correct. The sequence of symbols
is not a formula, because it does not conform to the grammar.
A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules (akin to the standard mathematical order of operations ) are assumed among the operators, making some operators more binding than others. For example, assuming the precedence (from most binding to least binding) 1. ¬ 2. → 3. ∧ 4. ∨. Then the formula
may be abbreviated as
This is, however, only a convention used to simplify the written representation of a formula. If the precedence was assumed, for example, to be left-right associative, in following order: 1. ¬ 2. ∧ 3. ∨ 4. →, then the same formula above (without parentheses) would be rewritten as
The definition of a formula in first-order logic Q S {\displaystyle {\mathcal {QS}}} is relative to the signature of the theory at hand. This signature specifies the constant symbols, predicate symbols, and function symbols of the theory at hand, along with the arities of the function and predicate symbols.
The definition of a formula comes in several parts. First, the set of terms is defined recursively. Terms, informally, are expressions that represent objects from the domain of discourse .
The next step is to define the atomic formulas .
Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds:
If a formula has no occurrences of ∃ x {\displaystyle \exists x} or ∀ x {\displaystyle \forall x} , for any variable x {\displaystyle x} , then it is called quantifier-free . An existential formula is a formula starting with a sequence of existential quantification followed by a quantifier-free formula.
An atomic formula is a formula that contains no logical connectives nor quantifiers , or equivalently a formula that has no strict subformulas.
The precise form of atomic formulas depends on the formal system under consideration; for propositional logic , for example, the atomic formulas are the propositional variables . For predicate logic , the atoms are predicate symbols together with their arguments, each argument being a term .
According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers. [ 15 ] This is not to be confused with a formula which is not closed.
A closed formula , also ground formula or sentence , is a formula in which there are no free occurrences of any variable . If A is a formula of a first-order language in which the variables v 1 , …, v n have free occurrences, then A preceded by ∀ v 1 ⋯ ∀ v n is a universal closure of A .
In earlier works on mathematical logic (e.g. by Church [ 16 ] ), formulas referred to any strings of symbols and among these strings, well-formed formulas were the strings that followed the formation rules of (correct) formulas.
Several authors simply say formula. [ 17 ] [ 18 ] [ 19 ] [ 20 ] Modern usages (especially in the context of computer science with mathematical software such as model checkers , automated theorem provers , interactive theorem provers ) tend to retain of the notion of formula only the algebraic concept and to leave the question of well-formedness , i.e. of the concrete string representation of formulas (using this or that symbol for connectives and quantifiers, using this or that parenthesizing convention , using Polish or infix notation, etc.) as a mere notational problem.
The expression "well-formed formulas" (WFF) also crept into popular culture. WFF is part of an esoteric pun used in the name of the academic game " WFF 'N PROOF : The Game of Modern Logic", by Layman Allen, [ 21 ] developed while he was at Yale Law School (he was later a professor at the University of Michigan ). The suite of games is designed to teach the principles of symbolic logic to children (in Polish notation ). [ 22 ] Its name is an echo of whiffenpoof , a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song and The Whiffenpoofs . [ 23 ]
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All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.
In mathematics , a binary relation R is called well-founded (or wellfounded or foundational [ 1 ] ) on a set or, more generally, a class X if every non-empty subset S ⊆ X has a minimal element with respect to R ; that is, there exists an m ∈ S such that, for every s ∈ S , one does not have s R m . In other words, a relation is well-founded if: ( ∀ S ⊆ X ) [ S ≠ ∅ ⟹ ( ∃ m ∈ S ) ( ∀ s ∈ S ) ¬ ( s R m ) ] . {\displaystyle (\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].} Some authors include an extra condition that R is set-like , i.e., that the elements less than any given element form a set.
Equivalently, assuming the axiom of dependent choice , a relation is well-founded when it contains no infinite descending chains , meaning there is no infinite sequence x 0 , x 1 , x 2 , ... of elements of X such that x n +1 R x n for every natural number n . [ 2 ] [ 3 ]
In order theory , a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order .
In set theory , a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x . The axiom of regularity , which is one of the axioms of Zermelo–Fraenkel set theory , asserts that all sets are well-founded.
A relation R is converse well-founded , upwards well-founded or Noetherian on X , if the converse relation R −1 is well-founded on X . In this case R is also said to satisfy the ascending chain condition . In the context of rewriting systems, a Noetherian relation is also called terminating .
An important reason that well-founded relations are interesting is because a version of transfinite induction can be used on them: if ( X , R ) is a well-founded relation, P ( x ) is some property of elements of X , and we want to show that
it suffices to show that:
That is, ( ∀ x ∈ X ) [ ( ∀ y ∈ X ) [ y R x ⟹ P ( y ) ] ⟹ P ( x ) ] implies ( ∀ x ∈ X ) P ( x ) . {\displaystyle (\forall x\in X)\;[(\forall y\in X)\;[y\mathrel {R} x\implies P(y)]\implies P(x)]\quad {\text{implies}}\quad (\forall x\in X)\,P(x).}
Well-founded induction is sometimes called Noetherian induction, [ 4 ] after Emmy Noether .
On par with induction, well-founded relations also support construction of objects by transfinite recursion . Let ( X , R ) be a set-like well-founded relation and F a function that assigns an object F ( x , g ) to each pair of an element x ∈ X and a function g on the initial segment { y : y R x } of X . Then there is a unique function G such that for every x ∈ X , G ( x ) = F ( x , G | { y : y R x } ) . {\displaystyle G(x)=F\left(x,G\vert _{\left\{y:\,y\mathrel {R} x\right\}}\right).}
That is, if we want to construct a function G on X , we may define G ( x ) using the values of G ( y ) for y R x .
As an example, consider the well-founded relation ( N , S ) , where N is the set of all natural numbers , and S is the graph of the successor function x ↦ x +1 . Then induction on S is the usual mathematical induction , and recursion on S gives primitive recursion . If we consider the order relation ( N , <) , we obtain complete induction , and course-of-values recursion . The statement that ( N , <) is well-founded is also known as the well-ordering principle .
There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers , the technique is called transfinite induction . When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction . When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction . See those articles for more details.
Well-founded relations that are not totally ordered include:
Examples of relations that are not well-founded include:
If ( X , <) is a well-founded relation and x is an element of X , then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example:
Let X be the union of the positive integers with a new element ω that is bigger than any integer. Then X is a well-founded set, but
there are descending chains starting at ω of arbitrary great (finite) length;
the chain ω, n − 1, n − 2, ..., 2, 1 has length n for any n .
The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X that is extensional, there exists a class C such that ( X , R ) is isomorphic to ( C , ∈) .
A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example, in the natural numbers with their usual order ≤, we have 1 ≥ 1 ≥ 1 ≥ ... . To avoid these trivial descending sequences, when working with a partial order ≤, it is common to apply the definition of well foundedness (perhaps implicitly) to the alternate relation < defined such that a < b if and only if a ≤ b and a ≠ b . More generally, when working with a preorder ≤, it is common to use the relation < defined such that a < b if and only if a ≤ b and b ≰ a . In the context of the natural numbers, this means that the relation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of a well-founded relation is changed from the definition above to include these conventions.
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A well-known URI is a Uniform Resource Identifier for URL path prefixes that start with /.well-known/ . They are implemented in webservers so that requests to the servers for well-known services or information are available at URLs consistent well-known locations across servers.
Well-known URIs are Uniform Resource Identifiers defined by the IETF in RFC 8615. [ 1 ] They are URL path prefixes that start with /.well-known/ . This implementation is in response to the common expectation for web-based protocols to require certain services or information be available at URLs consistent across servers, regardless of the way URL paths are organized on a particular host. The URIs are implemented in webservers so that requests to the servers for well-known services or information are available at URLs consistently in well-known locations across servers.
The IETF has defined a simple way for web servers to hold metadata that any user agent (e.g., web browser ) can request. The metadata is useful for various tasks, including directing a web user to use a mobile app instead of the website or indicating the different ways that the site can be secured. The well-known locations are used by web servers to share metadata with user agents; sometimes these are files and sometimes these are requests for information from the web server software itself. The way to declare the different metadata requests that can be provided is standardized by the IETF so that other developers know how to find and use this information.
The path well-known URI begins with the characters /.well-known/ , and whose scheme is "HTTP", "HTTPS", or another scheme that has explicitly been specified to use well-known URIs. As an example, if an application hosts the service "example", the corresponding well-known URIs on https://www.example.com/ would start with https://www.example.com/.well-known/example . [ 1 ]
Information shared by a web site as a well-known service is expected to meet a specific standard. Specifications that need to define a resource for such site-wide metadata can register their use with the Internet Assigned Numbers Authority (IANA) to avoid collisions and minimize impingement upon sites' URI space.
The list below describes known standards for .well-known services that a web server can implement.
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All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.
In mathematics , a well-order (or well-ordering or well-order relation ) on a set S is a total ordering on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the ordering is then called a well-ordered set (or woset ). [ 1 ] In some academic articles and textbooks these terms are instead written as wellorder , wellordered , and wellordering or well order , well ordered , and well ordering .
Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element , has a unique successor (next element), namely the least element of the subset of all elements greater than s . There may be elements, besides the least element, that have no predecessor (see § Natural numbers below for an example). A well-ordered set S contains for every subset T with an upper bound a least upper bound , namely the least element of the subset of all upper bounds of T in S .
If ≤ is a non-strict well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a well-founded strict total order . The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible.
Every well-ordered set is uniquely order isomorphic to a unique ordinal number , called the order type of the well-ordered set. The well-ordering theorem , which is equivalent to the axiom of choice , states that every set can be well ordered. If a set is well ordered (or even if it merely admits a well-founded relation ), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.
The observation that the natural numbers are well ordered by the usual less-than relation is commonly called the well-ordering principle (for natural numbers).
Every well-ordered set is uniquely order isomorphic to a unique ordinal number , called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of counting , to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements, cardinal number ) of a finite set is equal to the order type. [ 2 ] Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite n , the expression " n -th element" of a well-ordered set requires context to know whether this counts from zero or one. In an expression " β -th element" where β can also be an infinite ordinal, it will typically count from zero.
For an infinite set, the order type determines the cardinality , but not conversely: sets of a particular infinite cardinality can have well-orders of many different types (see § Natural numbers , below, for an example). For a countably infinite set, the set of possible order types is uncountable.
The standard ordering ≤ of the natural numbers is a well ordering and has the additional property that every non-zero natural number has a unique predecessor.
Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:
This is a well-ordered set of order type ω + ω . Every element has a successor (there is no largest element). Two elements lack a predecessor: 0 and 1.
Unlike the standard ordering ≤ of the natural numbers , the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element.
The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds:
This relation R can be visualized as follows:
R is isomorphic to the ordinal number ω + ω .
Another relation for well ordering the integers is the following definition: x ≤ z y {\displaystyle x\leq _{z}y} if and only if
This well order can be visualized as follows:
This has the order type ω .
The standard ordering ≤ of any real interval is not a well ordering, since, for example, the open interval ( 0 , 1 ) ⊆ [ 0 , 1 ] {\displaystyle (0,1)\subseteq [0,1]} does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice ) one can show that there is a well order of the reals. Also Wacław Sierpiński proved that ZF + GCH (the generalized continuum hypothesis ) imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals. [ 3 ] However it is consistent with ZFC that a definable well ordering of the reals exists—for example, it is consistent with ZFC that V=L , and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.
An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well order: Suppose X is a subset of R {\displaystyle \mathbb {R} } well ordered by ≤ . For each x in X , let s ( x ) be the successor of x in ≤ ordering on X (unless x is the last element of X ). Let A = { ( x , s ( x ) ) | x ∈ X } {\displaystyle A=\{(x,s(x))\,|\,x\in X\}} whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an injective function from A to Q . {\displaystyle \mathbb {Q} .} There is an injection from X to A (except possibly for a last element of X , which could be mapped to zero later). And it is well known that there is an injection from Q {\displaystyle \mathbb {Q} } to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from X to the natural numbers, which means that X is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard ≤ . For example,
Examples of well orders:
If a set is totally ordered , then the following are equivalent to each other:
Every well-ordered set can be made into a topological space by endowing it with the order topology .
With respect to this topology there can be two kinds of elements:
For subsets we can distinguish:
A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum that is also maximum of the whole set.
A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal to ω 1 ( omega-one ), that is, if and only if the set is countable or has the smallest uncountable order type.
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https://en.wikipedia.org/wiki/Well-order
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In mathematics , the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element . [ 1 ] In other words, the set of nonnegative integers is well-ordered by its "natural" or "magnitude" order in which x {\displaystyle x} precedes y {\displaystyle y} if and only if y {\displaystyle y} is either x {\displaystyle x} or the sum of x {\displaystyle x} and some nonnegative integer (other orderings include the ordering 2 , 4 , 6 , . . . {\displaystyle 2,4,6,...} ; and 1 , 3 , 5 , . . . {\displaystyle 1,3,5,...} ).
The phrase "well-ordering principle" is sometimes taken to be synonymous with the " well-ordering theorem ", according to which every set can be well-ordered. On other occasions it is understood to be the proposition that the set of integers { … , − 2 , − 1 , 0 , 1 , 2 , 3 , … } {\displaystyle \{\ldots ,-2,-1,0,1,2,3,\ldots \}} contains a well-ordered subset, called the natural numbers , in which every nonempty subset contains a least element.
Depending on the framework in which the natural numbers are introduced, this (second-order) property of the set of natural numbers is either an axiom or a provable theorem. For example:
In the second sense, this phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form: to prove that every natural number belongs to a specified set S {\displaystyle S} , assume the contrary, which implies that the set of counterexamples is non-empty and thus contains a smallest counterexample . Then show that for any counterexample there is a still smaller counterexample, producing a contradiction. This mode of argument is the contrapositive of proof by complete induction . It is known light-heartedly as the " minimal criminal " method [ 2 ] and is similar in its nature to Fermat's method of " infinite descent ".
Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern Algebra that this property, like the least upper bound axiom for real numbers, is non-algebraic; i.e., it cannot be deduced from the algebraic properties of the integers (which form an ordered integral domain ).
The well-ordering principle can be used in the following proofs.
Theorem: Every integer greater than one can be factored as a product of primes. This theorem constitutes part of the Prime Factorization Theorem .
Proof (by well-ordering principle). Let C {\displaystyle C} be the set of all integers greater than one that cannot be factored as a product of primes. We show that C {\displaystyle C} is empty.
Assume for the sake of contradiction that C {\displaystyle C} is not empty. Then, by the well-ordering principle, there is a least element n ∈ C {\displaystyle n\in C} ; n {\displaystyle n} cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, n {\displaystyle n} has factors a , b {\displaystyle a,b} , where a , b {\displaystyle a,b} are integers greater than one and less than n {\displaystyle n} . Since a , b < n {\displaystyle a,b<n} , they are not in C {\displaystyle C} as n {\displaystyle n} is the smallest element of C {\displaystyle C} . So, a , b {\displaystyle a,b} can be factored as products of primes, where a = p 1 p 2 . . . p k {\displaystyle a=p_{1}p_{2}...p_{k}} and b = q 1 q 2 . . . q l {\displaystyle b=q_{1}q_{2}...q_{l}} , meaning that n = p 1 p 2 . . . p k ⋅ q 1 q 2 . . . q l {\displaystyle n=p_{1}p_{2}...p_{k}\cdot q_{1}q_{2}...q_{l}} , a product of primes. This contradicts the assumption that n ∈ C {\displaystyle n\in C} , so the assumption that C {\displaystyle C} is nonempty must be false. [ 3 ]
Theorem: 1 + 2 + 3 + . . . + n = n ( n + 1 ) 2 {\displaystyle 1+2+3+...+n={\frac {n(n+1)}{2}}} for all positive integers n {\displaystyle n} .
Proof . Suppose for the sake of contradiction that the above theorem is false. Then, there exists a non-empty set of positive integers C = { n ∈ N ∣ 1 + 2 + 3 + . . . + n ≠ n ( n + 1 ) 2 } {\displaystyle C=\{n\in \mathbb {N} \mid 1+2+3+...+n\neq {\frac {n(n+1)}{2}}\}} . By the well-ordering principle, C {\displaystyle C} has a minimum element c {\displaystyle c} such that when n = c {\displaystyle n=c} , the equation is false, but true for all positive integers less than c {\displaystyle c} . The equation is true for n = 1 {\displaystyle n=1} , so c > 1 {\displaystyle c>1} ; c − 1 {\displaystyle c-1} is a positive integer less than c {\displaystyle c} , so the equation holds for c − 1 {\displaystyle c-1} as it is not in C {\displaystyle C} . Therefore,
1 + 2 + 3 + . . . + ( c − 1 ) = ( c − 1 ) c 2 1 + 2 + 3 + . . . + ( c − 1 ) + c = ( c − 1 ) c 2 + c = c 2 − c 2 + 2 c 2 = c 2 + c 2 = c ( c + 1 ) 2 {\begin{aligned}1+2+3+...+(c-1)&={\frac {(c-1)c}{2}}\\1+2+3+...+(c-1)+c&={\frac {(c-1)c}{2}}+c\\&={\frac {c^{2}-c}{2}}+{\frac {2c}{2}}\\&={\frac {c^{2}+c}{2}}\\&={\frac {c(c+1)}{2}}\end{aligned}} ,
which shows that the equation holds for c {\displaystyle c} , a contradiction. So, the equation must hold for all positive integers. [ 3 ]
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In mathematics , the well-ordering theorem , also known as Zermelo's theorem , states that every set can be well-ordered . A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents ). [ 1 ] [ 2 ] Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. [ 3 ] One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction , which is considered by mathematicians to be a powerful technique. [ 3 ] One famous consequence of the theorem is the Banach–Tarski paradox .
Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought". [ 4 ] However, it is considered difficult or even impossible to visualize a well-ordering of R {\displaystyle \mathbb {R} } , the set of all real numbers ; such a visualization would have to incorporate the axiom of choice. [ 5 ] In 1904, Gyula Kőnig claimed to have proven that such a well-ordering cannot exist. A few weeks later, Felix Hausdorff found a mistake in the proof. [ 6 ] It turned out, though, that in first-order logic the well-ordering theorem is equivalent to the axiom of choice, in the sense that the Zermelo–Fraenkel axioms with the axiom of choice included are sufficient to prove the well-ordering theorem, and conversely, the Zermelo–Fraenkel axioms without the axiom of choice but with the well-ordering theorem included are sufficient to prove the axiom of choice. (The same applies to Zorn's lemma .) In second-order logic , however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem. [ 7 ]
There is a well-known joke about the three statements, and their relative amenability to intuition:
The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma ? [ 8 ]
The well-ordering theorem follows from the axiom of choice as follows. [ 9 ]
Let the set we are trying to well-order be A {\displaystyle A} , and let f {\displaystyle f} be a choice function for the family of non-empty subsets of A {\displaystyle A} . For every ordinal α {\displaystyle \alpha } , define an element a α {\displaystyle a_{\alpha }} that is in A {\displaystyle A} by setting a α = f ( A ∖ { a ξ ∣ ξ < α } ) {\displaystyle a_{\alpha }\ =\ f(A\smallsetminus \{a_{\xi }\mid \xi <\alpha \})} if this complement A ∖ { a ξ ∣ ξ < α } {\displaystyle A\smallsetminus \{a_{\xi }\mid \xi <\alpha \}} is nonempty, or leaves a α {\displaystyle a_{\alpha }} undefined if it is. That is, a α {\displaystyle a_{\alpha }} is chosen from the set of elements of A {\displaystyle A} that have not yet been assigned a place in the ordering (or undefined if the entirety of A {\displaystyle A} has been successfully enumerated). Then the order < {\displaystyle <} on A {\displaystyle A} defined by a α < a β {\displaystyle a_{\alpha }<a_{\beta }} if and only if α < β {\displaystyle \alpha <\beta } (in the usual well-order of the ordinals) is a well-order of A {\displaystyle A} as desired, of order type sup { α ∣ a α is defined } + 1 {\displaystyle \sup\{\alpha \mid a_{\alpha }{\text{ is defined}}\}+1} .
The axiom of choice can be proven from the well-ordering theorem as follows.
An essential point of this proof is that it involves only a single arbitrary choice, that of R {\displaystyle R} ; applying the well-ordering theorem to each member S {\displaystyle S} of E {\displaystyle E} separately would not work, since the theorem only asserts the existence of a well-ordering, and choosing for each S {\displaystyle S} a well-ordering would require just as many choices as simply choosing an element from each S {\displaystyle S} . Particularly, if E {\displaystyle E} contains uncountably many sets, making all uncountably many choices is not allowed under the axioms of Zermelo-Fraenkel set theory without the axiom of choice.
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All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.
In mathematics , specifically order theory , a well-quasi-ordering or wqo on a set X {\displaystyle X} is a quasi-ordering of X {\displaystyle X} for which every infinite sequence of elements x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\ldots } from X {\displaystyle X} contains an increasing pair x i ≤ x j {\displaystyle x_{i}\leq x_{j}} with i < j . {\displaystyle i<j.}
Well-founded induction can be used on any set with a well-founded relation , thus one is interested in when a quasi-order is well-founded. (Here, by abuse of terminology, a quasiorder ≤ {\displaystyle \leq } is said to be well-founded if the corresponding strict order x ≤ y ∧ y ≰ x {\displaystyle x\leq y\land y\nleq x} is a well-founded relation.) However the class of well-founded quasiorders is not closed under certain operations—that is, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, this quasiorder is found to be not well-founded. By placing stronger restrictions on the original well-founded quasiordering one can hope to ensure that our derived quasiorderings are still well-founded.
An example of this is the power set operation. Given a quasiordering ≤ {\displaystyle \leq } for a set X {\displaystyle X} one can define a quasiorder ≤ + {\displaystyle \leq ^{+}} on X {\displaystyle X} 's power set P ( X ) {\displaystyle P(X)} by setting A ≤ + B {\displaystyle A\leq ^{+}B} if and only if for each element of A {\displaystyle A} one can find some element of B {\displaystyle B} that is larger than it with respect to ≤ {\displaystyle \leq } . One can show that this quasiordering on P ( X ) {\displaystyle P(X)} needn't be well-founded, but if one takes the original quasi-ordering to be a well-quasi-ordering, then it is.
A well-quasi-ordering on a set X {\displaystyle X} is a quasi-ordering (i.e., a reflexive , transitive binary relation ) such that any infinite sequence of elements x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\ldots } from X {\displaystyle X} contains an increasing pair x i ≤ x j {\displaystyle x_{i}\leq x_{j}} with i < j {\displaystyle i<j} . The set X {\displaystyle X} is said to be well-quasi-ordered , or shortly wqo .
A well partial order , or a wpo , is a wqo that is a proper ordering relation, i.e., it is antisymmetric .
Among other ways of defining wqo's, one is to say that they are quasi-orderings which do not contain infinite strictly decreasing sequences (of the form x 0 > x 1 > x 2 > ⋯ {\displaystyle x_{0}>x_{1}>x_{2}>\cdots } ) [1] nor infinite sequences of pairwise incomparable elements. Hence a quasi-order ( X , ≤) is wqo if and only if ( X , <) is well-founded and has no infinite antichains .
Let X {\displaystyle X} be well partially ordered. A (necessarily finite) sequence ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} of elements of X {\displaystyle X} that contains no pair x i ≤ x j {\displaystyle x_{i}\leq x_{j}} with i < j {\displaystyle i<j} is usually called a bad sequence . The tree of bad sequences T X {\displaystyle T_{X}} is the tree that contains a vertex for each bad sequence, and an edge joining each nonempty bad sequence ( x 1 , … , x n − 1 , x n ) {\displaystyle (x_{1},\ldots ,x_{n-1},x_{n})} to its parent ( x 1 , … , x n − 1 ) {\displaystyle (x_{1},\ldots ,x_{n-1})} . The root of T X {\displaystyle T_{X}} corresponds to the empty sequence. Since X {\displaystyle X} contains no infinite bad sequence, the tree T X {\displaystyle T_{X}} contains no infinite path starting at the root. [ citation needed ] Therefore, each vertex v {\displaystyle v} of T X {\displaystyle T_{X}} has an ordinal height o ( v ) {\displaystyle o(v)} , which is defined by transfinite induction as o ( v ) = lim w c h i l d o f v ( o ( w ) + 1 ) {\displaystyle o(v)=\lim _{w\mathrm {\ child\ of\ } v}(o(w)+1)} . The ordinal type of X {\displaystyle X} , denoted o ( X ) {\displaystyle o(X)} , is the ordinal height of the root of T X {\displaystyle T_{X}} .
A linearization of X {\displaystyle X} is an extension of the partial order into a total order. It is easy to verify that o ( X ) {\displaystyle o(X)} is an upper bound on the ordinal type of every linearization of X {\displaystyle X} . De Jongh and Parikh [ 1 ] proved that in fact there always exists a linearization of X {\displaystyle X} that achieves the maximal ordinal type o ( X ) {\displaystyle o(X)} .
Let X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} be two disjoint wpo sets. Let Y = X 1 ∪ X 2 {\displaystyle Y=X_{1}\cup X_{2}} , and define a partial order on Y {\displaystyle Y} by letting y 1 ≤ Y y 2 {\displaystyle y_{1}\leq _{Y}y_{2}} if and only if y 1 , y 2 ∈ X i {\displaystyle y_{1},y_{2}\in X_{i}} for the same i ∈ { 1 , 2 } {\displaystyle i\in \{1,2\}} and y 1 ≤ X i y 2 {\displaystyle y_{1}\leq _{X_{i}}y_{2}} . Then Y {\displaystyle Y} is wpo, and o ( Y ) = o ( X 1 ) ⊕ o ( X 2 ) {\displaystyle o(Y)=o(X_{1})\oplus o(X_{2})} , where ⊕ {\displaystyle \oplus } denotes natural sum of ordinals. [ 1 ]
Given wpo sets X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} , define a partial order on the Cartesian product Y = X 1 × X 2 {\displaystyle Y=X_{1}\times X_{2}} , by letting ( a 1 , a 2 ) ≤ Y ( b 1 , b 2 ) {\displaystyle (a_{1},a_{2})\leq _{Y}(b_{1},b_{2})} if and only if a 1 ≤ X 1 b 1 {\displaystyle a_{1}\leq _{X_{1}}b_{1}} and a 2 ≤ X 2 b 2 {\displaystyle a_{2}\leq _{X_{2}}b_{2}} . Then Y {\displaystyle Y} is wpo (this is a generalization of Dickson's lemma ), and o ( Y ) = o ( X 1 ) ⊗ o ( X 2 ) {\displaystyle o(Y)=o(X_{1})\otimes o(X_{2})} , where ⊗ {\displaystyle \otimes } denotes natural product of ordinals. [ 1 ]
Given a wpo set X {\displaystyle X} , let X ∗ {\displaystyle X^{*}} be the set of finite sequences of elements of X {\displaystyle X} , partially ordered by the subsequence relation. Meaning, let ( x 1 , … , x n ) ≤ X ∗ ( y 1 , … , y m ) {\displaystyle (x_{1},\ldots ,x_{n})\leq _{X^{*}}(y_{1},\ldots ,y_{m})} if and only if there exist indices 1 ≤ i 1 < ⋯ < i n ≤ m {\displaystyle 1\leq i_{1}<\cdots <i_{n}\leq m} such that x j ≤ X y i j {\displaystyle x_{j}\leq _{X}y_{i_{j}}} for each 1 ≤ j ≤ n {\displaystyle 1\leq j\leq n} . By Higman's lemma , X ∗ {\displaystyle X^{*}} is wpo. The ordinal type of X ∗ {\displaystyle X^{*}} is [ 1 ] [ 5 ] o ( X ∗ ) = { ω ω o ( X ) − 1 , o ( X ) finite ; ω ω o ( X ) + 1 , o ( X ) = ε α + n for some α and some finite n ; ω ω o ( X ) , otherwise . {\displaystyle o(X^{*})={\begin{cases}\omega ^{\omega ^{o(X)-1}},&o(X){\text{ finite}};\\\omega ^{\omega ^{o(X)+1}},&o(X)=\varepsilon _{\alpha }+n{\text{ for some }}\alpha {\text{ and some finite }}n;\\\omega ^{\omega ^{o(X)}},&{\text{otherwise}}.\end{cases}}}
Given a wpo set X {\displaystyle X} , let T ( X ) {\displaystyle T(X)} be the set of all finite rooted trees whose vertices are labeled by elements of X {\displaystyle X} . Partially order T ( X ) {\displaystyle T(X)} by the tree embedding relation . By Kruskal's tree theorem , T ( X ) {\displaystyle T(X)} is wpo. This result is nontrivial even for the case | X | = 1 {\displaystyle |X|=1} (which corresponds to unlabeled trees), in which case o ( T ( X ) ) {\displaystyle o(T(X))} equals the small Veblen ordinal . In general, for o ( X ) {\displaystyle o(X)} countable, we have the upper bound o ( T ( X ) ) ≤ ϑ ( Ω ω o ( X ) ) {\displaystyle o(T(X))\leq \vartheta (\Omega ^{\omega }o(X))} in terms of the ϑ {\displaystyle \vartheta } ordinal collapsing function . (The small Veblen ordinal equals ϑ ( Ω ω ) {\displaystyle \vartheta (\Omega ^{\omega })} in this ordinal notation.) [ 6 ]
In practice, the wqo's one manipulates are quite often not orderings (see examples above), and the theory is technically smoother [ citation needed ] if we do not require antisymmetry, so it is built with wqo's as the basic notion. On the other hand, according to Milner 1985, no real gain in generality is obtained by considering quasi-orders rather than partial orders... it is simply more convenient to do so.
Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes induced by the kernel of the wqo. For example, if we order Z {\displaystyle \mathbb {Z} } by divisibility, we end up with n ≡ m {\displaystyle n\equiv m} if and only if n = ± m {\displaystyle n=\pm m} , so that ( Z , | ) ≈ ( N , | ) {\displaystyle (\mathbb {Z} ,|)\approx (\mathbb {N} ,|)} .
If ( X , ≤ ) {\displaystyle (X,\leq )} is wqo then every infinite sequence x 0 , x 1 , x 2 , … , {\displaystyle x_{0},x_{1},x_{2},\ldots ,} contains an infinite increasing subsequence x n 0 ≤ x n 1 ≤ x n 2 ≤ ⋯ {\displaystyle x_{n_{0}}\leq x_{n_{1}}\leq x_{n_{2}}\leq \cdots } (with n 0 < n 1 < n 2 < ⋯ {\displaystyle n_{0}<n_{1}<n_{2}<\cdots } ). Such a subsequence is sometimes called perfect .
This can be proved by a Ramsey argument : given some sequence ( x i ) i {\displaystyle (x_{i})_{i}} , consider the set I {\displaystyle I} of indexes i {\displaystyle i} such that x i {\displaystyle x_{i}} has no larger or equal x j {\displaystyle x_{j}} to its right, i.e., with i < j {\displaystyle i<j} . If I {\displaystyle I} is infinite, then the I {\displaystyle I} -extracted subsequence contradicts the assumption that X {\displaystyle X} is wqo. So I {\displaystyle I} is finite, and any x n {\displaystyle x_{n}} with n {\displaystyle n} larger than any index in I {\displaystyle I} can be used as the starting point of an infinite increasing subsequence.
The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering, leading to an equivalent notion.
^ Here x < y means: x ≤ y {\displaystyle x\leq y} and x ≠ y . {\displaystyle x\neq y.}
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Well drainage means drainage of agricultural lands by wells. Agricultural land is drained by pumped dry wells (vertical drainage) to improve the soils by controlling water table levels and soil salinity.
Subsurface ( groundwater ) drainage for water table and soil salinity in agricultural land can be done by horizontal and vertical drainage systems. Horizontal drainage systems are drainage systems using open ditches ( trenches ) or buried pipe drains. Vertical drainage systems are drainage systems using pumped wells, either open dug wells or tube wells.
Both systems serve the same purposes , namely water table control and soil salinity control . Both systems can facilitate the reuse of drainage water (e.g. for irrigation), but wells offer more flexibility. Reuse is only feasible if the quality of the groundwater is acceptable and the salinity is low.
Although one well may be sufficient to solve groundwater and soil salinity problems in a few hectares, one usually needs a number of wells, because the problems may be widely spread. The wells may be arranged in a triangular, square or rectangular pattern. The design of the well field concerns depth, capacity, discharge, and spacing of the wells. [ 1 ]
The determination of the optimum depth of the water table is the realm of drainage research .
The basic, steady state , equation for flow to fully penetrating wells (i.e. wells reaching the impermeable base) in a regularly spaced well field in a uniform unconfined (phreatic) aquifer with a hydraulic conductivity that is isotropic is: [ 1 ]
where Q = safe well discharge - i.e. the steady state discharge at which no overdraught or groundwater depletion occurs - (m 3 /day), K = uniform hydraulic conductivity of the soil (m/day), D = depth below soil surface, D b {\displaystyle D_{b}} = depth of the bottom of the well equal to the depth of the impermeable base (m), D m {\displaystyle D_{m}} = depth of the watertable midway between the wells (m), D w {\displaystyle D_{w}} is the depth of the water level inside the well (m), R i {\displaystyle R_{i}} = radius of influence of the well (m) and R w {\displaystyle R_{w}} is the radius of the well (m).
The radius of influence of the wells depends on the pattern of the well field, which may be triangular, square, or rectangular. It can be found as:
where A t {\displaystyle A_{t}} = total surface area of the well field (m 2 )and N = number of wells in the well field.
The safe well discharge (Q) can also be found from:
where q is the safe yield or drainable surplus of the aquifer (m/day) and F w {\displaystyle F_{w}} is the operation intensity of the wells (hours/24 per day). Thus the basic equation can also be written as:
With a well spacing equation one can calculate various design alternatives to arrive at the most attractive or economical solution for watertable control in agricultural land.
The basic flow equation cannot be used for determining the well spacing in a partially penetrating well-field in a non-uniform and anisotropic aquifer, but one needs a numerical solution of more complicated equations. [ 3 ]
The costs of the most attractive solution can be compared with the costs of a horizontal drainage system - for which the drain spacing can be calculated with a drainage equation - serving the same purpose, to decide which system deserves preference.
The well design proper is described in [ 1 ]
An illustration of the parameters involved is shown in the figure. The hydraulic conductivity can be found from an aquifer test .
The numerical computer program WellDrain [ 3 ] for well spacing calculations takes into account fully and partially penetrating wells, layered aquifers, anisotropy (different vertical and horizontal hydraulic conductivity or permeability) and entrance resistance.
With a groundwater model that includes the possibility to introduce wells, one can study the impact of a well drainage system on the hydrology of the project area. There are also models that give the opportunity to evaluate the water quality .
SahysMod [ 4 ] is such a polygonal groundwater model permitting to assess the use of well water for irrigation , the effects on soil salinity and on depth of the water table .
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Well poisoning is the act of malicious manipulation of potable water resources in order to cause illness or death, or to deny an opponent access to fresh water resources.
Well poisoning has been historically documented as a strategy during wartime since antiquity , and was used both offensively (as a terror tactic to disrupt and depopulate a target area) and defensively (as a scorched earth tactic to deny an invading army sources of clean water). Rotting corpses (both animal and human) thrown down wells were the most common implementation; in one of the earliest examples of biological warfare , corpses known to have died from common transmissible diseases of the Pre-Modern era such as bubonic plague or tuberculosis were especially favored for well-poisoning.
Well poisoning has been used as an important scorched earth tactic at least since medieval times. In 1462, for example, Prince Vlad III the Impaler of Wallachia utilized this method to delay his pursuing adversaries. Nearly 500 years later during the Winter War , the Finns rendered wells unusable by putting animal carcasses or feces in them in order to passively combat invading Soviet forces. [ 1 ]
During the 20th century, the practice of poisoning wells has lost most of its potency and practicality against an organized force as modern military logistics ensure secure and decontaminated supplies and resources. Nevertheless, German forces during First World War poisoned wells in France as part of Operation Alberich . [ 2 ]
After World War 2 Nakam , a paramilitary organisation of about fifty Holocaust survivors , sought revenge for the murder of six million Jews during the Holocaust . The group's leader Abba Kovner went to Mandatory Palestine in order to secure large quantities of poison for poisoning water mains to kill large numbers of Germans. His followers infiltrated the water system of Nuremberg . However, Kovner was arrested upon arrival in the British zone of occupied Germany and had to throw the poison overboard. [ 3 ]
Israel poisoned the wells and water supplies of certain Palestinian towns and villages as part of their biological warfare program during the 1948 Palestine war , including a successful operation that caused a typhoid epidemic in Acre in early May 1948, and an unsuccessful attempt in Gaza that was foiled by the Egyptians in late May. [ 4 ]
In the late 20th century, accusations of well-poisoning were brought up, most notoriously in relation to the Kosovo Liberation War. [ 5 ] [ 6 ] [ 7 ] In the 21st century, Israeli settlers have been condemned due to suspicions of poisoning wells of villages in the occupied Palestine . [ 8 ] [ 9 ] [ 10 ]
Despite some vague understanding of how diseases could spread, the existence of viruses and bacteria was unknown in medieval times, and the outbreak of disease could not be scientifically explained. Any sudden deterioration of health was often blamed on poisoning. Europe was hit by several waves of the Black Death throughout the late Middle Ages. Crowded cities were especially hard hit by the disease, with death tolls as high as 50% of the population. In their distress, emotionally distraught survivors searched desperately for an explanation. The city-dwelling Jews of the Middle Ages, living in walled-up, segregated ghetto districts, aroused suspicion . [ 11 ] An outbreak of plague thus became the trigger for Black Death persecutions , with hundreds of Jews burned at the stake, or rounded up in synagogues and private houses that were then set aflame.
Walter Laqueur writes in his book The Changing Face of Anti-Semitism: From Ancient Times to the Present Day :
There were no mass attacks against "Jewish poisoners" after the period of the Black Death, but the accusation became part and parcel of antisemitic dogma and language. It appeared again in early 1953 in the form of the " doctors' plot " in Stalin's last days, when hundreds of Jewish physicians in the Soviet Union were arrested and some of them killed on the charge of having caused the death of prominent Communist leaders... Similar charges were made in the 1980s and 1990s in radical Arab nationalist and Muslim fundamentalist propaganda that accused the Jews of spreading AIDS and other infectious diseases . [ 12 ]
Allegations of well poisoning entwined with antisemitism have also emerged in the discourse around modern epidemics and pandemics such as swine flu , Ebola , avian flu , SARS , and COVID-19 . [ 13 ] [ better source needed ]
In his address to the European Parliament on 23 June 2016, in Brussels, Palestinian Authority president and PLO chairman Mahmoud Abbas made an unsubstantiated allegation, "accusing rabbis of poisoning Palestinian wells". [ 14 ] This was based on false media reports saying Israeli rabbis were inciting the poisoning of water of Palestinians, led by a rabbi Shlomo Mlma or Mlmad from the Council of Rabbis in the West Bank settlements. A rabbi by that name could not be located, nor is such an organization listed. [ 15 ]
Abbas said: "Only a week ago, a number of rabbis in Israel announced, and made a clear announcement, demanding that their government poison the water to kill the Palestinians ... Isn't that clear incitement to commit mass killings against the Palestinian people?" [ 16 ] The speech received a standing ovation. [ 14 ] [ 15 ] [ 17 ] The speech was described as "echoing anti-Semitic claims". [ 17 ] A day later, on Saturday 26 June, Abbas admitted that "his claims at the EU were baseless". [ 18 ] [ 19 ] Abbas' further said that he "didn't intend to do harm to Judaism or to offend Jewish people around the world." [ 20 ] Israeli Prime Minister Benjamin Netanyahu stated in reaction, that Abbas had spread a " blood libel " in his European Parliament address. [ 20 ] [ 21 ]
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In hydrology , a well test is conducted to evaluate the amount of water that can be pumped from a particular water well . More specifically, a well test will allow prediction of the maximum rate at which water can be pumped from a well, and the distance that the water level in the well will fall for a given pumping rate and duration of pumping.
Well testing differs from aquifer testing in that the behaviour of the well is primarily of concern in the former, while the characteristics of the aquifer (the geological formation or unit that supplies water to the well) are quantified in the latter.
When water is pumped from a well the water level in the well falls. This fall is called drawdown . The amount of water that can be pumped is limited by the drawdown produced. Typically, drawdown also increases with the length of time that the pumping continues.
The components of observed drawdown in a pumping well were first described by Jacob (1947), and the test was refined independently by Hantush (1964) and Bierschenk (1963) as consisting of two related components,
where s is drawdown (units of length e.g., m), Q {\displaystyle Q} is the pumping rate (units of volume flowrate e.g., m³/day), B {\displaystyle B} is the aquifer loss coefficient (which increases with time — as predicted by the Theis solution ) and C {\displaystyle C} is the well loss coefficient (which is constant for a given flow rate).
The first term of the equation ( B Q {\displaystyle BQ} ) describes the linear component of the drawdown; i.e., the part in which doubling the pumping rate doubles the drawdown.
The second term ( C Q 2 {\displaystyle CQ^{2}} ) describes what is often called the 'well losses'; the non-linear component of the drawdown. To quantify this it is necessary to pump the well at several different flow rates (commonly called steps ). Rorabaugh (1953) added to this analysis by making the exponent an arbitrary power (usually between 1.5 and 3.5).
To analyze this equation, both sides are divided by the discharge rate ( Q {\displaystyle Q} ), leaving s / Q {\displaystyle s/Q} on the left side, which is commonly referred to as specific drawdown . The right hand side of the equation becomes that of a straight line. Plotting the specific drawdown after a set amount of time ( Δ t {\displaystyle \Delta t} ) since the beginning of each step of the test (since drawdown will continue to increase with time) versus pumping rate should produce a straight line.
Fitting a straight line through the observed data, the slope of the best fit line will be C {\displaystyle C} (well losses) and the intercept of this line with Q = 0 {\displaystyle Q=0} will be B {\displaystyle B} (aquifer losses). This process is fitting an idealized model to real world data, and seeing what parameters in the model make it fit reality best. The assumption is then made that these fitted parameters best represent reality (given the assumptions that went into the model are true).
The relationship above is for fully penetrating wells in confined aquifers (the same assumptions used in the Theis solution for determining aquifer characteristics in an aquifer test ).
Often the well efficiency is determined from this sort of test, this is a percentage indicating the fraction of total observed drawdown in a pumping well which is due to aquifer losses (as opposed to being due to flow through the well screen and inside the borehole). A perfectly efficient well, with perfect well screen and where the water flows inside the well in a frictionless manner would have 100% efficiency. Unfortunately well efficiency is hard to compare between wells because it depends on the characteristics of the aquifer too (the same amount of well losses compared to a more transmissive aquifer would give a lower efficiency).
Specific capacity is a quantity that which a water well can produce per unit of drawdown. It is normally obtained from a step drawdown test. Specific capacity is expressed as:
where
The specific capacity of a well is also a function of the pumping rate it is determined at. Due to non-linear well losses the specific capacity will decrease with higher pumping rates. This complication makes the absolute value of specific capacity of little use; though it is useful for comparing the efficiency of the same well through time (e.g., to see if the well requires rehabilitation).
Additional references on pumping test analysis methods other than the one described above (typically referred to as the Hantush-Bierschenk method) can be found in the general references on aquifer tests and hydrogeology .
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https://en.wikipedia.org/wiki/Well_test
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Weller's theorem [ 1 ] is a theorem in economics . It says that a heterogeneous resource ("cake") can be divided among n partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a cake fairly without compromising on economic efficiency.
Moreover, Weller's theorem says that there exists a price such that the allocation and the price are a competitive equilibrium (CE) with equal incomes (EI). Thus, it connects two research fields which were previously unrelated: fair cake-cutting and general equilibrium .
Fair cake-cutting has been studied since the 1940s. There is a heterogeneous divisible resource, such as a cake or a land-estate. There are n partners, each of whom has a personal value-density function over the cake. The value of a piece to a partner is the integral of his value-density over that piece (this means that the value is a nonatomic measure over the cake). The envy-free cake-cutting problem is to partition the cake to n disjoint pieces, one piece per agent, such for each agent, the value of his piece is weakly larger than the values of all other pieces (so no agent envies another agent's share).
A corollary of the Dubins–Spanier convexity theorem (1961) is that there always exists a "consensus partition" – a partition of the cake to n pieces such that every agent values every piece as exactly 1 / n {\displaystyle 1/n} . A consensus partition is of course EF, but it is not PE. Moreover, another corollary of the Dubins–Spanier convexity theorem is that, when at least two agents have different value measures, there exists a division that gives each agent strictly more than 1 / n {\displaystyle 1/n} . This means that the consensus partition is not even weakly PE.
Envy-freeness, as a criterion for fair allocation, were introduced into economics in the 1960s and studied intensively during the 1970s. Varian's theorems study it in the context of dividing homogeneous goods . Under mild restrictions on the agents' utility functions, there exist allocations which are both PE and EF. The proof uses a previous result on the existence of a competitive equilibrium from equal incomes (CEEI). David Gale proved a similar existence result for agents with linear utilities .
Cake-cutting is more challenging than homogeneous good allocation, since a cake is heterogeneous. In a sense, a cake is a continuum of goods: each point in the cake is a different good. This is the topic of Weller's theorem.
The cake is denoted by C {\displaystyle C} . The number of partners is denoted by n {\displaystyle n} .
A cake partition , denoted by X {\displaystyle X} , is an n -tuple X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} of subsets of C {\displaystyle C} ; X i {\displaystyle X_{i}} is the piece given to partner i {\displaystyle i} .
A partition is called PEEF if it satisfies the following two conditions:
A partition X {\displaystyle X} and a price-measure P {\displaystyle P} on C {\displaystyle C} are called CEEI if they satisfy the following two conditions (where V i {\displaystyle V_{i}} is agent i {\displaystyle i} 's value measure and P {\displaystyle P} is the price measure):
CEEI is much stronger than PEEF: every CEEI allocation is PEEF, but there are many PEEF allocations which are not CEEI.
Weller's theorem proves the existence of a CEEI allocation, which implies the existence of a PEEF allocation.
The presentation below is based on Weller's paper and partly on [ 2 ] : 341–351 .
Weller's proof relies on weighted-utilitarian-maximal (WUM) cake divisions . A WUM division is a division maximizing a function of the following form:
where i {\displaystyle i} is an index of an agent, V i {\displaystyle V_{i}} is agent i {\displaystyle i} 's value measure, X i {\displaystyle X_{i}} is the piece given to i {\displaystyle i} , and w i {\displaystyle w_{i}} is a positive weight.
A corollary of the Dubins–Spanier compactness theorem is that, for every weight-vector w {\displaystyle w} , WUM allocations exist. Intuitively, each tiny piece of cake Z {\displaystyle Z} should be given to the person i {\displaystyle i} for whom V i ( Z ) w i {\displaystyle {V_{i}(Z) \over w_{i}}} is largest. If there are two or more people for whom this value is the same, then every arbitrary division of that piece between them results in a WUM division (WUM allocations can also be defined using the Radon–Nikodym set . Each weight-vector w {\displaystyle w} , as a point on the ( n − 1 ) {\displaystyle (n-1)} -dimensional unit simplex , defines a partition of that simplex. This partition induces an allocation of the Radon–Nikodym set of the cake, which induces one or more allocations of the cake) .
Every WUM division is obviously PE. However, a WUM division can be very unfair; for example, if w i {\displaystyle w_{i}} is very large, then agent i {\displaystyle i} might get only a small fraction of the cake (the weight-vector w {\displaystyle w} is very close to agent i {\displaystyle i} 's vertex of the unit-simplex, which means that i {\displaystyle i} will get only points of the Radon–Nikodym set that are very near its vertex) . In contrast, if w i {\displaystyle w_{i}} is very small, then agent i {\displaystyle i} might get the entire cake.
Weller proves that there exists a vector of weights for which the WUM division is also EF. This is done by defining several functions:
1. The function Par {\displaystyle \operatorname {Par} } : for every positive weight vector w = [ w 1 , … , w n ] {\displaystyle w=[w_{1},\dots ,w_{n}]} , Par ( w ) {\displaystyle \operatorname {Par} (w)} is the set of WUM partitions with weights w {\displaystyle w} . The function Par {\displaystyle \operatorname {Par} } is a set-valued function from the unit-simplex-interior into the space of sets of PE cake-partitions.
2. The function Val {\displaystyle \operatorname {Val} } : for every partition X = X 1 , … , X n {\displaystyle X=X_{1},\dots ,X_{n}} , Val ( X ) {\displaystyle \operatorname {Val} (X)} is a vector proportional to the values of the partners: Val ( X ) = [ V 1 ( X 1 ) , … , V n ( X n ) ] V 1 ( X 1 ) + ⋯ + V n ( X n ) {\displaystyle \operatorname {Val} (X)={\frac {[V_{1}(X_{1}),\dots ,V_{n}(X_{n})]}{V_{1}(X_{1})+\cdots +V_{n}(X_{n})}}} . The function Val {\displaystyle \operatorname {Val} } maps the space of cake-partitions into the unit-simplex.
3. The function Wel = Val ∘ Par {\displaystyle \operatorname {Wel} =\operatorname {Val} \circ \operatorname {Par} } : for every positive weight-vector w {\displaystyle w} , Wel ( w ) {\displaystyle \operatorname {Wel} (w)} is a set of new weight-vectors. This is a set-valued function from the interior of the unit-simplex into the set of subsets of the unit-simplex. The vectors in Wel ( w ) {\displaystyle \operatorname {Wel} (w)} are, in a sense, opposite to w {\displaystyle w} : if w i {\displaystyle w_{i}} is small, then the partitions in Par ( w ) {\displaystyle \operatorname {Par} (w)} give agent i {\displaystyle i} a large value and its weight in Wel ( w ) {\displaystyle \operatorname {Wel} (w)} is large. In contrast, if w i {\displaystyle w_{i}} is large then the partitions in Par ( w ) {\displaystyle \operatorname {Par} (w)} give agent i {\displaystyle i} a small value and its weight in Wel ( w ) {\displaystyle \operatorname {Wel} (w)} is large. This hints that, if Wel {\displaystyle \operatorname {Wel} } has a fixed-point, then this fixed-point corresponds to the PEEF partition that we look for.
To prove that the function Wel {\displaystyle \operatorname {Wel} } has a fixed-point, we would like to use the Kakutani fixed-point theorem . However, there is a technical issue that should be handled: the function Wel {\displaystyle \operatorname {Wel} } is defined only on the interior of the unit-simplex, while its range is the entire unit-simplex. Fortunately, it is possible to extend Wel {\displaystyle \operatorname {Wel} } to the boundary of the unit-simplex, in a way that will guarantee that any fixed-point will NOT be on the boundary. [ 2 ] : 343–344 The extended function, Wel ′ {\displaystyle \operatorname {Wel} '} , is indeed a function from the unit-simplex to subsets of the unit-simplex. Wel ′ {\displaystyle \operatorname {Wel} '} satisfies the requirements of Kakutani' fixed-point theorem, since: [ 2 ] : 345–349
Therefore, Wel ′ {\displaystyle \operatorname {Wel} '} has a fixed-point – a vector W {\displaystyle W} in the unit-simplex such that W ∈ Wel ′ ( W ) {\displaystyle W\in \operatorname {Wel} '(W)} . By the construction of Wel ′ {\displaystyle \operatorname {Wel} '} , it is possible to show that the fixed-point W {\displaystyle W} must be in the unit-simplex-interior, where Wel ′ ≡ Wel {\displaystyle \operatorname {Wel} '\equiv \operatorname {Wel} } . Hence:
By definition of Wel {\displaystyle \operatorname {Wel} } , W ∈ Val ( Par ( W ) ) {\displaystyle W\in \operatorname {Val} (\operatorname {Par} (W))} , so there exists a partition X {\displaystyle X} such that:
X {\displaystyle X} is clearly PE since it is WUM (with weight-vector W). It is also EF, since:
Combining the last two inequalities gives, for every two agents i , j {\displaystyle i,j} :
which is exactly the definition of envy-freeness.
Once we have a PEEF allocation X {\displaystyle X} , a price measure P {\displaystyle P} can be calculated as follows:
It is possible to prove that the pair X , P {\displaystyle X,P} satisfy the conditions of competitive equilibrium with equal incomes (CEEI). Specifically, the income of every agent, under the price measure P {\displaystyle P} , is exactly 1, since
As an illustration, consider a cake with two parts: chocolate and vanilla, and two partners: Alice and George, with the following valuations:
Since there are two agents, the vector w {\displaystyle w} can be represented by a single number – the ratio of the weight of Alice to the weight of George:
Berliant, Thomson and Dunz [ 3 ] introduced the criterion of group envy-freeness , which generalizes both Pareto-efficiency and envy-freeness. They proved the existence of group-envy-free allocations with additive utilities . Later, Berliant and Dunz [ 4 ] studied some natural non-additive utility functions, motivated by the problem of land division. When utilities are not additive, a CEEI allocation is no longer guaranteed to exist, but it does exist under certain restrictions.
More related results can be found in Efficient cake-cutting and Utilitarian cake-cutting .
Weller's theorem is purely existential. Some later works studied the algorithmic aspects of finding a CEEI partition. These works usually assume that the value measures are piecewise-constant , i.e, the cake can divided to homogeneous regions in which the value-density of each agent is uniform.
The first algorithm for finding a CEEI partition in this case was developed by Reijnierse and Potters. [ 5 ]
A more computationally-efficient algorithm was developed by Aziz and Ye. [ 6 ]
In fact, every CEEI cake-partition maximizes the product of utilities, and vice versa – every partition that maximizes the product of utilities is a CEEI. [ 7 ] Therefore, a CEEI can be found by solving a convex program maximizing the sum of the logarithms of utilities.
For two agents, the adjusted winner procedure can be used to find a PEEF allocation that is also equitable (but not necessarily a CEEI).
All the above algorithms can be generalized to value-measures that are Lipschitz continuous . Since such functions can be approximated as piecewise-constant functions "as close as we like", the above algorithms can also approximate a PEEF allocation "as close as we like". [ 5 ]
In the CEEI partition guaranteed by Weller, the piece allocated to each partner may be disconnected. Instead of a single contiguous piece, each partner may receive a pile of "crumbs". Indeed, when the pieces must be connected, CEEI partitions might not exist. Consider the following piecewise-constant valuations:
The CE condition implies that all peripheral slices must have the same price (say, p ) and both central slices must have the same price (say q ). The EI condition implies that the total cake-price should be 2, so q + 2 p = 1 {\displaystyle q+2p=1} . The EI condition again implies that, in any connected CEEI division, the cake is cut in the middle. Both Alice and George receive two peripheral slices and one central slice. The CE condition for Alice implies that q = p {\displaystyle q=p} but the CE condition on George implies that q = 4 p {\displaystyle q=4p} , which is a contradiction.
While the CEEI condition may be unattainable with connected pieces, the weaker PEEF condition is always attainable when there are two partners. This is because with two partners, envy-freeness is equivalent to proportionality, and proportionality is preserved under Pareto-improvements. However, when there are three or more partners, even the weaker PEEF condition may be unattainable. Consider the following piecewise-constant valuations: [ 8 ] : Example 5.1
EF implies that Bob receives at least some of his 7-valued slice (PE then implies that he receives all of it).
By connectivity, there are three options:
Hence, no allocation is PEEF.
In the above example, if we consider the cake to be a "pie" (i.e, if a piece is allowed to go around the cake boundary to the other boundary), then a PEEF allocation exists; however, Stromquist [ 9 ] showed a more sophisticated example where a PEEF allocation does not exist even in a pie.
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https://en.wikipedia.org/wiki/Weller's_theorem
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The Wellman Group is a group of manufacturing companies that make boilers and advanced defence equipment. It is one of the main boilermakers in the UK, if not the most common for large-scale industrial applications, having taken over many well-known boiler companies.
The main company began as the Wellman Smith Owen Engineering Corporation , a large conglomerate British engineering company. It derived from an English company of Samuel T. Wellman , a steel industry pioneer. It was based at Parnell House on Wilton Road in London, next to London Victoria station . It supplied furnaces to the British steel industry.
The site at Oldbury has been making boilers since 1862.
In August 1965 it split in several subsidiaries, including Wellman Machines (at Darlaston), Wellman Incandescent Furnace Company (at Darlaston , Staffordshire), Wellman Steelworks Engineering, and Wellman Incandescent International.
Wellman became a private company in December 1997, when bought by Alchemy Partners for £82 million, by setting up a nominal transitional consortium company called Newmall (an anagram of Wellman). [ 1 ] In 2003 it formed a commercial alliance with Loos International , a German boiler maker. [ 2 ] In August 2005, Alchemy split the company into two – Newmall, for the US subsidiaries and Really Newmall for the other subsidiaries; this became Wellman Group Ltd, owned by Kwikpower International, in the Kwikpower Wellman division. [ 3 ] In May 2009 the company formed an alliance with Wulff Energy Technologies GmbH of Husum to form Wellman Wulff. [ 4 ]
Robey of Lincoln was an agricultural firm that went into making boilers in 1870. It was bought by Babcock International in July 1985 when Robey had a turnover of £7 million.
It is headquartered on the A457 on the western side of Oldbury, next to the Birmingham Canal . It has three subsidiaries.
Wellman Graham merged with Hunt Thermal Engineering Ltd to form Wellman Hunt-Graham in 2012. It works in the heat transfer industry. It is the UK's largest manufacturer of shell and tube heat exchangers .
Wellman Graham began in 1956 as Heat Transfer Ltd. The company changed name to Graham Manufacturing Ltd in 1977, It was sold to Wellman Group in 1995 and Changed name to Wellman Graham Ltd, and was based in Gloucester before moving its design and manufacturing facility to Oldbury in the West Midlands.
Wellman Hunt Graham was acquired by Corac Group plc in 2012 and renamed to Hunt Graham Ltd, and in 2013 changed its name to Hunt Thermal Technologies. Corac Group plc was renamed as TP Group plc in 2015 and Hunt Thermal Technologies has since been renamed as TPG Engineering.
This makes industrial furnaces. A division of the company, Wellman Process Engineering, makes evaporators and crystallisers . The company makes boilers for combined heat and power schemes. Wellman-Robey boilers are made at Oldbury.
This started as the research division of John Brown Engineers and Constructors Ltd in 1957. It became part of Wellman Group in 1996. [ 5 ] Its main significance is that it developed the equipment for purified air that allows the Royal Navy ' s nuclear submarines to be submerged for months at a time – Submarine Atmosphere Control. This uses an electrolyser . Carbon dioxide from the submarine reacts with hydrogen from the electrolyser and is removed.
It also supplies oxygen generation equipment to other countries such as France for the new Barracuda -class submarine .
Wellman Defence was acquired by Corac Group plc in 2012 and renamed to Atmosphere Control International.
Corac Group plc was renamed as TP Group plc in 2015 and Atmosphere Control International has since been renamed as TPG Maritime
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https://en.wikipedia.org/wiki/Wellman_Group
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The Wellman–Lord process is a regenerable process to remove sulfur dioxide from flue gas ( flue-gas desulfurization ) without creating a throwaway sludge product.
In this process, sulfur dioxide from flue gas is absorbed in a sodium sulfite solution in water forming sodium bisulfite ; other components of flue gas are not absorbed. After lowering the temperature the bisulfite is converted to the sodium pyrosulfite which precipitates.
Upon heating, the two previously described chemical reactions are reversed, and sodium pyrosulfite is converted to a concentrated stream of sulfur dioxide and sodium sulfite. The sulfur dioxide can be used for further reactions (e.g. the production of sulfuric acid), and the sulfite is reintroduced into the process. [ 1 ] [ 2 ]
In its initial version ( Crane Station, Maryland , 1968) the process was based on potassium sulfite , but the economic prognosis was poor. Interest in the process occurred because of the worldwide shortage of sulfur in 1967 and resulting high prices; power-plant flue gas was viewed as an additional source of sulfur to relieve the shortage. the later version used sodium sulfite and was installed (as a demonstration system funded by USEPA ) at Mitchell Station, Indiana in 1974. It was coupled with the Allied reduction (by natural gas) process to make elemental sulfur which can be shipped anywhere, for example to a sulfuric acid plant. Additional installations of W-L were made in New Mexico. The process has been offered commercially by Davy Powergas in Lakeland, Florida . Because of side reactions forming thiosulfate (nonregenerable), there is a small makeup requirement in the form of trona (sodium carbonate).
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https://en.wikipedia.org/wiki/Wellman–Lord_process
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The Wells effect describes an empirical disconnect between people's judgment of guilt in a trial setting, and both the mathematical and subjective probability involving guilt. This finding shows that evidence that makes a defendant's guilt more or less probable will not necessarily make a guilty verdict more or less likely, which suggests that the judgments made in courts are not governed by rational decision making.
This behavioral effect was first established in a series of experiments by psychologist Gary L. Wells . [ 1 ] This study examined the difference between how mock jurors judged naked statistics (statistical evidence that is unrelated to the specific case) and other forms of evidence, and found that a simple probability-threshold model (i.e., that jurors decide guilt when the subjective probability of guilt crosses a threshold value) cannot account for juror behavior.
The experiments were based on variants of the hypothetical Blue Bus Case, which first appeared in the legal literature to describe the unsuitability of naked statistics in trial. [ 2 ] In Wells's studies, participants were asked to rule on a case in which a woman had watched her dog get struck by a bus and killed, but was unable to identify the bus. One group of participants (in the rate of traffic case ) was presented with evidence that the Blue Bus Company was responsible for 80% of the traffic on the road, and the competing Gray Bus Company was responsible for the other 20%; a second group (in the weight attendant case ) was presented with the testimony of a weight attendant who made a record indicating that a Blue Bus was on the road at a time corresponding to the accident, and that of a second witness who testified that this record was known to be incorrect 20% of the time. When asked to guess the probability that the Blue Bus Company was responsible for the accident, participants from both groups correctly reported an average 80% chance. However, when asked to make a determination of guilt in the case, those in the first group made a judgment against the Blue Bus Company only 8.2% of the time, while those in the second group found the Blue Bus Company liable in 67.1% of the cases.
The original study found evidence for a process Wells described as "fact-to-evidence reasoning". A juror engaged in such reasoning would ignore evidence unless the evidence itself can (or cannot) be supported by the judgment of the ultimate fact. I.e., jurors would ignore naked statistics (such as the rate of traffic) because the identity of the responsible bus wouldn't prove or disprove the statistics, but jurors would consider the weight attendant's testimony because his testimony could be supported (or not) by the identity of the responsible bus. In two additional experiments, participants were presented with the tire tracks case or the tire tracks-belief case . Both cases relied on the testimony of a transportation official who examined the prints of tire tracks and compared them to the buses from the two companies. In the tire tracks case , the official testified that the tracks matched 80% of the Blue Bus Company buses, and 20% of the Gray Bus Company buses. In the tire tracks-belief case , the official testified that he used a tire-matching technique that provides correct results 80% of the time, and he believed that a specific blue bus was responsible for the accident. As before, participants reported an average 80% probability that a blue bus struck the dog in both cases. However, the tire tracks participants tended not to judge against the Blue Bus Company, while the tire tracks-belief participants did.
Subsequent research has led to the proposal of an "ease-of-simulation" mechanism being responsible for the effect. [ 3 ] Niedermeier, Kerr, and Messé (1999) argued that jurors in Wells's experiments ruled in favor of the Blue Bus Company when they had an easier time imagining that the Gray Bus Company was responsible for the accident. They replicated the Wells Effect but also included manipulations that were meant to make it easier to imagine that a gray bus was responsible. For instance, their partial-match/simulation case was similar to Wells's tire tracks-belief case , except the witness was cross-examined by the defense and admitted that it was possible that a gray bus caused the accident. The researchers also probed participants with new questions meant to measure the ease of this mental simulation (e.g., "on an 11-point scale, how easy it was to imagine that a Grey Bus Company bus ran over the dog"). In these experiments, participants were less likely to make a judgment against the Blue Bus Company in cases where they had an easier time imagining that a gray bus was responsible for the accident, even though they reported an identical probability that a blue bus was responsible.
More recent work has endorsed a model of juror decision making that includes subjective probability of guilt as only one of its inputs. [ 4 ] Arkes, Shoots-Reinhard, & Mayes (2012) identified factors that influenced verdicts only by influencing subjective probability, and factors that influenced verdicts without changing subjective probability. For example, the addition of a non-diagnostic witness (whose testimony was shown to be unreliable during cross-examination) influenced verdicts but not subjective probability. On the other hand, negative evidence (i.e., evidence that they Gray Bus Company was not responsible) caused participants to erroneously make lower probability judgements that a blue bus struck the dog, and resulted in a concordant decrease in guilty verdicts. Across three experiments, the researchers also measured participants' level of agreement with the statement, "it is unfair to blame the Blue Bus Company unless you can prove that they hit the dog; just stating what is likely isn’t enough evidence, the plaintiff must show that they were directly involved in the accident." Levels of agreement were strongly correlated with verdicts in favor of the Blue Bus Company across case variants. This body of work also supports an earlier non-empirical argument positing that jurors would object to the use of naked statistics on the grounds of morality, because doing so would deny the autonomy of the defendant; [ 5 ] the morality of basing a decision on statistical evidence can influence verdicts without influencing the subjective probability of guilt.
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https://en.wikipedia.org/wiki/Wells_effect
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The Wells turbine is a low-pressure air turbine that rotates continuously in one direction independent of the direction of the air flow. Its blades feature a symmetrical airfoil with its plane of symmetry in the plane of rotation and perpendicular to the air stream.
It was developed for use in Oscillating Water Column wave power plants, in which a rising and falling water surface moving in an air compression chamber produces an oscillating air current. The use of this bidirectional turbine avoids the need to rectify the air stream by delicate and expensive check valve systems.
Its efficiency is lower than that of a turbine with constant air stream direction and asymmetric airfoil. One reason for the lower efficiency is that symmetric airfoils have a higher drag coefficient than asymmetric ones, even under optimal conditions. Also, in the Wells turbine, the symmetric airfoil runs partly under high angle of attack (i.e., low blade speed / air speed ratio), which occurs during the air velocity maxima of the oscillating flow. A high angle of attack causes a condition known as " stall " in which the airfoil loses lift. The efficiency of the Wells turbine in oscillating flow reaches values between 0.4 and 0.7.
The Wells turbine was developed by Prof. Alan Arthur Wells of Queen's University Belfast in the late 1970s.
Another solution of the problem of stream direction independent turbine is the Darrieus wind turbine (Darrieus rotor).
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https://en.wikipedia.org/wiki/Wells_turbine
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The Wellsite Information Transfer Specification (WITS) is a specification for the transfer of drilling rig -related data. This petroleum industry standard is recognized by a number of companies internationally and is supported by many hardware devices and software applications. [ citation needed ]
WITS is a multi-layered specification:
Though still in active use as of 2013, the specification has been superseded by the XML-based WITSML .
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https://en.wikipedia.org/wiki/Wellsite_Information_Transfer_Specification
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Welteislehre ( WEL ; "World Ice Theory" or "World Ice Doctrine"), also known as Glazial-Kosmogonie ( Glacial Cosmogony ), is a discredited cosmological concept proposed by Hanns Hörbiger , an Austrian engineer and inventor. According to his ideas, ice was the basic substance of all cosmic processes, and ice moons, ice planets, and the "global ether " (also made of ice) had determined the entire development of the universe. [ 1 ] Hörbiger did not arrive at his ideas through research, but said that he had received it in a "vision" in 1894. He published a book about the theory in 1912 and heavily promoted it in subsequent years, through lectures, magazines and associations.
By his own account, Hörbiger was observing the Moon when he was struck by the notion that the brightness and roughness of its surface were due to ice. Shortly after, he dreamt that he was floating in space watching the swinging of a pendulum which grew longer and longer until it broke. "I knew that Newton had been wrong and that the sun's gravitational pull ceases to exist at three times the distance of Neptune ", he concluded. [ 2 ] He worked out his concepts in collaboration with amateur astronomer and schoolteacher Philipp Fauth whom he met in 1898, and published it as Glazial-Kosmogonie in 1912. Fauth had previously produced a large (if somewhat inaccurate) lunar map and had a considerable following, which lent Hörbiger's ideas some respectability. [ 3 ]
It did not receive a great deal of attention at the time, but following World War I Hörbiger changed his strategy by promoting the new "cosmic truth" not only to people at universities and academies, but also to the general public. Hörbiger thought that if "the masses" accepted his ideas, then they might put enough pressure on the academic establishment to force his ideas into the mainstream. No effort was spared in popularising the ideas: "cosmotechnical" societies were founded, which offered public lectures that attracted large audiences, there were cosmic ice movies and radio programs, and even cosmic ice journals and novels. [ 4 ]
During this period, the name was changed from the Graeco-Latin Glazial-Kosmogonie to the Germanic Welteislehre [WEL] ("World ice theory"). The followers of WEL exerted a great deal of public pressure on behalf of the ideas. [ citation needed ] The movement published posters, pamphlets, books, and even a newspaper The Key to World Events . Companies owned by adherents would only hire people who declared themselves convinced of the WEL's truth. [ citation needed ] Some followers even attended astronomical meetings to heckle, shouting, "Out with astronomical orthodoxy! Give us Hörbiger!" [ 5 ]
Supporters of the idea were Houston Stewart Chamberlain , the leading theorist behind the early development of the National Socialist Party in Germany in 1923, and later both Hitler and Himmler . [ 6 ] [ 7 ] Esoteric and pseudo-scientific views were quite popular among the Nazi elite at the time, and WEL appealed to them because it represented a "Germanic" all-encompassing alternative to a natural science viewed as Jewish and soulless. [ 7 ]
Despite Hitler's claim that the WEL constituted an "Aryan" theory, a number of Jewish intellectuals supported the theory: for example, Viennese author Egon Friedell , who explained the World Ice Theory in his 1930 Cultural History of the Modern Age . [ 8 ] [ 9 ] Hans Schindler Bellamy , a Jewish member of the Austrian Social Democratic Party , was also a proponent. [ 10 ] He continued to advocate the viewpoint after he had fled Vienna following the Anschluss . On the left wing Raoul Hausmann also supported the theory, and corresponded with Hörbiger. [ 11 ]
Two organizations were set up in Vienna concerned with the idea: the Kosmotechnische Gesellschaft and the Hörbiger Institute . [ 7 ] The first was formed in 1921 by a group of enthusiastic adherents of the idea, which included engineers, physicians, civil servants, and businessmen. Most had been personally acquainted with Hörbiger and had attended his many lectures. [ 8 ]
According to the idea, the Solar System had its origin in a gigantic star into which a smaller, dead, waterlogged star fell. This impact caused a huge explosion that flung fragments of the smaller star out into interstellar space where the water condensed and froze into giant blocks of ice. A ring of such blocks formed, that we now call the Milky Way , as well as a number of solar systems among which was our own, but with many more planets than currently exist.
Interplanetary space is filled with traces of hydrogen gas, which cause the planets to slowly spiral inwards, along with ice blocks. The outer planets are large mainly because they have swallowed a large number of ice blocks, but the inner planets have not swallowed nearly as many. One can see ice blocks on the move in the form of meteors, and when one collides with Earth, it produces hailstorms over an area of many square kilometers, while when one falls into the Sun, it produces a sunspot and gets vaporized, making "fine ice", that covers the innermost planets.
It was also claimed that Earth had had several satellites before it acquired the Moon; they began as planets in orbits of their own, but over long spans of time were captured one by one and slowly spiralled in towards Earth until they disintegrated and their debris became part of Earth's structure. One can supposedly identify the rock strata of several geological eras with the impacts of these satellites. It was believed that the destruction of earlier ice-moons were responsible for the Flood . [ 1 ]
The last such impact, of the "Tertiary" or " Cenozoic Moon" and the capture of our present Moon, is supposedly remembered through myths and legends. This was worked out in detail by Hörbiger's English follower Hans Schindler Bellamy ; Bellamy recounted how as a child he would often dream about a large moon that would spiral closer and closer in until it burst, making the ground beneath roll and pitch, awakening him and giving him a very sick feeling. When he looked at the Moon's surface through a telescope, he found its surface looking troublingly familiar. When he learned of Hörbiger's idea in 1921, he found it a description of his dream. He explained the mythological support he found in such books as Moons, Myths, and Man , In the Beginning God , and The Book of Revelation is History . It was believed that our current Moon was the sixth since Earth began and that a new collision was inevitable. Believers argued that the great flood described in the Bible and the destruction of Atlantis were caused by the fall of previous moons.
Hörbiger had various responses to the criticism that he received. If it was pointed out to him that his assertions did not work mathematically, he responded: "Calculation can only lead you astray." If it was pointed out that there existed photographic evidence that the Milky Way was composed of millions of stars, he responded that the pictures had been faked by "reactionary" astronomers. He responded in a similar way when it was pointed out that the surface temperature of the Moon had been measured in excess of 100 °C in the daytime, writing to rocket expert Willy Ley : "Either you believe in me and learn, or you will be treated as the enemy." [ 12 ]
Astronomers generally dismissed his views and the following they acquired as a "carnival". [ citation needed ] As Martin Gardner argued in Chapter Three of his Fads and Fallacies in the Name of Science , Hörbiger's ideas have much in common with those of Immanuel Velikovsky .
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In geometric probability theory, Wendel's theorem , named after James G. Wendel, gives the probability that N points distributed uniformly at random on an ( n − 1 ) {\displaystyle (n-1)} -dimensional hypersphere all lie on the same "half" of the hypersphere. In other words, one seeks the probability that there is some half-space with the origin on its boundary that contains all N points. Wendel's theorem says that the probability is [ 1 ]
The statement is equivalent to p n , N {\displaystyle p_{n,N}} being the probability that the origin is not contained in the convex hull of the N points and holds for any probability distribution on R n that is symmetric around the origin. In particular this includes all distribution which are rotationally invariant around the origin.
This is essentially a probabilistic restatement of Schläfli 's theorem that N {\displaystyle N} hyperplanes in general position in R n {\displaystyle \mathbb {R} ^{n}} divides it into 2 ∑ k = 0 n − 1 ( N − 1 k ) {\displaystyle 2\sum _{k=0}^{n-1}{\binom {N-1}{k}}} regions. [ 2 ]
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Wender Taxol total synthesis in organic chemistry describes a Taxol total synthesis (one of six to date) by the group of Paul Wender at Stanford University published in 1997. [ 1 ] [ 2 ] This synthesis has much in common with the Holton Taxol total synthesis in that it is a linear synthesis starting from a naturally occurring compound with ring construction in the order A,B,C,D. The Wender effort is shorter by approximately 10 steps.
Raw materials for the preparation of Taxol by this route include verbenone , prenyl bromide , allyl bromide , propiolic acid , Gilman reagent , and Eschenmoser's salt .
The taxol synthesis started from the terpene verbenone 1 in Scheme 1 , which is the oxidation product of naturally occurring α-pinene and forming ring A. Construction of ring B started with abstraction of the pendant methyl group proton by potassium tert -butoxide (conjugated anion is formed) followed by nucleophilic displacement of the bromine atom in prenyl bromide 2 to form diene 3 . Ozonolysis of the prenyl group (more electron-rich than the internal double bond) formed aldehyde 4 , which, after isomerization or photorearrangement to the chrysanthenone 5 , was reacted with the lithium salt (via LDA ) of the ethyl ester of propiolic acid 6 in a nucleophilic addition to the alcohol 7 . This compound was not isolated but trapped in situ with trimethylsilyl chloride to the silyl ether 9 . In the next step, Gilman reagent 8 is a methylating reagent in nucleophilic conjugate addition through the alkyne group to the ketone group, which formed the alcohol 10 . The silyl ether protective group was removed by reaction with acetic acid to alcohol 11 , which was then oxidized to the ketone 12 with RuCl 2 (PPh 3 ) 3 and NMO as the sacrificial catalyst . The acyloin group in 13 was introduced by KHMDS and Davis’ oxaziridine (see Holton Taxol total synthesis for another use of this system) and its hydroxyl group together with the ester group were reduced by lithium aluminium hydride to tetrol 14 . Finally, the primary alcohol group was protected as a tert -butyldimethylsilyl ether by the corresponding silylchloride and imidazole in triol 15 .
In the second part ( Scheme 2 ) the procedures are still confined to rings A and B. More protective groups were added to triol 15 as reaction with PPTS and 2-methoxypropene gives the acetonide 16 . At this point the double bond in ring A was epoxidized with m -CPBA and sodium carbonate to epoxide 17 and a Grob fragmentation (also present in the Holton effort) initiated by DABCO opened up the AB ring system in alcohol 18 , which was not isolated but protected as a TIPS silyl ether 19 with triisopropylsilyl triflate and 2,6-lutidine . The C1 position was next oxidized by the phosphite ester , P(OEt) 3 and the strong base KO t -Bu , and oxygen to alcohol 20 (the stereochemistry controlled by bowl-shaped AB ring with hydroxylation from unhindered convex direction), the primary alcohol group was deprotected with ammonium chloride in methanol to diol 21 and two reductions first with NaBH 4 to triol 22 and then hydrogen gas and Crabtree's catalyst give triol 23 . These positions were protected by trimethylsilyl chloride and pyridine to 24 and then triphosgene to 25 in order to facilitate the oxidation of the primary alcohol group to the aldehyde 26 by PCC .
The next part constructed the C ring starting from aldehyde 26 , which was extended by one carbon atom to homologue 27 in a Wittig reaction with methoxymethylenetriphenylphosphine ( Scheme 3 ). The acetonide group was removed by dilute hydrochloric acid and sodium iodide in dioxane and one hydroxyl group in the resulting diol 28 was protected as the triethylsilyl ether (TES) 29 with the corresponding silyl chloride and pyridine enabling oxidation of the remaining hydroxyl group to the ketone 30 with the Dess-Martin periodinane . Reaction with Eschenmoser's salt placed a methylene group (C20 in the Taxol framework) in the alpha position of the aldehyde to 31 and the next reaction introduced (the still lacking) C6 and C7 as the Grignard reagent of allyl bromide in a nucleophilic addition aided by zinc(II) chloride , which blocked the Grignard from attack on carbonate group, to alcohol 32 . The newly formed alcohol was protected as the BOM ether 33 with BOMCl and N,N-diisopropylethylamine . After removal of the TES protecting group with ammonium fluoride , the carbonate group in 34 was converted to a hydroxybenzoate group by action of phenyllithium and the secondary alcohol to the acetate 35 by in situ reaction with acetic anhydride and DMAP . In the next step the acyloin group had its positions swapped by reaction with triazabicyclodecene (other amine bases fail) forming 36 and in the final steps ring closure of ring C was accomplished by ozonolysis at the allyl group to 37 and Aldol reaction with 4-pyrrolidinopyridine to 38 .
The final part dealt with the construction of oxetane ring D starting with protection of the alcohol group in 38 ( Scheme 4 ). as a TROC alcohol 39 with 2,2,2-trichloroethyl chloroformate and pyridine . The OBOM group was replaced by a bromine group in three steps: deprotection to 40 with hydrochloric acid and sodium iodide , mesylation to 41 with mesyl chloride , DMAP and pyridine and nucleophilic substitution with inversion of configuration with lithium bromide to bromide 42 . Because the oxidation of the alkene group to the diol 43 with osmium tetroxide was accompanied by the undesired migration of the benzoate group, this step was taken to completion with imidazole as 44 . Two additional countermeasures were required: reprotection of the diol as the carbonate ester 45 with triphosgene and removal of the benzoate group (KCN) to alcohol 46 in preparation of the actual ring closure to the oxetane 47 with N,N-diisopropylethylamine . In the final steps the tertiary alcohol was acylated in 48 , the TIPS group removed in 49 and the benzoate group re-introduced in 50 .
Tail addition of the Ojima lactam 51 was not disclosed in detail but finally taxol 52 was formed in several steps similar to the other efforts.
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Wendy B. Young is an advisor at Google Ventures and a former senior vice president of small molecule drug discovery at Genentech .
Young received her B.S. and M.S. from Wake Forest University , working with Prof. Huw Davies . [ 1 ] She was co-author on an early application of Davies' rhodium(II) carbenoid insertion - Cope rearrangement chemistry, leading to the total synthesis of three small tropane natural products. [ 2 ] Young received her Ph.D. from Princeton in 1993, working with Edward C. Taylor on heterocycles [ 3 ] derived from natural pigments, one of which ultimately became pemetrexed [ 4 ] (Alimta), [ 5 ] an oncology treatment. In her postdoctoral fellowship with Samuel Danishefsky , Young was among one of a handful of groups in the mid-1990s to synthesize paclitaxel (Taxol), [ 6 ] a highly-oxygenated terpenoid natural product used to treat cancer.
Despite multiple employment offers on the East Coast of the United States, [ 1 ] Young chose to remain in the San Francisco Bay Area for her professional career. From 1995 to 2006, Young worked at Celera Genomics , studying inhibitor compounds of human plasma proteins [ 7 ] such as kallikrein and Factors VIIa and IXa . She was recruited to Genentech in 2006, and in 2018 was promoted to Senior Vice President of Small Molecule drug discovery. [ 1 ] One of her major research successes was development of a chemistry campaign against Bruton's tyrosine kinase , leading to molecules to potentially treat rheumatoid arthritis and B-cell lymphomas. [ 8 ] Her team developed fenebrutinib, currently in Phase II clinical trials for several autoimmune disorders. [ 9 ] In 2023, she became an advisor at Google Ventures .
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The Wenker synthesis is an organic reaction converting a beta amino alcohol to an aziridine with the help of sulfuric acid . It is used industrially for the synthesis of aziridine itself. [ 1 ]
The original Wenker synthesis of aziridine itself takes place in two steps. In the first step ethanolamine is reacted with sulfuric acid at high temperatures (250 °C) to form the sulfate monoester. This salt is then reacted with sodium hydroxide in the second step forming aziridine. The base abstracts an amine proton enabling it to displace the sulfate group. A modification of this reaction involving lower reaction temperatures (140–180 °C) and therefore reduced charring increases the yield of the intermediate. [ 2 ]
The Wenker synthesis protocol using trans -2-aminocyclooctanol, available from reaction of ammonia with the epoxide of cyclooctene , gives a mixture of cyclooctenimine (the Wenker aziridine product) and cyclooctanone (a competing Hofmann elimination product). [ 3 ]
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Werner Arber (born 3 June 1929 in Gränichen , Aargau ) [ 1 ] is a Swiss microbiologist and geneticist . Along with American researchers Hamilton Smith and Daniel Nathans , Werner Arber shared the 1978 Nobel Prize in Physiology or Medicine for the discovery of restriction endonucleases . Their work would lead to the development of recombinant DNA technology.
Arber studied chemistry and physics at the Swiss Federal Institute of Technology in Zürich from 1949 to 1953. Late in 1953, he took an assistantship for electron microscopy at the University of Geneva , in time left the electron microscope, went on to research bacteriophages and write his dissertation on defective lambda prophage mutants. In his Nobel Autobiography, he writes:
In the summer of 1956, we learned about experiments made by Larry Morse and Esther and Joshua Lederberg on the lambda-mediated transduction (gene transfer from one bacterial strain to another by a bacteriophage serving as vector) of bacterial determinants for galactose fermentation. Since these investigators had encountered defective lysogenic strains among their transductants, we felt that such strains should be included in the collection of lambda prophage mutants under study in our laboratory. Very rapidly, thanks to the stimulating help by Jean Weigle and Grete Kellenberger , this turned out to be extremely fruitful. ... This was the end of my career as an electron microscopist and in chosing [ sic ] genetic and physiological approaches I became a molecular geneticist.
Arber received his doctorate in 1958 from the University of Geneva. He then worked at the University of Southern California in phage genetics with Gio ("Joe") Bertani starting in the summer of 1958. [ 2 ] Late in 1959 he accepted an offer to return to Geneva at the beginning of 1960, but only after spending "several very fruitful weeks" [ 1 ] at each of the laboratories of Gunther Stent ( University of California, Berkeley ), Joshua Lederberg and Esther Lederberg [ 3 ] ( Stanford University ) and Salvador Luria ( Massachusetts Institute of Technology ). Arber notes that it was in 1963, while he was a researcher in Stent's Berkeley lab, when experiments produced the first evidence that modification in E. coli B and K is brought about by nucleotide methylation. [ 4 ]
Back at the University of Geneva, Arber worked in a laboratory in the basement of the Physics Institute, where he carried out productive research and hosted "a number of first class graduate students, postdoctoral fellows and senior scientists." including Daisy Roulland Dussoix , [ 1 ] whose work helped him to later obtain the Nobel Prize. [ 5 ] In 1965, the University of Geneva promoted him to Extraordinary Professor for Molecular Genetics. In 1971, after spending a year as a visiting Miller Professor in the Department of Molecular Biology at Berkeley , Arber moved to the University of Basel . In Basel, he was one of the first persons to work in the newly constructed Biozentrum , which housed the departments of biophysics, biochemistry, microbiology, structural biology, cell biology and pharmacology and was thus conducive to interdisciplinary research.
On 27 occasions since 1981, Werner Arber has shared his expertise and passion for science with young scientists at the Lindau Nobel Laureate Meetings . [ 6 ]
Werner Arber is member of the World Knowledge Dialogue Scientific Board and of the Pontifical Academy of Sciences since 1981. In 1981, Arber became a founding member of the World Cultural Council . [ 7 ] He was elected a Fellow of the American Academy of Arts and Sciences in 1984. [ 8 ] Pope Benedict XVI appointed him as President of the Pontifical Academy of Sciences in January 2011, making him the first Protestant to hold the position. [ 9 ] In 2017, Arber retired as President of the Pontifical Academy of Sciences and was replaced by German scientist Joachim von Braun . [ 10 ]
Arber is married and has two daughters, including Silvia Arber .
Arber is a Christian and theistic evolutionist , stating "The most primitive cells may require at least several hundred different specific biological macromolecules. How such already quite complex structures may have come together, remains a mystery to me. The possibility of the existence of a Creator, of God, represents to me a satisfactory solution to this problem." [ 11 ] In addition, he has affirmed: "I know that the concept of God helped me to master many questions in life; it guides me in critical situations, and I see it confirmed in many deep insights into the beauty of the functioning of the world." [ 12 ]
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Werner Kutzelnigg (September 10, 1933 – November 24, 2019 in Bochum ) was a prominent Austrian-born theoretical chemist and professor in the Chemistry Faculty, Ruhr-Universität Bochum , Germany .
Kutzelnigg was born in Vienna . His most significant contributions were in the following fields: relativistic quantum chemistry , coupled cluster methods, theoretical calculation of NMR chemical shifts , explicitly correlated wavefunctions. He was a member of the International Academy of Quantum Molecular Science .
Werner Kutzelnigg studied chemistry in Bonn and Freiburg i. Br. and was awarded his doctorate in 1960 for his experimental work "Untersuchungen zur Zuordnung der Normalschwingungen und Aufklärung der Struktur organischer Ionen". [ 1 ] He then turned to theoretical chemistry and became a postdoc with Bernard Pullman and Gaston Berthier in Paris from 1960 to 1963 and with Per-Olov Löwdin at Uppsala University from 1963 to 1964. [ 1 ] In 1967 Kutzelnigg habilitated at the University of Göttingen under Werner A. Bingel. [ 1 ] From 1970 to 1973 he was professor at the University of Karlsruhe and then full professor at the Chair of Theoretical Chemistry at the Ruhr University Bochum from 1972 until his retirement in 1998. [ 1 ]
Kutzelnigg has published papers on various topics in quantum chemistry: methods of treating electron correlation, magnetic properties of molecules (especially chemical shift), relativistic quantum chemistry, theory of chemical bonding and theory of intermolecular forces. [ 1 ]
Kutzelnigg also became known for his standard work Einführung in die theoretische Chemie. [ 1 ]
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In physics , The Werthamer–Helfand–Hohenberg (WHH) theory was proposed in 1966 by N. Richard Werthamer, Eugene Helfand and Pierre Hohenberg [ 1 ] to go beyond BCS theory of superconductivity and it provides predictions of upper critical field ( H c2 ) in type-II superconductors . [ 1 ] [ 2 ] The theory predicts the upper critical field ( H c2 ) at 0 K from T c and the slope of H c2 at T c .
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West is one of the four cardinal directions or points of the compass . It is the opposite direction from east and is the direction in which the Sun sets on the Earth .
The word "west" is a Germanic word passed into some Romance languages ( ouest in French, oest in Catalan, ovest in Italian, vest in Romanian, oeste in Spanish and Portuguese). As in other languages, the word formation stems from the fact that west is the direction of the setting sun in the evening: 'west' derives from the Indo-European root *wes reduced from *wes-pero 'evening, night', cognate with Ancient Greek ἕσπερος hesperos 'evening; evening star; western' and Latin vesper 'evening; west'. [ 1 ] Examples of the same formation in other languages include Latin occidens 'west' from occidō 'to go down, to set' and Hebrew מַעֲרָב (maarav) 'west' from עֶרֶב (erev) 'evening'.
West is sometimes abbreviated as W .
To go west using a compass for navigation (in a place where magnetic north is the same direction as true north) one needs to set a bearing or azimuth of 270°.
West is the direction opposite that of the Earth 's rotation on its axis, and is therefore the general direction towards which the Sun appears to constantly progress and eventually set. This is not true on the planet Venus , which rotates in the opposite direction from the Earth ( retrograde rotation ). To an observer on the surface of Venus, the Sun would rise in the west and set in the east [ 2 ] although Venus's opaque clouds prevent observing the Sun from the planet's surface. [ 3 ]
In a map with north at the top, west is on the left.
Moving continuously west is following a circle of latitude .
Due to the direction of the Earth's rotation, the prevailing wind in many places in the middle latitudes (i.e. between 35 and 65 degrees latitude ) is from the west, known as the westerlies . [ 4 ] [ 5 ]
The phrase "the West" is often spoken in reference to the Western world , which includes the European Union (also the EFTA countries), the United Kingdom, the Americas, Israel, Australia, New Zealand and (in part) South Africa.
The concept of the Western part of the earth has its roots in the Western Roman Empire and the Western Christianity . During the Cold War "the West" was often used to refer to the NATO camp as opposed to the Warsaw Pact and non-aligned nations . The expression survives, with an increasingly ambiguous meaning.
In Chinese Buddhism , the West represents movement toward the Buddha or enlightenment (see Journey to the West ). The ancient Aztecs believed that the West was the realm of the great goddess of water , mist, and maize . In Ancient Egypt , the West was considered to be the portal to the netherworld , and is the cardinal direction regarded in connection with death , though not always with a negative connotation. Ancient Egyptians also believed that the Goddess Amunet was a personification of the West. [ 6 ] The Celts believed that beyond the western sea off the edges of all maps lay the Otherworld , or Afterlife.
In Judaism , west is seen to be toward the Shekinah (presence) of God, as in Jewish history the Tabernacle and subsequent Jerusalem Temple faced east, with God's Presence in the Holy of Holies up the steps to the west. According to the Bible , the Israelites crossed the Jordan River westward into the Promised Land .
In Islam , cardinal directions carry spiritual significance, but the emphasis is often on the omnipresence of God rather than symbolic geography. The West, like all directions, is encompassed by the divine presence. The Qur’an states: "To God belong the East and the West. Wheresoever you turn, there is the Face of God. God is All-Encompassing, All-Knowing." (2:115). This verse underscores the idea that spiritual truth transcends direction, offering a contrast to traditions that attach specific symbolic meanings to the West.
In American literature (e.g., in The Great Gatsby ) moving West has sometimes symbolized gaining freedom , perhaps as an association with the settling of the Wild West (see also the American frontier and Manifest Destiny ).
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The West Coast Computer Faire was an annual computer industry conference and exposition most often associated with San Francisco , its first and most frequent venue. The first fair was held in 1977 and was organized by Jim Warren and Bob Reiling . At the time, it was the biggest computer show in the world, intended to popularize the personal computer in the home. The West Coast PC Faire was formed to provide a more specialized show. However, Apple Inc. stopped exhibiting at the West Coast Computer Faire, refusing to exhibit at any show other than COMDEX that also had PC-based exhibits.
In 1983, Warren sold the rights to the Faire for US$3 million to Prentice Hall , who later sold it to Sheldon Adelson , the owner of Interface Group and COMDEX. In total, sixteen shows were held, with the last in 1991. After Warren sold the show, it had a few more good years, and then declined rapidly. [ according to whom? ]
The first fair took place on April 15–17, 1977, [ 1 ] [ 2 ] in San Francisco Civic Auditorium , and saw the debut of the Commodore PET , presented by Chuck Peddle , and the Apple II , [ 3 ] presented by then-22-year-old Steve Jobs and 26-year-old Steve Wozniak . At the exhibition, Jobs introduced the Apple II to Japanese textile maker Toshio Mizushima, who became the first authorized Apple dealer in Japan. [ 4 ] [ 5 ] Other visitors included Tomio Gotō who developed the TK-80 and PC-8001 , and Kazuhiko Nishi who produced the MSX . [ 6 ] There were about 180 exhibitors, among them Intel , MITS , and Digital Research .
When the first fair opened, almost twice as many people arrived as Warren anticipated, and thousands of people were waiting to get into the auditorium. More than 12,000 people visited the fair.
As Jim Warren later recalled: “We had these lines running all around the fucking building and nobody was irritated. Nobody was pushy. We didn’t know what we were doing and the exhibitors didn’t know what they were doing and the attendees didn’t know what was going on, but everybody was excited and congenial and undemanding and it was a tremendous turn-on. People just stood and talked—‘Oh, you’ve got an Altair ? Far out!’ ‘You solved this problem?’ And nobody was irritated.” ... The first Computer Faire was to the hardware hackers an event comparable to Woodstock in the movement of the sixties. Like the concert at Max Yasgur’s farm, this was both a cultural vindication and a signal that the movement had gotten so big that it no longer belonged to its progenitors.
The 2nd West Coast Computer Faire was held March 3–5, 1978, [ 8 ] at what was then the San Jose Convention Center . This event had the first-ever micro computer chess tournament, won by Sargon .
The 3rd West Coast Computer Faire was held on November 3–5, 1978, at the Los Angeles Convention Center . [ 9 ]
The 4th West Coast Computer Faire [ 10 ] [ 11 ] [ 12 ] returned to San Francisco in May 1979 at Brooks Hall and Civic Auditorium. Dan Bricklin demonstrated VisiCalc , the first spreadsheet program for personal computers. [ 13 ] [ 14 ]
At the 5th West Coast Computer Faire, held in March 1980, Microsoft announced their first hardware product, the Z-80 SoftCard , which gave the Apple II CP/M capabilities. [ 15 ] [ 16 ] [ 17 ] [ pages needed ] [ 18 ]
The 6th West Coast Computer Faire was held on April 3–5, 1981, notable for being the venue where Adam Osborne introduced the Osborne 1 .
The 7th West Coast Computer Faire saw the introduction of the 5 MB Winchester disk drive for IBM PCs by Davong Systems . It was held on March 19–21, 1982, in San Francisco. That year's conference also featured a Saturday breakout session, titled "THE IBM PERSONAL COMPUTER", with eight talks delivered in a three-hour period. One of these was (as listed in the program):
At its peak, all available spaces for exhibits were rented out, including the balcony of Civic Auditorium, and the hallway to the restrooms in Brooks Hall (where Bob Wallace ("Quicksoft") introduced " PC-Write ").
The 8th West Coast Computer Faire was held March 18–20, 1983.
Subsequent West Coast Computer Faires were held in Moscone Center in San Francisco. After the 10th Faire, Bruce Webster wrote that "Warren sold out just in time. The Faire is shrinking. It may not be dying, but it is no longer the important trade show it was two short years ago. Without the giant booths from IBM, Apple, and AT&T, the Faire would have looked like any other small, local, end-user show. The move to the Moscone Center didn't help that impression; a large chunk of the main floor was unused, adding to the impression of the Faire's shrunken size". [ 19 ]
The 12th West Coast Computer Faire was held in March 1987. [ 20 ]
The 16th West Coast Computer Faire was held from May 30 to June 2, 1991, at Moscone Center.
First West Coast IBM PC Faire, August 26–28, 1983 in San Francisco, CA, was presented by Computer Faire, Inc., Redwood City, CA. [ 21 ]
Third Personal Computer Faire September 5–7, 1985 in San Francisco, CA was presented by Computer Faire, Inc., Newton, MA. [ 22 ]
Fourth Personal Computer Faire, in San Francisco, was presented September 25–27, 1986, by The Interface Group, Needham, Mass. [ 23 ] [ 24 ]
The Northeast Computer Faire in Boston, was presented by Computer Faire Inc., Newton, Mass., a subsidiary of Prentice-Hall. [ 25 ] [ 26 ]
The Eighth Northeast Computer Faire, September 26–29, 1985, Bayside Exposition Center. Boston. MA. was presented by Computer Faire Inc., Newton, MA. [ 27 ] [ 22 ]
The 11th Northeast Computer Faire , which ran October 27-29, 1988, was presented by The Interface Group and Boston Computer Society in Boston. [ 28 ]
Southern California Computer Faire was presented by Computer Faire Inc., Newton, Mass., a subsidiary of Prentice-Hall. [ 25 ]
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The West Melbourne Gasworks was a coal gasification plant in West Melbourne , Victoria, Australia. [ 1 ]
Melbourne was settled in 1835 and by the early 1850s, the gold rushes had led to rapid population growth. The City of Melbourne Gas and Coke Company leased 5 acres (2 hectares) of Crown Land in 1854 to construct a plant for the gasification of black coal. The works operated until 1962, importing coal via a dedicated wharf and a system of narrow gauge tracks drawn by steam locomotives. [ 2 ] Several of these locomotives have been preserved and are now used on the Puffing Billy Railway .
In 1962, the works were upgraded to incorporate a catalytic oil gas facility, but this was short-lived. In 1970, the works was closed down and the remaining structures were all demolished by 1974. [ 3 ]
The works were gradually expanded between 1900 and 1910, eventually covering 8 hectares (20 acres) in an area now extending from Waterview Walk to the Yarra River, with the new Collins Street extension and Harbour Esplanade running through the site. [ citation needed ]
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The West PC-800 was a home computer introduced by Norwegian company West Computer AS in 1984. The computer was designed as an alarm center allowing use of several CPUs (6502, Z80, 8086, 68000) and operating systems. The company introduced an IBM PC compatible in early 1986 and the West PC-800 line was phased out.
West Computer AS was founded in late 1983 by Tov Westby, Terje Holen and Geir Ståle Sætre. [ 1 ] : 12 In early 1984, the company presented its computer then called Sherlock at the Mikrodata'84 fair. The new computer had both 6502 and Z80 CPUs, promised rich expansion capabilities and included two rather unusual features: a wireless keyboard and an alarm device, which could report fire, flood or burglary via phone and the built-in modem. [ 2 ] The machine was released in Autumn 1984 at the Sjølyst "Home and Hobby" fair. [ 3 ] The West PC 800 did not sell as well as expected, probably due to weak Apple II position in Norway, and West Computer AS announced in late 1985 the IBM PC compatible West PC 1600. [ 4 ]
In March 1985, the price of the basic computer was NOK10,200. An additional package with one floppy disk drive (200 KB unformatted capacity), 3 applications and 3 games was available for NOK3,750 and another floppy disk drive for NOK3,300. [ 5 ] : 62 [ nb 1 ]
West Computer designed its computer primarily as an alarm center with emphasis that it could also function as a games machine (thanks to it having Apple II compatibility). [ 5 ] : 57 From ca. serial no. 100 the machine became Apple II Plus compatible due to an updated BIOS . [ citation needed ] Built-in software included two BASIC variants (one for 6502, one for Z80), [ 5 ] : 58 but available was only an old BASIC variant for 6502 (for full Applesoft BASIC compatibility). [ 1 ] : 9 Disk drives are controlled by West DOS (similar to Apple DOS ), whose commands are accessible directly from BASIC. [ 5 ] : 58 However, ProDOS - at the time of the machine introduction - was not compatible with the West DOS. [ 1 ] : 10
A Z80 CPU was available for CP/M compatibility. As access to the Z80 is via 6502, its performance is crippled by design. The company offered additional CPU cards (e.g. Z80B 6 MHz) to improve the performance. [ 5 ] : 58
The alarm system is independent on the machine and has its own CPU and memory. [ 1 ] : 11 A Supplied 300/300 baud modem can work as an autodial modem, Which includes a telephone number database. The modem can be connected to sensors and during an alarm situation, the machine will dial selected number(s). The alarm system works also with a wearable " panic button " with an infrared transmitter, and the computer may even dial another number, if the first desired number is not responding. [ 5 ] : 58
The Wireless keyboard offers 20 function keys and Caps Lock , with another key to turn the keyboard ON and OFF. It is able to operate up to 12–15 meters from the machine [ 1 ] : 9 for about three hours, and recharging takes about 16 hours. [ 5 ] : 57
The West PC-800 can take several CPU cards including a MS-DOS compatibility package (NOK3,000) and Motorola 68000 (NOK7-12,000) expansion cards. [ 5 ] : 57 There was even a Motorola 6809 CPU card for OS-9 compatibility. [ 5 ] : 59
The computer allows cassette and floppy disk drive data storage. The standard floppy disk drive (FDD) had a 142 KB formatted capacity (Apple II compatible) and there were several other storage options e.g. additional FDD 655 KB, 128 KB RAM disk or hard disk drives up to 20 MB . [ 1 ] : 10
The West PC-800 offers rich expansion capabilities thanks to its Apple II compatible expansion bus [ 1 ] : 10 with 7 expansion slots, but some are occupied in the standard configuration (e.g. by the alarm card or RF modulator ). [ 5 ] : 57
The West PC-800 was well received by the press. [ 7 ] Especially lauded were its alarm features [ 1 ] : 11 and high flexibility of the machine's design. [ 5 ] : 59 On the other hand, its graphics capabilities were found dated by 1985 standards [ 5 ] : 58 and support for some of the platforms was rather rudimentary (e.g. supplied only an old MS-DOS version, issues with Z80 speed without a dedicated Z80 CPU card, limited data transfer on the available floppy disk drive). [ 5 ] : 59 A Review in Hjemme-Data magazine concluded, "it is hard to judge the computer, as it stands too outside of the regular market." [ 5 ] : 62
West Computers choose the advertising agency Næss og Mørch with Jørgen Gulvik as Creative Director for the introduction campaign for this new home computer before the Christmas sales 1984. Together with Founder Tov Westby and CEO Fredrik Stange they designed this ad, which won an award from the Norwegian Advertising Association as the best advertising for consumer products in 1984. Apple would use the same picture in their advertising for the Think Different campaign in 1997. [ citation needed ]
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The West number is an empirical parameter used to characterize the performance of Stirling engines and other Stirling systems. It is very similar to the Beale number where a larger number indicates higher performance; however, the West number includes temperature compensation. The West number is often used to approximate of the power output of a Stirling engine. The average value is (0.25) for a wide variety of engines, although it may range up to (0.35), [ 1 ] particularly for engines operating with a high temperature differential.
The West number may be defined as:
where:
When the Beale number is known, but the West number is not known, it is possible to calculate it. First calculate the West number at the temperatures T H and T K for which the Beale number is known, and then use the resulting West number to calculate output power for other temperatures.
To estimate the power output of a new engine design, nominal values are assumed for the West number, pressure, swept volume and frequency, and the power is calculated as follows: [ 2 ]
For example, with an absolute temperature ratio of 2, the portion of the equation representing temperature correction equals 1/3. With a temperature ratio of 3, the temperature term is 1/2. This factor accounts for the difference between the West equation, and the Beale equation in which this temperature term is taken as a constant. Thus, the Beale number is typically in the range of 0.10 to 0.15, which is about 1/3 to 1/2 the value of the West number.
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The Western Ghats Ecology Expert Panel ( WGEEP ), also known as the Gadgil Commission after its chairman Madhav Gadgil , was an environmental research commission appointed by the Ministry of Environment and Forests of India. The commission submitted the report to the Government of India on 31 August 2011. The Expert Panel approached the project through a set of tasks, such as:
Certain sections of people in Kerala, strongly protested the implementation of the report since most of the farmers obtained their livelihood from the hilly regions in Wayanad . [ 2 ] [ 3 ] During the 20th century, a very large number of people had migrated from southern Kerala and acquired forest land in Wayanad and other areas. The Gadgil Commission report was criticised for being excessively environment-friendly and not in tune with the ground realities. [ 4 ]
The report was considered by UNESCO , which added the 39 serial sites of the Western Ghats on the World Heritage List . [ 6 ]
The Kasturirangan Commission has sought to balance the two concerns of development and environment protection, by watering down the environmental regulation regime proposed by the Western Ghats Ecology Experts Panel’s Gadgil report in 2012. The Kasturirangan report seeks to bring just 37% of the Western Ghats under the Ecologically Sensitive Area (ESA) zones — down from the 64% suggested by the Gadgil report. [ 7 ] Dr. V.S. Vijayan, member of the Western Ghats Ecology Expert Panel (WGEEP) said recommendations of the Kasturirangan report are undemocratic and anti-environmental. [ 4 ] [ 8 ]
A crucial report on Western Ghats prepared by K. Kasturirangan -led high-level working group (HLWG) has recommended prohibition on development activities in 60,000 km 2 ecologically sensitive area spread over Gujarat, Karnataka, Maharashtra, Goa, Kerala and Tamil Nadu. [ citation needed ]
The 10-member panel, constituted to examine the Western Ghats ecology expert panel report prepared under the leadership of environmentalist Madhav Gadgil, has also moved away from the suggestions of the Gadgil panel. [ citation needed ]
The Gadgil panel had recommended a blanket approach consisting of guidelines for sector-wise activities, which could be permitted in the ecologically sensitive zones. [ citation needed ]
"Environmentally sound development cannot preclude livelihood and economic options for this region... The answer (to the question of how to manage and conserve the Ghats) will not lie in removing these economic options, but in providing better incentives to move them towards greener and more sustainable practices," the HLWG report says. [ citation needed ]
The panel submitted the report to Environment Minister Jayanthi Natarajan. [ citation needed ]
Roughly 37 per cent of the total area defined as the boundary of the Western Ghats is ecologically sensitive. Over this area of some 60,000 km 2 , spread over the States of Gujarat, Maharashtra, Goa, Karnataka, Kerala and Tamil Nadu, the working group has recommended a prohibitory regime on those activities with maximum interventionist and destructive impact on the environment, the panel says in its report.
The Working Group was constituted to advise the government on the recommendations of an earlier report of ecologist Madhav Gadgil-led "Western Ghats Ecology Expert Panel" (WGEEP).
The WGEEP had recommended 64% of Western Ghats to be declared as an ecologically sensitive area. It had suggested three levels of categorization where regulatory measures for protection would be imposed and had recommended the establishment of the Western Ghats Ecology Authority for management of the Ghats.
The 10-member Working Group, headed by Planning Commission member Kasturirangan, has environmental experts and other professionals as its members.
" The Western Ghats is a biological treasure trove that is endangered, and it needs to be protected and regenerated, indeed celebrated for its enormous wealth of endemic species and natural beauty ", the report says.
Natarajan said that the recommendations would be looked into urgently so that action can be taken to address these challenges.
Kasturirangan said, " The message of this report is serious, alarming and urgent. It is imperative that we protect, manage and regenerate the lands now remaining in the Western Ghats as biologically rich, diverse, natural landscapes ".
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Several 8-bit character sets (encodings) were designed for binary representation of common Western European languages ( Italian , Spanish , Portuguese , French , German , Dutch , English , Danish , Swedish , Norwegian , and Icelandic ), which use the Latin alphabet , a few additional letters and ones with precomposed diacritics , some punctuation , and various symbols (including some Greek letters ). These character sets also happen to support many other languages such as Malay , Swahili , and Classical Latin .
This material is technically obsolete, having been functionally replaced by Unicode . However it continues to have historical interest.
The ISO-8859 series of 8-bit character sets encodes all Latin character sets used in Europe , albeit that the same code points have multiple uses that caused some difficulty (including mojibake , or garbled characters, and communication issues). The arrival of Unicode , with a unique code point for every glyph , resolved these issues.
The earlier seven- bit U.S. American Standard Code for Information Interchange ('ASCII') encoding has characters sufficient to properly represent only a few languages such as English, Latin, Malay and Swahili. It is missing some letters and letter-diacritic combinations used in other Latin-alphabet languages. However, since there was no other choice on most US-supplied computer platforms, use of ASCII was unavoidable except where there was a strong national computing industry. There was the ISO 646 group of encodings which replaced some of the symbols in ASCII with local characters, but space was very limited, and some of the symbols replaced were quite common in things like programming languages.
Most computers internally used eight-bit bytes but communication (seen as inherently unreliable) used seven data bits plus one parity bit . In time, it became common to use all eight bits for data, creating space for another 128 characters. In the early days most of these were system specific, but gradually the ISO/IEC 8859 standards emerged to provide some cross-platform similarity to enable information interchange.
Towards the end of the 20th century, as storage and memory costs fell, the issues associated with multiple meanings of a given eight-bit code (there are seven ISO-Latin code sets alone) have ceased to be justified. All major operating systems have moved to Unicode as their main internal representation. However, as Windows did not support the UTF-8 method of encoding Unicode (preferring UTF-16 ), many applications continued to be restricted to these legacy character sets.
The introduction of the euro and its associated euro sign ( € ) introduced significant pressure on computer systems developers to support this new symbol, and most 8-bit character sets had to be adapted in some way.
Whilst these decisions had limited effect for documents that were only used within a single computer (or at least within a single vendor's " digital ecosystem "), it meant that documents containing a euro sign would fail to render as expected when interchanged between ecosystems.
All of these issues have been resolved as operating systems have been upgraded to support Unicode as standard, which encodes the euro sign at U+20AC (decimal 8364).
Code points U+ 0000 to U+007F are not shown in this table currently, as they are directly mapped in all character sets listed here. The ASCII coding standard defines the original specification for the mapping of the first 0-127 characters.
The table is arranged by Unicode code point. Character sets are referred to here by their IANA names in upper case .
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The Western Palaearctic or Western Palearctic is part of the Palaearctic realm , one of the eight biogeographic realms dividing the Earth's surface. Because of its size, the Palaearctic is often divided for convenience into two, with Europe, North Africa, northern and central parts of the Arabian Peninsula , and part of temperate Asia, roughly to the Ural Mountains forming the western zone, and the rest of temperate Asia becoming the Eastern Palaearctic. Its exact boundaries differ depending on the authority in question, but the Handbook of the Birds of Europe, the Middle East, and North Africa: The Birds of the Western Palearctic ( BWP ) definition is widely used, and is followed by the most popular Western Palearctic checklist, that of the Association of European Rarities Committees (AERC). The Western Palearctic realm includes mostly boreal and temperate climate ecoregions. [ 1 ]
The Palaearctic region has been recognised as a natural zoogeographic region since Sclater proposed it in 1858. The oceans to the north and west, and the Sahara to the south are obvious natural boundaries with other realms, but the eastern boundary is more arbitrary, since it merges into another part of the same realm, and the mountain ranges used as markers are less effective biogeographic separators. [ 2 ] The climate differences across the Western Palearctic region can cause behavioural differences within the same species across geographical distance, such as in the sociality of behaviour for bees of the species Lasioglossum malachurum . [ 3 ]
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https://en.wikipedia.org/wiki/Western_Palaearctic
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The Western Russian fortresses are a system of fortifications built by the Russian Empire in Eastern Europe in the early 19th century. The fortifications were constructed in three chains at strategic locations along Russia's western border, primarily to combat the threat of Prussia (later Germany ) and Austria-Hungary , and to establish Russian rule in new western territories. By the late 19th century the fortifications were obsolete and the system became defunct by the collapse of the Russian Empire in 1917.
During 1830–1831, the Russian Empire under the rule of Tsar Nicholas I crushed the November Uprising , a Polish revolt against Russian authority over the Kingdom of Poland , at the time Russia's westernmost territory that shared borders with other powerful European empires such as Austria-Hungary and Prussia . The Kingdom of Poland, which until then maintained a large degree of autonomy , had its constitution abolished and was placed under the direct rule of Russia. To maintain secure control over the lands and to suppress any future revolts that might occur here, Nicholas I assigned his prominent military engineers to design a reliable system of fortifications in this part of Europe. The endorsed project included construction of new fortifications and reconstruction of the old fortresses within 10 to 15 years.
The project included three lines of fortresses:
The extensive size of the Russian system led to high costs of construction and maintenance, and work on the fortifications slowed in the 1840s, leading to some fortress never being completed. The importance of the forts as military garrisons declined over the following decades, with some being used as prisons or warehouses in addition to barracks .
The Franco-Prussian War from 1870 to 1871 demonstrated to the Russians the vulnerability of the fortification system when the French cities of Paris , Metz , and Sedan were taken by the Prussians despite being protected by similar system. New rifled artillery with longer range, greater accuracy, and explosive shells of greater destructive power effectively rendered the Russian fortifications obsolete .
When relations between Germany and Russia deteriorated in the 1880s the fortifications saw a resurgence of importance, with the Russians modernizing some of them and adding new modern fortresses in between the old ones. In 1905, the defeat of Russia in the Russo-Japanese War caused a rethinking of military strategy, in particular the idea of concentrating forces in the interior away from the borders before hostilities began to gain popularity, eliminating the need for a chain of border fortresses.
In 1909, General Vladimir Sukhomlinov , the new War Minister for the Russian Empire, planned to demolish the western fortress system believing the forts were obsolete. Sukhomlinov's plan was overruled by a vote in the Imperial Duma , instead it was decided to strengthen and expand the system instead, and construction of the new forts was still happening at the outbreak of World War I in 1914. When Russia was invaded by Germany the following year, construction of the forts was rushed with the intention of being holdouts behind German lines, but many of the fortifications were quickly captured by German troops. The collapse of the Russian Empire in 1917 rendered the fortress system effectively useless as much of it was now located in independent countries such as the Second Polish Republic and the Baltic states .
This Russian military article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Western_Russian_fortresses
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The Western Society of Naturalists is a scientific organization with a strong focus on promoting the study of marine biology . Most of its members are on the Pacific coast of North America . Originally established in 1910 as the Biological Society of the Pacific, it changed its name in 1916. [ 1 ] It held its first meeting under the new name in San Diego on August 10, 1916, where papers on zoology and botany were presented. [ 2 ]
The Naturalist of the Year Award was established in 1999 at the suggestion of Paul Dayton to "recognize those unsung heroes who define our future by inspiring young people with the wonders and sheer joy of natural history". [ 3 ]
This article about a biology organization is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Western_Society_of_Naturalists
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The western blot (sometimes called the protein immunoblot ), or western blotting , is a widely used analytical technique in molecular biology and immunogenetics to detect specific proteins in a sample of tissue homogenate or extract. [ 1 ] Besides detecting the proteins, this technique is also utilized to visualize, distinguish, and quantify the different proteins in a complicated protein combination. [ 2 ]
Western blot technique uses three elements to achieve its task of separating a specific protein from a complex: separation by size, transfer of protein to a solid support, and marking target protein using a primary and secondary antibody to visualize. [ 1 ] A synthetic or animal-derived antibody (known as the primary antibody ) is created that recognizes and binds to a specific target protein. The electrophoresis membrane is washed in a solution containing the primary antibody, before excess antibody is washed off. [ 3 ] A secondary antibody is added which recognizes and binds to the primary antibody. The secondary antibody is visualized through various methods such as staining , immunofluorescence , and radioactivity, allowing indirect detection of the specific target protein. [ 3 ]
Other related techniques include dot blot analysis, quantitative dot blot , immunohistochemistry and immunocytochemistry , where antibodies are used to detect proteins in tissues and cells by immunostaining , and enzyme-linked immunosorbent assay (ELISA).
The name western blot is a play on the Southern blot , a technique for DNA detection named after its inventor, English biologist Edwin Southern . Similarly, detection of RNA is termed as northern blot . [ 4 ] The term western blot was given by W. Neal Burnette in 1981, [ 5 ] although the method, but not the name, was independently invented in 1979 by Jaime Renart, Jakob Reiser, and George Stark , [ 6 ] and by Harry Towbin, Theophil Staehelin, and Julian Gordon at the Friedrich Miescher Institute in Basel , Switzerland . [ 7 ] The Towbin group also used secondary antibodies for detection, thus resembling the actual method that is almost universally used today. Between 1979 and 2019 "it has been mentioned in the titles, abstracts, and keywords of more than 400,000 PubMed -listed publications" and may still be the most-used protein-analytical technique. [ 8 ]
The western blot is extensively used in biochemistry for the qualitative detection of single proteins and protein-modifications (such as post-translational modifications ). At least 8–9% of all protein-related publications are estimated to apply western blots. [ 8 ] It is used as a general method to identify the presence of a specific single protein within a complex mixture of proteins. A semi-quantitative estimation of a protein can be derived from the size and colour intensity of a protein band on the blot membrane. In addition, applying a dilution series of a purified protein of known concentrations can be used to allow a more precise estimate of protein concentration. The western blot is routinely used for verification of protein production after cloning . It is also used in medical diagnostics, e.g., in the HIV test or BSE -Test. [ 9 ]
The confirmatory HIV test employs a western blot to detect anti-HIV antibody in a human serum sample. Proteins from known HIV -infected cells are separated and blotted on a membrane as above. Then, the serum to be tested is applied in the primary antibody incubation step; free antibody is washed away, and a secondary anti-human antibody linked to an enzyme signal is added. The stained bands then indicate the proteins to which the patient's serum contains antibody. [ 10 ] A western blot is also used as the definitive test for variant Creutzfeldt–Jakob disease , a type of prion disease linked to the consumption of contaminated beef from cattle with bovine spongiform encephalopathy (BSE, commonly referred to as 'mad cow disease'). [ 11 ] Another application is in the diagnosis of tularemia . An evaluation of the western blot's ability to detect antibodies against F. tularensis revealed that its sensitivity is almost 100% and the specificity is 99.6%. [ 12 ] Some forms of Lyme disease testing employ western blotting. [ 13 ] A western blot can also be used as a confirmatory test for Hepatitis B infection and HSV-2 (Herpes Type 2) infection. [ 14 ] [ 15 ] In veterinary medicine, a western blot is sometimes used to confirm FIV + status in cats. [ 16 ]
Further applications of the western blot technique include its use by the World Anti-Doping Agency (WADA). Blood doping is the misuse of certain techniques and/or substances to increase one's red blood cell mass, which allows the body to transport more oxygen to muscles and therefore increase stamina and performance. There are three widely known substances or methods used for blood doping, namely, erythropoietin (EPO), synthetic oxygen carriers and blood transfusions. Each is prohibited under WADA's List of Prohibited Substances and Methods. The western blot technique was used during the 2014 FIFA World Cup in the anti-doping campaign for that event. [ 17 ] In total, over 1000 samples were collected and analysed by Reichel, et al. [ 18 ] in the WADA accredited Laboratory of Lausanne , Switzerland . Recent research utilizing the western blot technique showed an improved detection of EPO in blood and urine based on novel Velum SAR precast horizontal gels optimized for routine analysis. [ 19 ] With the adoption of the horizontal SAR-PAGE in combination with the precast film-supported Velum SAR gels the discriminatory capacity of micro-dose application of rEPO was significantly enhanced.
For medication development, the identification of therapeutic targets, and biological research, it is essential to comprehend where proteins are located within a cell. [ 2 ] [ 20 ] The subcellular locations of proteins inside the cell and their functions are closely related. The relationship between protein function and localization suggests that when proteins move, their functions may change or acquire new characteristics. A protein's subcellular placement can be determined using a variety of methods. Numerous efficient and reliable computational tools and strategies have been created and used to identify protein subcellular localization. [ 21 ] With the aid of subcellular fractionation methods, WB continues to be an important fundamental method for the investigation and comprehension of protein localization. [ 2 ]
Due to their various epitopes, antibodies have gained interest in both basic and clinical research. The foundation of antibody characterization and validation is epitope mapping. The procedure of identifying an antibody's binding sites (epitopes) on the target protein is referred to as "epitope mapping." Finding the binding epitope of an antibody is essential for the discovery and creation of novel vaccines, diagnostics, and therapeutics. [ 2 ] As a result, various methods for mapping antibody epitopes have been created. At this point, western blotting's specificity is the main feature that sets it apart from other epitope mapping techniques. There are several application of western blot for epitope mapping on human skin samples, hemorrhagic disease virus. [ 2 ] [ 22 ] [ 23 ]
The western blot method is composed of gel electrophoresis to separate native proteins by 3-D structure or denatured proteins by the length of the polypeptide, followed by an electrophoretic transfer onto a membrane (mostly PVDF or nitrocellulose ) and an immunostaining procedure to visualize a certain protein on the blot membrane.
Sodium dodecyl sulfate–polyacrylamide gel electrophoresis (SDS-PAGE) is generally used for the denaturing electrophoretic separation of proteins. Sodium dodecyl sulfate (SDS) is generally used as a buffer (as well as in the gel) in order to give all proteins present a uniform negative charge, since proteins can be positively, negatively, or neutrally charged. Prior to electrophoresis, protein samples are often boiled to denature the proteins present. This ensures that proteins are separated based on size and prevents proteases (enzymes that break down proteins) from degrading samples. Following electrophoretic separation, the proteins are transferred to a membrane (typically nitrocellulose or PVDF). The membrane is often then stained with Ponceau S in order to visualize the proteins on the blot and ensure a proper transfer occurred. Next the proteins are blocked with milk (or other blocking agents) to prevent non-specific antibody binding, and then stained with antibodies specific to the target protein. [ 7 ] [ 6 ] Lastly, the membrane will be stained with a secondary antibody that recognizes the first antibody staining, which can then be used for detection by a variety of methods. The gel electrophoresis step is included in western blot analysis to resolve the issue of the cross-reactivity of antibodies.
As a significant step in conducting a western blot, sample preparation has to be done effectively since the interpretation of this assay is influenced by the protein preparation, which is composed of protein extraction and purification processes. [ 24 ] [ 3 ] To achieve efficient protein extraction, a proper homogenization method needs to be chosen due to the fact that it is responsible for bursting the cell membrane and releasing the intracellular components. [ 3 ] [ 25 ] Besides that, the ideal lysis buffer is needed to acquire substantial amounts of target protein content because the buffer is leading the process of protein solubilization and preventing protein degradation. After completing the sample preparation, the protein content is ready to be separated by the utilization of gel electrophoresis. [ 3 ]
The proteins of the sample are separated using gel electrophoresis . Separation of proteins may be by isoelectric point (pI), molecular weight , electric charge, or a combination of these factors. The nature of the separation depends on the treatment of the sample and the nature of the gel.
By far the most common type of gel electrophoresis employs polyacrylamide gels and buffers loaded with sodium dodecyl sulfate (SDS). SDS-PAGE (SDS-polyacrylamide gel electrophoresis) maintains polypeptides in a denatured state once they have been treated with strong reducing agents to remove secondary and tertiary structure (e.g. disulfide bonds [S-S] to sulfhydryl groups [SH and SH]) and thus allows separation of proteins by their molecular mass . Sampled proteins become covered in the negatively charged SDS, effectively becoming anionic , and migrate towards the positively charged (higher voltage) anode (usually having a red wire) through the acrylamide mesh of the gel. Smaller proteins migrate faster through this mesh, and the proteins are thus separated according to size (usually measured in kilodaltons, kDa ). The concentration of acrylamide determines the resolution of the gel – the greater the acrylamide concentration, the better the resolution of lower molecular weight proteins. The lower the acrylamide concentration, the better the resolution of higher molecular weight proteins. Proteins travel only in one dimension along the gel for most blots.
Samples are loaded into wells in the gel. One lane is usually reserved for a marker or ladder , which is a commercially available mixture of proteins of known molecular weights, typically stained so as to form visible, coloured bands. When voltage is applied along the gel, proteins migrate through it at different speeds dependent on their size. These different rates of advancement (different electrophoretic mobilities ) separate into bands within each lane . Protein bands can then be compared to the ladder bands, allowing estimation of the protein's molecular weight.
It is also possible to use a two-dimensional gel which spreads the proteins from a single sample out in two dimensions. Proteins are separated according to isoelectric point ( pH at which they have a neutral net charge) in the first dimension, and according to their molecular weight in the second dimension.
To make the proteins accessible to antibody detection, they are moved from within the gel onto a membrane, a solid support, which is an essential part of the process. There are two types of membrane: nitrocellulose (NC) or polyvinylidene difluoride (PVDF ). NC membrane has high affinity for protein and its retention abilities. However, NC is brittle, and does not allow the blot to be used for re-probing, whereas PVDF membrane allows the blot to be re-probed. [ 1 ] The most commonly used method for transferring the proteins is called electroblotting . Electroblotting uses an electric current to pull the negatively charged proteins from the gel towards the positively charged anode, and into the PVDF or NC membrane. The proteins move from within the gel onto the membrane while maintaining the organization they had within the gel. An older method of transfer involves placing a membrane on top of the gel, and a stack of filter papers on top of that. The entire stack is placed in a buffer solution which moves up the paper by capillary action , bringing the proteins with it. In practice this method is not commonly used due to the lengthy procedure time.
As a result of either transfer process, the proteins are exposed on a thin membrane layer for detection. Both varieties of membrane are chosen for their non-specific protein binding properties (i.e. binds all proteins equally well). Protein binding is based upon hydrophobic interactions, as well as charged interactions between the membrane and protein. Nitrocellulose membranes are cheaper than PVDF, but are far more fragile and cannot withstand repeated probings.
Total protein staining allows the total protein that has been successfully transferred to the membrane to be visualised, allowing the user to check the uniformity of protein transfer and to perform subsequent normalization of the target protein with the actual protein amount per lane. Normalization with the so-called "loading control" was based on immunostaining of housekeeping proteins in the classical procedure, but is heading toward total protein staining recently, due to multiple benefits. [ 26 ] At least seven different approaches for total protein staining have been described for western blot normalization: Ponceau S , stain-free techniques, Sypro Ruby, Epicocconone , Coomassie R-350 , Amido Black , and Cy5 . [ 26 ] In order to avoid noise of signal, total protein staining should be performed before blocking of the membrane. Nevertheless, post-antibody stainings have been described as well. [ 27 ]
Since the membrane has been chosen for its ability to bind protein and as both antibodies and the target are proteins, steps must be taken to prevent the interactions between the membrane and the antibody used for detection of the target protein. Blocking of non-specific binding is achieved by placing the membrane in a dilute solution of protein – typically 3–5% bovine serum albumin (BSA) or non-fat dry milk (both are inexpensive) in tris-buffered saline (TBS) or I-Block, with a minute percentage (0.1%) of detergent such as Tween 20 or Triton X-100 . Although non-fat dry milk is preferred due to its availability, an appropriate blocking solution is needed as not all proteins in milk are compatible with all the detection bands. [ 1 ] The protein in the dilute solution attaches to the membrane in all places where the target proteins have not attached. Thus, when the antibody is added, it cannot bind to the membrane, and therefore the only available binding site is the specific target protein. This reduces background in the final product of the western blot, leading to clearer results, and eliminates false positives.
During the detection process, the membrane is "probed" for the protein of interest with a modified antibody which is linked to a reporter enzyme; when exposed to an appropriate substrate, this enzyme drives a colorimetric reaction and produces a colour. For a variety of reasons, this traditionally takes place in a two-step process, although there are now one-step detection methods available for certain applications.
The primary antibodies are generated when a host species or immune cell culture is exposed to the protein of interest (or a part thereof). Normally, this is part of the immune response, whereas here they are harvested and used as sensitive and specific detection tools that bind the protein directly.
After blocking, a solution of primary antibody (generally between 0.5 and 5 micrograms/mL) diluted in either PBS or TBST wash buffer is incubated with the membrane under gentle agitation for typically an hour at room temperature, or overnight at 4 ° C. It can also be incubated at different temperatures, with lesser temperatures being associated with more binding, both specific (to the target protein, the "signal") and non-specific ("noise"). Following incubation, the membrane is washed several times in wash buffer to remove unbound primary antibody, and thereby minimize background. [ 1 ] Typically, the wash buffer solution is composed of buffered saline solution with a small percentage of detergent, and sometimes with powdered milk or BSA.
After rinsing the membrane to remove unbound primary antibody, the membrane is exposed to another antibody known as the secondary antibody . Antibodies come from animal sources (or animal sourced hybridoma cultures). The secondary antibody recognises and binds to the species-specific portion of the primary antibody. Therefore, an anti-mouse secondary antibody will bind to almost any mouse-sourced primary antibody, and can be referred to as an 'anti-species' antibody (e.g. anti-mouse, anti-goat etc.). To allow detection of the target protein, the secondary antibody is commonly linked to biotin or a reporter enzyme such as alkaline phosphatase or horseradish peroxidase . This means that several secondary antibodies will bind to one primary antibody and enhance the signal, allowing the detection of proteins of a much lower concentration than would be visible by SDS-PAGE alone.
Horseradish peroxidase is commonly linked to secondary antibodies to allow the detection of the target protein by chemiluminescence . The chemiluminescent substrate is cleaved by horseradish peroxidase, resulting in the production of luminescence . Therefore, the production of luminescence is proportional to the amount of horseradish peroxidase-conjugated secondary antibody, and therefore, indirectly measures the presence of the target protein. A sensitive sheet of photographic film is placed against the membrane, and exposure to the light from the reaction creates an image of the antibodies bound to the blot. A cheaper but less sensitive approach utilizes a 4-chloronaphthol stain with 1% hydrogen peroxide ; the reaction of peroxide radicals with 4-chloronaphthol produces a dark purple stain that can be photographed without using specialized photographic film.
As with the ELISPOT and ELISA procedures, the enzyme can be provided with a substrate molecule that will be converted by the enzyme to a coloured reaction product that will be visible on the membrane (see the figure below with blue bands).
Another method of secondary antibody detection utilizes a near-infrared fluorophore-linked antibody. The light produced from the excitation of a fluorescent dye is static, making fluorescent detection a more precise and accurate measure of the difference in the signal produced by labeled antibodies bound to proteins on a western blot. Proteins can be accurately quantified because the signal generated by the different amounts of proteins on the membranes is measured in a static state, as compared to chemiluminescence, in which light is measured in a dynamic state. [ 28 ]
A third alternative is to use a radioactive label rather than an enzyme coupled to the secondary antibody, such as labeling an antibody-binding protein like Staphylococcus Protein A or Streptavidin with a radioactive isotope of iodine. Since other methods are safer, quicker, and cheaper, this method is now rarely used; however, an advantage of this approach is the sensitivity of auto-radiography-based imaging, which enables highly accurate protein quantification when combined with optical software (e.g. Optiquant).
Historically, the probing process was performed in two steps because of the relative ease of producing primary and secondary antibodies in separate processes. This gives researchers and corporations huge advantages in terms of flexibility, reduction of cost, and adds an amplification step to the detection process. Given the advent of high-throughput protein analysis and lower limits of detection, however, there has been interest in developing one-step probing systems that would allow the process to occur faster and with fewer consumables. This requires a probe antibody which both recognizes the protein of interest and contains a detectable label, probes which are often available for known protein tags . The primary probe is incubated with the membrane in a manner similar to that for the primary antibody in a two-step process, and then is ready for direct detection after a series of wash steps.
After the unbound probes are washed away, the western blot is ready for detection of the probes that are labeled and bound to the protein of interest. In practical terms, not all westerns reveal protein only at one band in a membrane. Size approximations are taken by comparing the stained bands to that of the marker or ladder loaded during electrophoresis. The process is commonly repeated for a structural protein, such as actin or tubulin , that should not change between samples. The amount of target protein is normalized to the structural protein to control between groups. A superior strategy is the normalization to the total protein visualized with trichloroethanol [ 29 ] [ 30 ] or epicocconone . [ 31 ] This practice ensures correction for the amount of total protein on the membrane in case of errors or incomplete transfers. (see western blot normalization )
The colorimetric detection method depends on incubation of the western blot with a substrate that reacts with the reporter enzyme (such as peroxidase ) that is bound to the secondary antibody. This converts the soluble dye into an insoluble form of a different colour that precipitates next to the enzyme and thereby stains the membrane. Development of the blot is then stopped by washing away the soluble dye. Protein levels are evaluated through densitometry (how intense the stain is) or spectrophotometry .
Chemiluminescent detection methods depend on incubation of the western blot with a substrate that will luminesce when exposed to the reporter on the secondary antibody. The light is then detected by CCD cameras which capture a digital image of the western blot or photographic film. The use of film for western blot detection is slowly disappearing because of non linearity of the image (non accurate quantification). The image is analysed by densitometry, which evaluates the relative amount of protein staining and quantifies the results in terms of optical density. Newer software allows further data analysis such as molecular weight analysis if appropriate standards are used.
Radioactive labels do not require enzyme substrates, but rather, allow the placement of medical X-ray film directly against the western blot, which develops as it is exposed to the label and creates dark regions which correspond to the protein bands of interest (see image above). The importance of radioactive detections methods is declining due to its hazardous radiation [ citation needed ] , because it is very expensive, health and safety risks are high, and ECL (enhanced chemiluminescence) provides a useful alternative.
The fluorescently labeled probe is excited by light and the emission of the excitation is then detected by a photosensor such as a CCD camera equipped with appropriate emission filters which captures a digital image of the western blot and allows further data analysis such as molecular weight analysis and a quantitative western blot analysis. Fluorescence is considered to be one of the best methods for quantification but is less sensitive than chemiluminescence. [ 32 ]
One major difference between nitrocellulose and PVDF membranes relates to the ability of each to support "stripping" antibodies off and reusing the membrane for subsequent antibody probes. While there are well-established protocols available for stripping nitrocellulose membranes, the sturdier PVDF allows for easier stripping, and for more reuse before background noise limits experiments. Another difference is that, unlike nitrocellulose, PVDF must be soaked in 95% ethanol, isopropanol or methanol before use. PVDF membranes also tend to be thicker and more resistant to damage during use. [ 33 ]
In order to ensure that the results of Western blots are reproducible, it is important to report the various parameters mentioned above, including specimen preparation, the concentration of protein used for loading, the percentage of gel and running condition, various transfer methods, attempting to block conditions, the concentration of antibodies, and identification and quantitative determination methods. Many of the articles that have been published don't cover all of these variables. Hence, it is crucial to describe different experimental circumstances or parameters in order to increase the repeatability and precision of WB. To increase WB repeatability, a minimum reporting criteria is thus required. [ 2 ] [ 34 ]
Two-dimensional SDS-PAGE uses the principles and techniques outlined above. 2-D SDS-PAGE, as the name suggests, involves the migration of polypeptides in 2 dimensions. For example, in the first dimension, polypeptides are separated according to isoelectric point , while in the second dimension, polypeptides are separated according to their molecular weight . The isoelectric point of a given protein is determined by the relative number of positively (e.g. lysine, arginine) and negatively (e.g. glutamate, aspartate) charged amino acids, with negatively charged amino acids contributing to a low isoelectric point and positively charged amino acids contributing to a high isoelectric point. Samples could also be separated first under nonreducing conditions using SDS-PAGE, and under reducing conditions in the second dimension, which breaks apart disulfide bonds that hold subunits together. SDS-PAGE might also be coupled with urea-PAGE for a 2-dimensional gel.
In principle, this method allows for the separation of all cellular proteins on a single large gel. A major advantage of this method is that it often distinguishes between different isoforms of a particular protein – e.g. a protein that has been phosphorylated (by addition of a negatively charged group). Proteins that have been separated can be cut out of the gel and then analysed by mass spectrometry , which identifies their molecular weight.
There may be a weak or absent signal in the band for a number of reasons related to the amount of antibody and antigen used. This problem might be resolved by using the ideal antigen and antibody concentrations and dilutions specified in the supplier's data sheet. Increasing the exposition period in the detection system's software can address weak bands caused by lower sample and antibody concentrations. [ 2 ]
When the protein is broken down by proteases, several bands other than predicted bands of low molecular weight might appear. The development of numerous bands can be prevented by properly preparing protein samples with enough protease inhibitors. Multiple bands might show up in the high molecular weight region because some proteins form dimers, trimers, and multimers; this issue might be solved by heating the sample for longer periods of time. Proteins with post-translational modifications (PTMs) or numerous isoforms cause several bands to appear at various molecular weight areas. PTMs can be removed from a specimen using specific chemicals, which also remove extra bands. [ 2 ]
Strong antibody concentrations, inadequate blocking, inadequate washing, and excessive exposure time during imaging can result in a high background in the blots. A high background in the blots could be avoided by fixing these issues. [ 2 ]
It has been claimed that a variety of odd and unequal bands, including black dots, white spots or bands, and curving bands, have occurred. The block dots are removed from the blots by effective blocking. White patches develop as a result of bubbles between the membrane and gel. White bands appear in the blots when main and secondary antibodies are present in significant concentrations. Because of the high voltage used during the gel run and the rapid protein migration, smiley bands appear in the blots. The strange bands in the blot are resolved by resolving these problems. [ 2 ]
During the western blotting, there could be several problems related to the different steps of this procedure. Those problems could originate from a protein analysis step such as the detection of low- or post-translationally modified proteins. Additionally, they can be based on the selection of antibodies since the quality of the antibodies plays a significant role in the detection of proteins specifically. [ 3 ] On account of the presence of these kinds of problems, a variety of improvements are being produced in the fields of preparation of cell lysate and blotting procedures to build up reliable results. Moreover, to achieve more sensitive analysis and overcome the problems associated with western blotting, several different techniques have been developed and utilized, such as far-western blotting , diffusion blotting, single-cell resolution western blotting , and automated microfluidic western blotting . [ 3 ]
Researchers use different software to process and align image-sections for elegant presentation of western blot results. Popular tools include Sciugo , Microsoft PowerPoint , Adobe Illustrator and GIMP .
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Normalization of Western blot data is an analytical step that is performed to compare the relative abundance of a specific protein across the lanes of a blot or gel under diverse experimental treatments, or across tissues or developmental stages. [ 1 ] [ 2 ] The overall goal of normalization is to minimize effects arising from variations in experimental errors, such as inconsistent sample preparation, unequal sample loading across gel lanes, or uneven protein transfer, which can compromise the conclusions that can be obtained from Western blot data. [ 1 ] Currently, there are two methods for normalizing Western blot data: (i) housekeeping protein normalization and (ii) total protein normalization. [ 1 ] [ 2 ] [ 3 ] [ 4 ]
Normalization occurs directly on either the gel or the blotting membrane. First, the stained gel or blot is imaged, a rectangle is drawn around the target protein in each lane, and the signal intensity inside the rectangle is measured. [ 1 ] The signal intensity obtained can then be normalized with respect to the signal intensity of the loading internal control detected on the same gel or blot. [ 1 ] When using protein stains, the membrane may be incubated with the chosen stain before or after immunodetection, depending on the type of stain. [ 5 ]
Housekeeping genes and proteins, including β-Actin , GAPDH , HPRT1 , and RPLP1 , are often used as internal controls in western blots because they are thought to be expressed constitutively, at the same levels, across experiments. [ 1 ] [ 2 ] [ 6 ] [ 7 ] However, recent studies have shown that expression of housekeeping proteins (HKPs) can change across different cell types and biological conditions. [ 1 ] [ 8 ] [ 9 ] [ 10 ] Therefore, scientific publishers and funding agencies now require that normalization controls be previously validated for each experiment to ensure reproducibility and accuracy of the results. [ 8 ] [ 9 ] [ 10 ]
When using fluorescent antibodies to image proteins in western blots, normalization requires that the user define the upper and lower limits of quantitation and characterize the linear relationship between signal intensity and the sample mass volume for each antigen. [ 1 ] Both the target protein and the normalization control need to fluoresce within the dynamic range of detection. [ 1 ] Many HKPs are expressed at high levels and are preferred for use with highly-expressed target proteins. [ 1 ] Lower expressing proteins are difficult to detect on the same blot. [ 1 ]
Fluorescent antibodies are commercially available, and fully characterized antibodies are recommended to ensure consistency of results. [ 11 ] [ 12 ] [ 13 ]
When fluorescent detection is not utilized, the loading control protein and the protein of interest must differ considerably in molecular weight so they are adequately separated by gel electrophoresis for accurate analysis. [ 1 ]
Membranes need to be stripped and re-probed using a new set of detection antibodies when detecting multiple protein targets on the same blot. [ 6 ] Ineffective stripping could result in a weak signal from the target protein. [ 6 ] To prevent loss of the antigen, only three stripping incubations are recommended per membrane. [ 6 ] It could be difficult to completely eliminate signal from highly-abundant proteins, so it is recommended that one detects lowly-expressed proteins first. [ 6 ]
Since HKP levels can be inconsistent between tissues, scientists can control for the protein of interest by spiking in a pure, exogenous protein of a known concentration within the linear range of the antibody. [ 8 ] [ 9 ] [ 10 ] Compared to HKP, a wider variety of proteins are available for spike-in controls. [ 14 ]
In total protein normalization (TPN), the abundance of the target protein is normalized to the total amount of protein in each lane. [ 3 ] [ 4 ] Because TPN is not dependent on a single loading control, validation of controls and stripping/reprobing of blots for detection of HKPs is not necessary. [ 6 ] [ 15 ] This can improve precision (down to 0.1 μg of total protein per lane), cost-effectiveness, and data reliability. [ 16 ]
Fluorescent stains and stain-free gels require special equipment to visualize the proteins on the gel/blot. [ 5 ] Stains may not cover the blot evenly; more stain might collect towards the edges of the blot than in the center. Non-uniformity in the image can result in inaccurate normalization. [ 1 ]
Anionic dyes such as Ponceau S and Coomassie brilliant blue , and fluorescent dyes like Sypro Ruby and Deep Purple, are used before antibodies are added because they do not affect downstream immunodetection. [ 17 ] [ 18 ] [ 19 ] [ 20 ]
Ponceau S is a negatively charged reversible dye that stains proteins a reddish pink color and is removed easily by washing in water. [ 21 ] [ 22 ] The intensity of Ponceau S staining decreases quickly over time, so documentation should be conducted rapidly. [ 5 ] A linear range of up to 140 μg is reported for Ponceau S with poor reproducibility due to its highly time-dependent staining intensity and low signal-to-noise ratio. [ 21 ] [ 22 ]
Fluorescent dyes like Sypro Ruby have a broad linear range and are more sensitive than anionic dyes. [ 22 ] They are permanent, photostable stains that can be visualized with a standard UV or blue-light transilluminator or a laser scan. [ 1 ] [ 22 ] Membranes can then be documented either on film or digitally using a charge-coupled device camera . [ 23 ] Sypro Ruby blot staining is time-intensive and tends to saturate above 50 μg of protein per lane. [ 22 ]
Amido black is a commonly used permanent post-antibody anionic stain that is more sensitive than Ponceau S. [ 24 ] This stain is applied after immunodetection. [ 24 ]
Stain-free technology employs an in-gel chemistry for imaging. [ 22 ] [ 25 ] [ 26 ] This chemical reaction does not affect protein transfer or downstream antibody binding. [ 27 ] Also, it does not involve staining/destaining steps, and the intensity of the bands remain constant over time. [ 28 ]
Stain-free technology cannot detect proteins that do not contain tryptophan residues. A minimum of two tryptophans is needed to enable detection. [ 5 ] The linear range for stain-free normalization is up to 80 μg of protein per lane for 18-well and up to 100 μg per lane for 12-well Criterion mid-sized gels. This range is compatible with typical protein loads in quantitative western blots and enables loading control calculations over a wide protein-loading range. [ 29 ] [ 4 ] A more efficient stain-free method has also recently become available. [ 30 ] [ 31 ] When using high protein loads, stain-free technology has demonstrated greater success than stains. [ 29 ]
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Microtus pennsylvanicus drummondii
The western meadow vole ( Microtus drummondii ) is a species of North American vole found in western North America , the midwestern United States , western Ontario, Canada , and formerly in Mexico . It was previously considered conspecific with the eastern meadow vole ( M. pennsylvanicus ), but genetic studies indicate that it is a distinct species. [ 1 ] [ 2 ] It is sometimes called the field mouse or meadow mouse , although these common names can also refer to other species.
It ranges from Ontario west to Alaska , and south to Missouri , north-central Nebraska , the northern half of Wyoming , and central Washington south through Idaho into north-central Utah . A disjunct subset of its range occurs from central Colorado to northwestern New Mexico . [ 3 ] An isolated population was formerly found in Chihuahua , Mexico, but has since been extirpated . [ 4 ] The United States portion of the Souris River is alternately known as the Mouse River because of the large numbers of field mice that lived along its banks.
In eastern Washington and northern Idaho, meadow voles are found in relative abundance in sedge ( Carex sp.) fens, but not in adjacent cedar ( Thuja sp.)- hemlock ( Tsuga sp.), Douglas-fir ( Pseudotsuga menziesii ), or ponderosa pine ( Pinus ponderosa ) forests. Meadow voles are also absent from fescue ( Festuca sp.)- snowberry ( Symphoricarpos sp.) associations. Moisture may be a major factor in habitat use; possibly the presence of free water is a deciding factor. [ 5 ] [ 6 ]
In southeastern Montana, western meadow voles were the second-most abundant small mammal (after deer mice, Peromyscus maniculatus ) in riparian areas within big sagebrush ( Artemisia tridentata )- buffalo grass ( Bouteloua dactyloides ) habitats. [ 5 ] Western meadow voles are listed as riparian-dependent vertebrates in the Snake River drainage of Wyoming. In a compilation of 11 studies [ 6 ] on small mammals, western meadow voles were reported in only three of 29 sites in subalpine forests of the central Rocky Mountains. Their range extensions were likely to be related to irrigation practices. [ 7 ] They are now common in hayfields, pastures, and along ditches in the Rocky Mountain states. [ 8 ] In Pipestone National Monument, Minnesota, western meadow voles were present in riparian shrublands, tallgrass prairie, and other habitats. [ 9 ]
In an Iowa prairie restoration project, meadow voles experienced an initial population increase during the initial stage of vegetation succession (old field dominated by foxtail grass ( Setaria spp.), red clover ( Trifolium pratense ), annual ragweed ( Ambrosia artemisiifolia ), alfalfa ( Medicago sativa ), and thistles ( Cirsium spp.). However, populations reached their peak abundance during the perennial grass stage of succession from old field to tallgrass prairie. [ 10 ] Meadow vole habitat devoid of tree cover and grasses dominated the herb layer. [ 11 ] with low tolerance for habitat variation (i. e., a species that is intolerant of variations in habitat, is restricted to few habitats, and/or uses habitats less evenly than tolerant species). [ 11 ]
In most areas, meadow voles clearly prefer habitat with dense vegetation. In tallgrass prairie at Pipestone National Monument, they were positively associated with dense vegetation and litter. [ 12 ] The variables important to meadow vole habitat in Virginia include vegetative cover reaching a height of 8 to 16 inches (20–41 cm) and presence of litter. [ 13 ] Meadow voles appeared to be randomly distributed within a grassland habitat in southern Quebec. [ 14 ] Grant and Morris [ 14 ] were not able to establish any association of meadow vole abundance with particular plant species. They were also unable to distinguish between food and cover as the determining factor in meadow vole association with dense vegetation. In South Dakota, meadow voles prefer grasslands to Rocky Mountain juniper ( Juniperus scopulorum ) woodlands. [ 15 ] In New Mexico, meadow voles were captured in stands of grasses, wild rose ( Rosa sp.), prickly pear ( Opuntia sp.), and various forbs; meadow voles were also captured in wet areas with tall marsh grasses. [ 16 ]
Open habitat with a thick mat of perennial grass favors voles. [ 17 ] In west-central Illinois, they were the most common small mammals on Indian grass ( Sorghastrum nutans )-dominated and switchgrass ( Panicum virgatum )-dominated study plots. They were present in very low numbers on orchard grass ( Dactylis glomerata )-dominated plots. The most stable population occurred on unburned big bluestem ( Andropogon gerardii )-dominated plots. [ 18 ] In Ontario, meadow voles and white-footed mice ( Peromyscus leucopus ) occur together in ecotones . Meadow voles were the most common small mammals in oak savanna/tallgrass prairie dominated by northern pin oak ( Quercus ellipsoidalis ) and grasses including bluejoint reedgrass ( Calamagrostis canadensis ), prairie cordgrass ( Sporobolus michauxianus ), big bluestem, switchgrass, and Indian grass.
In Michigan, strip clearcuts in a conifer swamp resulted in an increase in the relative abundance of meadow voles. They were most abundant in clearcut strip interiors and least abundant in uncut strip interiors. Slash burning did not appear to affect meadow vole numbers about 1.5 years after treatment. [ 19 ]
Birds not usually considered predators of mice do take voles; examples include gulls ( Larus sp.), northern shrikes ( Larius borealis ), black-billed magpies ( Pica hudsonica ), common ravens ( Corvus corax ), American crows ( C. brachyrhynchos ), great blue herons ( Ardea herodias ), and American bitterns ( Botaurus lentiginosus ). [ 20 ]
Major mammalian predators include the badger ( Taxidea taxus ), striped skunk ( Mephitis mephitis ), weasels ( Mustela and Neogale sp.), martens ( Martes americana and M. caurina ), domestic dogs ( Canis familiaris ), domestic cats ( Felis catus ) and mountain lions ( Puma concolor ). Other animals reported to have ingested voles include trout ( Salmo sp.), Pacific giant salamanders ( Dicampton ensatus ), garter snakes ( Thamnophis sp.), yellow-bellied racers ( Coluber constrictor flaviventris ), gopher snakes ( Pituophis melanoleucas ), plains rattlesnakes ( Crotalus viridis ), and rubber boas ( Charina bottae ). [ 21 ]
In northern prairie wetlands, meadow voles are a large portion of the diets of red foxes ( Vulpes vulpes ), American mink ( Neogale vison ), short-eared owls ( Asio flammeus ), and northern harriers ( Circus cyaneus ). [ 22 ] Voles are frequently taken by racers ( Coluber sp.) since both often use the same burrows. [ 23 ]
In forest plantations in British Columbia, an apparently abundant (not measured) meadow vole population was associated with a high rate of "not sufficient regeneration"; damage to tree seedlings was attributed to meadow voles and lemmings ( Synaptomys sp.). [ 24 ]
The cycle of meadow vole abundance is an important proximate factor affecting the life histories of its major predators. Meadow voles are usually the most abundant small mammals in northern prairie wetlands, often exceeding 40% of all individual small mammals present. [ 25 ] Numbers of short-eared owls, northern harriers, rough-legged hawks ( Buteo lagopus ), coyotes ( Canis latrans ), and red foxes were directly related to large numbers of meadow voles in a field in Wisconsin. [ 26 ] Predator numbers are positively associated with meadow vole abundance. [ 27 ] [ 28 ]
The species depends heavily on mesic habitats, and in areas on the periphery of its range, which contain distinctive and divergent subspecies, populations may be lost if the wetness of the habitats changes. [ 2 ] A distinct Pleistocene relict subspecies, M. d . chihuahuensis , the Chihuahuan vole , was also found in Chihuahua, Mexico, but has not been recorded since 1988 after its habitat was degraded by recreational activities and especially overgrazing , and eventually the marsh was completely drained by the early 2000s. This subspecies displayed notable divergence from other populations and was highly isolated from any others, and would be considered a distinctive subspecies. [ 4 ] In addition, two other populations in New Mexico appear to have been extirpated in recent times, likely as a consequence of climate change -induced drying and overgrazing. Due to the heavy association between meadow voles and mesic habitats, they are especially at risk from drying trends in areas at the peripheries of their range, leaving many of these populations at heavy risk of extirpation. [ 2 ]
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The Westphalen–Lettré rearrangement is a classic organic reaction in organic chemistry describing a rearrangement reaction of cholestane-3β,5α,6β-triol diacetate with acetic anhydride and sulfuric acid . In this reaction one equivalent of water is lost, a double bond is formed at C10–C11 and importantly the methyl group at the C10 position migrates to the C5 position. [ 1 ] [ 2 ] [ 3 ]
The reaction is first-order in steroid in the presence of an excess of sulfuric acid [ 4 ] and the first reaction step in the reaction mechanism is likely the formation of a sulfate ester followed by that of a carbocation at C5 after which the actual re-arrangement takes place. [ citation needed ]
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Wet infrastructure is the spectrum of water-related projects relating to water supply, treatment and storage, water resource management, flood management, coastal restoration, hydropower and renewable energy facilities. [ 1 ] Common examples of wet infrastructure include new construction as well as renovations and maintenance of locks, weirs, storm-surge barriers, guiding structures, pumping plants, culverts , bridges, controlling systems, operating systems, and tunnel installations. [ 2 ]
This hydrology article is a stub . You can help Wikipedia by expanding it .
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A wet lab , or experimental lab , is a type of laboratory where it is necessary to handle various types of chemicals and potential "wet" hazards, so the room has to be carefully designed, constructed, and controlled to avoid spillage and contamination .
A dry lab might have large experimental equipment but minimal chemicals, or instruments for analyzing data produced elsewhere. [ 1 ]
A wet lab is a type of laboratory in which a wide range of experiments are performed, for example, characterizing of enzymes in biology , titration in chemistry , diffraction of light in physics , etc. - all of which may sometimes involve dealing with hazardous substances . [ 2 ] Due to the nature of these experiments, the proper appropriate arrangement of safety equipment are of great importance. [ 3 ]
The researchers (the occupants) are required to know basic laboratory techniques including safety procedures and techniques related to the experiments that they perform. [ 4 ]
At the present, lab design tends to focus on increasing the interactions between researchers through the use of open plans, allowing the space and opportunity for researchers to exchange ideas, share equipment, and share storage space; increasing productivity and efficiency of experiments. [ 5 ] This style of design has been proposed to support team-based work, though more compartmentalised or individual spaces are still important for some types of processes which require separate/isolated space such as electron microscopes , tissue cultures , work/workers that may be disturbed by noise levels, etc. [ 5 ]
Flexibility of laboratory design should also be promoted, for example, the wall and ceiling should be removable in case of expansion or contraction, the pipes, tubes and fume hoods should also be removable for future expansion, reallocation and change of use. A well thought-through design will ensure that a lab can be adjusted for any future use. [ 6 ] The sustainability of resources is also a concern, so the amount of resources and energy used in the lab should be reduced where possible to save the environment, but still yield the same products. [ 7 ]
As a laboratory consists of many areas such as wet lab, dry lab and office areas, wet labs should be separated from other spaces using controlling devices or dividers to prevent cross-contamination or spillage. [ 8 ]
Due to the nature of processes used in wet labs, the environmental conditions may need to be carefully considered and controlled using a cleanroom system.
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Wet nanotechnology (also known as wet nanotech ) involves working up to large masses from small ones. [ 1 ]
Wet nanotechnology requires water in which the process occurs. [ 1 ] The process also involves chemists and biologists trying to reach larger scales by putting together individual molecules. [ 1 ] While Eric Drexler put forth the idea of nano-assemblers working dry, wet nanotech appears to be the likely first area in which something like a nano-assembler may achieve economic results. [ 2 ] Pharmaceuticals and bioscience are central features of most nanotech start-ups. [ 2 ] Richard A.L. Jones calls nanotechnology that steals bits of natural nanotechnology and puts them in a synthetic structure biokleptic nanotechnology . [ 3 ] He calls building with synthetic materials according to nature's design principles biomimetic nanotechnology . [ 3 ]
Using these guiding principles could lead to trillions of nanotech robots, that resemble bacteria in structural properties, entering a person's blood stream to do medical treatments. [ 3 ]
Wet nanotechnology is an anticipated new sub-discipline of nanotech that is going to mostly be dominated by the different forms of wet engineering . The processes that will be used are going to take place in aqueous solutions and are very close to that of biotechnology manufacturing / bio-molecular manufacturing which is largely concerned with the production of biomolecules like proteins and DNA / RNA . [ 4 ] There is some overlap of Biotechnology and Wet nanotechnology because living things are inherently bottom-up engineered and any exploitation of this by biotechnologists means they dabble in bottom-up engineering (though mostly at the level of producing macromolecules like proteins and nucleic acids from there monomer units. Wet nanotech, however, seeks to analyse living things and their components as engineering systems and aims to understand them completely to have complete control of the behavior of the system and to derive principles and methods that can be applied more broadly to bottom up manufacturing, to manipulate matter on the atomic and molecular scales and to creating machines or devices at the nanometer and microscopic scales. Biotech is mostly about exploiting living systems in any way possible. Molecular Biology and related disciplines compare the mechanism of function of proteins in particular - and nucleic acids to a lesser extent - as like "molecular machines". In order for engineers to mimic these nanoscale machines in a way that they could be produced with some efficiency, they must look into bottom-up manufacturing . Bottom-up manufacturing deals with manipulating individual atoms during the manufacturing process, so that there is absolute control of their placement and interactions. [ 5 ]
Then from the atomic scale, nanomachines could be made and even be designed to self-replicate themselves as long as they are designed in an environment with copious amount of the needed materials. Because individual atoms are being manipulated in the process, bottom-up manufacturing is often referred to as “atom by atom” manufacturing. [ 5 ] If the manufacturing of nanomachines can be made more readily available through improved techniques, there could be a large economic and social impact. This would start with improvements in making microelectromechanical systems and then would allow for the creation of nanoscale biological sensors along with things that have not been thought of yet. [ 4 ] This is because “wet” nanotech is only in the beginning of its life. Scientists and engineers alike feel that biomimetics is a great way to start looking at creating nanoscale machines. [ 5 ] Humans have only had a few thousand years to try to learn about the mechanics of things at really small scales. However, nature has been working on perfecting the design and functionality of nanomachines for millions of years. This is why there are already nanomachines, such as ATP synthase , working in our bodies that have an unheard of 95% efficiency. [ 6 ]
Wet nanotechnology is a form of wet engineering as opposed to dry engineering. [ 4 ] There are different fields that deal with those two types of engineering. Biologists, from the point of view of nanotechnology, deal with wet engineering. They study processes that happen in life , and for the most part those processes take place in aqueous environments. Our bodies are made up mostly of water.
Electrical and mechanical engineers are on the other side of the line in dry engineering. [ 4 ] They are involved with processes and manufacturing that does not occur in aqueous environments.
For the most part, wet engineering deals with “soft” materials that allow for flexibility which is vital at the nanoscale in biological manufacturing. Dry engineers mostly handle things with rigid structures and parts. [ 5 ] These differences stem for the fact that the forces that the two types of engineers must deal with are very different. At a larger scale, most things are dominated by Newtonian physics . However, when one looks at the nanoscale, especially in biological matters, the dominating force is Brownian motion . [ 5 ]
Because nanotechnology in the new age is going to most likely deal with both dry and wet in conjunction with each other, there is going to have to be a change in the way society looks at engineering and manufacturing. People will have to be not only well educated in engineering but also in biology because the integration of the two is how there will be the largest improvements in nanotechnology. [ 4 ]
With the existence of natural nanomachines, “a complex precision microscopic-sized machine that fits the standard definition of a machine”, [ 6 ] such as ATP synthase and T4 bacteriophage , scientists and biologists know that they are capable of making similar types of machines at the same scale. [ 5 ] However, nature has had a long time to perfect the building and creation of these nanomachines and humankind has only just begun to look into them with greater interest.
This interest may have been sparked because of the existence of nanomachines such as ATP synthase ( adenosine triphosphate ), which is the “second in importance only to DNA”. [ 6 ] ATP is the main energy converter that our bodies contain and without it, life as we know it would not be able to flourish or even survive. [ 6 ]
Brownian motion is a random, constantly fluctuating force that acts on a body in environments that are at a microscale. [ 5 ] This force is one that mechanical engineers and physicists are not used to dealing with because, at the larger scale that humankind tends to think of things, this force is not one that needs to be taken into account. People think of gravity, inertia, and other physics based forces that act on us all the time, however at the nanoscale those forces are mostly “negligible”. [ 5 ]
In order for nanomachines to be recreated by humans, either there will need to be discoveries that allow us to understand how to “exploit” Brownian motion as nature does or find a way to work around it by using materials that are rigid enough to stand up to these forces. The way that nature has been able to exploit Brownian motion is through self-assembly . This force pushes and pulls all of the proteins and amino acids around in our bodies and sticks them together in all sorts of combinations. The combinations that do not work separate and continue with their random attachment however, the combinations that do work produce things like ATP synthase. [ 5 ] Through this process nature has been able to make a nanomachine that is 95% efficient, which is a feat that humans have not been able to accomplish yet. This is all because nature does not try to work around the forces; it uses them at its advantage.
Growing cells in culture to take advantage of their internal chemical synthesis machinery can be considered a form of nanotechnology but this machinery has also been manipulated outside of living cells. [ 7 ]
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https://en.wikipedia.org/wiki/Wet_nanotechnology
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Wet oxidation is a form of hydrothermal treatment. It is the oxidation of dissolved or suspended components in water using oxygen as the oxidizer . It is referred to as wet air oxidation (WAO) when air is used. The oxidation reactions occur in superheated water at a temperature above the normal boiling point of water (100 °C), but below the critical point (374 °C).
The system must be maintained under pressure to avoid excessive evaporation of water. This is done to control energy consumption due to the latent heat of vaporization . It is also done because liquid water is necessary for most of the oxidation reactions to occur. Compounds oxidize under wet oxidation conditions that would not oxidize under dry conditions at the same temperature and pressure.
Wet oxidation has been used commercially for around 60 years. It is used predominantly for treating wastewater. It is often referred to as Zimpro (from ZIMmerman PROcess), after Fred J. Zimmermann who commercialized it in the mid 20th century. [ 1 ]
Commercial systems typically use a bubble column reactor , where air is bubbled through a vertical column that is liquid full of the hot and pressurized wastewater. Fresh wastewater enters the bottom of the column and oxidized wastewater exits the top. The heat released during the oxidation is used to maintain the operating temperature.
WAO is a liquid phase reaction using dissolved oxygen in water to oxidize wastewater contaminants. The dissolved oxygen is typically supplied using pressurized air, but pure oxygen can also be used. The oxidation reaction generally occurs at moderate temperatures of 150°-320 °C and at pressures from 10 to 220 bar. The process converts organic contaminants to carbon dioxide, water, and biodegradable short chain organic acids. Inorganic constituents such as sulfides and cyanides are converted to non-reactive inorganic compounds.
In the WAO reaction, complex organic molecules, including biological refractory compounds, are broken into simpler organic compounds or to a complete mineralized state (CO 2 , NH 3 , Cl − , SO 4 −2 , PO 4 −3 ) . Simple organic compounds such as low molecular weight carboxylic acids and mineralized reaction products may be present in the WAO effluents. Because of this, the WAO effluent generally requires post treatment prior to discharge. WAO effluents are typically readily biodegradable and exhibit high values for BOD : COD ratios. Standard treatment techniques such as activated sludge biotreatment are typically used with WAO for complete treatment. [ 2 ]
Catalyst can be used in the WAO system to enhance treatment and achieve a higher COD destruction. Heterogeneous and homogenous catalysts have been used. Heterogeneous catalysts are based on precious metals deposited on a stable substrate. Homogenous catalysts are dissolved transition metals. Several processes, such as Ciba-Geigy, LOPROX, and ATHOS utilize a homogenous catalyst. [ 3 ] [ 4 ] Mixed metal catalysts, such a Ce/Mn, Co/Ce, Ag/Ce, have also been effective in improving the treatment achieved in a WAO system. [ 5 ]
A special type of wet oxidation process was the so-called "VerTech process" system. A system of this type operated in Apeldoorn , Netherlands between 1994 and 2004. The system was installed in a below-ground pressure vessel (also called a gravity pressure vessel or GPV). The pressure was supplied by feeding the material to a reactor with a depth of 1,200 metres (3,900 ft). The deep shaft reactor also served as a heat exchanger, so no pre-heating was required. The operating temperature was about 270 °C with a pressure of about 100 bars (1,500 psi). The installation was eventually shut down due to operational problems. [ 6 ] [ 7 ]
The majority of commercial wet oxidation systems are used to treat industrial wastewater, such as sulfide laden spent caustic streams from ethylene and LPG production as well as naphthenic and cresylic spent caustics from refinery applications.
(Pressure)
(2-10 bar)
(20-45 bar)
(45-100 bar)
Typical classification of WAO treatment systems. [ 8 ]
Low temperature WAO systems oxidize sulfides to thiosulfate and sulfate but high concentrations of thiosulfate are present in the treated effluent. The mid temperature systems fully oxidize sulfides to sulfate and mercaptans are oxidized to sulfonic acids. For sulfidic spent caustics, this results in a high chemical oxygen demand (COD) destruction (>90%). High temperature systems are used to oxidize organic compounds that are present in naphthenic and cresylic spent caustics.
Almost as many systems are also used for treating biosolids , in order to pasteurize and to decrease volume of material for disposal. The thermal conditioning occurs at temperatures of 210 – 240 °C. A 4% dry solid slurry can be processed in a WAO system where it is disinfected and the treated effluent can be dewatered to 55% dry solids using a filter press. [ 4 ]
Wet air oxidation has also been used to treat a variety of other industrial process waters and wastewaters which include:
· Hazardous Waste [ 9 ]
· Kinetic Hydrate Inhibitors (KHI) from produced water [ 10 ]
· Polyol ether/styrene monomer (POSM) wastewater [ 11 ]
· Ammonium sulfate crystallizer mother liquor [ 11 ]
· Pharmaceutical wastewater [ 11 ]
· Cyanide Wastewater [ 11 ]
· Powdered Activated Carbon regeneration [ 11 ]
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https://en.wikipedia.org/wiki/Wet_oxidation
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A retention basin, sometimes called a retention pond, wet detention basin , or storm water management pond (SWMP), is an artificial pond with vegetation around the perimeter and a permanent pool of water in its design. [ 1 ] [ 2 ] [ 3 ] It is used to manage stormwater runoff , for protection against flooding , for erosion control , and to serve as an artificial wetland and improve the water quality in adjacent bodies of water.
It is distinguished from a detention basin , sometimes called a "dry pond", which temporarily stores water after a storm, but eventually empties out at a controlled rate to a downstream water body. It also differs from an infiltration basin which is designed to direct stormwater to groundwater through permeable soils.
Wet ponds are frequently used for water quality improvement, groundwater recharge , flood protection, aesthetic improvement, or any combination of these. Sometimes they act as a replacement for the natural absorption of a forest or other natural process that was lost when an area is developed. As such, these structures are designed to blend into neighborhoods and viewed as an amenity. [ 4 ]
In urban areas, impervious surfaces (roofs, roads) reduce the time spent by rainfall before entering into the stormwater drainage system. If left unchecked, this will cause widespread flooding downstream. The function of a stormwater pond is to contain this surge and release it slowly. This slow release mitigates the size and intensity of storm-induced flooding on downstream receiving waters. Stormwater ponds also collect suspended sediments, which are often found in high concentrations in stormwater water due to upstream construction and sand applications to roadways.
Storm water is typically channeled to a retention basin through a system of street and/or parking lot storm drains , and a network of drain channels or underground pipes. The basins are designed to allow relatively large flows of water to enter, but discharges to receiving waters are limited by outlet structures that function only during very large storm events.
Retention ponds are often landscaped with a variety of grasses , shrubs , and/or aquatic plants to provide bank stability and aesthetic benefits. Vegetation also provides water quality benefits by removing soluble nutrients through uptake. [ 5 ] In some areas the ponds can attract nuisance types of wildlife like ducks or Canada geese , particularly where there is minimal landscaping and grasses are mowed. This reduces the ability of foxes , coyotes , and other predators to approach their prey unseen. Such predators tend to hide in the cattails and other tall, thick grass surrounding natural water features.
Proper depth of retention ponds is important for removal of pollutants and maintenance of fish populations. Urban fishing continues to be one of the fastest growing fishing segments as new suburban neighborhoods are built around these aquatic areas. [ citation needed ]
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https://en.wikipedia.org/wiki/Wet_pond
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Wet Processing Engineering is one of the major streams in Textile Engineering or Textile manufacturing which refers to the engineering of textile chemical processes and associated applied science . [ 1 ] The other three streams in textile engineering are yarn engineering, fabric engineering, and apparel engineering. The processes of this stream are involved or carried out in an aqueous stage. Hence, it is called a wet process which usually covers pre-treatment, dyeing , printing , and finishing . [ 2 ]
The wet process is usually done in the manufactured assembly of interlacing fibers, filaments and yarns, having a substantial surface (planar) area in relation to its thickness, and adequate mechanical strength giving it a cohesive structure. In other words, the wet process is done on manufactured fiber , yarn and fabric .
All of these stages require an aqueous medium which is created by water. A massive amount of water is required in these processes per day. It is estimated that, on an average, almost 50–100 liters of water is used to process only 1 kilogram of textile goods, depending on the process engineering and applications. [ 3 ] Water can be of various qualities and attributes. Not all water can be used in the textile processes; it must have some certain properties, quality, color and attributes of being used. This is the reason why water is a prime concern in wet processing engineering. [ 4 ]
Water consumption and discharge of wastewater are the two major concerns . The textile industry uses a large amount of water in its varied processes especially in wet operations such as pre-treatment, dyeing, and printing. Water is required as a solvent of various dyes and chemicals and it is used in washing or rinsing baths in different steps. Water consumption depends upon the application methods, processes, dyestuffs , equipment/machines and technology which may vary mill to mill and material composition . Longer processing sequences, processing of extra dark colors and reprocessing lead to extra water consumption. And process optimization and right first-time [ 5 ] production may save much water. [ 6 ]
Water hardness can be removed by the boiling process, liming process, sodalime process, base exchange process, or synthetic ion exchange process. Recently, some companies have started harvesting rainwater for use in wet processes as it is less likely to cause the problems associated with water hardness.
Textile mills including carpet manufacturers, generate wastewater from a wide variety of processes, including wool cleaning and finishing, yarn manufacturing and fabric finishing (such as bleaching , dyeing , resin treatment, waterproofing and retardant flameproofing ). Pollutants generated by textile mills include BOD, SS, oil and grease, sulfide, phenols, and chromium. Insecticide residues in fleeces are a particular problem in treating waters generated in wool processing. Animal fats may be present in the wastewater, which, if not contaminated, can be recovered for the production of tallow or further rendering.
Textile dyeing plants generate wastewater that contains synthetic (e.g., reactive dyes, acid dyes, basic dyes, disperse dyes, vat dyes, sulfur dyes, mordant dyes, direct dyes, ingrain dyes, solvent dyes, pigment dyes) and natural dyestuff, gum thickener (guar) and various wetting agents, pH buffers and dye retardants or accelerators. Following treatment with polymer-based flocculants and settling agents, typical monitoring parameters include BOD, COD, color (ADMI), sulfide, oil and grease, phenol, TSS and heavy metals (chromium, zinc , lead, copper).
Wet process engineering is the most significant division in textile preparation and processing. It is a major stream in textile engineering, which is under the section of textile chemical processing and applied science. Textile manufacturing covers everything from fiber to apparel; covering with yarn, fabric, fabric dyeing , printing, finishing, garments, or apparel manufacturing. There are many variable processes available at the spinning and fabric-forming stages coupled with the complexities of the finishing and coloration processes to the production of a wide range of products.
In the textile industry, wet process engineering plays a vital role in the area of pre-treatment, dyeing , printing , and finishing of both fabrics and apparel. Coloration in fiber stage or yarn stage is also included in the wet processing division.
All the processes of this stream are carried out in an aqueous state or aqueous medium. The main processes of this section include;
The process of singeing is carried out for the purpose of removing the loose hairy fibers protruding from the surface of the cloth, thereby giving it a smooth, even and clean looking face. Singeing is an essential process for the goods or textile material which will be subjected to mercerizing, dyeing and printing to obtain best results from these processes.
The fabric passes over the brushes to raise the fibers, then passes over a plate heated by gas flames. When done to fabrics containing cotton , this results in increased water affinity, better dyeing characteristics, improved reflection, no "frosty" appearance, a smoother surface, better clarity in printing, improved visibility of the fabric structure, less pilling and decreased contamination through the removal of fluff and lint.
Singeing machines can be of three types: plate singeing, roller singeing, or gas singeing. Gas singeing is widely used in the textile industry. In gas singeing, a flame comes into direct contact to the fabric and burn the protruding fiber. Here, flame height and fabric speed is the main concern to minimize the fabric damage.
Singeing is performed only in the woven fabric. But in case of knit fabric, similar process of singeing is known as bio-polishing where enzyme is used to remove the protruding fibers.
Desizing is the process of removing sizing materials from the fabric, which is applied in order to increase the strength of the yarn which can withstand with the friction of loom . Fabric which has not been desized is very stiff and causes difficulty in its treatment with a different solution in subsequent processes.
After singeing operation the sizing material is removed by making it water-soluble and washing it with warm water. Desizing can be done by either the hydrolytic method (rot steep, acid steep, enzymatic steep) or the oxidative method (chlorine, chloride , bromite , hydrogen peroxide )
Depending on the sizing materials that has been used, the cloth may be steeped in a dilute acid and then rinsed, or enzymes may be used to break down the sizing material. Enzymes are applied in the desizing process if starch is used as sizing materials. Carboxymethyl cellulose (CMC) and Poly vinyl alcohol (PVA) are often used as sizing materials.
Scouring is a chemical washing process carried out on cotton fabric to remove natural wax and non-fibrous impurities (e.g. The remains of seed fragments) from the fibers and any added soiling or dirt. Scouring is usually carried in iron vessels called kiers . The fabric is boiled in an alkali, which forms a soap with free fatty acids ( saponification ). A kier is usually enclosed, so the solution of sodium hydroxide can be boiled under pressure, excluding oxygen which would degrade the cellulose in the fiber. If the appropriate reagents are used, scouring will also remove size from the fabric although desizing often precedes scouring and is considered to be a separate process known as fabric preparation. Preparation and scouring are prerequisites to most of the other finishing processes. Even the most naturally white cotton fibers are yellowish at this stage, thus at the next process, bleaching, is required.
The three main processes involved in the scouring are saponification, emulsification and detergency.
The main chemical reagent used in the cotton scouring is sodium hydroxide, which converts saponifiable fats and oils into soaps, dissolves mineral matter and converts pectose and pectin into their soluble salts.
Another scouring chemical is a detergent which is an emulsifying agent and removes dust and dirt particles from the fabric.
Since damage can be caused to the cotton substrate by sodium hydroxide. Due to this, and in order to reduce the alkali content in the effluent, Bio-scouring is introduced in the scouring process in which biological agent is used, such as an enzyme.
Bleaching improves whiteness by removing natural coloration and remaining trace impurities from the cotton; the degree of bleaching necessary is determined by the required whiteness and absorbency. Cotton being a vegetable fiber will be bleached using an oxidizing agent , such as dilute sodium hypochlorite or dilute hydrogen peroxide. If the fabric is to be dyed a deep shade, then lower levels of bleaching are acceptable. However, for white bedsheets and medical applications, the highest levels of whiteness and absorbency are essential.
Reductive bleaching is also carried out, using sodium hydrosulphite . Fibers like polyamide , polyacrylics and polyacetates can be bleached using reductive bleaching technology.
After scouring and bleaching, optical brightening agents (OBA), are applied to make the textile material appear more white. These OBAs are available in different tints such as blue, violet and red.
Mercerization is a treatment for cotton fabric and thread that gives fabric or yarns a lustrous appearance and strengthens them. The process is applied to cellulosic materials like cotton or hemp. A further possibility is mercerizing during which the fabric is treated with a sodium hydroxide solution to cause swelling of the fibers. This results in improved luster, strength, and dye affinity. [ 7 ] Cotton is mercerized under tension, and all alkalis must be washed out before the tension is released or shrinkage will take place. Mercerizing can take place directly on grey cloth, or after bleaching.
Dyeing is the application of dyes or pigments on textile materials such as fibers , yarns , and fabrics with the goal of achieving color with desired color fastness . Dyeing is normally done in a special solution containing dyes and particular chemical material. Dye molecules are fixed to the fiber by absorption, diffusion, or bonding with temperature and time being key controlling factors. The bond between dye molecule and fiber may be strong or weak, depending on the dye used. Dyeing and printing are different applications; in printing, color is applied to a localized area with desired patterns. In dyeing, it is applied to the entire textile.
Solution dyeing, also known as dope or spun dyeing, is the process of adding pigments or insoluble dyes to the spinning solution before the solution is extruded through the spinneret. Only manufactured fibers can be solution dyed. It is used for difficult-to-dye fibers such as olefin fibers, and for dyeing fibers for end uses that require excellent colorfastness properties. Because the color pigments become a part of the fiber, solution dyed materials have excellent colorfastness to light, washing, crocking (rubbing), perspiration, and bleach. Dyeing at the solution stage is more expensive since the equipment has to be cleaned thoroughly each time a different color is produced. Thus, the variety of colors and shades produced are limited. In addition, it is difficult to stock the inventory for each color. Decisions regarding color have to be made very early in the manufacturing process. Thus, this stage of dyeing is usually not used for apparel fabrics. [ 8 ] [ 9 ]
Filament fibers that are produced using the wet spinning method can be dyed while the fibers are still in the coagulating bath. The dye penetration at this stage is high as the fibers are still soft. This method is known as gel dyeing.
Stock dyeing, top dyeing, and tow dyeing are used to dye fibers at various stages of the manufacturing process prior to the fibers being spun into yarns. The names refer to the stage at which the fiber is when it is dyed. All three are included under the broad category of fiber dyeing.
Stock dyeing is dyeing raw fibers, also called stock, before they are aligned, blended, and spun into yarns.
Top dyeing is dyeing worsted wool fibers after they have been combed to straighten and remove the short fibers. The wool fiber at this stage is known as top. Top dyeing is preferred for worsted wools as the dye does not have to be wasted on the short fibers that are removed during the combing process.
Tow dyeing is dyeing filament fibers before they are cut into short staple fibers. The filament fibers at this stage are known as tow.
The dye penetration is excellent in fiber dyeing, therefore the amount of dye used to dye at this stage is also higher. Fiber dyeing is comparatively more costly than yarn, fabric, and product dyeing. The decision regarding the selection of colors has to be made early in the manufacturing process. Fiber dyeing is typically used to dye wool and other fibers that are used to produce yarns with two or more colors. Fibers for tweeds and fabrics with a "heather" look are often fiber dyed.
Yarn dyeing adds color at the yarn stage. Skein, package, beam, and space dyeing methods are used to dye yarns.
In skein dyeing the yarns are loosely wound into hanks or skein and then dyed. The yarns have good dye penetration, but the process is slow and comparatively more expensive.
In package dyeing yarns that have been wound on perforated spools are dyed in a pressurized tank. The process is comparatively faster, but the dye uniformity may not be as good as that of skein dyed yarn.
In beam dyeing a perforated warp beam is used instead of the spools used in package dyeing.
Space dyeing is used to produce yarns with multiple colors.
In general, yarn dyeing provides adequate color absorption and penetration for most materials. Thick and highly twisted yarns may not have good dye penetration. This process is typically used when different colored yarns are used in the construction of fabrics (e.g. plaids, checks, iridescent fabrics).
Fabric dyeing, also known as piece dyeing, is dyeing fabric after it has been constructed. It is economical and the most common method of dyeing solid-colored fabrics. The decision regarding color can be made after the fabric has been manufactured. Thus, it is suitable for quick response orders. Dye penetration may not be good in thicker fabrics, so yarn dyeing is sometimes used to dye thick fabrics in solid colors. Various types of dyeing machines are used for piece dyeing. The selection of the equipment is based on factors such as dye and fabric characteristics, cost, and the intended end-use.
Union dyeing is "a method of dyeing a fabric containing two or more types of fibers or yarns to the same shade so as to achieve the appearance of a solid-colored fabric". [ 10 ] Fabrics can be dyed using a single or multiple step process. Union dyeing is used to dye solid colored blends and combination fabrics commonly used for apparel and home furnishings.
Cross dyeing is "a method of dyeing blend or combination fabrics to two or more shades by the use of dyes with different affinities for the different fibers". [ 10 ] The cross dyeing process can be used to create heather effects, and plaid, check, or striped fabrics. Cross dyed fabrics may be mistaken for fiber or yarn-dyed materials as the fabric is not a solid color, a characteristic considered typical of piece-dyed fabrics. [ 11 ] It is not possible to visually differentiate between cross-dyed fabrics and those dyed at the fiber or yarn stage. An example is cross dyeing blue worsted wool fabric with polyester pinstripes. When dyed, the wool yarns are dyed blue, whereas the polyester yarns remain white.
Cross dyeing is commonly used with piece or fabric dyed materials. However, the same concept is applicable to yarn and product dyeing. For example, silk fabric embroidered with white yarn can be embroidered prior to dyeing and product dyed when an order is placed.
Product dyeing, also known as garment dyeing, is the process of dyeing products such as hosiery , sweaters, and carpet after they have been produced. This stage of dyeing is suitable when all components dye the same shade (including threads). This method is used to dye sheer hosiery since it is knitted using tubular knitting machines and then stitched prior to dyeing. Tufted carpets, with the exception of carpets produced using solution dyed fibers, are often dyed after they have been tufted. This method is not suitable for apparel with many components such as lining, zippers, and sewing thread, as each component may dye differently. The exception is tinting jeans with pigments for a "vintage" look. In tinting, color is used, whereas in other treatments such as acid-wash and stone-wash, chemical or mechanical processes are used. After garment construction, these products are given the "faded" or "used" look by finishing methods as opposed to dyeing.
Dyeing at this stage is ideal for a quick response. Many t-shirts, sweaters, and other types of casual clothing are product dyed for maximum response to fashion's demand for certain popular colors. Thousands of garments are constructed from prepared-for-dye (PFD) fabric, and then dyed to colors that sell best.
Acid dyes are water-soluble anionic dyes that are applied to fibers such as silk, wool, nylon, and modified acrylic fibers using neutral to acid dye baths. Attachment to the fiber is attributed, at least partly, to salt formation between anionic groups in the dyes and cationic groups in the fiber. Acid dyes are not substantive to cellulosic fibers.
Basic dyes are water-soluble cationic dyes that are mainly applied to acrylic fibers but find some use for wool and silk. Usually acetic acid is added to the dyebath to help the uptake of the dye onto the fiber.
Direct or substantive dyeing is normally carried out in a neutral or slightly alkaline dyebath, at or near boiling point, with the addition of either sodium chloride, sodium sulfate or sodium carbonate. Direct dyes are used on cotton, paper, leather, wool, silk, and nylon.
Mordant dyes require a mordant, which improves the fastness of the dye against water, light and perspiration. The choice of mordant is very important as different mordants can change the final color significantly. Most natural dyes are mordant dyes and there is therefore a large literature base describing dyeing techniques. The most important mordant dyes are the synthetic mordant dyes, or chrome dyes, used for wool; these comprise some 30% of dyes used for wool and are especially useful for black and navy shades. The mordant, potassium dichromate , is applied as an after-treatment. Many mordants, particularly those in the heavy metal category, can be hazardous to health and extreme care must be taken in using them.
Vat dyes are essentially insoluble in water and incapable of dyeing fibers directly. However, reduction in alkaline liquor produces the water-soluble alkali metal salt of the dye, which, in this leuco form, has an affinity for the textile fiber. Subsequent oxidation reforms the original insoluble dye. The color of denim is due to indigo , the original vat dye.
Reactive dyes utilize a chromophore attached to a substituent that is capable of directly reacting with the fiber substrate. The covalent bonds that attach reactive dye to natural fibers make them among the most permanent of dyes. "Cold" reactive dyes, such as Procion MX, Cibacron F, and Drimarene K, are very easy to use because the dye can be applied at room temperature. Reactive dyes are by far the best choice for dyeing cotton and other cellulose fibers at home or in the art studio.
Disperse dyes were originally developed for the dyeing of cellulose acetate , and are water-insoluble. The dyes are finely ground in the presence of a dispersing agent and sold as a paste, or spray-dried and sold as a powder. Their main use is to dye polyester but they can also be used to dye nylon, cellulose triacetate, and acrylic fibers. In some cases, a dyeing temperature of 130 °C is required, and a pressurized dyebath is used. The very fine particle size gives a large surface area that aids dissolution to allow uptake by the fiber. The dyeing rate can be significantly influenced by the choice of dispersing agent used during the grinding.
Azoic dyeing is a technique in which an insoluble azo dye is produced directly onto or within the fiber. This is achieved by treating a fiber with both diazoic and coupling components. With suitable adjustment of dyebath conditions the two components react to produce the required insoluble azo dye. This technique of dyeing is unique, in that the final color is controlled by the choice of the diazoic and coupling components. This method of dyeing cotton is declining in importance due to the toxic nature of the chemicals used.
Sulfur dyes are two-part "developed" dyes used to dye cotton with dark colors. The initial bath imparts a yellow or pale chartreuse color, This is after–treated with a sulfur compound in place to produce the dark black we are familiar with in socks for instance. Sulfur Black 1 is the largest selling dye by volume.
Textile printing is referred as localized dyeing. It is the application of color in the form of a paste or ink to the surface of a fabric, in a predetermined pattern. Printing designs onto already dyed fabric is also possible. In properly printed fabrics, the color is bonded with the fiber, so as to resist washing and friction. Textile printing is related to dyeing but, whereas in dyeing proper the whole fabric is uniformly covered with one color, in printing one or more colors are applied to it in certain parts only, and in sharply defined patterns. In printing, wooden blocks, stencils , engraved plates, rollers, or silkscreens can be used to place colors on the fabric. Colorants used in printing contain dyes thickened to prevent the color from spreading by capillary attraction beyond the limits of the pattern or design.
Textile finishing is the term used for a series of processes to which all bleached, dyed, printed, and certain grey fabrics are subjected before they put on the market. The object of textile finishing is to render textile goods fit for their purpose or end-use and/or improve serviceability of the fabric.
Finishing on fabric is carried out for both aesthetic and functional purposes to improve the quality and look of a fabric. Fabric may receive considerable added value by applying one or more finishing processes. Finishing processes include
Calendering is an operation carried out on a fabric to improve its aesthetics. The fabric passes through a series of calender rollers by wrapping; the face in contact with a roller alternates from one roller to the next. An ordinary calender consists of a series of hard and soft (resilient) bowls (rollers) placed in a definite order. The soft roller may be compressed with either cotton or wool-paper, linen paper or flax paper. The hard metal bowl is either of chilled iron or cast iron or steel. The calender may consist of 3, 5, 6, 7 and 10 rollers. The sequence of the rollers is that no two hard rollers are in contact with each other. Pressure may be applied by compound levers and weights, or hydraulic pressure may be used as an alternative. The pressure and heat applied in calendering depend on the type of the finish required.
The purposes of calendering are to upgrade the fabric hand and to impart a smooth, silky touch to the fabric, to compress the fabric and reduce its thickness, to improve the opacity of the fabric, to reduce the air permeability of the fabric by changing its porosity, to impart different degree of luster of the fabric, and to reduce the yarn slippage.
An important and oldest textile finishing is brushing or raising. Using this process a wide variety of fabrics including blankets, flannelettes, and industrial fabrics can be produced. The process of raising consists of lifting from the body of the fabric a layer of fibers which stands out from the surface which is termed as "pile". The formation of the pile on a fabric results in a "lofty" handle and may also subdue the weave or pattern and color of the cloth. [ 12 ]
There are two types of raising machines; the Teasel machine and the Card-wire machine. The speed of the card-wire raising machine varies from 12-15 yards per minute, which is 20-30% higher than that of teasel-raising. That is why the card-wire raising machine is widely used.
Crease formation in woven or knitted fabric composed of cellulose during washing or folding is the main drawback of cotton fabrics. The molecular chains of the cotton fibers are attached to each other by weak hydrogen bonds. During washing or folding, the hydrogen bonds break easily, and after drying new hydrogen bonds form with the chains in their new position and the crease is stabilized. If crosslink between the polymer chains can be introduced by cross-linking chemicals, then it reinforces the cotton fibers and prevents the permanent displacement of the polymer chains when the fibers are stressed. It is therefore much more difficult for creases to form or for the fabric to shrink on washing.
Crease-resist finishing of cotton includes the following steps:
The catalyst allows the reaction to be carried out 130-180 degree temperature range usually employed in the textile industry and within the usual curing time (within 3 minutes, maximum).
Mainly three classes of catalysts are commonly used now a day.
The purpose of the additives is to offset or counterbalance partly or completely the adverse effect of the crosslinking agent. Thus softening and smoothing agents are applied not only to improve the handle but also to compensate as much as possible for losses in tear strength and abrasion resistance.
Every resin finish recipe contains surfactants as emulsifiers, wetting agents and stabilizers. these surface-active substances are necessary to ensure that the fabric is wet rapidly and thoroughly during padding and the components are stable in the liquor.
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https://en.wikipedia.org/wiki/Wet_process_engineering
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The term wet scrubber describes a variety of devices that remove pollutants from a furnace flue gas or from other gas streams. In a wet scrubber, the polluted gas stream is brought into contact with the scrubbing liquid, by spraying it with the liquid, by forcing it through a pool of liquid, or by some other contact method, so as to remove the pollutants.
Wet scrubbers capture relatively small dust particles with the wet scrubber's large liquid droplets. In most wet scrubbing systems, droplets produced are generally larger than 50 micrometres (in the 150 to 500 micrometres range). As a point of reference, human hair ranges in diameter from 50 to 100 micrometres. The size distribution of particles to be collected is source specific.
For example, particles produced by mechanical means (crushing or grinding) tend to be large (above 10 micrometres); whereas, particles produced from combustion or a chemical reaction will have a substantial portion of small (less than 5 micrometres) and submicrometre particles.
The most critical sized particles are those in the 0.1 to 0.5 micrometres range because they are the most difficult for wet scrubbers to collect.
The design of wet scrubbers or any air pollution control device depends on the industrial process conditions and the nature of the air pollutants involved. Inlet gas characteristics and dust properties (if particles are present) are of primary importance. Scrubbers can be designed to collect particulate matter and/or gaseous pollutants. The versatility of wet scrubbers allow them to be built in numerous configurations, all designed to provide good contact between the liquid and polluted gas stream.
Wet scrubbers remove dust particles by capturing them in liquid droplets. The droplets are then collected, the liquid dissolving or absorbing the pollutant gases. Any droplets that are in the scrubber inlet gas must be separated from the outlet gas stream by means of another device referred to as a mist eliminator or entrainment separator (these terms are interchangeable). Also, the resultant scrubbing liquid must be treated prior to any ultimate discharge or being reused in the plant.
A wet scrubber's ability to collect small particles is often directly proportional to the power input into the scrubber. Low energy devices such as spray towers are used to collect particles larger than 5 micrometers. To obtain high efficiency removal of 1 micrometer (or less) particles generally requires high-energy devices such as venturi scrubbers or augmented devices such as condensation scrubbers. Additionally, a properly designed and operated entrainment separator or mist eliminator is important to achieve high removal efficiencies. The greater the number of liquid droplets that are not captured by the mist eliminator, the higher the potential emission levels.
Wet scrubbers that remove gaseous pollutants are referred to as absorbers . Good gas-to-liquid contact is essential to obtain high removal efficiencies in absorbers. Various wet-scrubber designs are used to remove gaseous pollutants, with the packed tower and the plate tower being the most common.
If the gas stream contains both particulate matter and gases, wet scrubbers are generally the only single air pollution control device that can remove both pollutants. Wet scrubbers can achieve high removal efficiencies for either particles or gases and, in some instances, can achieve a high removal efficiency for both pollutants in the same system. However, in many cases, the best operating conditions for particles collection are the poorest for gas removal.
In general, obtaining high simultaneous gas and particulate removal efficiencies requires that one of them be easily collected (i.e., that the gases are very soluble in the liquid or that the particles are large and readily captured), or by the use of a scrubbing reagent such as lime or sodium hydroxide .
The "cleaned" gases are normally passed through a mist eliminator (demister pads) to remove water droplets from the gas stream. The dirty water from the scrubber system is either cleaned and discharged or recycled to the scrubber. Dust is removed from the scrubber in a clarification unit or a drag chain tank. In both systems solid material settles on the bottom of the tank. A drag chain conveyor system removes the sludge and deposits in into a dumpster or stockpile.
Droplets are produced by several methods:
These droplets collect particles by using one or more of several collection mechanisms such as impaction, direct interception, diffusion , electrostatic attraction , condensation , centrifugal force and gravity . However, impaction and diffusion are the main ones.
In a wet scrubbing system, dust particles will tend to follow the streamlines of the exhaust stream . However, when liquid droplets are introduced into the exhaust stream, particles cannot always follow these streamlines as they diverge around the droplet (Figure 1). The particle's mass causes it to break away from the streamlines and impact or hit the droplet.
Impaction increases as the diameter of the particle increases and as the relative velocity between the particle and droplets increases. As particles get larger they are less likely to follow the gas streamlines around droplets. Also, as particles move faster relative to the liquid droplet, there is a greater chance that the particle will hit a droplet. Impaction is the predominant collection mechanism for scrubbers having gas stream velocities greater than 0.3 m/s (1 ft/s) ( Perry 1973 ).
Most scrubbers operate with gas stream velocities well above 0.3 m/s. Therefore, at these velocities, particles having diameters greater than 1.0 μm are collected by this mechanism. Impaction also increases as the size of the liquid droplet decreases because the presence of more droplets within the vessel increases the probability that particles will impact on the droplets.
Very small particles (less than 0.1 μm in diameter) experience random movement in an exhaust stream. These particles are so tiny that they are bumped by gas molecules as they move in the exhaust stream. This bumping, or bombardment, causes them to first move one way and then another in a random manner, or to diffuse , through the gas. This irregular motion can cause the particles to collide with a droplet and be collected (Figure 2). Because of this, diffusion is the primary collection mechanism in wet scrubbers for particles smaller than 0.1 μm.
The rate of diffusion depends on the following:
For both impaction and diffusion, collection efficiency increases with an increase in relative velocity (liquid- or gas-pressure input) and a decrease in liquid-droplet size.
However, collection by diffusion increases as particle size decreases. This mechanism enables certain scrubbers to effectively remove the very tiny particles (less than 0.1 μm).
In the particle size range of approximately 0.1 to 1.0 μm, neither of these two collection mechanisms (impaction or diffusion) dominates. This relationship is illustrated in Figure 3.
In recent years, some scrubber manufacturers have utilized other collection mechanisms such as electrostatic attraction and condensation to enhance particle collection without increasing power consumption.
In electrostatic attraction , particles are captured by first inducing a charge on them. Then, the charged particles are either attracted to each other, forming larger, easier-to-collect particles, or they are collected on a surface.
Condensation of water vapor on particles promotes collection by adding mass to the particles. Other mechanisms such as gravity , centrifugal force , and direct interception slightly affect particle collection. [ 1 ]
For particulate control, wet scrubbers (also referred to as wet collectors) are evaluated against fabric filters and electrostatic precipitators (ESPs). Some advantages of wet scrubbers over these devices are as follows:
Some disadvantages of wet scrubbers include corrosion, the need for entrainment separation or mist removal to obtain high efficiencies and the need for treatment or reuse of spent liquid.
Wet scrubbers have been used in a variety of industries such as acid plants , fertilizer plants, steel mills , asphalt plants, and large power plants .
Some components that are specific to the wet scrubbing process include:
A system may include one or multiple of these components in addition to various supporting components such as:
A typical wet scrubbing process can be described as follows:
Wet scrubbers can be categorized by the manner in which the gas and liquid phases are brought into contact. Scrubbers are designed to use power, or energy, from the gas stream or the liquid stream, or some other method to bring the pollutant gas stream into contact with the liquid. These categories are given in the table below. [ 2 ]
There is a large variety of wet scrubbers; however, all have one of three basic configurations:
1. Gas-humidification - The gas-humidification process agglomerates fine particles, increasing the bulk, making collection easier.
2. Gas-liquid contact - This is one of the most important factors affecting collection efficiency. The particle and droplet come into contact by four primary mechanisms:
3. Gas-liquid separation - Regardless of the contact mechanism used, as much liquid and dust as possible must be removed. Once contact is made, dust particulates and water droplets combine to form agglomerates. As the agglomerates grow larger, they settle into a collector.
Since wet scrubbers vary greatly in complexity and method of operation, devising categories into which all of them neatly fit is extremely difficult. Scrubbers for particle collection are usually categorized by the gas-side pressure drop of the system. Gas-side pressure drop refers to the pressure difference, or pressure drop , that occurs as the exhaust gas is pushed or pulled through the scrubber, disregarding the pressure that would be used for pumping or spraying the liquid into the scrubber.
Spray-tower scrubber wet scrubbers may be categorized by pressure drop as follows:
However, most scrubbers operate over a wide range of pressure drops , depending on their specific application, thereby making this type of categorization difficult.
Due to the large number of commercial scrubbers available, it is not possible to describe each individual type here. However, the following sections provide examples of typical scrubbers in each category.
In the simple, gravity-spray-tower scrubber, liquid droplets formed by liquid atomized in spray nozzles fall through rising exhaust gases. Dirty water is drained at the bottom.
These scrubbers operated at pressure drops of 1 to 2 in. water gauge (¼ to ½ kPa) and are approximately 70% efficient on 10 μm particles. Their efficiency is poor below 10 μm. However, they are capable of treating relatively high dust concentrations without becoming plugged.
Wet cyclones use centrifugal force to spin the dust particles (similar to a cyclone), and throw the particulates upon the collector's wetted walls. Water introduced from the top to wet the cyclone walls carries these particles away. The wetted walls also prevent dust reentrainment.
Pressure drops for these collectors range from 2 to 8 in. water (½ to 2 kPa), and the collection efficiency is good for 5 μm particles and above.
Packed-bed scrubbers consist of beds of packing elements, such as coke, broken rock, rings, saddles, or other manufactured elements. The packing breaks down the liquid flow into a high-surface-area film so that the dusty gas streams passing through the bed achieve maximum contact with the liquid film and become deposited on the surfaces of the packing elements. These scrubbers have a good collection efficiency for respirable dust.
Three types of packed-bed scrubbers are:
Efficiency can be greatly increased by minimizing target size, i.e., using 0.003 in. (0.076 mm) diameter stainless steel wire and increasing gas velocity to more than 1,800 ft/min (9.14 m/s).
Venturi scrubbers consist of a venturi-shaped inlet and separator. The dust-laden gases venturi scrubber enter through the venturi and are accelerated to speeds between 12,000 and 36,000 ft/min (60.97-182.83 m/s). These high-gas velocities immediately atomize the coarse water spray, which is injected radially into the venturi throat, into fine droplets. High energy and extreme turbulence promote collision between water droplets and dust particulates in the throat. The agglomeration process between particle and droplet continues in the diverging section of the venturi. The large agglomerates formed in the venturi are then removed by an inertial separator.
Venturi scrubbers achieve very high collection efficiencies for respirable dust. Since efficiency of a venturi scrubber depends on pressure drop, some manufacturers supply a variable-throat venturi to maintain pressure drop with varying gas flows.
Another way to classify wet scrubbers is by their use - to primarily collect either particulates or gaseous pollutants. Again, this distinction is not always clear since scrubbers can often be used to remove both types of pollutants.
Corrosion can be a prime problem associated with chemical industry scrubbing systems. Fibre-reinforced plastic and dual keys are often used as most dependable materials of construction.
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Wet stacking is a condition in diesel engines in which unburned fuel passes on into the exhaust system . [ 1 ] The word "stacking" comes from the term "stack" for exhaust pipe or chimney stack . The oily exhaust pipe is therefore a "wet stack".
This condition can have several causes. The most common cause is idling the engine for long intervals, which does not generate enough heat in the cylinder for a complete burn. "Idling" may be running at full rated operating speed, but with very little load applied. Another is excessive fueling. That may be caused by weak or leaky injectors, fuel settings turned up too high or over fueling for the given rpms. Cold weather running or other causes that prevent the engine from reaching proper operating temperature can cause a buildup of fuel due to incomplete burn that can result in 'wet stacking'. [ 2 ] [ 3 ] In diesel generators , it is usually because the diesel engine is running at only a small percentage of its rated output. For efficient combustion, a diesel engine should not be run under at least 60 percent of its rated power output. [ 4 ]
Wet stacking is detectable by the presence of a black ooze around the exhaust manifold, piping and turbocharger, if fitted. It can be mistaken for lubricating oil in some cases, but it consists of the "heavy ends" of the diesel fuel which do not burn when combustion temperature is too low. The heavier, more oily components of diesel fuel contain more stored energy than a comparable quantity of gasoline, but diesel requires an adequate loading of the engine in order to keep combustion temperature high enough to make use of it. Often, one can hear a slight miss in the engine due to fuel buildup. When the engine is first placed under a load after long periods of idling and wet stacking, it may blow some black exhaust out as it burns that excess fuel off. Continuous black exhaust from the stack when under a constant load is also an indication that some of the fuel is not being burned. [ 5 ] Additionally, wet stacking can result in a build up of diesel fuel in the engine which does not combust due to the low temperature in the engine. This results in a reduced fuel economy. This fuel leaks through the cylinders and dilutes the engine oil. If not frequently changed, this diluted oil can lead to increased wear on the cylinder and premature engine failure. [ 4 ]
In order to avoid failures from either type of wetstacking, the conventional advice is that the engine should never be operated for extended periods of time at less than half-load. In order to assure that the engine operates at a load of at least 50%, artificial loads are sometimes applied... Wetstacking also occurs when solid carbon or nonvolatile liquids accumulate along the cylinder wall or in the piston ring grooves and inhibit expansion and sealing of the ring against the cylinder wall. The cylinder can become glazed which further contributes to poor charge air compression heating and poor combustion of fuel. The performance of the engine continues to deteriorate until charge air compression heating is no longer sufficient for the engine to operate.
It is already well known that when the diesel engine is operated for long periods at a partial load, a condition known as ‘wet-stacking’ occurs (Donaldson, 2005). This is mainly attributed to incomplete combustion of fuel when the engine runs at low operating temperature. This results in reduced fuel efficiency and, simultaneously, shortening of engine operating life, and the time interval between routine maintenance calls.
Operating the engine without an electrical load for prolonged periods of time (one hour or more) will eventually cause damage to the diesel engine (sometimes called wet stacking which is NOT covered by the engine manufacturer’s warranty) or damage to the rings, fuel oil entering the lubricating oil and excessive vibration. Breaking in the diesel engine can require 5 to 10 hours of operation. It is extremely important to have electrical loads of at least 25% to 50% of the rated capacity of the generator’s nameplate rating to have the valves properly seat.
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Wet storage stain , more commonly known as white rust or white corrosion , is a type of zinc corrosion . It is called wet storage stain because it occurs when a fresh zinc surface is stored in a wet environment with limited oxygen and carbon dioxide sources; the restriction in air is usually due to the items being stacked on one another or otherwise stored in close quarters. This type of corrosion does not usually occur to zinc surfaces that have had time to form their normal layers of corrosion protection. [ 1 ]
Wet storage stain is a white, crumbly, and porous substance that is a mixture of three chemical compounds : 2ZnCO 3 ·3Zn(OH) 2 , ZnO, and β-Zn(OH) 2 . Underneath the white coating is usually a dark gray surface. The corrosion product is very voluminous; it is approximately 100 times greater in volume than the zinc consumed. Because of this the corrosion is not usually detrimental to the usability of the item, unless the zinc surface is only a thin coating, such as zinc electroplating . [ 1 ] [ 2 ]
Wet storage stain only occurs in situations where there is a lack of oxygen or carbon dioxide, because it usually forms zinc oxide and zinc hydroxide in open air environments. These oxides are usually present on zinc surfaces, but do not protect them from wet storage stain because they are only loosely adherent to the surface and any moisture can attack the oxides from underneath. Also, chlorides and sulfates accelerate the formation of corrosion. [ 1 ] [ 3 ]
To stop the corrosion from continuing the object just needs to be aired out to remove any moisture and allow the normal layer of protection to form. Washing and a wire brush will remove most of the corrosion. For complete removal, 10% acetic acid or a mixture of polishing chalk and 20–40% NaOH can be used. Both require a thorough water rinsing afterward and do not restore lustrous surface finish if one was previously present. [ 3 ]
Wet storage stain can be prevented for a limited amount of time by coating in a light oil , chromate conversion coatings , or phosphate conversion coatings . A more permanent solution is to paint the surface. [ 3 ]
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The wet sulfuric acid process ( WSA process ) is a gas desulfurization process. After Danish company Haldor Topsoe introduced this technology in 1987, it has been recognized as a process for recovering sulfur from various process gases in the form of commercial quality sulfuric acid (H 2 SO 4 ) with the simultaneous production of high-pressure steam. The WSA process can be applied in all industries where sulfur removal presents an issue.
The wet catalysis process is used for processing sulfur-containing streams, such as: [ 1 ]
The energy released by the above-mentioned reactions is used for steam production. Approximately 2–3 tons of high-pressure steam are produced per ton of acid.
Industries where WSA process plants are installed:
The acid gas coming from a Rectisol -, Selexol -, amine gas treating or similar unit installed after the gasifier contains H 2 S, COS and hydrocarbons in addition to CO 2 . These gases were previously vented to the atmosphere , but now the acid gas requires purification in order not to affect the environment with SO 2 emission. The WSA process provides a high sulfur recovery and recovers heat for steam production. The heat recovery rate is high, and the cooling water consumption is low, which saves resources. [ 2 ]
Example 1:
Example 2:
A sulfur plant in China will be built in connection with an ammonia plant, producing 500 kilotons/year of ammonia for fertilizer production [ 3 ]
The WSA process can also be used for the production of sulfuric acid from sulfur burning or regeneration of the spent acid from e.g., alkylation plants. Wet catalysis processes differ from other contact sulfuric acid processes in that the feed gas contains excess moisture when it comes into contact with the catalyst. The sulfur trioxide formed by catalytic oxidation of the sulfur dioxide reacts instantly with the moisture to produce sulfuric acid in the vapor phase to an extent determined by the temperature . Liquid acid is subsequently formed by condensation of the sulfuric acid vapor and not by the absorption of the sulfur trioxide in concentrated sulfuric acid, as in contact processes based on dry gases.
The concentration of the product acid depends on the H 2 O:SO 3 ratio in the catalytically converted gases and on the condensation temperature. [ 4 ] [ 5 ]
The combustion gases are cooled to the converter inlet temperature of about 420–440 °C. Processing these wet gases in a conventional cold-gas contact process (DCDA) plant would necessitate cooling and drying of the gas to remove all moisture. Therefore, the WSA process is, in most cases, a more cost-efficient way of producing sulfuric acid.
About 80% to 85% of the world’s sulfur production is used to manufacture sulfuric acid. 50% of the world’s sulfuric acid production is used in fertilizer production, mainly to convert phosphates to water-soluble forms. according to the Fertilizer Manual published jointly by the United Nations Industrial Development Organization (UNIDO) and the International Fertilizer Development Center [ 6 ]
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A wet wing (also referred to as integral fuel tanks [ 1 ] ) is an aerospace engineering technique where an aircraft's wing structure is sealed and used as a fuel tank .
The use of wet wings has become common among civilian designs, from large transport aircraft, such as airliners , to small general aviation aircraft. Several military aircraft, such as airlifters and aerial refueling tankers , have incorporated the technique as well. [ 2 ] [ 3 ] A number of strike aircraft, such as the Grumman A-6 Intruder , have also been furnished with wet wings. [ 4 ] While it is technically feasible, studies have found it generally impractical to convert aircraft between wet wing and non-wet wing fuel storage. [ 5 ]
The wet wing offers several advantages. By eliminating the need for separate bladders, tanks, or other containers to house the fuel, weight savings are achieved, improving operational efficiency and performance. [ 6 ] In comparison with other methods, the wet wing maximises the structural volume available within the wings, while alternative approaches are less space-efficient. [ 7 ] There are benefits from a safety point of view, as fuel would be discharged externally in the event of a leak, rather than within a potentially populated section of the aircraft. The thickness of the wing is typically greater than that of an individual bladder or tank, a factor which decreases the likelihood of damage-related leaks, particularly in the event of a crash. [ 8 ]
A disadvantage of the wet wing is that every rivet, bolt, nut plate , hose and tube that penetrates the wing must be sealed to prevent fuel from leaking or seeping around these hardware components. This sealant must allow for expansion and contraction due to rapid temperature changes (such as when cold fuel is pumped into a warm wing tank) and must retain its sealing properties when submerged in fuel and when left dry for long periods of time. Because the tanks form an integral part of the structure, they cannot be removed without considerable disassembly of the overall aircraft; several access panels are also necessary to perform maintenance activities and permit inspections. [ 9 ]
Beyond the complications in the design and manufacture of the aircraft, a wet wing necessitates ongoing maintenance activities throughout its operating life. Commonly, the sealant will need to be replaced; the removal of old sealant (and the application of fresh) can be considerably difficult when working on a relatively small wing tank. Without appropriate maintenance, wet wings will commonly start leaking after a while, usually due to seal deterioration; however, resealing work may not be immediately successful and require multiple applications. [ 9 ] Improved methods of sealing have been devised, reportedly extending the interval between resealing. [ 10 ]
Notable accidents in which the wet wing design and its drawbacks were causative include Chalk's Ocean Airways Flight 101 and the 1961 Goldsboro B-52 crash . [ 11 ] [ 12 ] Multiple aircraft have also sustained considerable structural damage due to improper wet wing maintenance. [ 13 ] Multiple instances of manufacturing-related debris, posing a threat to aircraft safety, have been discovered on both civilian and military aircraft. [ 14 ] [ 15 ]
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The wetted perimeter is the perimeter of the cross sectional area that is "wet". [ 1 ] The length of line of the intersection of channel wetted surface with a cross sectional plane normal to the flow direction. The term wetted perimeter is common in civil engineering , environmental engineering , hydrology , geomorphology , and heat transfer applications; it is associated with the hydraulic diameter or hydraulic radius . Engineers commonly cite the cross sectional area of a river.
The wetted perimeter can be defined mathematically as
where l i is the length of each surface in contact with the aqueous body.
In open channel flow , the wetted perimeter is defined as the surface of the channel bottom and sides in direct contact with the aqueous body. Friction losses typically increase with an increasing wetted perimeter, resulting in a decrease in head . [ 1 ] In a practical experiment, one is able to measure the wetted perimeter with a tape measure weighted down to the river bed to get a more accurate measurement .
When a channel is much wider than it is deep, the wetted perimeter approximates the channel width. [ 1 ]
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In electrical and electronics engineering , wetting current is the minimum electric current needing to flow through a contact to break through the surface film resistance at a contact. [ 1 ] It is typically far below the contact's nominal maximum current rating. [ 2 ]
A thin film of oxidation, or an otherwise passivated layer, tends to form in most environments, particularly those with high humidity , and, along with surface roughness, contributes to the contact resistance at an interface. [ 3 ] Providing a sufficient amount of wetting current is a crucial step in designing circuits that use switches with low contact pressure . Failing to do this might result in switches remaining electrically "open" when pressed, due to contact oxidation. [ 4 ] [ 5 ]
In some low voltage applications, where switching current is below the manufacturer's wetting current specification, a capacitor discharge method may be employed by placing a small capacitor across the switch contacts to boost the current through contact surface upon contact closure. [ 4 ] [ 6 ]
A related term sealing current (aka wetting current or fritting current ) is widely used in the telecommunication industry describing a small constant DC current (typically 1-20 mA) in copper wire loops in order to avoid contact oxidation of contacts and splices . It is defined in ITU-T G.992.3 for "all digital mode ADSL" as a current flowing from the ATU-C ( ADSL Linecard ) via the phone lines to the ATU-R ( CPE ). [ 7 ] [ 8 ] Carbon brushes develop high resistance glaze when they're used without current flow for an extended period. A special circuit is utilized for turbines and generators to introduce current through the brushes into the shaft to prevent this contact fritting . [ 9 ]
Contact cleaner can be applied to the contact surfaces to inhibit the formation of resistive surface films and/or to ameliorate existing films. [ 10 ]
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Wetting solutions are liquids containing active chemical compounds that minimise the distance between two immiscible phases by lowering the surface tension to induce optimal spreading. The two phases, known as an interface , can be classified into five categories, namely, solid-solid, solid-liquid, solid-gas, liquid-liquid and liquid-gas. [ 1 ]
Although wetting solutions have a long history of acting as detergents for four thousand plus years, the fundamental chemical mechanism was not fully discovered until 1913 by the pioneer McBain. [ 2 ] [ 3 ] Since then, diverse studies have been conducted to reveal the underlying mechanism of micelle formation and working principle of wetting solutions, broadening the area of applications.
The addition of wetting solution to an aqueous droplet leads to the formation of a thin film due to its intrinsic spreading property. This property favours the formation of micelles which are specific chemical structures consisting of a cluster of surfactant molecules that has a hydrophobic core and a hydrophilic surface that can lower the surface tension between two different phases. [ 4 ]
In addition, wetting solutions can be further divided into four classes; non-ionic, anionic, cationic and zwitterionic. [ 5 ]
The spreading property may be examined by adding a drop of the liquid onto an oily surface. If the liquid is not a wetting solution, the droplet will remain intact. If the liquid is a wetting solution, the droplet will spread uniformly on the oily surface because the formation of the micelles lowers the surface tension of the liquid. [ 6 ]
Wetting solutions can be applied in pharmaceuticals , [ 7 ] cosmetics [ 8 ] and agriculture . [ 9 ] Albeit a number of practical uses of wetting solutions, the presence of wetting solution can be a hindrance to water purification in industrial membrane distillation . [ 10 ]
Wetting agent was used as soap for cleansing purposes for thousands of years. The oldest evidence of wetting solution went back to 2800 BC in ancient Babylon . [ 2 ] The earliest credible reference of soap is in the writings of Galen , the Greek physician, around 200 AD. [ 11 ] Over the following centuries, wetting solutions mainly functioned as detergents due to their wetting properties. Despite the extensive use of wetting solutions, the underlying chemical mechanism remained unknown until the emergence of McBain's proposed theory in 1913. Founded on his research on how the electrical conductivity of a solution of surfactant molecules changed with concentration , he raised the possibility of surfactant molecules in the form of self-assembled aggregates . [ 3 ] Not until Debye published his original hypothesis in 1949 did he described the reason of micelle formation and the existence of finite-shaped micelles. [ 12 ] [ 13 ] McBain's discovery sparked numerous studies by Hobbs, [ 14 ] Ooshika, [ 15 ] Reich [ 16 ] and Halsey [ 17 ] from 1950 to 1956. These scholars intended to correct some of the foundational theories of the description of an equilibrium system, as well as emphasising the role of surface energy which was overlooked in Debye's prototype. In 1976, the fundamental theory for understanding the mechanism of micelle formation was developed by Tanford's free energy model. [ 18 ] Apart from integrating all relevant physicochemical elements and explaining the growth of micells, he provided a comprehensive reasoning of why micelles are finite in terms of opposing interactional forces. [ 19 ] [ 20 ]
The chemical structure of wetting solution molecules consist of a hydrophilic head and a long hydrophobic tail. Its distinct amphiphilicity allows it to bury its hydrophilic head in an aqueous bulk phase and hydrophobic part in the organic bulk phase respectively. [ 6 ] Wetting solution molecules break the intermolecular forces between each molecule in the organic phase and each water molecule in the aqueous phase by displacement . [ 5 ] Due to the lowered attractive forces, the surface tension is reduced.
Upon adding more wetting solution, the elevated concentration of wetting solution molecules leads to a further decrease in surface tension and makes the molecules at the surfaces become more crowded. The molecules will be forced to remain in the aqueous phase when there are no more vacancies for them to stay on the surface. At this point, the surface tension is maximally lowered and is termed as the critical micelle concentration (CMC) . [ 21 ] The lower the CMC, the more efficient the wetting solution is in reducing surface tension. Any additional wetting solution molecules will undergo self-aggregation into several special structures called micelles . Micelles are spheres with a hydrophobic core formed by the non-polar tail of wetting solution molecules and are surrounded by a hydrophilic layer arising from the molecules’ polar heads. [ 4 ] Extra wetting solution molecules will be forced to form micelles instead of adhering to the surface, hence the surface tension remains constant. Due to the minimised surface tension, the droplet can now spread thoroughly and form a thin film on the surface. [ 4 ]
Generally, the wetting solution molecules consist of a hydrophilic head and a long hydrophobic tail. The hydrophobic region usually contains saturated or unsaturated hydrocarbon chains, heterocyclic rings or aromatic rings . [ 5 ] Despite the similar amphiphilic composition, the molecules can be divided into four classes with respect to the nature of the hydrophilic group, namely, non-ionic, anionic, cationic and zwitterionic.
The following table shows the composition, special features of the corresponding classes and common examples of various forms of the respective wetting solutions.
Generally, wetting solution is applied in pharmaceuticals , [ 7 ] cosmetics [ 8 ] and agriculture . [ 9 ] McBain’s research on maximising the application of wetting solutions have an important role in enabling a range of options to both manufacturers and consumers and improving product performance in the respective areas of application, such as modifying the stability of pharmaceuticals, delivery of drugs, effectiveness of cleansing products and water retention in soils.
Specific properties of different wetting solutions are able to alternate drug delivery which is beneficial in improving drug safety and patients' experiences . For example, solulan C-24, a non-ionic wetting solution, forms large bilayers of wetting solution molecules known as discosomes that have a lower risk of causing systemic adverse effects . [ 7 ] [ 25 ] Non-ionic wetting solutions are found to have a wider usage and are more efficient in reducing surface tension compared to ionic wetting solutions which have higher toxicity and CMC value in general. [ 7 ] To ensure the safety, efficacy and quality of the preparations, toxicity and interaction profiles of the choice of wetting solutions are carefully investigated. [ 7 ]
Suspension preparation is a liquid dosage form that contains insoluble solid drug particles. [ 7 ] The suspension preparation is ideal if solid particles that have become compacted together during storage can re-disperse throughout the liquid vehicle readily with gentle shaking for a period of time that is sufficient for measuring the required dosage. [ 26 ]
Solid particles have a natural tendency to aggregate and eventually cause caking due to the presence of air film coating. [ 7 ] A solution to this is using a wetting solution as the liquid vehicle for suspension preparation. [ 7 ] Wetting solution increases the dispersal ability of the solid particles by replacing the air film to increase steric hindrance and minimise interactions between solid particles and resulting in a decreased rate of aggregation. [ 7 ] [ 27 ]
Wetting solutions lowers the surface tension of topical ophthalmic solutions and induces instant spreading when applied onto the cornea by increasing the interaction between the two. [ 7 ] The instant spreading increases the amount of drug molecules that are exposed to the cornea for absorption and therefore a quicker onset of action . [ 7 ] The increased interaction allow the topical ophthalmic solutions to remain on the corneal surface for a longer period of time to maximise the amount of drug that can diffuse from the applied topical ophthalmic solution layer to the corneal epithelium through tear film, the protective layer of the cornea from the external environment. [ 7 ] [ 28 ]
Skin cleansing products including facial cleanser, body wash and shampoo consist of wetting solutions. [ 8 ] Wetting solutions allow efficient spreading and wetting of the surface of skin and scalp by reducing the surface tension between the hydrophobic sebum secreted by the sebaceous gland in our skin. [ 8 ] An efficient wetting solution penetrates the skin and clears any topical applications, body fluids including sebum secreted via openings of hair follicles , dead skin cells and microbes . [ 8 ]
Non-ionic wetting solutions have a low risk of causing skin irritation and are efficient in reducing surface tension between different ingredients, for example, fragrance and essential oils extracted from plants, in skin cleansing products to produce a consistent liquid formula. [ 8 ] However, non-ionic wetting solutions are of higher cost than the other types of wetting solutions hence are less favourable for commercial products. [ 8 ]
Cationic wetting solutions cause more severe skin irritation problems hence are not used in skin cleansing products. [ 8 ] They are used in hair conditioners that are only applied to the second half hair length and washed off after a short period of time. [ 8 ]
Anionic and amphoteric wetting solutions are often used as a mixture in body wash and shampoo. [ 8 ] The anionic wetting solutions formulated into skin cleansing products have often undergone chemical modification as they often contain sulphur which triggers skin irritation by causing collagen in skin cells to swell and sometimes cell death. [ 8 ] [ 29 ] Examples of modified anionic wetting solutions include ammonium laureth sulphate and modified sulfosuccinates , both reported to exhibit low skin irritation. [ 8 ] [ 29 ]
Wetting solutions are widely used in Agriculture to increase crop yield which is affected by the degree of infiltration and penetration of water, nutrients and chemicals such as fertilisers and pesticides . [ 9 ] [ 30 ] Wetting solutions reduce surface runoff of water and nutrients and enhance water infiltration in water repelling soil by reducing surface tension. [ 31 ] Wetting-solution-treated soil has shown to retain high water content and an even distribution of nutrients in the root zone that are in deep soil areas, benefiting crop yield and improving water efficiency . [ 31 ] Examples of wetting solutions used in agriculture are modified alkylated polyol , mixture of polyether polyol and glycol ether and mixture of poloxalene , 2-butoxyethanol . [ 30 ]
Membrane distillation is a water purification process that utilises a hydrophobic membrane with pores to separate water vapour from contaminants , for example, oil and unwanted chemicals. [ 10 ] The filtration efficiency and stability of the membrane can be diminished by wetting. [ 10 ] [ 32 ] Wetting of the hydrophobic membrane is resulted from the presence of wetting solution in sewage due to its increasing large variety of usage in different fields, for example, pharmaceuticals, cosmetics and agriculture. [ 10 ] A possible solution is to pretreat the sewage to remove wetting solutions, limiting the amount of wetting solution in contact with the membrane. [ 32 ] Other possible solutions to lengthen durability of the membrane include modification of the membrane material repellent to water and oil, air-backwashing and membrane surface geometry modification. [ 10 ] [ 32 ] [ 33 ] These solutions are costly and require further research and development to optimise the durability and efficiency of membrane distillation. [ 32 ]
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A wetting transition (Cassie–Wenzel transition) may occur during the process of wetting of a solid (or liquid) surface with a liquid. The transition corresponds to a certain change in contact angle , the macroscopic parameter characterizing wetting. [ 1 ] Various contact angles can co-exist on the same solid substrate. Wetting transitions may occur in a different way depending on whether the surface is flat or rough.
When a liquid drop is put onto a flat surface, two situations may result. If the contact angle is zero, the situation is referred to as complete wetting. If the contact angle is between 0 and 180°, the situation is called partial wetting. A wetting transition is a surface phase transition from partial wetting to complete wetting. [ 2 ]
The situation on rough surfaces is much more complicated. The main characteristic of the wetting properties of rough surfaces is the so-called apparent contact angle (APCA). It is well known that the APCA usually measured are different from those predicted by the Young equation . Two main hypotheses were proposed in order to explain this discrepancy, namely the Wenzel and Cassie wetting models. [ 3 ] [ 4 ] [ 5 ] According to the traditional Cassie model, air can remain trapped below the drop, forming "air pockets". Thus, the hydrophobicity of the surface is strengthened because the drop sits partially on air. On the other hand, according to the Wenzel model the roughness increases the area of a solid surface, which also geometrically modifies the wetting properties of this surface. [ 1 ] [ 3 ] [ 4 ] [ 5 ] Transition from Cassie to Wenzel regime is also called wetting transition. [ 6 ] [ 7 ] [ 8 ] Under certain external stimuli, such as pressure or vibration, the Cassie air trapping wetting state could be converted into the Wenzel state. [ 6 ] [ 9 ] [ 10 ] [ 11 ] Apart from external stimuli, intrinsic contact angle of the liquid (below or above 90 degree), liquid volatility, structure of cavities (reentrant or non-reentrant, connected or unconnected) are known to be important factors determining the rate of wetting transition. [ 12 ] Complex structures such as hierarchical structures, and concave structures are reported to prevent wetting transition. [ 13 ] It is well accepted that the Cassie air trapping wetting regime corresponds to a higher energetic state, and the Cassie–Wenzel transition is irreversible. [ 14 ] However, the mechanism of the transition remains unclear. It was suggested that the Cassie–Wenzel transition occurs via a nucleation mechanism starting from the drop center. [ 15 ] On the other hand, recent experiments showed that the Cassie–Wenzel transition is more likely to be due to the displacement of a triple line under an external stimulus. [ 9 ] [ 10 ] [ 11 ] The existence of so-called impregnating Cassie wetting state also has to be considered. [ 11 ] Understanding wetting transitions is of a primary importance for design of superhydrophobic surfaces. [ 16 ] [ 17 ]
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https://en.wikipedia.org/wiki/Wetting_transition
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The term wetware is used to describe the protocols and molecular devices used in molecular biology and synthetic biology .
Where biological components and systems are treated in a similar manner to software , and similar development models and methodologies are applied, the term 'wetware' can be used to imply an approach to their problems as 'bugs' and their beneficial aspects as 'features'. In this manner, genetic code can be subjected to Version Control Systems such as Git , for the development of improvements and new gene edits, therapeutic components and therapies .
The National Science Foundation (NSF) funded Wiki project Open Wetware (OWW) provides a resource for reagent, project and laboratory notebook sharing. [ 1 ]
A somewhat related NSF consortium Synthetic Biology Engineering Research Center (SynBERC) constructed and distributed wetware. [ 2 ] It was succeeded in 2016 by the Engineering Biology Research Consortium .
This biology article is a stub . You can help Wikipedia by expanding it .
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The Weyburn-Midale Carbon Dioxide Project (or IEA GHG Weyburn-Midale CO 2 Monitoring and Storage Project) was, as of 2008, the world's largest carbon capture and storage project. [ 1 ] It has since been overtaken in terms of carbon capture capacity by projects such as the Shute Creek project and the Century Plant. [ 2 ] [ 3 ] It is located in Midale, Saskatchewan , Canada.
The IEAGHG Weyburn-Midale CO 2 Monitoring and Storage Project is an international collaborative scientific study to assess the technical feasibility of CO 2 storage in geological formations with a focus on oil reservoirs , together with the development of world leading best practices for project implementation. The project itself began in 2000 and runs until the end of 2011 when a best practices manual for the transitioning of CO 2 -EOR operations into long-term storage operations will be released.
The research project accesses data from the actual CO 2 - enhanced oil recovery operations in the Weyburn oil field (formerly operated by Cenovus Energy of Calgary before its Saskatchewan operations were sold to Whitecap Resources in 2017 [ 4 ] ), and after the year 2005 from the adjacent Midale field (operated by Apache Canada). These EOR operations are independent of the research program. Cenovus Energy's only contribution to the IEAGHG Weyburn-Midale CO 2 Monitoring and Storage Project was to allow access to the fields for measurement, monitoring and verification of the CO 2 for the global scientists and researchers involved in the project.
The Weyburn and Midale oil fields were discovered in 1954 near Midale, Saskatchewan.
The Weyburn Oilfield covers an area of some 52,000 acres (210 km 2 ) and has a current oil production rate of ~3,067 m3/day. Original oil-in-place is estimated to be 1.4 billion barrels (220,000,000 m 3 ). The oil is produced from a total of 963 active wells made up of 534 vertical wells, 138 horizontal wells, and 171 injection systems . There are also 146 enclosed wells. Current production consists primarily of medium-gravity crude oil with a low gas-to-oil ratio.
The Midale oil field is about 102 square miles (260 km 2 ) in size, and has 515 million barrels (81,900,000 m 3 ) of oil-in-place. It began injecting CO 2 in 2005.
Various enhanced oil recovery techniques were used in the Weyburn field prior to the introduction of CO 2 , between the 1970s and 1990s. These include additional vertical drilling, the introduction of horizontal drilling , and the use of waterfloods to increase pressure in the reservoir. In October 2000, Cenovus (formerly Pan Canadian, Encana ) began injecting significant amounts of carbon dioxide into the Weyburn field in order to boost oil production. Cenovus was the operator and held the largest share of the 37 current partners in the oilfield prior to the sale of local assets to Whitecap in 2017. [ 4 ]
Initial CO 2 injection rates in the Weyburn field amounted to ~5,000 tonnes/day or 95 million scf/day (2.7 million m3/d); this would otherwise have been vented to the atmosphere from the Dakota Gasification facility . At one point, CO 2 injection by Cenovus at Weyburn was at ~6,500 tonnes per day. Apache Canada is injecting approximately 1,500 tonnes/day into the Midale field.
Overall, it is anticipated that some 40 Mt of carbon dioxide will be permanently sequestered over the lifespan of the project in the Weyburn and Midale fields. The gas is being supplied via a 320 kilometre mile long pipeline (completed in 1999) from the lignite -fired Dakota Gasification Company synfuels plant site in Beulah, North Dakota (See attached image). The company is a subsidiary of Basin Electric Power Co-operative . At the plant, CO 2 is produced from a Rectisol unit in the gas cleanup train. The CO 2 project adds about $30 million of gross revenue to the gasification plant's cash flow each year. Approximately 8000 tonnes/day of compressed CO 2 (in liquid form) is provided to the Weyburn and Midale fields via the pipeline.
During its life, the Weyburn and Midale fields combined are expected to produce at least 220 million additional barrels of incremental oil, through miscible or near-miscible displacement with CO 2 , from a fields that have already produced over 500 million barrels (79,000,000 m 3 ) since discovery in 1954. This will extend the life of the Weyburn field by approximately 20–25 years. It is estimated that ultimate oil recovery will increase to 34% of the oil-in-place. It has been estimated that, on a full life-cycle basis, the oil produced at Weyburn by CO 2 EOR will release only two-thirds as much CO 2 to the atmosphere compared to oil produced using conventional technology.
This is the first instance of cross-border transfer of CO 2 from the US to Canada and highlights the ability for international cooperation with GHG mitigation technologies. Whilst there are emissions trading projects being developed within countries such as Canada, the Weyburn project is essentially the first international project where physical quantities of CO 2 are being sold commercially for enhanced oil recovery, with the added benefit of carbon sequestration .
The First Phase of the IEAGHG Weyburn CO 2 Monitoring and Storage Project (the Midale oil field did not join the research project until the Final phase research) which began in 2000 and ended in 2004, verified the ability of an oil reservoir to securely store CO 2 for significant lengths of time. This was done through a comprehensive analysis of the various process factors as well as monitoring/modeling methods designed to measure, monitor and track the CO 2 . Research was conducted into geological characterization of both the geosphere (the geological layers deeper than near surface) and biosphere (basically from the depths of groundwater up). As well, prediction, monitoring and verification techniques were used to examine the movements of the CO 2 . Finally, both the economic and geologic limits of the CO 2 storage capacity were predicted, and a long-term risk assessment developed for storage of CO 2 permanently in the formation.
A critical part of the First Phase was the accumulation of baseline surveys for both CO 2 soil content, and water wells in the area. These baselines were identified in 2001 and have helped to confirm through comparison with more recent readings that CO 2 is not leaking from the reservoir into the biosphere in the study area.
The Final Phase of the IEAGHG Weyburn-Midale CO 2 Monitoring and Storage Project is utilizing scientific experts from most of the world's leading carbon capture and storage research organizations and universities to further develop and build upon the most scrutinized CO 2 geological storage data set in the world.
The project's major technical research "themes" can be broadly broken out into four areas:
Technical Components:
Ultimately, the goal of the final phase of the project is to produce a best practices manual that can be used by other jurisdictions and organizations to help transition CO 2 -EOR operations into long-term storage projects. The research of the project's final phase should be complete in 2011, with the Best Practices Manual issued before the end of that year.
A report [ 5 ] of CO 2 leaks above the project was released in January 2011 [ 6 ] by an advocacy group on behalf of owners of land above the project. They reported ponds fizzing with bubbles, dead animals found near those ponds, sounds of explosions which they attributed to gas blowing out holes in the walls of a quarry . The report said that carbon dioxide levels in the soil averaged about 23,000 parts per million, several times higher than is normal for the area. "The ... source of the high concentrations of CO2 in the soils of the Kerr property is clearly the anthropogenic CO2 injected into the Weyburn reservoir... The survey also demonstrates that the overlying thick cap rock of anhydrite over the Weyburn reservoir is not an impermeable barrier to the upward movement of light hydrocarbons and CO2 as is generally thought." said the report. [ 7 ]
The PTRC posted an extensive rebuttal of the Petro-Find report, stating that the isotopic signatures of the CO 2 , claimed by Mr. Lafleur to be indicative of the manmade CO 2 being injected into the reservoir, were in fact, according to studies of CO 2 conducted by the British Geological Survey and two other European Union geological groups prior to CO 2 being injected at Weyburn, occurring naturally in several locations near the Kerr farm. Subsequent soil surveys after injection in 2002 to 2005 found CO 2 levels dropped in these same regions. In addition, prior to injection occurring into the oil field, these samplings were found to be as high as 125,000 parts per million and averaging 25,000 ppm CO 2 across the region, even more than the average and largest readings from the Kerr's property that were being claimed as unusually high. The report also questions, based on seismic imaging conducted over ten years, that any active faults exist or that the caprock is compromised to allow pathways for the CO 2 to reach the surface. [ 8 ] The PTRC acknowledged that they do not monitor the entire site for leaks, rather primarily above the part of the Weyburn field where CO 2 is injected and key locations outside it, but the organization did monitor the Kerr's well between 2002 and 2006, finding no appreciable difference in water quality . [ 9 ] They have also acknowledged that PTRC is a research organisation rather than a regulator, and manage the IEA GHG Weyburn-Midale CO 2 Monitoring and Storage Project on behalf of the International Energy Agency 's Greenhouse Gas R&D Programme, which includes some 30 international research groups. [ 10 ]
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In linear algebra , Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.
Let A , B {\textstyle A,B} be Hermitian on inner product space V {\textstyle V} with dimension n {\textstyle n} , with spectrum ordered in descending order λ 1 ≥ . . . ≥ λ n {\textstyle \lambda _{1}\geq ...\geq \lambda _{n}} . Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices). [ 1 ]
Weyl inequality — λ i + j − 1 ( A + B ) ≤ λ i ( A ) + λ j ( B ) ≤ λ i + j − n ( A + B ) {\displaystyle \lambda _{i+j-1}(A+B)\leq \lambda _{i}(A)+\lambda _{j}(B)\leq \lambda _{i+j-n}(A+B)}
By the min-max theorem , it suffices to show that any W ⊂ V {\textstyle W\subset V} with dimension i + j − 1 {\textstyle i+j-1} , there exists a unit vector w {\textstyle w} such that ⟨ w , ( A + B ) w ⟩ ≤ λ i ( A ) + λ j ( B ) {\textstyle \langle w,(A+B)w\rangle \leq \lambda _{i}(A)+\lambda _{j}(B)} .
By the min-max principle, there exists some W A {\textstyle W_{A}} with codimension ( i − 1 ) {\textstyle (i-1)} , such that λ i ( A ) = max x ∈ W A ; ‖ x ‖ = 1 ⟨ x , A x ⟩ {\displaystyle \lambda _{i}(A)=\max _{x\in W_{A};\|x\|=1}\langle x,Ax\rangle } Similarly, there exists such a W B {\textstyle W_{B}} with codimension j − 1 {\textstyle j-1} . Now W A ∩ W B {\textstyle W_{A}\cap W_{B}} has codimension ≤ i + j − 2 {\textstyle \leq i+j-2} , so it has nontrivial intersection with W {\textstyle W} . Let w ∈ W ∩ W A ∩ W B {\textstyle w\in W\cap W_{A}\cap W_{B}} , and we have the desired vector.
The second one is a corollary of the first, by taking the negative.
Weyl's inequality states that the spectrum of Hermitian matrices is stable under perturbation. Specifically, we have: [ 1 ]
Corollary (Spectral stability) — λ k ( A + B ) − λ k ( A ) ∈ [ λ n ( B ) , λ 1 ( B ) ] {\displaystyle \lambda _{k}(A+B)-\lambda _{k}(A)\in [\lambda _{n}(B),\lambda _{1}(B)]} | λ k ( A + B ) − λ k ( A ) | ≤ ‖ B ‖ o p {\displaystyle |\lambda _{k}(A+B)-\lambda _{k}(A)|\leq \|B\|_{op}} where ‖ B ‖ o p = max ( | λ 1 ( B ) | , | λ n ( B ) | ) {\displaystyle \|B\|_{op}=\max(|\lambda _{1}(B)|,|\lambda _{n}(B)|)} is the operator norm.
In jargon, it says that λ k {\displaystyle \lambda _{k}} is Lipschitz-continuous on the space of Hermitian matrices with operator norm.
Let A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} have singular values σ 1 ( A ) ≥ ⋯ ≥ σ n ( A ) ≥ 0 {\displaystyle \sigma _{1}(A)\geq \cdots \geq \sigma _{n}(A)\geq 0} and eigenvalues ordered so that | λ 1 ( A ) | ≥ ⋯ ≥ | λ n ( A ) | {\displaystyle |\lambda _{1}(A)|\geq \cdots \geq |\lambda _{n}(A)|} . Then
For k = 1 , … , n {\displaystyle k=1,\ldots ,n} , with equality for k = n {\displaystyle k=n} . [ 2 ]
Assume that R {\displaystyle R} is small in the sense that its spectral norm satisfies ‖ R ‖ 2 ≤ ϵ {\displaystyle \|R\|_{2}\leq \epsilon } for some small ϵ > 0 {\displaystyle \epsilon >0} . Then it follows that all the eigenvalues of R {\displaystyle R} are bounded in absolute value by ϵ {\displaystyle \epsilon } . Applying Weyl's inequality, it follows that the spectra of the Hermitian matrices M and N are close in the sense that [ 3 ]
Note, however, that this eigenvalue perturbation bound is generally false for non-Hermitian matrices (or more accurately, for non-normal matrices). For a counterexample, let t > 0 {\displaystyle t>0} be arbitrarily small, and consider
whose eigenvalues μ 1 = μ 2 = 0 {\displaystyle \mu _{1}=\mu _{2}=0} and ν 1 = + 1 / t , ν 2 = − 1 / t {\displaystyle \nu _{1}=+1/t,\nu _{2}=-1/t} do not satisfy | μ i − ν i | ≤ ‖ R ‖ 2 = 1 {\displaystyle |\mu _{i}-\nu _{i}|\leq \|R\|_{2}=1} .
Let M {\displaystyle M} be a p × n {\displaystyle p\times n} matrix with 1 ≤ p ≤ n {\displaystyle 1\leq p\leq n} . Its singular values σ k ( M ) {\displaystyle \sigma _{k}(M)} are the p {\displaystyle p} positive eigenvalues of the ( p + n ) × ( p + n ) {\displaystyle (p+n)\times (p+n)} Hermitian augmented matrix
Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values. [ 1 ] This result gives the bound for the perturbation in the singular values of a matrix M {\displaystyle M} due to an additive perturbation Δ {\displaystyle \Delta } :
where we note that the largest singular value σ 1 ( Δ ) {\displaystyle \sigma _{1}(\Delta )} coincides with the spectral norm ‖ Δ ‖ 2 {\displaystyle \|\Delta \|_{2}} .
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In number theory , Weyl's inequality , named for Hermann Weyl , states that if M , N , a and q are integers, with a and q coprime , q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies
for some t greater than or equal to 1, then for any positive real number ε {\displaystyle \scriptstyle \varepsilon } one has
This inequality will only be useful when
for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as ≤ N {\displaystyle \scriptstyle \leq \,N} provides a better bound.
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https://en.wikipedia.org/wiki/Weyl's_inequality_(number_theory)
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In relativistic cosmology , Weyl's postulate stipulates that in the Friedmann model of the universe (a fluid cosmological model), the wordlines of fluid particles (modeling galaxies) should be hypersurface orthogonal . Meaning, they should form a 3-bundle of non-intersecting geodesics orthogonal to a series of spacelike hypersurfaces (hyperslices). [ 1 ]
Sometimes, the additional hypothesis is added that the world lines form timelike geodesics .
The ADM formalism introduced a family of spatial hyperslices. This allows us to think of the geometry of "space" as evolving over "time". This is an attractive viewpoint, but in general no such family of hyperslices will be physically preferred. The Weyl hypothesis can be understood as the assumption that we should consider only cosmological models in which there is such a preferred slicing, namely the one given by taking the unique hyperslices orthogonal to the world lines of the fluid particles.
One consequence of this hypothesis is that if it holds true, we can introduce a comoving chart such that the metric tensor contains no terms of form dt dx , dt dy , or dt dz .
The additional hypothesis that the world lines of the fluid particles be geodesics is equivalent to assuming that no body forces act within the fluid. In other words, the fluid has zero pressure, so that we are considering a dust solution .
The condition that the congruence corresponding to the fluid particles should be hypersurface orthogonal is by no means assured. A generic congruence does not possess this property, which is in fact mathematically equivalent to stipulating that the congruence of world lines should be vorticity-free . That is, they should not be twisting about one another, or in other words, the fluid elements should not be swirling about their neighbors in the manner of the fluid particles in a stirred cup of some liquid. (Nonzero vorticity model is presented in https://arxiv.org/ftp/arxiv/papers/1210/1210.4091.pdf or Nurgaliev I.S. «Singularities Are Averted by Vortices». Gravitation and Cosmology, 2010, Vol. 16, No. 4, pp. 313–315.)
This physical cosmology -related article is a stub . You can help Wikipedia by expanding it .
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In philosophy , Weyl's tile argument , introduced by Hermann Weyl in 1949, is an argument against the notion that physical space is "discrete", as if composed of a number of finite sized units or tiles . [ 1 ] The argument purports to show a distance function approximating Pythagoras' theorem on a discrete space cannot be defined and, since the Pythagorean theorem has been confirmed to be approximately true in nature, physical space is not discrete. [ 2 ] [ 3 ] [ 4 ] Academic debate on the topic continues, with counterarguments proposed in the literature. [ 5 ] [ 6 ] [ 7 ]
The tile argument appears in Weyl's 1949 book Philosophy of Mathematics and Natural Sciences , where he writes:
If a square is built up of miniature tiles, then there are as many tiles along the diagonal as there are along the side; thus the diagonal should be equal in length to the side. [ 1 ]
A demonstration of Weyl's argument proceeds by constructing a square tiling of the plane representing a discrete space. A discretized triangle, n units tall and n units long, can be constructed on the tiling. The hypotenuse of the resulting triangle will be n tiles long. However, by the Pythagorean theorem, a corresponding triangle in a continuous space—a triangle whose height and length are n —will have a hypotenuse measuring n 2 {\displaystyle n{\sqrt {2}}} units long. To show that the former result does not converge to the latter for arbitrary values of n , one can examine the percent difference between the two results: n 2 − n n 2 = 1 − 1 2 . {\displaystyle {\frac {n{\sqrt {2}}-n}{n{\sqrt {2}}}}=1-{\frac {1}{\sqrt {2}}}.} Since n cancels out, the two results never converge, even in the limit of large n . The argument can be constructed for more general triangles, but, in each case, the result is the same. Thus, a discrete space does not even approximate the Pythagorean theorem.
In response, Kris McDaniel has argued the Weyl tile argument depends on accepting a "size thesis" which posits that the distance between two points is given by the number of tiles between the two points. However, as McDaniel points out, the size thesis is not accepted for continuous spaces. Thus, we might have reason not to accept the size thesis for discrete spaces. [ 5 ]
This philosophy -related article is a stub . You can help Wikipedia by expanding it .
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In physics , particularly in quantum field theory , the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions . The equation is named after Hermann Weyl . The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions .
None of the elementary particles in the Standard Model are Weyl fermions. Previous to the confirmation of the neutrino oscillations , it was considered possible that the neutrino might be a Weyl fermion (it is now expected to be either a Dirac or a Majorana fermion). In condensed matter physics , some materials can display quasiparticles that behave as Weyl fermions, leading to the notion of Weyl semimetals .
Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term. [ 1 ]
The Dirac equation was published in 1928 by Paul Dirac , and was first used to model spin-1/2 particles in the framework of relativistic quantum mechanics . [ 2 ] Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation. [ 2 ] [ 3 ] Wolfgang Pauli wrote in 1933 against Weyl's equation because it violated parity . [ 4 ] However, three years before, Pauli had predicted the existence of a new elementary fermion , the neutrino , to explain the beta decay , which eventually was described using the Weyl equation.
In 1937, Conyers Herring proposed that Weyl fermions may exist as quasiparticles in condensed matter. [ 5 ]
Neutrinos were experimentally observed in 1956 as particles with extremely small masses (and historically were even sometimes thought to be massless). [ 4 ] The same year the Wu experiment showed that parity could be violated by the weak interaction , addressing Pauli's criticism. [ 6 ] This was followed by the measurement of the neutrino's helicity in 1958. [ 4 ] As experiments showed no signs of a neutrino mass, interest in the Weyl equation resurfaced. Thus, the Standard Model was built under the assumption that neutrinos were Weyl fermions. [ 4 ]
While Italian physicist Bruno Pontecorvo had proposed in 1957 the possibility of neutrino masses and neutrino oscillations , [ 4 ] it was not until 1998 that Super-Kamiokande eventually confirmed the existence of neutrino oscillations, and their non-zero mass. [ 4 ] This discovery confirmed that Weyl's equation cannot completely describe the propagation of neutrinos, as the equations can only describe massless particles. [ 2 ]
In 2015, the first Weyl semimetal was demonstrated experimentally in crystalline tantalum arsenide (TaAs) by the collaboration of M.Z. Hasan 's ( Princeton University ) and H. Ding's ( Chinese Academy of Sciences ) teams. [ 5 ] Independently, the same year, M. Soljačić team ( Massachusetts Institute of Technology ) also observed Weyl-like excitations in photonic crystals . [ 5 ]
The Weyl equation comes in two forms. The right-handed form can be written as follows: [ 7 ] [ 8 ] [ 9 ]
Expanding this equation, and inserting c {\displaystyle c} for the speed of light , it becomes
where
is a vector whose components are the 2×2 identity matrix I 2 {\displaystyle I_{2}} for μ = 0 {\displaystyle \mu =0} and the Pauli matrices for μ = 1 , 2 , 3 , {\displaystyle \mu =1,2,3,} and ψ {\displaystyle \psi } is the wavefunction – one of the Weyl spinors . The left-handed form of the Weyl equation is usually written as:
where
The solutions of the right- and left-handed Weyl equations are different: they have right- and left-handed helicity , and thus chirality , respectively. It is convenient to indicate this explicitly, as follows: σ μ ∂ μ ψ R = 0 {\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0} and σ ¯ μ ∂ μ ψ L = 0 . {\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}
The plane-wave solutions to the Weyl equation are referred to as the left and right handed Weyl spinors, each is with two components. Both have the form
where
is a momentum-dependent two-component spinor which satisfies
or
By direct manipulation, one obtains that
and concludes that the equations correspond to a particle that is massless . As a result, the magnitude of momentum p {\displaystyle \mathbf {p} } relates directly to the wave-vector k {\displaystyle \mathbf {k} } by the de Broglie relations as:
The equation can be written in terms of left and right handed spinors as:
The left and right components correspond to the helicity λ {\displaystyle \lambda } of the particles, the projection of angular momentum operator J {\displaystyle \mathbf {J} } onto the linear momentum p {\displaystyle \mathbf {p} } :
Here λ = ± 1 2 . {\textstyle \lambda =\pm {\frac {1}{2}}~.}
Both equations are Lorentz invariant under the Lorentz transformation x ↦ x ′ = Λ x {\displaystyle x\mapsto x^{\prime }=\Lambda x} where Λ ∈ S O ( 1 , 3 ) . {\displaystyle \Lambda \in \mathrm {SO} (1,3)~.} More precisely, the equations transform as
where S † {\displaystyle S^{\dagger }} is the Hermitian transpose , provided that the right-handed field transforms as
The matrix S ∈ S L ( 2 , C ) {\displaystyle S\in SL(2,\mathbb {C} )} is related to the Lorentz transform by means of the double covering of the Lorentz group by the special linear group S L ( 2 , C ) {\displaystyle \mathrm {SL} (2,\mathbb {C} )} given by
Thus, if the untransformed differential vanishes in one Lorentz frame, then it also vanishes in another. Similarly
provided that the left-handed field transforms as
Proof: Neither of these transformation properties are in any way "obvious", and so deserve a careful derivation. Begin with the form
for some unknown R ∈ S L ( 2 , C ) {\displaystyle R\in \mathrm {SL} (2,\mathbb {C} )} to be determined. The Lorentz transform, in coordinates, is
or, equivalently,
This leads to
In order to make use of the Weyl map
a few indexes must be raised and lowered. This is easier said than done, as it invokes the identity
where η = diag ( + 1 , − 1 , − 1 , − 1 ) {\displaystyle \eta ={\mbox{diag}}(+1,-1,-1,-1)} is the flat-space Minkowski metric . The above identity is often used to define the elements Λ ∈ S O ( 1 , 3 ) . {\displaystyle \Lambda \in \mathrm {SO} (1,3).} One takes the transpose:
to write
One thus regains the original form if S − 1 R = 1 , {\displaystyle S^{-1}R=1,} that is, R = S . {\displaystyle R=S.} Performing the same manipulations for the left-handed equation, one concludes that
with L = ( S † ) − 1 . {\displaystyle L=\left(S^{\dagger }\right)^{-1}.} [ a ]
The Weyl equation is conventionally interpreted as describing a massless particle . However, with a slight alteration, one may obtain a two-component version of the Majorana equation . [ 10 ] This arises because the special linear group S L ( 2 , C ) {\displaystyle \mathrm {SL} (2,\mathbb {C} )} is isomorphic to the symplectic group S p ( 2 , C ) . {\displaystyle \mathrm {Sp} (2,\mathbb {C} )~.} The symplectic group is defined as the set of all complex 2×2 matrices that satisfy
where
The defining relationship can be rewritten as ω S ∗ = ( S † ) − 1 ω {\displaystyle \omega S^{*}=\left(S^{\dagger }\right)^{-1}\omega } where S ∗ {\displaystyle S^{*}} is the complex conjugate . The right handed field, as noted earlier, transforms as
and so the complex conjugate field transforms as
Applying the defining relationship, one concludes that
which is exactly the same Lorentz covariance property noted earlier. Thus, the linear combination, using an arbitrary complex phase factor η = e i ϕ {\displaystyle \eta =e^{i\phi }}
transforms in a covariant fashion; setting this to zero gives the complex two-component Majorana equation . The Majorana equation is conventionally written as a four-component real equation, rather than a two-component complex equation; the above can be brought into four-component form (see that article for details). Similarly, the left-chiral Majorana equation (including an arbitrary phase factor ζ {\displaystyle \zeta } ) is
As noted earlier, the left and right chiral versions are related by a parity transformation. The skew complex conjugate ω ψ ∗ = i σ 2 ψ {\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi } can be recognized as the charge conjugate form of ψ . {\displaystyle \psi ~.} Thus, the Majorana equation can be read as an equation that connects a spinor to its charge-conjugate form. The two distinct phases on the mass term are related to the two distinct eigenvalues of the charge conjugation operator; see charge conjugation and Majorana equation for details.
Define a pair of operators, the Majorana operators,
where K {\displaystyle K} is a short-hand reminder to take the complex conjugate. Under Lorentz transformations, these transform as
whereas the Weyl spinors transform as
just as above. Thus, the matched combinations of these are Lorentz covariant, and one may take
as a pair of complex 2-spinor Majorana equations.
The products D L D R {\displaystyle D_{\rm {L}}D_{\rm {R}}} and D R D L {\displaystyle D_{\rm {R}}D_{\rm {L}}} are both Lorentz covariant. The product is explicitly
Verifying this requires keeping in mind that ω 2 = − 1 {\displaystyle \omega ^{2}=-1} and that K i = − i K . {\displaystyle Ki=-iK~.} The RHS reduces to the Klein–Gordon operator provided that η ζ ∗ = 1 {\displaystyle \eta \zeta ^{*}=1} , that is η = ζ . {\displaystyle \eta =\zeta ~.} These two Majorana operators are thus "square roots" of the Klein–Gordon operator.
The equations are obtained from the Lagrangian densities
By treating the spinor and its conjugate (denoted by † {\displaystyle \dagger } ) as independent variables, the relevant Weyl equation is obtained.
The term Weyl spinor is also frequently used in a more general setting, as an element of a Clifford module . This is closely related to the solutions given above, and gives a natural geometric interpretation to spinors as geometric objects living on a manifold . This general setting has multiple strengths: it clarifies their interpretation as fermions in physics, and it shows precisely how to define spin in General Relativity , or, indeed, for any Riemannian manifold or pseudo-Riemannian manifold . This is informally sketched as follows.
The Weyl equation is invariant under the action of the Lorentz group. This means that, as boosts and rotations are applied, the form of the equation itself does not change. However, the form of the spinor ψ {\displaystyle \psi } itself does change. Ignoring spacetime entirely, the algebra of the spinors is described by a (complexified) Clifford algebra . The spinors transform under the action of the spin group . This is entirely analogous to how one might talk about a vector, and how it transforms under the rotation group , except that now, it has been adapted to the case of spinors.
Given an arbitrary pseudo-Riemannian manifold M {\displaystyle M} of dimension ( p , q ) {\displaystyle (p,q)} , one may consider its tangent bundle T M {\displaystyle TM} . At any given point x ∈ M , {\displaystyle x\in M,} the tangent space T x M {\displaystyle T_{x}M} is a ( p , q ) {\displaystyle (p,q)} dimensional vector space . Given this vector space, one can construct the Clifford algebra C l ( p , q ) {\displaystyle \mathrm {Cl} (p,q)} on it. If { e i } {\displaystyle \{e_{i}\}} are a vector space basis on T x M {\displaystyle T_{x}M} , one may construct a pair of Weyl spinors as [ 11 ]
and
When properly examined in light of the Clifford algebra, these are naturally anti-commuting , that is, one has that w j w m = − w m w j . {\displaystyle w_{j}w_{m}=-w_{m}w_{j}~.} This can be happily interpreted as the mathematical realization of the Pauli exclusion principle , thus allowing these abstractly defined formal structures to be interpreted as fermions. For ( p , q ) = ( 1 , 3 ) {\displaystyle (p,q)=(1,3)} dimensional Minkowski space-time , there are only two such spinors possible, by convention labelled "left" and "right", as described above. A more formal, general presentation of Weyl spinors can be found in the article on the spin group .
The abstract, general-relativistic form of the Weyl equation can be understood as follows: given a pseudo-Riemannian manifold M , {\displaystyle M,} one constructs a fiber bundle above it, with the spin group as the fiber. The spin group S p i n ( p , q ) {\displaystyle \mathrm {Spin} (p,q)} is a double cover of the special orthogonal group S O ( p , q ) {\displaystyle \mathrm {SO} (p,q)} , and so one can identify the spin group fiber-wise with the frame bundle over M . {\displaystyle M~.} When this is done, the resulting structure is called a spin structure .
Selecting a single point on the fiber corresponds to selecting a local coordinate frame for spacetime; two different points on the fiber are related by a (Lorentz) boost/rotation, that is, by a local change of coordinates. The natural inhabitants of the spin structure are the Weyl spinors, in that the spin structure completely describes how the spinors behave under (Lorentz) boosts/rotations.
Given a spin manifold , the analog of the metric connection is the spin connection ; this is effectively "the same thing" as the normal connection, just with spin indexes attached to it in a consistent fashion. The covariant derivative can be defined in terms of the connection in an entirely conventional way. It acts naturally on the Clifford bundle ; the Clifford bundle is the space in which the spinors live. The general exploration of such structures and their relationships is termed spin geometry .
For even n {\displaystyle n} , the even subalgebra C l 0 ( n ) {\displaystyle \mathbb {C} l^{0}(n)} of the complex Clifford algebra C l ( n ) {\displaystyle \mathbb {C} l(n)} is isomorphic to E n d ( C N / 2 ) ⊕ E n d ( C N / 2 ) =: Δ n + ⊕ Δ n − {\displaystyle \mathrm {End} (\mathbb {C} ^{N/2})\oplus \mathrm {End} (\mathbb {C} ^{N/2})=:\Delta _{n}^{+}\oplus \Delta _{n}^{-}} , where N = 2 n / 2 {\displaystyle N=2^{n/2}} . A left-handed (respectively, right-handed) complex Weyl spinor in n {\displaystyle n} -dimensional space is an element of Δ n + {\displaystyle \Delta _{n}^{+}} (respectively, Δ n − {\displaystyle \Delta _{n}^{-}} ).
There are three important special cases that can be constructed from Weyl spinors. One is the Dirac spinor , which can be taken to be a pair of Weyl spinors, one left-handed, and one right-handed. These are coupled together in such a way as to represent an electrically charged fermion field. The electric charge arises because the Dirac field transforms under the action of the complexified spin group S p i n C ( p , q ) . {\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).} This group has the structure
where S 1 ≅ U ( 1 ) {\displaystyle S^{1}\cong \mathrm {U} (1)} is the circle, and can be identified with the U ( 1 ) {\displaystyle \mathrm {U} (1)} of electromagnetism . The product × Z 2 {\displaystyle \times _{\mathbb {Z} _{2}}} is just fancy notation denoting the product S p i n ( p , q ) × S 1 {\displaystyle \mathrm {Spin} (p,q)\times S^{1}} with opposite points ( s , u ) = ( − s , − u ) {\displaystyle (s,u)=(-s,-u)} identified (a double covering).
The Majorana spinor is again a pair of Weyl spinors, but this time arranged so that the left-handed spinor is the charge conjugate of the right-handed spinor. The result is a field with two less degrees of freedom than the Dirac spinor. It is unable to interact with the electromagnetic field, since it transforms as a scalar under the action of the s p i n C {\displaystyle \mathrm {spin} ^{\mathbb {C} }} group. That is, it transforms as a spinor, but transversally, such that it is invariant under the U ( 1 ) {\displaystyle \mathrm {U} (1)} action of the spin group.
The third special case is the ELKO spinor , constructed much as the Majorana spinor, except with an additional minus sign between the charge-conjugate pair. This again renders it electrically neutral, but introduces a number of other quite surprising properties.
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In physics , the Weyl expansion , also known as the Weyl identity or angular spectrum expansion , expresses an outgoing spherical wave as a linear combination of plane waves . In a Cartesian coordinate system , it can be denoted as [ 1 ] [ 2 ]
where k x {\displaystyle k_{x}} , k y {\displaystyle k_{y}} and k z {\displaystyle k_{z}} are the wavenumbers in their respective coordinate axes:
The expansion is named after Hermann Weyl , who published it in 1919. [ 3 ] The Weyl identity is largely used to characterize the reflection and transmission of spherical waves at planar interfaces; it is often used to derive the Green's functions for Helmholtz equation in layered media. The expansion also covers evanescent wave components. It is often preferred to the Sommerfeld identity when the field representation is needed to be in Cartesian coordinates. [ 1 ]
The resulting Weyl integral is commonly encountered in microwave integrated circuit analysis and electromagnetic radiation over a stratified medium; as in the case for Sommerfeld integral, it is numerically evaluated . [ 4 ] As a result, it is used in calculation of Green's functions for method of moments for such geometries. [ 5 ] Other uses include the descriptions of dipolar emissions near surfaces in nanophotonics , [ 6 ] [ 7 ] [ 8 ] holographic inverse scattering problems , [ 9 ] Green's functions in quantum electrodynamics [ 10 ] and acoustic or seismic waves . [ 11 ]
This electromagnetism -related article is a stub . You can help Wikipedia by expanding it .
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In mathematics , the Weyl integral (named after Hermann Weyl ) is an operator defined, as an example of fractional calculus , on functions f on the unit circle having integral 0 and a Fourier series . In other words there is a Fourier series for f of the form
with a 0 = 0.
Then the Weyl integral operator of order s is defined on Fourier series by
where this is defined. Here s can take any real value, and for integer values k of s the series expansion is the expected k -th derivative, if k > 0, or (− k )th indefinite integral normalized by integration from θ = 0.
The condition a 0 = 0 here plays the obvious role of excluding the need to consider division by zero. The definition is due to Hermann Weyl (1917).
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In general relativity , the Weyl metrics (named after the German-American mathematician Hermann Weyl ) [ 1 ] are a class of static and axisymmetric solutions to Einstein's field equation . Three members in the renowned Kerr–Newman family solutions, namely the Schwarzschild , nonextremal Reissner–Nordström and extremal Reissner–Nordström metrics, can be identified as Weyl-type metrics.
The Weyl class of solutions has the generic form [ 2 ] [ 3 ]
where ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} and γ ( ρ , z ) {\displaystyle \gamma (\rho ,z)} are two metric potentials dependent on Weyl's canonical coordinates { ρ , z } {\displaystyle \{\rho \,,z\}} . The coordinate system { t , ρ , z , ϕ } {\displaystyle \{t,\rho ,z,\phi \}} serves best for symmetries of Weyl's spacetime (with two Killing vector fields being ξ t = ∂ t {\displaystyle \xi ^{t}=\partial _{t}} and ξ ϕ = ∂ ϕ {\displaystyle \xi ^{\phi }=\partial _{\phi }} ) and often acts like cylindrical coordinates , [ 2 ] but is incomplete when describing a black hole as { ρ , z } {\displaystyle \{\rho \,,z\}} only cover the horizon and its exteriors.
Hence, to determine a static axisymmetric solution corresponding to a specific stress–energy tensor T a b {\displaystyle T_{ab}} , we just need to substitute the Weyl metric Eq(1) into Einstein's equation (with c = G =1):
and work out the two functions ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} and γ ( ρ , z ) {\displaystyle \gamma (\rho ,z)} .
One of the best investigated and most useful Weyl solutions is the electrovac case, where T a b {\displaystyle T_{ab}} comes from the existence of (Weyl-type) electromagnetic field (without matter and current flows). As we know, given the electromagnetic four-potential A a {\displaystyle A_{a}} , the anti-symmetric electromagnetic field F a b {\displaystyle F_{ab}} and the trace-free stress–energy tensor T a b {\displaystyle T_{ab}} ( T = g a b T a b = 0 ) {\displaystyle (T=g^{ab}T_{ab}=0)} will be respectively determined by
which respects the source-free covariant Maxwell equations:
Eq(5.a) can be simplified to:
in the calculations as Γ b c a = Γ c b a {\displaystyle \Gamma _{bc}^{a}=\Gamma _{cb}^{a}} . Also, since R = − 8 π T = 0 {\displaystyle R=-8\pi T=0} for electrovacuum, Eq(2) reduces to
Now, suppose the Weyl-type axisymmetric electrostatic potential is A a = Φ ( ρ , z ) [ d t ] a {\displaystyle A_{a}=\Phi (\rho ,z)[dt]_{a}} (the component Φ {\displaystyle \Phi } is actually the electromagnetic scalar potential ), and together with the Weyl metric Eq(1), Eqs(3)(4)(5)(6) imply that
where R = 0 {\displaystyle R=0} yields Eq(7.a), R t t = 8 π T t t {\displaystyle R_{tt}=8\pi T_{tt}} or R φ φ = 8 π T φ φ {\displaystyle R_{\varphi \varphi }=8\pi T_{\varphi \varphi }} yields Eq(7.b), R ρ ρ = 8 π T ρ ρ {\displaystyle R_{\rho \rho }=8\pi T_{\rho \rho }} or R z z = 8 π T z z {\displaystyle R_{zz}=8\pi T_{zz}} yields Eq(7.c), R ρ z = 8 π T ρ z {\displaystyle R_{\rho z}=8\pi T_{\rho z}} yields Eq(7.d), and Eq(5.b) yields Eq(7.e). Here ∇ 2 = ∂ ρ ρ + 1 ρ ∂ ρ + ∂ z z {\displaystyle \nabla ^{2}=\partial _{\rho \rho }+{\frac {1}{\rho }}\,\partial _{\rho }+\partial _{zz}} and ∇ = ∂ ρ e ^ ρ + ∂ z e ^ z {\displaystyle \nabla =\partial _{\rho }\,{\hat {e}}_{\rho }+\partial _{z}\,{\hat {e}}_{z}} are respectively the Laplace and gradient operators. Moreover, if we suppose ψ = ψ ( Φ ) {\displaystyle \psi =\psi (\Phi )} in the sense of matter-geometry interplay and assume asymptotic flatness, we will find that Eqs(7.a-e) implies a characteristic relation that
Specifically in the simplest vacuum case with Φ = 0 {\displaystyle \Phi =0} and T a b = 0 {\displaystyle T_{ab}=0} , Eqs(7.a-7.e) reduce to [ 4 ]
We can firstly obtain ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} by solving Eq(8.b), and then integrate Eq(8.c) and Eq(8.d) for γ ( ρ , z ) {\displaystyle \gamma (\rho ,z)} . Practically, Eq(8.a) arising from R = 0 {\displaystyle R=0} just works as a consistency relation or integrability condition .
Unlike the nonlinear Poisson's equation Eq(7.b), Eq(8.b) is the linear Laplace equation ; that is to say, superposition of given vacuum solutions to Eq(8.b) is still a solution. This fact has a widely application, such as to analytically distort a Schwarzschild black hole .
We employed the axisymmetric Laplace and gradient operators to write Eqs(7.a-7.e) and Eqs(8.a-8.d) in a compact way, which is very useful in the derivation of the characteristic relation Eq(7.f). In the literature, Eqs(7.a-7.e) and Eqs(8.a-8.d) are often written in the following forms as well:
and
Considering the interplay between spacetime geometry and energy-matter distributions, it is natural to assume that in Eqs(7.a-7.e) the metric function ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} relates with the electrostatic scalar potential Φ ( ρ , z ) {\displaystyle \Phi (\rho ,z)} via a function ψ = ψ ( Φ ) {\displaystyle \psi =\psi (\Phi )} (which means geometry depends on energy), and it follows that
Eq(B.1) immediately turns Eq(7.b) and Eq(7.e) respectively into
which give rise to
Now replace the variable ψ {\displaystyle \psi } by ζ := e 2 ψ {\displaystyle \zeta :=e^{2\psi }} , and Eq(B.4) is simplified to
Direct quadrature of Eq(B.5) yields ζ = e 2 ψ = Φ 2 + C ~ Φ + B {\displaystyle \zeta =e^{2\psi }=\Phi ^{2}+{\tilde {C}}\Phi +B} , with { B , C ~ } {\displaystyle \{B,{\tilde {C}}\}} being integral constants. To resume asymptotic flatness at spatial infinity, we need lim ρ , z → ∞ Φ = 0 {\displaystyle \lim _{\rho ,z\to \infty }\Phi =0} and lim ρ , z → ∞ e 2 ψ = 1 {\displaystyle \lim _{\rho ,z\to \infty }e^{2\psi }=1} , so there should be B = 1 {\displaystyle B=1} . Also, rewrite the constant C ~ {\displaystyle {\tilde {C}}} as − 2 C {\displaystyle -2C} for mathematical convenience in subsequent calculations, and one finally obtains the characteristic relation implied by Eqs(7.a-7.e) that
This relation is important in linearize the Eqs(7.a-7.f) and superpose electrovac Weyl solutions.
In Weyl's metric Eq(1), e ± 2 ψ = ∑ n = 0 ∞ ( ± 2 ψ ) n n ! {\textstyle e^{\pm 2\psi }=\sum _{n=0}^{\infty }{\frac {(\pm 2\psi )^{n}}{n!}}} ; thus in the approximation for weak field limit ψ → 0 {\displaystyle \psi \to 0} , one has
and therefore
This is pretty analogous to the well-known approximate metric for static and weak gravitational fields generated by low-mass celestial bodies like the Sun and Earth, [ 5 ]
where Φ N ( ρ , z ) {\displaystyle \Phi _{N}(\rho ,z)} is the usual Newtonian potential satisfying Poisson's equation ∇ L 2 Φ N = 4 π ϱ N {\displaystyle \nabla _{L}^{2}\Phi _{N}=4\pi \varrho _{N}} , just like Eq(3.a) or Eq(4.a) for the Weyl metric potential ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} . The similarities between ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} and Φ N ( ρ , z ) {\displaystyle \Phi _{N}(\rho ,z)} inspire people to find out the Newtonian analogue of ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} when studying Weyl class of solutions; that is, to reproduce ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} nonrelativistically by certain type of Newtonian sources. The Newtonian analogue of ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} proves quite helpful in specifying particular Weyl-type solutions and extending existing Weyl-type solutions. [ 2 ]
The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq( 8 ) are given by [ 2 ] [ 3 ] [ 4 ]
where
From the perspective of Newtonian analogue, ψ S S {\displaystyle \psi _{SS}} equals the gravitational potential produced by a rod of mass M {\displaystyle M} and length 2 M {\displaystyle 2M} placed symmetrically on the z {\displaystyle z} -axis; that is, by a line mass of uniform density σ = 1 / 2 {\displaystyle \sigma =1/2} embedded the interval z ∈ [ − M , M ] {\displaystyle z\in [-M,M]} . (Note: Based on this analogue, important extensions of the Schwarzschild metric have been developed, as discussed in ref. [ 2 ] )
Given ψ S S {\displaystyle \psi _{SS}} and γ S S {\displaystyle \gamma _{SS}} , Weyl's metric Eq( 1 ) becomes
and after substituting the following mutually consistent relations
one can obtain the common form of Schwarzschild metric in the usual { t , r , θ , ϕ } {\displaystyle \{t,r,\theta ,\phi \}} coordinates,
The metric Eq( 14 ) cannot be directly transformed into Eq( 16 ) by performing the standard cylindrical-spherical transformation ( t , ρ , z , ϕ ) = ( t , r sin θ , r cos θ , ϕ ) {\displaystyle (t,\rho ,z,\phi )=(t,r\sin \theta ,r\cos \theta ,\phi )} , because { t , r , θ , ϕ } {\displaystyle \{t,r,\theta ,\phi \}} is complete while ( t , ρ , z , ϕ ) {\displaystyle (t,\rho ,z,\phi )} is incomplete. This is why we call { t , ρ , z , ϕ } {\displaystyle \{t,\rho ,z,\phi \}} in Eq( 1 ) as Weyl's canonical coordinates rather than cylindrical coordinates, although they have a lot in common; for example, the Laplacian ∇ 2 := ∂ ρ ρ + 1 ρ ∂ ρ + ∂ z z {\displaystyle \nabla ^{2}:=\partial _{\rho \rho }+{\frac {1}{\rho }}\partial _{\rho }+\partial _{zz}} in Eq( 7 ) is exactly the two-dimensional geometric Laplacian in cylindrical coordinates.
The Weyl potentials generating the nonextremal Reissner–Nordström solution ( M > | Q | {\displaystyle M>|Q|} ) as solutions to Eqs( 7 ) are given by [ 2 ] [ 3 ] [ 4 ]
where
Thus, given ψ R N {\displaystyle \psi _{RN}} and γ R N {\displaystyle \gamma _{RN}} , Weyl's metric becomes
and employing the following transformations
one can obtain the common form of non-extremal Reissner–Nordström metric in the usual { t , r , θ , ϕ } {\displaystyle \{t,r,\theta ,\phi \}} coordinates,
The potentials generating the extremal Reissner–Nordström solution ( M = | Q | {\displaystyle M=|Q|} ) as solutions to Eqs( 7 ) are given by [ 4 ] (Note: We treat the extremal solution separately because it is much more than the degenerate state of the nonextremal counterpart.)
Thus, the extremal Reissner–Nordström metric reads
and by substituting
we obtain the extremal Reissner–Nordström metric in the usual { t , r , θ , ϕ } {\displaystyle \{t,r,\theta ,\phi \}} coordinates,
Mathematically, the extremal Reissner–Nordström can be obtained by taking the limit Q → M {\displaystyle Q\to M} of the corresponding nonextremal equation, and in the meantime we need to use the L'Hospital rule sometimes.
Remarks: Weyl's metrics Eq( 1 ) with the vanishing potential γ ( ρ , z ) {\displaystyle \gamma (\rho ,z)} (like the extremal Reissner–Nordström metric) constitute a special subclass which have only one metric potential ψ ( ρ , z ) {\displaystyle \psi (\rho ,z)} to be identified. Extending this subclass by canceling the restriction of axisymmetry, one obtains another useful class of solutions (still using Weyl's coordinates), namely the conformastatic metrics, [ 6 ] [ 7 ]
where we use λ {\displaystyle \lambda } in Eq( 22 ) as the single metric function in place of ψ {\displaystyle \psi } in Eq( 1 ) to emphasize that they are different by axial symmetry ( ϕ {\displaystyle \phi } -dependence).
Weyl's metric can also be expressed in spherical coordinates that
which equals Eq( 1 ) via the coordinate transformation ( t , ρ , z , ϕ ) ↦ ( t , r sin θ , r cos θ , ϕ ) {\displaystyle (t,\rho ,z,\phi )\mapsto (t,r\sin \theta ,r\cos \theta ,\phi )} (Note: As shown by Eqs( 15 )( 21 )( 24 ), this transformation is not always applicable.) In the vacuum case, Eq( 8.b ) for ψ ( r , θ ) {\displaystyle \psi (r,\theta )} becomes
The asymptotically flat solutions to Eq( 28 ) is [ 2 ]
where P n ( cos θ ) {\displaystyle P_{n}(\cos \theta )} represent Legendre polynomials , and a n {\displaystyle a_{n}} are multipole coefficients. The other metric potential γ ( r , θ ) {\displaystyle \gamma (r,\theta )} is given by [ 2 ]
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Weyl semimetals are semimetals or metals whose quasiparticle excitation is the Weyl fermion , a particle that played a crucial role in quantum field theory but has not been observed as a fundamental particle in vacuum. [ 1 ] In these materials, electrons have a linear dispersion relation, making them a solid-state analogue of relativistic massless particles. [ 2 ]
Weyl fermions are massless chiral fermions embodying the mathematical concept of a Weyl spinor . Weyl spinors in turn play an important role in quantum field theory and the Standard Model , where they are a building block for fermions in quantum field theory. Weyl spinors are a solution to the Dirac equation derived by Hermann Weyl , called the Weyl equation . [ 3 ] For example, one-half of a charged Dirac fermion of a definite chirality is a Weyl fermion. [ 4 ]
Weyl fermions may be realized as emergent quasiparticles in a low-energy condensed matter system. This prediction was first proposed by Conyers Herring in 1937, in the context of electronic band structures of solid state systems such as electronic crystals. [ 5 ] [ 6 ] Topological materials in the vicinity of band inversion transition became a primary target in search of topologically protected bulk electronic band crossings. [ 7 ]
The first (non-electronic) liquid state which is suggested, has similarly emergent but neutral excitation and theoretically interpreted superfluid 's chiral anomaly as observation of Fermi points is in Helium-3 A superfluid phase. [ 8 ] [ non-primary source needed ] Crystalline tantalum arsenide (TaAs) is the first discovered topological Weyl fermion semimetal which exhibits topological surface Fermi arcs where Weyl fermion is electrically charged along the line of original suggestion by Herring. [ 6 ] [ 9 ] An electronic Weyl fermion is not only charged but stable at room temperature where there is no such superfluid or liquid state known. [ citation needed ]
A Weyl semimetal is a solid state crystal whose low energy excitations are Weyl fermions that carry electrical charge even at room temperatures. [ 11 ] [ 12 ] [ 13 ] A Weyl semimetal enables realization of Weyl fermions in electronic systems. [ 9 ] It is a topologically nontrivial phase of matter, together with Helium-3 A superfluid phase, that broadens the topological classification beyond topological insulators. [ 14 ] The Weyl fermions at zero energy correspond to points of bulk band degeneracy, the Weyl nodes (or Fermi points) that are separated in momentum space . Weyl fermions have distinct chiralities, either left handed or right handed.
In a Weyl semimetal crystal, the chiralities associated with the Weyl nodes (Fermi points) can be understood as topological charges, leading to monopoles and anti-monopoles of Berry curvature in momentum space , which (the splitting) serve as the topological invariant of this phase. [ 11 ] Comparable to the Dirac fermions in graphene or on the surface of topological insulators , Weyl fermions in a Weyl semimetal are the most robust electrons and do not depend on symmetries except the translation symmetry of the crystal lattice. Hence the Weyl fermion quasiparticles in a Weyl semimetal possess a high degree of mobility. Due to the nontrivial topology, a Weyl semimetal is expected to demonstrate Fermi arc electron states on its surface. [ 9 ] [ 11 ] These arcs are discontinuous or disjoint segments of a two dimensional Fermi contour, which are terminated onto the projections of the Weyl fermion nodes on the surface. A 2012 theoretical investigation of superfluid Helium-3 [ 15 ] suggested Fermi arcs in neutral superfluids.
On 16 July 2015 the first experimental observations of Weyl fermion semimetal and topological Fermi arcs in an inversion symmetry-breaking single crystal material tantalum arsenide (TaAs) were made. [ 9 ] Both Weyl fermions and Fermi arc surface states were observed using direct electronic imaging using ARPES , which established its topological character for the first time. [ 9 ] This discovery was built upon previous theoretical predictions proposed in November 2014 by a team led by Bangladeshi scientist M Zahid Hasan . [ 16 ] [ 17 ]
Weyl points (Fermi points) were also observed in non-electronic systems such as photonic crystals, in fact even before their experimental observation in electronic systems [ 18 ] [ 19 ] [ 20 ] and Helium-3 superfluid quasiparticle spectrum (neutral fermions). [ 21 ] Note that while these systems are different from electronic condensed matter systems, the basic physics is very similar.
TaAs is the first discovered Weyl semimetal (conductor). Large-size (~1 cm), high-quality TaAs single crystals [ 22 ] can be obtained by
chemical vapor transport method using iodine as the transport agent.
TaAs crystallizes in a body-centered tetragonal unit cell with lattice constants a = 3.44 Å and c = 11.64 Å and space group I41md (No. 109). Ta and As atoms are six coordinated to each other. This structure lacks a horizontal mirror plane and thus inversion symmetry, which is essential to realize Weyl semimetal.
TaAs single crystals have shiny facets, which can be divided into three groups: the two truncated surfaces are {001}, the trapezoid or isosceles triangular surfaces are {101}, and the rectangular ones {112}. TaAs belongs to point group 4mm, the equivalent {101} and {112} planes should form a ditetragonal appearance. The observed morphology can be vary of degenerated cases of the ideal form.
Beside the initial discovery of TaAs as Weyl semimetal, many other materials such as Co 2 TiGe, MoTe 2 , WTe 2 , LaAlGe and PrAlGe have been identified to exhibit Weyl semimetallic behavior. [ 23 ] [ 24 ]
The Weyl fermions in the bulk and the Fermi arcs on the surfaces of Weyl semimetals are of interest in physics and materials technology. [ 3 ] [ 25 ] The high mobility of charged Weyl fermions may find use in electronics and computing.
In 2017, [ 26 ] a research team from Vienna University of Technology carrying out experimental work to develop new materials, and a team at Rice University carrying out theoretical work, have produced material which they term Weyl–Kondo semimetals. [ 27 ]
A group of international researchers led by a team from Boston College discovered in 2019 that the Weyl semimetal Tantalum Arsenide delivers the largest intrinsic conversion of light to electricity of any material, more than ten times larger than previously achieved. [ 28 ]
2D Weyl semimetals are spin-polarized analogues of graphene that promise access to topological properties of Weyl fermions in (2+1)-dim spacetime. In 2024, an intrinsic 2D Weyl semimetal with spin-polarized Weyl cones and topological Fermi string edge states was discovered in epitaxial monolayer bismuthene by a team from University of Missouri, National Cheng Kung University, and Oak Ridge National Laboratory . [ 29 ]
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In mathematics , a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl : [ 1 ]
The sequence of all multiples of an irrational α ,
In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval [0, 1).
In computing , an integer version of this sequence is often used to generate a discrete uniform distribution rather than a continuous one. Instead of using an irrational number, which cannot be calculated on a digital computer, the ratio of two integers is used in its place. An integer k is chosen, relatively prime to an integer modulus m . In the common case that m is a power of 2, this amounts to requiring that k is odd.
The sequence of all multiples of such an integer k ,
That is, the sequence of the remainders of each term when divided by m will be uniformly distributed in the interval [0, m ).
The term appears to originate with George Marsaglia ’s paper " Xorshift RNGs". [ 3 ] The following C code generates what Marsaglia calls a "Weyl sequence":
In this case, the odd integer is 362437, and the results are computed modulo m = 2 32 because d is a 32-bit quantity. The results are equidistributed modulo 2 32 .
This mathematics -related article is a stub . You can help Wikipedia by expanding it .
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