text stringlengths 2.13k 184k | source stringlengths 31 108 |
|---|---|
In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications.
In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules:
- Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa
- The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match)
- The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match)
- The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if
$$
j > i
$$
are indices from the first sequence, then there must not be two indices
$$
l > k
$$
in the other sequence, such that index
$$
i
$$
is matched with index
$$
l
$$
and index
$$
j
$$
is matched with index
$$
k
$$
, and vice versa
We can plot each match between the sequences
$$
1:M
$$
and
$$
1:N
$$
as a path in a
$$
M\times N
$$
matrix from
$$
(1, 1)
$$
to
$$
(M, N)
$$
, such that each step is one of
$$
(0, 1), (1, 0), (1, 1)
$$
. In this formulation, we see that the number of possible matches is the Delannoy number.
The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values.
The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold.
In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section.
This is conceptually very similar to the Needleman–Wunsch algorithm.
## Implementation
This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, `d(x, y)` is a distance between the symbols, e.g. `d(x, y)` =
$$
| x - y |
$$
.
int DTWDistance(s: array [1..n], t: array [1..m]) {
DTW := array [0..n, 0..m]
for i := 0 to n
for j := 0 to m
DTW[i, j] := infinity
DTW[0, 0] := 0
for i := 1 to n
for j := 1 to m
cost := d(s[i], t[j])
DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion
DTW[i , j-1], // deletion
DTW[i-1, j-1]) // match
return DTW[n, m]
}
where `DTW[i, j]` is the distance between `s[1:i]` and `t[1:j]` with the best alignment.
We sometimes want to add a locality constraint. That is, we require that if `s[i]` is matched with `t[j]`, then
$$
| i - j |
$$
is no larger than w, a window parameter.
We can easily modify the above algorithm to add a locality constraint (differences marked).
However, the above given modification works only if
$$
| n - m |
$$
is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that
$$
| n - m | \le w
$$
(see the line marked with (*) in the code).
int DTWDistance(s: array [1..n], t: array [1..m], w: int) {
DTW := array [0..n, 0..m]
w := max(w, abs(n-m)) // adapt window size (*)
for i := 0 to n
for j:= 0 to m
DTW[i, j] := infinity
DTW[0, 0] := 0
for i := 1 to n
for j := max(1, i-w) to min(m, i+w)
DTW[i, j] := 0
for i := 1 to n
for j := max(1, i-w) to min(m, i+w)
cost := d(s[i], t[j])
DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion
DTW[i , j-1], // deletion
DTW[i-1, j-1]) // match
return DTW[n, m]
}
## Warping properties
The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements.
## Complexity
The time complexity of the DTW algorithm is
$$
O(NM)
$$
, where
$$
N
$$
and
$$
M
$$
are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in
$$
O({N^2}/\log \log N)
$$
time and space for two input sequences of length
$$
N
$$
.
This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form
$$
O(N^{2-\epsilon})
$$
for some
$$
\epsilon > 0
$$
cannot exist unless the Strong exponential time hypothesis fails.
While the dynamic programming algorithm for DTW requires
$$
O(NM)
$$
space in a naive implementation, the space consumption can be reduced to
$$
O(\min(N,M))
$$
using Hirschberg's algorithm.
## Fast computation
Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW.Thomas Prätzlich, Jonathan Driedger, and Meinard Müller (2016). Memory-Restricted Multiscale Dynamic Time Warping. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 569—573.
A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective.
In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time.
## Average sequence
Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences.
NLAAF is an exact method to average two sequences using DTW.
For more than two sequences, the problem is related to that of multiple alignment and requires heuristics.
DBA is currently a reference method to average a set of sequences consistently with DTW.
COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process.
## Supervised learning
A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure.
## Amerced Dynamic Time Warping
Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks.
## Alternative approaches
In functional data analysis, time series are regarded as discretizations of smooth (differentiable) functions of time. By viewing the observed samples at smooth functions, one can utilize continuous mathematics for analyzing data. Smoothness and monotonicity of time warp functions may be obtained for instance by integrating a time-varying radial basis function, thus being a one-dimensional diffeomorphism. Optimal nonlinear time warping functions are computed by minimizing a measure of distance of the set of functions to their warped average. Roughness penalty terms for the warping functions may be added, e.g., by constraining the size of their curvature. The resultant warping functions are smooth, which facilitates further processing. This approach has been successfully applied to analyze patterns and variability of speech movements.
Another related approach are hidden Markov models (HMM) and it has been shown that the Viterbi algorithm used to search for the most likely path through the HMM is equivalent to stochastic DTW.
DTW and related warping methods are typically used as pre- or post-processing steps in data analyses. If the observed sequences contain both random variation in both their values, shape of observed sequences and random temporal misalignment, the warping may overfit to noise leading to biased results. A simultaneous model formulation with random variation in both values (vertical) and time-parametrization (horizontal) is an example of a nonlinear mixed-effects model. In human movement analysis, simultaneous nonlinear mixed-effects modeling has been shown to produce superior results compared to DTW.
## Open-source software
- The tempo C++ library with Python bindings implements Early Abandoned and Pruned DTW as well as Early Abandoned and Pruned ADTW and DTW lower bounds LB_Keogh, LB_Enhanced and LB_Webb.
- The UltraFastMPSearch Java library implements the UltraFastWWSearch algorithm for fast warping window tuning.
- The lbimproved C++ library implements Fast Nearest-Neighbor Retrieval algorithms under the GNU General Public License (GPL). It also provides a C++ implementation of dynamic time warping, as well as various lower bounds.
- The FastDTW library is a Java implementation of DTW and a FastDTW implementation that provides optimal or near-optimal alignments with an O(N) time and memory complexity, in contrast to the O(N2) requirement for the standard DTW algorithm. FastDTW uses a multilevel approach that recursively projects a solution from a coarser resolution and refines the projected solution.
- FastDTW fork (Java) published to Maven Central.
- time-series-classification (Java) a package for time series classification using DTW in Weka.
- The DTW suite provides Python (dtw-python) and R packages (dtw) with a comprehensive coverage of the DTW algorithm family members, including a variety of recursion rules (also called step patterns), constraints, and substring matching.
- The mlpy Python library implements DTW.
- The pydtw Python library implements the Manhattan and Euclidean flavoured DTW measures including the LB_Keogh lower bounds.
- The cudadtw C++/CUDA library implements subsequence alignment of Euclidean-flavoured DTW and z-normalized Euclidean distance similar to the popular UCR-Suite on CUDA-enabled accelerators.
- The JavaML machine learning library implements DTW.
- The ndtw C# library implements DTW with various options.
- Sketch-a-Char uses Greedy DTW (implemented in JavaScript) as part of LaTeX symbol classifier program.
- The MatchBox implements DTW to match mel-frequency cepstral coefficients of audio signals.
- Sequence averaging: a GPL Java implementation of DBA.
- The Gesture Recognition Toolkit|GRT C++ real-time gesture-recognition toolkit implements DTW.
- The PyHubs software package implements DTW and nearest-neighbour classifiers, as well as their extensions (hubness-aware classifiers).
- The simpledtw Python library implements the classic O(NM) Dynamic Programming algorithm and bases on Numpy. It supports values of any dimension, as well as using custom norm functions for the distances. It is licensed under the MIT license.
- The tslearn Python library implements DTW in the time-series context.
- The cuTWED CUDA Python library implements a state of the art improved Time Warp Edit Distance using only linear memory with phenomenal speedups.
- DynamicAxisWarping.jl Is a Julia implementation of DTW and related algorithms such as FastDTW, SoftDTW, GeneralDTW and DTW barycenters.
- The Multi_DTW implements DTW to match two 1-D arrays or 2-D speech files (2-D array).
- The dtwParallel (Python) package incorporates the main functionalities available in current DTW libraries and novel functionalities such as parallelization, computation of similarity (kernel-based) values, and consideration of data with different types of features (categorical, real-valued, etc.).
## Applications
### Spoken-word recognition
Due to different speaking rates, a non-linear fluctuation occurs in speech pattern versus time axis, which needs to be eliminated. DP matching is a pattern-matching algorithm based on dynamic programming (DP), which uses a time-normalization effect, where the fluctuations in the time axis are modeled using a non-linear time-warping function. Considering any two speech patterns, we can get rid of their timing differences by warping the time axis of one so that the maximal coincidence is attained with the other. Moreover, if the warping function is allowed to take any possible value, distinction can be made between words belonging to different categories. So, to enhance the distinction between words belonging to different categories, restrictions were imposed on the warping function slope.
### Correlation power analysis
Unstable clocks are used to defeat naive power analysis. Several techniques are used to counter this defense, one of which is dynamic time warping.
### Finance and econometrics
Dynamic time warping is used in finance and econometrics to assess the quality of the prediction versus real-world data. | https://en.wikipedia.org/wiki/Dynamic_time_warping |
In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume).
The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions.
Under a Poisson distribution with the expectation of λ events in a given interval, the probability of k events in the same interval is:
$$
\frac{\lambda^k e^{-\lambda}}{k!} .
$$
For instance, consider a call center which receives an average of λ = 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next one will arrive. Under these assumptions, the number k of calls received during any minute has a Poisson probability distribution. Receiving k = 1 to 4 calls then has a probability of about 0.77, while receiving 0 or at least 5 calls has a probability of about 0.23.
A classic example used to motivate the Poisson distribution is the number of radioactive decay events during a fixed observation period.
## History
The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837). The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus . This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.
In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.
A further practical application was made by Ladislaus Bortkiewicz in 1898. Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution..
## Definitions
### Probability mass function
A discrete random variable is said to have a Poisson distribution with parameter
$$
\lambda>0
$$
if it has a probability mass function given by:
$$
f(k; \lambda) = \Pr(X{=}k)= \frac{\lambda^k e^{-\lambda}}{k!},
$$
where
- is the number of occurrences (
$$
k = 0, 1, 2, \ldots
$$
)
- is Euler's number (
$$
e = 2.71828\ldots
$$
)
- k! = k(k–1) ··· (3)(2)(1) is the factorial.
The positive real number is equal to the expected value of and also to its variance.
$$
\lambda = \operatorname{E}(X) = \operatorname{Var}(X).
$$
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution.
The equation can be adapted if, instead of the average number of events
$$
\lambda,
$$
we are given the average rate
$$
r
$$
at which events occur. Then
$$
\lambda = r t,
$$
and:
$$
P(k \text{ events in interval } t) = \frac{(rt)^k e^{-rt}}{k!}.
$$
### Examples
The Poisson distribution may be useful to model events such as:
- the number of meteorites greater than 1-meter diameter that strike Earth in a year;
- the number of laser photons hitting a detector in a particular time interval;
- the number of students achieving a low and high mark in an exam; and
- locations of defects and dislocations in materials.
Examples of the occurrence of random points in space are: the locations of asteroid impacts with earth (2-dimensional), the locations of imperfections in a material (3-dimensional), and the locations of trees in a forest (2-dimensional).
### Assumptions and validity
The Poisson distribution is an appropriate model if the following assumptions are true:
- , a nonnegative integer, is the number of times an event occurs in an interval.
- The occurrence of one event does not affect the probability of a second event.
- The average rate at which events occur is independent of any occurrences.
- Two events cannot occur at exactly the same instant.
If these conditions are true, then is a Poisson random variable; the distribution of is a Poisson distribution.
The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals divided by the number of trials, as the number of trials approaches infinity (see
## Related distributions
).
#### Examples of probability for Poisson distributions
On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate.
Because the average event rate is one overflow flood per 100 years, = 1
$$
P(k \text{ overflow floods in 100 years}) = \frac{\lambda^k e^{-\lambda}}{k!} = \frac{1^k e^{-1}}{k!}
$$
$$
P(k = 0 \text{ overflow floods in 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{e^{-1}}{1} \approx 0.368
$$
$$
P(k = 1 \text{ overflow flood in 100 years}) = \frac{1^1 e^{-1}}{1!} = \frac{e^{-1}}{1} \approx 0.368
$$
$$
P(k = 2 \text{ overflow floods in 100 years}) = \frac{1^2 e^{-1}}{2!} = \frac{e^{-1}}{2} \approx 0.184
$$
{| class="wikitable"
|-
! !! ( overflow floods in 100 years)
|-
| 0|| 0.368
|-
| 1|| 0.368
|-
| 2|| 0.184
|-
| 3|| 0.061
|-
| 4|| 0.015
|-
| 5|| 0.003
|-
| 6|| 0.0005
|}
The probability for 0 to 6 overflow floods in a 100-year period.
In this example, it is reported that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate.
Because the average event rate is 2.5 goals per match, = 2.5 .
$$
P(k \text{ goals in a match}) = \frac{2.5^k e^{-2.5}}{k!}
$$
$$
P(k = 0 \text{ goals in a match}) = \frac{2.5^0 e^{-2.5}}{0!} = \frac{e^{-2.5}}{1} \approx 0.082
$$
$$
P(k = 1 \text{ goal in a match}) = \frac{2.5^1 e^{-2.5}}{1!} = \frac{2.5 e^{-2.5}}{1} \approx 0.205
$$
$$
P(k = 2 \text{ goals in a match}) = \frac{2.5^2 e^{-2.5}}{2!} = \frac{6.25 e^{-2.5}}{2} \approx 0.257
$$
{| class="wikitable"
|-
! !! ( goals in a World Cup soccer match)
|-
| 0|| 0.082
|-
| 1|| 0.205
|-
| 2|| 0.257
|-
| 3|| 0.213
|-
| 4|| 0.133
|-
| 5|| 0.067
|-
| 6|| 0.028
|-
| 7|| 0.010
|}
The probability for 0 to 7 goals in a match.
#### Once in an interval events: The special case of = 1 and = 0
Suppose that astronomers estimate that large meteorites (above a certain size) hit the earth on average once every 100 years ( event per 100 years), and that the number of meteorite hits follows a Poisson distribution. What is the probability of meteorite hits in the next 100 years?
$$
P(k = \text{0 meteorites hit in next 100 years}) = \frac{1^0 e^{-1}}{0!} = \frac{1}{e} \approx 0.37.
$$
Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years.
In an example above, an overflow flood occurred once every 100 years The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation.
In general, if an event occurs on average once per interval ( = 1), and the events follow a Poisson distribution, then In addition, as shown in the table for overflow floods.
### Examples that violate the Poisson assumptions
The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups). The non-constant arrival rate may be modeled as a mixed Poisson distribution, and the arrival of groups rather than individual students as a compound Poisson process.
The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution, if one large earthquake increases the probability of aftershocks of similar magnitude.
Examples in which at least one event is guaranteed are not Poisson distributed; but may be modeled using a zero-truncated Poisson distribution.
Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a zero-inflated model.
## Properties
### Descriptive statistics
- The expected value of a Poisson random variable is .
- The variance of a Poisson random variable is also .
- The coefficient of variation is
$$
\lambda^{-1/2},
$$
while the index of dispersion is 1.
- The mean absolute deviation about the mean is
$$
\operatorname{E}[\ |X-\lambda|\ ]= \frac{2 \lambda^{\lfloor\lambda\rfloor + 1} e^{-\lambda}}{\lfloor\lambda\rfloor!}.
$$
- The mode of a Poisson-distributed random variable with non-integer is equal to
$$
\lfloor \lambda \rfloor,
$$
which is the largest integer less than or equal to . This is also written as floor(). When is a positive integer, the modes are and − 1.
- All of the cumulants of the Poisson distribution are equal to the expected value . The th factorial moment of the Poisson distribution is .
- The expected value of a Poisson process is sometimes decomposed into the product of intensity and exposure (or more generally expressed as the integral of an "intensity function" over time or space, sometimes described as "exposure").
### Median
Bounds for the median (
$$
\nu
$$
) of the distribution are known and are sharp:
$$
\lambda - \ln 2 \le \nu < \lambda + \frac{1}{3}.
$$
### Higher moments
The higher non-centered moments of the Poisson distribution are Touchard polynomials in :
$$
m_k = \sum_{i=0}^k \lambda^i \begin{Bmatrix} k \\ i \end{Bmatrix},
$$
where the braces { } denote Stirling numbers of the second kind. In other words,
$$
E[X] = \lambda, \quad E[X(X-1)] = \lambda^2, \quad E[X(X-1)(X-2)] = \lambda^3, \cdots
$$
When the expected value is set to λ = 1, Dobinski's formula implies that the ‑th moment is equal to the number of partitions of a set of size .
A simple upper bound is:
$$
m_k = E[X^k] \le \left(\frac{k}{\log(k/\lambda+1)}\right)^k \le \lambda^k \exp\left(\frac{k^2}{2\lambda}\right).
$$
### Sums of Poisson-distributed random variables
If
$$
X_i \sim \operatorname{Pois}(\lambda_i)
$$
for
$$
i=1,\dotsc,n
$$
are independent, then
$$
\sum_{i=1}^n X_i \sim \operatorname{Pois}\left(\sum_{i=1}^n \lambda_i\right).
$$
A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.
### Maximum entropy
It is a maximum-entropy distribution among the set of generalized binomial distributions
$$
B_n(\lambda)
$$
with mean
$$
\lambda
$$
and
$$
n\rightarrow\infty
$$
, where a generalized binomial distribution is defined as a distribution of the sum of N independent but not identically distributed Bernoulli variables.
### Other properties
- The Poisson distributions are infinitely divisible probability distributions.
- The directed Kullback–Leibler divergence of
$$
P=\operatorname{Pois}(\lambda)
$$
from
$$
P_0=\operatorname{Pois}(\lambda_0)
$$
is given by
$$
\operatorname{D}_{\text{KL}}(P\parallel P_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.
$$
- If
$$
\lambda \geq 1
$$
is an integer, then
$$
Y\sim \operatorname{Pois}(\lambda)
$$
satisfies
$$
\Pr(Y \geq E[Y]) \geq \frac{1}{2}
$$
and
$$
\Pr(Y \leq E[Y]) \geq \frac{1}{2}.
$$
- Bounds for the tail probabilities of a Poisson random variable
$$
X \sim \operatorname{Pois}(\lambda)
$$
can be derived using a Chernoff bound argument.
$$
P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda,
$$
$$
P(X \leq x) \leq \frac{(e \lambda)^x e^{-\lambda} }{x^x}, \text{ for } x < \lambda.
$$
- The upper tail probability can be tightened (by a factor of at least two) as follows:
$$
P(X \geq x) \leq \frac{e^{-\operatorname{D}_{\text{KL}}(Q\parallel P)}}{\max{(2, \sqrt{4\pi\operatorname{D}_{\text{KL}}(Q\parallel P)}})}, \text{ for } x > \lambda,
$$
where
$$
\operatorname{D}_{\text{KL}}(Q\parallel P)
$$
is the Kullback–Leibler divergence of
$$
Q=\operatorname{Pois}(x)
$$
from
$$
P=\operatorname{Pois}(\lambda)
$$
.
- Inequalities that relate the distribution function of a Poisson random variable
$$
X \sim \operatorname{Pois}(\lambda)
$$
to the Standard normal distribution function
$$
\Phi(x)
$$
are as follows:
$$
\Phi\left(\operatorname{sign}(k-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(Q_-\parallel P)}\right) < P(X \leq k) < \Phi\left(\operatorname{sign}(k+1-\lambda)\sqrt{2\operatorname{D}_{\text{KL}}(Q_+\parallel P)}\right), \text{ for } k > 0,
$$
where
$$
\operatorname{D}_{\text{KL}}(Q_-\parallel P)
$$
is the Kullback–Leibler divergence of
$$
Q_-=\operatorname{Pois}(k)
$$
from
$$
P=\operatorname{Pois}(\lambda)
$$
and
$$
\operatorname{D}_{\text{KL}}(Q_+\parallel P)
$$
is the Kullback–Leibler divergence of
$$
Q_+=\operatorname{Pois}(k+1)
$$
from
$$
P
$$
.
### Poisson races
Let
$$
X \sim \operatorname{Pois}(\lambda)
$$
and
$$
Y \sim \operatorname{Pois}(\mu)
$$
be independent random variables, with
$$
\lambda < \mu,
$$
then we have that
$$
\frac{e^{-(\sqrt{\mu} -\sqrt{\lambda})^2 }}{(\lambda + \mu)^2} - \frac{e^{-(\lambda + \mu)}}{2\sqrt{\lambda \mu}} - \frac{e^{-(\lambda + \mu)}}{4\lambda \mu} \leq P(X - Y \geq 0) \leq e^{- (\sqrt{\mu} -\sqrt{\lambda})^2}
$$
The upper bound is proved using a standard Chernoff bound.
The lower bound can be proved by noting that
$$
P(X-Y\geq0\mid X+Y=i)
$$
is the probability that
$$
Z \geq \frac{i}{2},
$$
where
$$
Z \sim \operatorname{Bin}\left(i, \frac{\lambda}{\lambda+\mu}\right),
$$
which is bounded below by
$$
\frac{1}{(i+1)^2} e^{-iD\left(0.5 \| \frac{\lambda}{\lambda+\mu}\right)},
$$
where
$$
D
$$
is relative entropy (See the entry on bounds on tails of binomial distributions for details). Further noting that
$$
X+Y \sim \operatorname{Pois}(\lambda+\mu),
$$
and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath et al.
Related distributions
### As a Binomial distribution with infinitesimal time-steps
The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore, it can be used as an approximation of the binomial distribution if is sufficiently large and p is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if ≥ 100 and ≤ 10. Letting
$$
F_{\mathrm B}
$$
and
$$
F_{\mathrm P}
$$
be the respective cumulative density functions of the binomial and Poisson distributions, one has:
$$
F_\mathrm{B}(k;n, p) \ \approx\ F_\mathrm{P}(k;\lambda=np).
$$
One derivation of this uses probability-generating functions. Consider a Bernoulli trial (coin-flip) whose probability of one success (or expected number of successes) is
$$
\lambda \leq 1
$$
within a given interval. Split the interval into n parts, and perform a trial in each subinterval with probability
$$
\tfrac{ \lambda }{n}
$$
. The probability of k successes out of n trials over the entire interval is then given by the binomial distribution whose generating function is:Taking the limit as n increases to infinity (with x fixed) and applying the product limit definition of the exponential function, this reduces to the generating function of the Poisson distribution:
### General
- If
$$
X_1 \sim \mathrm{Pois}(\lambda_1)\,
$$
and
$$
X_2 \sim \mathrm{Pois}(\lambda_2)\,
$$
are independent, then the difference
$$
Y = X_1 - X_2
$$
follows a Skellam distribution.
- If
$$
X_1 \sim \mathrm{Pois}(\lambda_1)\,
$$
and
$$
X_2 \sim \mathrm{Pois}(\lambda_2)\,
$$
are independent, then the distribution of
$$
X_1
$$
conditional on
$$
X_1+X_2
$$
is a binomial distribution. Specifically, if
$$
X_1+X_2=k,
$$
then
$$
X_1| X_1+X_2=k\sim \mathrm{Binom}(k, \lambda_1/(\lambda_1+\lambda_2)).
$$
More generally, if X1, X2, ..., X are independent Poisson random variables with parameters 1, 2, ..., then
- : given
$$
\sum_{j=1}^n X_j=k,
$$
it follows that
$$
X_i\Big|\sum_{j=1}^n X_j=k \sim \mathrm{Binom}\left(k, \frac{\lambda_i}{\sum_{j=1}^n \lambda_j}\right).
$$
In fact,
$$
\{X_i\} \sim \mathrm{Multinom}\left(k, \left\{\frac{\lambda_i}{\sum_{j=1}^n\lambda_j}\right\}\right).
$$
- If
$$
X \sim \mathrm{Pois}(\lambda)\,
$$
and the distribution of
$$
Y
$$
conditional on X = is a binomial distribution,
$$
Y \mid (X = k) \sim \mathrm{Binom}(k, p),
$$
then the distribution of Y follows a Poisson distribution
$$
Y \sim \mathrm{Pois}(\lambda \cdot p).
$$
In fact, if, conditional on
$$
\{X = k\},
$$
$$
\{Y_i\}
$$
follows a multinomial distribution,
$$
\{Y_i\} \mid (X = k) \sim \mathrm{Multinom}\left(k, p_i\right),
$$
then each
$$
Y_i
$$
follows an independent Poisson distribution
$$
Y_i \sim \mathrm{Pois}(\lambda \cdot p_i), \rho(Y_i, Y_j) = 0.
$$
- The Poisson distribution is a special case of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter. The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a special case of a compound Poisson distribution.
- For sufficiently large values of , (say >1000), the normal distribution with mean and variance (standard deviation
$$
\sqrt{\lambda}
$$
) is an excellent approximation to the Poisson distribution. If is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed, i.e., if , where x is a non-negative integer, is replaced by .
$$
F_\mathrm{Poisson}(x;\lambda) \approx F_\mathrm{normal}(x;\mu=\lambda,\sigma^2=\lambda)
$$
- Variance-stabilizing transformation: If
$$
X \sim \mathrm{Pois}(\lambda),
$$
then
$$
Y = 2 \sqrt{X} \approx \mathcal{N}(2\sqrt{\lambda};1),
$$
and
$$
Y = \sqrt{X} \approx \mathcal{N}(\sqrt{\lambda};1/4).
$$
Under this transformation, the convergence to normality (as
$$
\lambda
$$
increases) is far faster than the untransformed variable. Other, slightly more complicated, variance stabilizing transformations are available, one of which is Anscombe transform. See Data transformation (statistics) for more general uses of transformations.
- If for every t > 0 the number of arrivals in the time interval follows the Poisson distribution with mean λt, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1/.
- The cumulative distribution functions of the Poisson and chi-squared distributions are related in the following ways:
$$
F_\text{Poisson}(k;\lambda) = 1-F_{\chi^2}(2\lambda;2(k+1)) \quad\quad \text{ integer } k,
$$
and
$$
P(X=k)=F_{\chi^2}(2\lambda;2(k+1)) -F_{\chi^2}(2\lambda;2k).
$$
### Poisson approximation
Assume
$$
X_1\sim\operatorname{Pois}(\lambda_1), X_2\sim\operatorname{Pois}(\lambda_2), \dots, X_n\sim\operatorname{Pois}(\lambda_n)
$$
where
$$
\lambda_1 + \lambda_2 + \dots + \lambda_n=1,
$$
then
$$
(X_1, X_2, \dots, X_n)
$$
is multinomially distributed
$$
(X_1, X_2, \dots, X_n) \sim \operatorname{Mult}(N, \lambda_1, \lambda_2, \dots, \lambda_n)
$$
conditioned on
$$
N = X_1 + X_2 + \dots X_n.
$$
This means, among other things, that for any nonnegative function
$$
f(x_1, x_2, \dots, x_n),
$$
if
$$
(Y_1, Y_2, \dots, Y_n)\sim\operatorname{Mult}(m, \mathbf{p})
$$
is multinomially distributed, then
$$
\operatorname{E}[f(Y_1, Y_2, \dots, Y_n)] \le e\sqrt{m}\operatorname{E}[f(X_1, X_2, \dots, X_n)]
$$
where
$$
(X_1, X_2, \dots, X_n)\sim\operatorname{Pois}(\mathbf{p}).
$$
The factor of
$$
e\sqrt{m}
$$
can be replaced by 2 if
$$
f
$$
is further assumed to be monotonically increasing or decreasing.
### Bivariate Poisson distribution
This distribution has been extended to the bivariate case. The generating function for this distribution is
$$
g( u, v ) = \exp[ ( \theta_1 - \theta_{12} )( u - 1 ) + ( \theta_2 - \theta_{12} )(v - 1) + \theta_{12} ( uv - 1 ) ]
$$
with
$$
\theta_1, \theta_2 > \theta_{ 12 } > 0
$$
The marginal distributions are Poisson(θ1) and Poisson(θ2) and the correlation coefficient is limited to the range
$$
0 \le \rho \le \min\left\{ \sqrt{ \frac{ \theta_1 }{ \theta_2 } }, \sqrt{ \frac{ \theta_2 }{ \theta_1 } } \right\}
$$
A simple way to generate a bivariate Poisson distribution
$$
X_1,X_2
$$
is to take three independent Poisson distributions
$$
Y_1,Y_2,Y_3
$$
with means
$$
\lambda_1,\lambda_2,\lambda_3
$$
and then set
$$
X_1 = Y_1 + Y_3, X_2 = Y_2 + Y_3.
$$
The probability function of the bivariate Poisson distribution is
$$
\Pr(X_1=k_1,X_2=k_2) =
\exp\left(-\lambda_1-\lambda_2-\lambda_3\right) \frac{\lambda_1^{k_1}}{k_1!} \frac{\lambda_2^{k_2}}{k_2!} \sum_{k=0}^{\min(k_1,k_2)} \binom{k_1}{k} \binom{k_2}{k} k! \left( \frac{\lambda_3}{\lambda_1\lambda_2}\right)^k
$$
### Free Poisson distribution
The free Poisson distribution with jump size
$$
\alpha
$$
and rate
$$
\lambda
$$
arises in free probability theory as the limit of repeated free convolution
$$
\left( \left(1-\frac{\lambda}{N}\right)\delta_0 + \frac{\lambda}{N}\delta_\alpha\right)^{\boxplus N}
$$
as .
In other words, let
$$
X_N
$$
be random variables so that
$$
X_N
$$
has value
$$
\alpha
$$
with probability
$$
\frac{\lambda}{N}
$$
and value 0 with the remaining probability. Assume also that the family
$$
X_1, X_2, \ldots
$$
are freely independent. Then the limit as
$$
N \to \infty
$$
of the law of
$$
X_1 + \cdots +X_N
$$
is given by the Free Poisson law with parameters
$$
\lambda,\alpha.
$$
This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.
The measure associated to the free Poisson law is given by
$$
\mu=\begin{cases}
(1-\lambda) \delta_0 + \nu,& \text{if } 0\leq \lambda \leq 1 \\
\nu, & \text{if }\lambda >1,
\end{cases}
$$
where
$$
\nu = \frac{1}{2\pi\alpha t}\sqrt{4\lambda \alpha^2 - ( t - \alpha (1+\lambda))^2} \, dt
$$
and has support
$$
[\alpha (1-\sqrt{\lambda})^2,\alpha (1+\sqrt{\lambda})^2].
$$
This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to
$$
\kappa_n=\lambda\alpha^n.
$$
#### Some transforms of this law
We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher
The R-transform of the free Poisson law is given by
$$
R(z)=\frac{\lambda \alpha}{1-\alpha z}.
$$
The Cauchy transform (which is the negative of the Stieltjes transformation) is given by
$$
G(z) = \frac{ z + \alpha - \lambda \alpha - \sqrt{ (z-\alpha (1+\lambda))^2 - 4 \lambda \alpha^2}}{2\alpha z}
$$
The S-transform is given by
$$
S(z) = \frac{1}{z+\lambda}
$$
in the case that
$$
\alpha = 1.
$$
### Weibull and stable count
Poisson's probability mass function
$$
f(k; \lambda)
$$
can be expressed in a form similar to the product distribution of a Weibull distribution and a variant form of the stable count distribution.
The variable
$$
(k+1)
$$
can be regarded as inverse of Lévy's stability parameter in the stable count distribution:
$$
f(k; \lambda) =
\int_0^\infty \frac{1}{u} \, W_{k+1} \left(\frac{\lambda}{u}\right)
\left[ (k+1) u^k \, \mathfrak{N}_{\frac{1}{k+1}} (u^{k+1}) \right] \, du ,
$$
where
$$
\mathfrak{N}_\alpha(\nu)
$$
is a standard stable count distribution of shape
$$
\alpha = 1/(k+1),
$$
and
$$
W_{k+1}(x)
$$
is a standard Weibull distribution of shape
$$
k+1.
$$
## Statistical inference
### Parameter estimation
Given a sample of measured values
$$
k_i \in \{0,1,\dots\},
$$
for we wish to estimate the value of the parameter of the Poisson population from which the sample was drawn. The maximum likelihood estimate is
$$
\widehat{\lambda}_\mathrm{MLE}=\frac{1}{n}\sum_{i=1}^n k_i\ .
$$
Since each observation has expectation , so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of . It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB). Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for .
To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample
$$
\mathbf{x}
$$
, called
$$
h(\mathbf{x})
$$
, and one that depends on the parameter
$$
\lambda
$$
and the sample
$$
\mathbf{x}
$$
only through the function
$$
T(\mathbf{x}).
$$
Then
$$
T(\mathbf{x})
$$
is a sufficient statistic for
$$
\lambda.
$$
$$
P(\mathbf{x})=\prod_{i=1}^n\frac{\lambda^{x_i} e^{-\lambda}}{x_i!}=\frac{1}{\prod_{i=1}^n x_i!} \times \lambda^{\sum_{i=1}^n x_i}e^{-n\lambda}
$$
The first term
$$
h(\mathbf{x})
$$
depends only on
$$
\mathbf{x}
$$
. The second term
$$
g(T(\mathbf{x})|\lambda)
$$
depends on the sample only through
$$
T(\mathbf{x})=\sum_{i=1}^n x_i.
$$
Thus,
$$
T(\mathbf{x})
$$
is sufficient.
To find the parameter that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:
$$
\begin{align}
\ell(\lambda) & = \ln \prod_{i=1}^n f(k_i \mid \lambda) \\
& = \sum_{i=1}^n \ln\!\left(\frac{e^{-\lambda}\lambda^{k_i}}{k_i!}\right) \\
& = -n\lambda + \left(\sum_{i=1}^n k_i\right) \ln(\lambda) - \sum_{i=1}^n \ln(k_i!).
\end{align}
$$
We take the derivative of
$$
\ell
$$
with respect to and compare it to zero:
$$
\frac{\mathrm{d}}{\mathrm{d}\lambda} \ell(\lambda) = 0 \iff -n + \left(\sum_{i=1}^n k_i\right) \frac{1}{\lambda} = 0. \!
$$
Solving for gives a stationary point.
$$
\lambda = \frac{\sum_{i=1}^n k_i}{n}
$$
So is the average of the i values. Obtaining the sign of the second derivative of L at the stationary point will determine what kind of extreme value is.
$$
\frac{\partial^2 \ell}{\partial \lambda^2} = -\lambda^{-2}\sum_{i=1}^n k_i
$$
Evaluating the second derivative at the stationary point gives:
$$
\frac{\partial^2 \ell}{\partial \lambda^2} = - \frac{n^2}{\sum_{i=1}^n k_i}
$$
which is the negative of times the reciprocal of the average of the ki. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.
For completeness, a family of distributions is said to be complete if and only if
$$
E(g(T)) = 0
$$
implies that
$$
P_\lambda(g(T) = 0) = 1
$$
for all
$$
\lambda.
$$
If the individual
$$
X_i
$$
are iid
$$
\mathrm{Po}(\lambda),
$$
then
$$
T(\mathbf{x})=\sum_{i=1}^n X_i\sim \mathrm{Po}(n\lambda).
$$
Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.
$$
E(g(T))=\sum_{t=0}^\infty g(t)\frac{(n\lambda)^te^{-n\lambda}}{t!} = 0
$$
For this equality to hold,
$$
g(t)
$$
must be 0. This follows from the fact that none of the other terms will be 0 for all
$$
t
$$
in the sum and for all possible values of
$$
\lambda.
$$
Hence,
$$
E(g(T)) = 0
$$
for all
$$
\lambda
$$
implies that
$$
P_\lambda(g(T) = 0) = 1,
$$
and the statistic has been shown to be complete.
### Confidence interval
The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation from a Poisson distribution with mean μ, a confidence interval for μ with confidence level is
$$
\tfrac {1}{2}\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac {1}{2} \chi^{2}(1-\alpha/2; 2k+2),
$$
or equivalently,
$$
F^{-1}(\alpha/2; k,1) \le \mu \le F^{-1}(1-\alpha/2; k+1,1),
$$
where
$$
\chi^{2}(p;n)
$$
is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with degrees of freedom and
$$
F^{-1}(p;n,1)
$$
is the quantile function of a gamma distribution with shape parameter n and scale parameter 1. This interval is 'exact' in the sense that its coverage probability is never less than the nominal .
When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation):
$$
k \left( 1 - \frac{1}{9k} - \frac{z_{\alpha/2}}{3\sqrt{k}}\right)^3 \le \mu \le (k+1) \left( 1 - \frac{1}{9(k+1)} + \frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^3,
$$
where
$$
z_{\alpha/2}
$$
denotes the standard normal deviate with upper tail area .
For application of these formulae in the same context as above (given a sample of measured values i each drawn from a Poisson distribution with mean ), one would set
$$
k=\sum_{i=1}^n k_i ,
$$
calculate an interval for and then derive the interval for .
### Bayesian inference
In Bayesian inference, the conjugate prior for the rate parameter of the Poisson distribution is the gamma distribution. Let
$$
\lambda \sim \mathrm{Gamma}(\alpha, \beta)
$$
denote that is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β:
$$
g(\lambda \mid \alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; \lambda^{\alpha-1} \; e^{-\beta\,\lambda} \qquad \text{ for } \lambda>0 \,\!.
$$
Then, given the same sample of measured values i as before, and a prior of Gamma(α, β), the posterior distribution is
$$
\lambda \sim \mathrm{Gamma}\left(\alpha + \sum_{i=1}^n k_i, \beta + n\right).
$$
Note that the posterior mean is linear and is given by
$$
E[ \lambda \mid k_1, \ldots, k_n ] = \frac{\alpha + \sum_{i=1}^n k_i}{\beta + n}.
$$
It can be shown that gamma distribution is the only prior that induces linearity of the conditional mean. Moreover, a converse result exists which states that if the conditional mean is close to a linear function in the
$$
L_2
$$
distance than the prior distribution of must be close to gamma distribution in Levy distance.
The posterior mean E[] approaches the maximum likelihood estimate
$$
\widehat{\lambda}_\mathrm{MLE}
$$
in the limit as
$$
\alpha\to 0, \beta \to 0,
$$
which follows immediately from the general expression of the mean of the gamma distribution.
The posterior predictive distribution for a single additional observation is a negative binomial distribution, sometimes called a gamma–Poisson distribution.
### Simultaneous estimation of multiple Poisson means
Suppose
$$
X_1, X_2, \dots, X_p
$$
is a set of independent random variables from a set of
$$
p
$$
Poisson distributions, each with a parameter
$$
\lambda_i,
$$
$$
i=1,\dots, p,
$$
and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss
$$
L(\lambda,{\hat \lambda})=\sum_{i=1}^p \lambda_i^{-1} ({\hat \lambda}_i-\lambda_i)^2,
$$
when
$$
p>1,
$$
then, similar as in Stein's example for the Normal means, the MLE estimator
$$
{\hat \lambda}_i = X_i
$$
is inadmissible.
In this case, a family of minimax estimators is given for any
$$
0 < c \leq 2(p-1)
$$
and
$$
b \geq (p-2+p^{-1})
$$
as
$$
{\hat \lambda}_i = \left(1 - \frac{c}{b + \sum_{i=1}^p X_i}\right) X_i, \qquad i=1,\dots,p.
$$
## Occurrence and applications
Some applications of the Poisson distribution to count data (number of events):
- telecommunication: telephone calls arriving in a system,
- astronomy: photons arriving at a telescope,
- chemistry: the molar mass distribution of a living polymerization,
- biology: the number of mutations on a strand of DNA per unit length,
- management: customers arriving at a counter or call centre,
- finance and insurance: number of losses or claims occurring in a given period of time,
- seismology: asymptotic Poisson model of risk for large earthquakes,
- radioactivity: decays in a given time interval in a radioactive sample,
- optics: number of photons emitted in a single laser pulse (a major vulnerability of quantum key distribution protocols, known as photon number splitting).
More examples of counting events that may be modelled as Poisson processes include:
- soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example was used in a book by Ladislaus Bortkiewicz (1868–1931),
- yeast cells used when brewing Guinness beer. This example was used by William Sealy Gosset (1876–1937),
- phone calls arriving at a call centre within a minute. This example was described by A.K. Erlang (1878–1929),
- goals in sports involving two competing teams,
- deaths per year in a given age group,
- jumps in a stock price in a given time interval,
- times a web server is accessed per minute (under an assumption of homogeneity),
- mutations in a given stretch of DNA after a certain amount of radiation,
- cells infected at a given multiplicity of infection,
- bacteria in a certain amount of liquid,
- photons arriving on a pixel circuit at a given illumination over a given time period,
- landing of V-1 flying bombs on London during World War II, investigated by R. D. Clarke in 1946.
In probabilistic number theory, Gallagher showed in 1976 that, if a certain version of the unproved prime r-tuple conjecture holds, then the counts of prime numbers in short intervals would obey a Poisson distribution.
### Law of rare events
The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the binomial one, given only the information of expected number of total events in the whole interval.
Let the total number of events in the whole interval be denoted by
$$
\lambda.
$$
Divide the whole interval into
$$
n
$$
subintervals
$$
I_1,\dots,I_n
$$
of equal size, such that
$$
n > \lambda
$$
(since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in each of the subintervals is equal to
$$
\lambda/n.
$$
Now we assume that the occurrence of an event in the whole interval can be seen as a sequence of Bernoulli trials, where the
$$
i
$$
-th Bernoulli trial corresponds to looking whether an event happens at the subinterval
$$
I_i
$$
with probability
$$
\lambda/n.
$$
The expected number of total events in
$$
n
$$
such trials would be
$$
\lambda,
$$
the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form
$$
\textrm{B}(n,\lambda/n).
$$
As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as
$$
n
$$
goes to infinity.
In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.
In several of the above examples — such as the number of mutations in a given sequence of DNA — the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is
$$
X \sim \textrm{B}(n,p).
$$
In such cases is very large and is very small (and so the expectation is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution
$$
X \sim \textrm{Pois}(np).
$$
This approximation is sometimes known as the law of rare events, since each of the individual Bernoulli events rarely occurs.
The name "law of rare events" may be misleading because the total count of success events in a Poisson process need not be rare if the parameter is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.
The variance of the binomial distribution is 1 − p times that of the Poisson distribution, so almost equal when p is very small.
The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.
### Poisson point process
The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if D is some region space, for example Euclidean space Rd, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if denotes the number of points in D, then
$$
P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .
$$
### Poisson regression and negative binomial regression
Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count of the number of events or occurrences in an interval.
### Biology
The Luria–Delbrück experiment tested against the hypothesis of Lamarckian evolution, which should result in a Poisson distribution.
Katz and Miledi measured the membrane potential with and without the presence of acetylcholine (ACh). When ACh is present, ion channels on the membrane would be open randomly at a small fraction of the time. As there are a large number of ion channels each open for a small fraction of the time, the total number of ion channels open at any moment is Poisson distributed. When ACh is not present, effectively no ion channels are open. The membrane potential is
$$
V = N_{\text{open}} V_{\text{ion}} + V_0 + V_{\text{noise}}
$$
. Subtracting the effect of noise, Katz and Miledi found the mean and variance of membrane potential to be
$$
8.5 \times 10^{-3}\; \mathrm{V}, (29.2 \times 10^{-6}\; \mathrm{V})^2
$$
, giving
$$
V_{\text{ion}} = 10^{-7}\;\mathrm{V}
$$
. (pp. 94-95)
During each cellular replication event, the number of mutations is roughly Poisson distributed. For example, the HIV virus has 10,000 base pairs, and has a mutation rate of about 1 per 30,000 base pairs, meaning the number of mutations per replication event is distributed as
$$
\mathrm{Pois}(1/3)
$$
. (p. 64)
### Other applications in science
In a Poisson process, the number of observed occurrences fluctuates about its mean with a standard deviation
$$
\sigma_k =\sqrt{\lambda}.
$$
These fluctuations are denoted as Poisson noise or (particularly in electronics) as shot noise.
The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, even if that contribution is too small to be detected directly. For example, the charge e on an electron can be estimated by correlating the magnitude of an electric current with its shot noise. If N electrons pass a point in a given time t on the average, the mean current is
$$
I=eN/t
$$
; since the current fluctuations should be of the order
$$
\sigma_I = e\sqrt{N}/t
$$
(i.e., the standard deviation of the Poisson process), the charge
$$
e
$$
can be estimated from the ratio
$$
t\sigma_I^2/I.
$$
An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced silver grains, not to the individual grains themselves. By correlating the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided).
In causal set theory the discrete elements of spacetime follow a Poisson distribution in the volume.
The Poisson distribution also appears in quantum mechanics, especially quantum optics. Namely, for a quantum harmonic oscillator system in a coherent state, the probability of measuring a particular energy level has a Poisson distribution.
## Computational methods
The Poisson distribution poses two different tasks for dedicated software libraries: evaluating the distribution
$$
P(k;\lambda)
$$
, and drawing random numbers according to that distribution.
### Evaluating the Poisson distribution
Computing
$$
P(k;\lambda)
$$
for given
$$
k
$$
and
$$
\lambda
$$
is a trivial task that can be accomplished by using the standard definition of
$$
P(k;\lambda)
$$
in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: and . The fraction of to ! can also produce a rounding error that is very large compared to e−, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as
$$
\!f(k; \lambda)= \exp \left[ k\ln \lambda - \lambda - \ln \Gamma (k+1) \right],
$$
which is mathematically equivalent but numerically stable. The natural logarithm of the Gamma function can be obtained using the `lgamma` function in the C standard library (C99 version) or R, the `gammaln` function in MATLAB or SciPy, or the `log_gamma` function in Fortran 2008 and later.
Some computing languages provide built-in functions to evaluate the Poisson distribution, namely
- R: function `dpois(x, lambda)`;
- Excel: function `POISSON( x, mean, cumulative)`, with a flag to specify the cumulative distribution;
- Mathematica: univariate Poisson distribution as `PoissonDistribution[
$$
\lambda
$$
]`, bivariate Poisson distribution as `MultivariatePoissonDistribution[
$$
\theta_{12},
$$
{
$$
\theta_1 - \theta_{12},
$$
$$
\theta_2 - \theta_{12}
$$
}]`,.
### Random variate generation
The less trivial task is to draw integer random variate from the Poisson distribution with given
$$
\lambda.
$$
Solutions are provided by:
- R: function `rpois(n, lambda)`;
- GNU Scientific Library (GSL): function gsl_ran_poisson
A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth:
algorithm poisson random number (Knuth):
init:
Let L ← e−λ, k ← 0 and p ← 1.
do:
k ← k + 1.
Generate uniform random number u in [0,1] and let p ← p × u.
while p > L.
return k − 1.
The complexity is linear in the returned value , which is on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see below.
For large values of , the value of = e− may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that e−STEP does not underflow:
algorithm poisson random number (Junhao, based on Knuth):
init:
Let Left ← , k ← 0 and p ← 1.
do:
k ← k + 1.
Generate uniform random number u in (0,1) and let p ← p × u.
while p < 1 and Left > 0:
if Left > STEP:
p ← p × eSTEP
Left ← Left − STEP
else:
p ← p × eLeft
Left ← 0
while p > 1.
return k − 1.
The choice of STEP depends on the threshold of overflow. For double precision floating point format the threshold is near e700, so 500 should be a safe STEP.
Other solutions for large values of include rejection sampling and using Gaussian approximation.
Inverse transform sampling is simple and efficient for small values of , and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.
algorithm Poisson generator based upon the inversion by sequential search:
init:
Let x ← 0, p ← e−λ, s ← p.
Generate uniform random number u in [0,1].
while u > s do:
x ← x + 1.
p ← p × / x.
s ← s + p.
return x. | https://en.wikipedia.org/wiki/Poisson_distribution |
The spectral test is a statistical test for the quality of a class of pseudorandom number generators (PRNGs), the linear congruential generators (LCGs). LCGs have a property that when plotted in 2 or more dimensions, lines or hyperplanes will form, on which all possible outputs can be found. The spectral test compares the distance between these planes; the further apart they are, the worse the generator is. As this test is devised to study the lattice structures of LCGs, it can not be applied to other families of PRNGs.
According to Donald Knuth, this is by far the most powerful test known, because it can fail LCGs which pass most statistical tests. The IBM subroutine RANDU LCG fails in this test for 3 dimensions and above.
Let the PRNG generate a sequence
$$
u_1, u_2, \dots
$$
. Let
$$
1/\nu_t
$$
be the maximal separation between covering parallel planes of the sequence
$$
\{(u_{n+1:n+t}) \mid n = 0, 1, \dots\}
$$
. The spectral test checks that the sequence
$$
\nu_2, \nu_3, \nu_4, \dots
$$
does not decay too quickly.
Knuth recommends checking that each of the following 5 numbers is larger than 0.01.
$$
\begin{aligned}
\mu_2 &= \pi \nu_2^2 / m, &
\mu_3 &= \frac{4}{3} \pi \nu_3^3 / m, &
\mu_4 &= \frac{1}{2} \pi^2 \nu_4^4 / m, \\[1ex]
& &
\mu_5 &= \frac{8}{15} \pi^2 \nu_5^5 / m, &
\mu_6 &= \frac{1}{6} \pi^3 \nu_6^6 / m,
\end{aligned}
$$
where
$$
m
$$
is the modulus of the LCG.
## Figures of merit
Knuth defines a figure of merit, which describes how close the separation
$$
1/\nu_t
$$
is to the theoretical minimum. Under Steele & Vigna's re-notation, for a dimension
$$
d
$$
, the figure
$$
f_d
$$
is defined as
$$
f_d(m, a) = \nu_d / \left(\gamma^{1/2}_d \sqrt[d]{m}\right),
$$
where
$$
a, m, \nu_d
$$
are defined as before, and
$$
\gamma_d
$$
is the Hermite constant of dimension d.
$$
\gamma^{1/2}_d \sqrt[d]{m}
$$
is the smallest possible interplane separation.
L'Ecuyer 1991 further introduces two measures corresponding to the minimum of
$$
f_d
$$
across a number of dimensions. Again under re-notation,
$$
\mathcal{M}^+_d(m, a)
$$
is the minimum
$$
f_d
$$
for a LCG from dimensions 2 to
$$
d
$$
, and
$$
\mathcal{M}^*_d(m, a)
$$
is the same for a multiplicative congruential pseudorandom number generator (MCG), i.e. one where only multiplication is used, or
$$
c = 0
$$
. Steele & Vigna note that the
$$
f_d
$$
is calculated differently in these two cases, necessitating separate values. They further define a "harmonic" weighted average figure of merit,
$$
\mathcal{H}^+_d(m, a)
$$
(and
$$
\mathcal{H}^*_d(m, a)
$$
).
## Examples
A small variant of the infamous RANDU, with
$$
x_{n+1} = 65539 \, x_n \bmod 2^{29}
$$
has:
d2 3 4 5 6 7 8 ν 536936458 118 116 116 116 μd 3.14 10−5 10−4 10−3 0.02 fd0.5202240.0189020.0841430.2071850.3688410.5522050.578329
The aggregate figures of merit are:
$$
\mathcal{M}^{*}_8(65539, 2^{29}) = 0.018902
$$
,
$$
\mathcal{H}^{*}_8(65539, 2^{29}) = 0.330886
$$
.
George Marsaglia (1972) considers
$$
x_{n+1} = 69069 \, x_n \bmod 2^{32}
$$
as "a candidate for the best of all multipliers" because it is easy to remember, and has particularly large spectral test numbers.
d2 3 4 5 6 7 8 ν 4243209856 2072544 52804 6990 242 μd 3.10 2.91 3.20 5.01 0.017 fd0.4624900.3131270.4571830.5529160.3767060.4966870.685247
The aggregate figures of merit are:
$$
\mathcal{M}^{*}_8(69069, 2^{32}) =0.313127
$$
,
$$
\mathcal{H}^{*}_8(69069, 2^{32}) = 0.449578
$$
.
Steele & Vigna (2020) provide the multipliers with the highest aggregate figures of merit for many choices of m = 2n and a given bit-length of a. They also provide the individual
$$
f_d
$$
values and a software package for calculating these values. For example, they report that the best 17-bit a for m = 232 is:
- For an LCG (c ≠ 0), (121525).
$$
\mathcal{M}^{+}_8=0.6403
$$
,
$$
\mathcal{H}^{+}_8 = 0.6588
$$
.
- For an MCG (c = 0), (125229).
$$
\mathcal{M}^{*}_8=0.6623
$$
,
$$
\mathcal{H}^{*}_8 = 0.7497
$$
.
## Additional illustration
## References
## Further reading
- – lists the
$$
f_d
$$
(notated as
$$
S_s
$$
in this text) of many well-known LCGs
- An expanded version of this work is available as:
Category:Pseudorandom number generators | https://en.wikipedia.org/wiki/Spectral_test |
A conscience is a cognitive process that elicits emotion and rational associations based on an individual's moral philosophy or value system. Conscience is not an elicited emotion or thought produced by associations based on immediate sensory perceptions and reflexive responses, as in sympathetic central nervous system responses. In common terms, conscience is often described as leading to feelings of remorse when a person commits an act that conflicts with their moral values. The extent to which conscience informs moral judgment before an action and whether such moral judgments are or should be based on reason has occasioned debate through much of modern history between theories of basics in ethic of human life in juxtaposition to the theories of romanticism and other reactionary movements after the end of the Middle Ages.
### Religious
views of conscience usually see it as linked to a morality inherent in all humans, to a beneficent universe and/or to divinity. The diverse ritualistic, mythical, doctrinal, legal, institutional and material features of religion may not necessarily cohere with experiential, emotive, spiritual or contemplative considerations about the origin and operation of conscience. Common secular or scientific views regard the capacity for conscience as probably genetically determined, with its subject probably learned or imprinted as part of a culture.
Commonly used metaphors for conscience include the "voice within", the "inner light", or even Socrates' reliance on what the Greeks called his "daimōnic sign", an averting (ἀποτρεπτικός apotreptikos) inner voice heard only when he was about to make a mistake. Conscience, as is detailed in sections below, is a concept in national and international law, is increasingly conceived of as applying to the world as a whole, has motivated numerous notable acts for the public good and been the subject of many prominent examples of literature, music and film.
## Views
Although humanity has no generally accepted definition of conscience or universal agreement about its role in ethical decision-making, three approaches have addressed it:
1. Religious views
1.
### Secular
views
1.
### Philosophical
views
Religious
In the literary traditions of the Upanishads, Brahma Sutras and the Bhagavad Gita, conscience is the label given to attributes composing knowledge about good and evil, that a soul acquires from the completion of acts and consequent accretion of karma over many lifetimes. According to Adi Shankara in his Vivekachudamani morally right action (characterised as humbly and compassionately performing the primary duty of good to others without expectation of material or spiritual reward), helps "purify the heart" and provide mental tranquility but it alone does not give us "direct perception of the Reality". This knowledge requires discrimination between the eternal and non-eternal and eventually a realization in contemplation that the true self merges in a universe of pure consciousness.
In the Zoroastrian faith, after death a soul must face judgment at the Bridge of the Separator; there, evil people are tormented by prior denial of their own higher nature, or conscience, and "to all time will they be guests for the House of the Lie." The Chinese concept of Ren, indicates that conscience, along with social etiquette and correct relationships, assist humans to follow The Way (Tao) a mode of life reflecting the implicit human capacity for goodness and harmony.
Conscience also features prominently in Buddhism. In the Pali scriptures, for example, Buddha links the positive aspect of conscience to a pure heart and a calm, well-directed mind. It is regarded as a spiritual power, and one of the "Guardians of the World". The Buddha also associated conscience with compassion for those who must endure cravings and suffering in the world until right conduct culminates in right mindfulness and right contemplation. Santideva (685–763 CE) wrote in the Bodhicaryavatara (which he composed and delivered in the great northern Indian Buddhist university of Nalanda) of the spiritual importance of perfecting virtues such as generosity, forbearance and training the awareness to be like a "block of wood" when attracted by vices such as pride or lust; so one can continue advancing towards right understanding in meditative absorption. Conscience thus manifests in Buddhism as unselfish love for all living beings which gradually intensifies and awakens to a purer awareness where the mind withdraws from sensory interests and becomes aware of itself as a single whole.
The Roman Emperor Marcus Aurelius wrote in his Meditations that conscience was the human capacity to live by rational principles that were congruent with the true, tranquil and harmonious nature of our mind and thereby that of the Universe: "To move from one unselfish action to another with God in mind. Only there, delight and stillness ... the only rewards of our existence here are an unstained character and unselfish acts."
The Islamic concept of Taqwa is closely related to conscience. In the Qur’ān verses 2:197 & 22:37 Taqwa refers to "right conduct" or "piety", "guarding of oneself" or "guarding against evil". Qur’ān verse 47:17 says that God is the ultimate source of the believer's taqwā which is not simply the product of individual will but requires inspiration from God. In Qur’ān verses 91:7–8, God the Almighty talks about how He has perfected the soul, the conscience and has taught it the wrong (fujūr) and right (taqwā). Hence, the awareness of vice and virtue is inherent in the soul, allowing it to be tested fairly in the life of this world and tried, held accountable on the day of judgment for responsibilities to God and all humans.
Qur’ān verse 49:13 states: "O humankind! We have created you out of male and female and constituted you into different groups and societies, so that you may come to know each other-the noblest of you, in the sight of God, are the ones possessing taqwā." In Islam, according to eminent theologians such as Al-Ghazali, although events are ordained (and written by God in al-Lawh al-Mahfūz, the Preserved Tablet), humans possess free will to choose between wrong and right and are thus responsible for their actions; the conscience being a dynamic personal connection to God enhanced by knowledge and practise of the Five Pillars of Islam, deeds of piety, repentance, self-discipline, and prayer; and disintegrated and metaphorically covered in blackness through sinful acts. Marshall Hodgson wrote the three-volume work: The Venture of Islam: Conscience and History in a World Civilization.
In the Protestant Christian tradition, Martin Luther insisted at the Diet of Worms that his conscience was captive to the Word of God, and it was neither safe nor right to go against conscience. To Luther, conscience falls within the ethical, rather than the religious, sphere. John Calvin saw conscience as a battleground: "the enemies who rise up in our conscience against his Kingdom and hinder his decrees prove that God's throne is not firmly established therein". Many Christians regard following one's conscience as important as, or even more important than, obeying human authority. According to the Bible, as enunciated in Romans 2:15, conscience is the one bearing witness, accusing or excusing one another, so we would know when we break the law written in our hearts; the guilt we feel when we do something wrong tells us that we need to repent." This can sometimes (as with the conflict between William Tyndale and Thomas More over the translation of the Bible into English) lead to moral quandaries: "Do I unreservedly obey my Church/priest/military/political leader or do I follow my own inner feeling of right and wrong as instructed by prayer and a personal reading of scripture?" Some contemporary Christian churches and religious groups hold the moral teachings of the Ten Commandments or of Jesus as the highest authority in any situation, regardless of the extent to which it involves responsibilities in law. In the Gospel of John (7:53–8:11, King James Version), Jesus challenges those accusing a woman of adultery: "'He that is without sin among you, let him first cast a stone at her.' And again he stooped down, and wrote on the ground. And they which heard it, being convicted by their own conscience, went out one by one" (see Jesus and the woman taken in adultery). Of note, however, the word 'conscience' is not in the original New Testament Greek and is not in the vast majority of Bible versions. In the Gospel of Luke (10:25–37), Jesus tells the story of how a despised and heretical Samaritan (see Parable of the Good Samaritan) who (out of compassion or pity; the word 'conscience' is not used) helps an injured stranger beside a road, qualifies better for eternal life by loving his neighbor than a priest who passes by on the other side.
This dilemma of obedience in conscience to divine or state law, was demonstrated dramatically in Antigone's defiance of King Creon's order against burying her brother an alleged traitor, appealing to the "unwritten law" and to a "longer allegiance to the dead than to the living".
Catholic theology sees conscience as the last practical "judgment of reason which at the appropriate moment enjoins [a person] to do good and to avoid evil". The Second Vatican Council (1962–65) describes: "Deep within his conscience man discovers a law which he has not laid upon himself but which he must obey. Its voice, ever calling him to love and to do what is good and to avoid evil, tells him inwardly at the right movement: do this, shun that. For man has in his heart a law inscribed by God. His dignity lies in observing this law, and by it he will be judged. His conscience is man’s most secret core, and his sanctuary. There he is alone with God whose voice echoes in his depths." Thus, conscience is not like the will, nor a habit like prudence, but "the interior space in which we can listen to and hear the truth, the good, the voice of God. It is the inner place of our relationship with Him, who speaks to our heart and helps us to discern, to understand the path we ought to take, and once the decision is made, to move forward, to remain faithful" In terms of logic, conscience can be viewed as the practical conclusion of a moral syllogism whose major premise is an objective norm and whose minor premise is a particular case or situation to which the norm is applied. Thus, Catholics are taught to carefully educate themselves as to revealed norms and norms derived therefrom, so as to form a correct conscience. Catholics are also to examine their conscience daily and with special care before confession. Catholic teaching holds that, "Man has the right to act according to his conscience and in freedom so as personally to make moral decisions. He must not be forced to act contrary to his conscience. Nor must he be prevented from acting according to his conscience, especially in religious matters". This right of Conscience allows one to form their Morality from sincere and traditional sources and form their opinions from therein. Thus, the Church teaches that one must form their morality and then follow it to the best of their ability. Nevertheless it is taught in more than one area, that the conscience can, and sometimes should, stand against the teaching of the Church. Thus the Church teaches that the Conscience is a supreme authority, even above that of the Popes, Bishops, and Priests. Thus while the Conscience does grant man a great degree of freedom, if one is going to disagree with conventional morality or with the teachings of the Church, it is absolutely necessary to make sure that one's conscience is well formed and certain of what it is claiming or not claiming. Joseph Ratzinger (later Pope Benedict XVI), commentary on Gaudium et Spes, in: Herbert Vorgrimler (ed.), Commentary on the Documents of Vatican II, vol. 5, Burns & Oates, 1969, p. 134Riley Clare Valentine, Overturning the Catechism: A Catholic Argument for AbortionCharles E. Curran, “Ten Years Later,” Commonweal, vol. 105, July 7, 1978, p. 429. A sincere conscience presumes one is diligently seeking moral truth from authentic sources, whether that be from the Church, or from Scripture, or from the numerous Church Fathers. Nevertheless, despite one's best effort, "[i]t can happen that moral conscience remains in ignorance and makes erroneous judgments about acts to be performed or already committed ... This ignorance can, but not always, be imputed to personal responsibility, This is the case when a man "takes little trouble to find out what is true and good", or in other words, puts forth very little effort and does not take the forming of the Conscience seriously. In such cases, the person is culpable for the wrong he commits." Not necessarily because of the error itself, but because of the bad faith or miniscule effort put forth by the one whos Conscience is in question. The Catholic Church has warned that "rejection of the Church's authority and her teaching ... can sometimes be at the source of errors in judgment in moral conduct". An example of someone following his conscience to the point of accepting the consequence of being condemned to death is Sir Thomas More (1478-1535). A theologian who wrote on the distinction between the 'sense of duty' and the 'moral sense', as two aspects of conscience, and who saw the former as some feeling that can only be explained by a divine Lawgiver, was John Henry Cardinal Newman. A well known saying of him is that he would first toast on his conscience and only then on the pope, since his conscience brought him to acknowledge the authority of the pope.
This relates to the concept of the different types of heresy as understood within Church teaching. The Church distinguishes between Material Heresy and Formal Heresy. Material Heresy occurs when an individual, after sincere and thorough study of the Church’s moral teachings and a genuine effort to form their conscience in accordance with those teachings, concludes—respectfully and in good faith—that the Church is mistaken on one or more moral issues. In such cases, if the individual maintains their personal belief despite their best efforts to understand and accept Church doctrine, they are considered a Material Heretic. However, because their error stems from a well-intentioned and conscientious process, no sin is imputed to them."Heresy". Catholic Encyclopedia. New Advent. 1912. Retrieved 6 March 2017. Public Domain This article incorporates text from this source, which is in the public domain.
Formal Heresy, by contrast, involves a willful and culpable rejection of Church teaching despite recognizing its truth. In this case, the individual acknowledges that the Church's doctrine is correct but chooses to reject it knowingly, often out of pride, defiance, malice, or other forms of vice. This rejection constitutes a grave moral fault because it entails acting against one’s own conscience and embracing falsehood knowingly. As such, Formal Heresy is considered a sin, as it reflects both an intentional departure from truth and a deliberate act of dishonesty. One must maintain the seperation between Material Heresy and Formal Heresy, simply for the fact that one is sinful, and the other is not.
Judaism arguably does not require uncompromising obedience to religious authority; the case has been made that throughout Jewish history, rabbis have circumvented laws they found unconscionable, such as capital punishment. Similarly, although an occupation with national destiny has been central to the Jewish faith (see Zionism) many scholars (including Moses Mendelssohn) stated that conscience as a personal revelation of scriptural truth was an important adjunct to the Talmudic tradition.Levi Meier (Ed.) Conscience and Autonomy within Judaism: A Special Issue of the Journal of Psychology and Judaism. Springer-Verlag. New York . The concept of inner light in the Religious Society of Friends or Quakers is associated with conscience. Freemasonry describes itself as providing an adjunct to religion and key symbols found in a Freemason Lodge are the square and compasses explained as providing lessons that Masons should "square their actions by the square of conscience", learn to "circumscribe their desires and keep their passions within due bounds toward all mankind." The historian Manning Clark viewed conscience as one of the comforters that religion placed between man and death but also a crucial part of the quest for grace encouraged by the Book of Job and the Book of Ecclesiastes, leading us to be paradoxically closest to the truth when we suspect that what matters most in life ("being there when everyone suddenly understands what it has all been for") can never happen. Leo Tolstoy, after a decade studying the issue (1877–1887), held that the only power capable of resisting the evil associated with materialism and the drive for social power of religious institutions, was the capacity of humans to reach an individual spiritual truth through reason and conscience. Many prominent religious works about conscience also have a significant philosophical component: examples are the works of Al-Ghazali, Avicenna, Aquinas, Joseph Butler and Dietrich Bonhoeffer (all discussed in the philosophical views section).
Secular
The secular approach to conscience includes psychological, physiological, sociological, humanitarian, and authoritarian views. Lawrence Kohlberg considered critical conscience to be an important psychological stage in the proper moral development of humans, associated with the capacity to rationally weigh principles of responsibility, being best encouraged in the very young by linkage with humorous personifications (such as Jiminy Cricket) and later in adolescents by debates about individually pertinent moral dilemmas. Erik Erikson placed the development of conscience in the 'pre-schooler' phase of his eight stages of normal human personality development. The psychologist Martha Stout terms conscience "an intervening sense of obligation based in our emotional attachments." Thus a good conscience is associated with feelings of integrity, psychological wholeness and peacefulness and is often described using adjectives such as "quiet", "clear" and "easy".
Sigmund Freud regarded conscience as originating psychologically from the growth of civilisation, which periodically frustrated the external expression of aggression: this destructive impulse being forced to seek an alternative, healthy outlet, directed its energy as a superego against the person's own "ego" or selfishness (often taking its cue in this regard from parents during childhood). According to Freud, the consequence of not obeying our conscience is guilt, which can be a factor in the development of neurosis; Freud claimed that both the cultural and individual super-ego set up strict ideal demands with regard to the moral aspects of certain decisions, disobedience to which provokes a 'fear of conscience'.
Antonio Damasio considers conscience an aspect of extended consciousness beyond survival-related dispositions and incorporating the search for truth and desire to build norms and ideals for behavior.
#### Conscience as a society-forming instinct
Michel Glautier argues that conscience is one of the instincts and drives which enable people to form societies: groups of humans without these drives or in whom they are insufficient cannot form societies and do not reproduce their kind as successfully as those that do.
Charles Darwin considered that conscience evolved in humans to resolve conflicts between competing natural impulses-some about self-preservation but others about safety of a family or community; the claim of conscience to moral authority emerged from the "greater duration of impression of social instincts" in the struggle for survival. In such a view, behavior destructive to a person's society (either to its structures or to the persons it comprises) is bad or "evil". Thus, conscience can be viewed as an outcome of those biological drives that prompt humans to avoid provoking fear or contempt in others; being experienced as guilt and shame in differing ways from society to society and person to person. A requirement of conscience in this view is the capacity to see ourselves from the point of view of another person. Persons unable to do this (psychopaths, sociopaths, narcissists) therefore often act in ways which are "evil".
Fundamental in this view of conscience is that humans consider some "other" as being in a social relationship. Thus, nationalism is invoked in conscience to quell tribal conflict and the notion of a Brotherhood of Man is invoked to quell national conflicts. Yet such crowd drives may not only overwhelm but redefine individual conscience. Friedrich Nietzsche stated: "communal solidarity is annihilated by the highest and strongest drives that, when they break out passionately, whip the individual far past the average low level of the 'herd-conscience.'" Jeremy Bentham noted that: "fanaticism never sleeps ... it is never stopped by conscience; for it has pressed conscience into its service." Hannah Arendt in her study of the trial of Adolf Eichmann in Jerusalem, notes that the accused, as with almost all his fellow Germans, had lost track of his conscience to the point where they hardly remembered it; this wasn't caused by familiarity with atrocities or by psychologically redirecting any resultant natural pity to themselves for having to bear such an unpleasant duty, so much as by the fact that anyone whose conscience did develop doubts could see no one who shared them: "Eichmann did not need to close his ears to the voice of conscience ... not because he had none, but because his conscience spoke with a "respectable voice", with the voice of the respectable society around him".
Sir Arthur Keith in 1948 developed the Amity-enmity complex. We evolved as tribal groups surrounded by enemies; thus conscience evolved a dual role; the duty to save and protect members of the in-group, and the duty to show hatred and aggression towards any out-group.
An interesting area of research in this context concerns the similarities between our relationships and those of animals, whether animals in human society (pets, working animals, even animals grown for food) or in the wild. One idea is that as people or animals perceive a social relationship as important to preserve, their conscience begins to respect that former "other", and urge actions that protect it. Similarly, in complex territorial and cooperative breeding bird communities (such as the Australian magpie) that have a high degree of etiquettes, rules, hierarchies, play, songs and negotiations, rule-breaking seems tolerated on occasions not obviously related to survival of the individual or group; behaviour often appearing to exhibit a touching gentleness and tenderness.
#### Evolutionary biology
Contemporary scientists in evolutionary biology seek to explain conscience as a function of the brain that evolved to facilitate altruism within societies. In his book The God Delusion, Richard Dawkins states that he agrees with Robert Hinde's Why Good is Good, Michael Shermer's The Science of Good and Evil, Robert Buckman's Can We Be Good Without God? and Marc Hauser's Moral Minds, that our sense of right and wrong can be derived from our Darwinian past. He subsequently reinforced this idea through the lens of the gene-centered view of evolution, since the unit of natural selection is neither an individual organism nor a group, but rather the "selfish" gene, and these genes could ensure their own "selfish" survival by, inter alia, pushing individuals to act altruistically towards its kin.
#### Neuroscience and artificial conscience
Numerous case studies of brain damage have shown that damage to areas of the brain (such as the anterior prefrontal cortex) results in the reduction or elimination of inhibitions, with a corresponding radical change in behaviour. When the damage occurs to adults, they may still be able to perform moral reasoning; but when it occurs to children, they may never develop that ability.Jorge Moll, Roland Zahn, Ricardo de Oliveira-Souza, Frank Krueger & Jordan Grafman. The Neural Basis of Human Moral Cognition . Vision Circle 10 October 2005 accessed 18 October 2009.
Attempts have been made by neuroscientists to locate the free will necessary for what is termed the 'veto' of conscience over unconscious mental processes (see Neuroscience of free will and Benjamin Libet) in a scientifically measurable awareness of an intention to carry out an act occurring 350–400 microseconds after the electrical discharge known as the 'readiness potential.'AC Grayling. "Do We Have a Veto?" Times Literary Supplement. 2000; 5076 (14 July): 4.
Jacques Pitrat claims that some kind of artificial conscience is beneficial in artificial intelligence systems to improve their long-term performance and direct their introspective processing.
Philosophical
The word "conscience" derives etymologically from the Latin conscientia, meaning "privity of knowledge"
or "with-knowledge". The English word implies internal awareness of a moral standard in the mind concerning the quality of one's motives, as well as a consciousness of our own actions. Thus conscience considered philosophically may be first, and perhaps most commonly, a largely unexamined "gut feeling" or "vague sense of guilt" about what ought to be or should have been done. Conscience in this sense is not necessarily the product of a process of rational consideration of the moral features of a situation (or the applicable normative principles, rules or laws) and can arise from parental, peer group, religious, state or corporate indoctrination, which may or may not be presently consciously acceptable to the person ("traditional conscience"). Conscience may be defined as the practical reason employed when applying moral convictions to a situation ("critical conscience"). In purportedly morally mature mystical people who have developed this capacity through daily contemplation or meditation combined with selfless service to others, critical conscience can be aided by a "spark" of intuitive insight or revelation (called marifa in Islamic Sufi philosophy and synderesis in medieval Christian scholastic moral philosophy).Langston, Douglas C. Conscience and Other Virtues: From Bonaventure to MacIntyre. The Pennsylvania State University Press, University Park, Pennsylvania, 2001. p. 34 Conscience is accompanied in each case by an internal awareness of 'inner light' and approbation or 'inner darkness' and condemnation as well as a resulting conviction of right or duty either followed or declined.
#### Medieval
The medieval Islamic scholar and mystic Al-Ghazali divided the concept of Nafs (soul or self (spirituality)) into three categories based on the Qur’an:
1. Nafs Ammarah (12:53) which "exhorts one to freely indulge in gratifying passions and instigates to do evil"
1. Nafs Lawammah (75:2) which is "the conscience that directs man towards right or wrong"
1. Nafs Mutmainnah (89:27) which is "a self that reaches the ultimate peace"
The medieval Persian philosopher and physician Muhammad ibn Zakariya al-Razi believed in a close relationship between conscience or spiritual integrity and physical health; rather than being self-indulgent, man should pursue knowledge, use his intellect and apply justice in his life. The medieval Islamic philosopher Avicenna, whilst imprisoned in the castle of Fardajan near Hamadhan, wrote his famous isolated-but-awake "Floating Man" sensory deprivation thought experiment to explore the ideas of human self-awareness and the substantiality of the soul; his hypothesis being that it is through intelligence, particularly the active intellect, that God communicates truth to the human mind or conscience. According to the Islamic Sufis conscience allows Allah to guide people to the marifa, the peace or "light upon light" experienced where a Muslim's prayers lead to a melting away of the self in the inner knowledge of God; this foreshadowing the eternal Paradise depicted in the Qur’ān.
Some medieval Christian scholastics such as Bonaventure made a distinction between conscience as a rational faculty of the mind (practical reason) and inner awareness, an intuitive "spark" to do good, called synderesis arising from a remnant appreciation of absolute good and when consciously denied (for example to perform an evil act), becoming a source of inner torment. Early modern theologians such as William Perkins and William Ames developed a syllogistic understanding of the conscience, where God's law made the first term, the act to be judged the second and the action of the conscience (as a rational faculty) produced the judgement. By debating test cases applying such understanding conscience was trained and refined (i.e. casuistry).
In the 13th century, St. Thomas Aquinas regarded conscience as the application of moral knowledge to a particular case (S.T. I, q. 79, a. 13). Thus, conscience was considered an act or judgment of practical reason that began with synderesis, the structured development of our innate remnant awareness of absolute good (which he categorised as involving the five primary precepts proposed in his theory of Natural Law) into an acquired habit of applying moral principles. According to Singer, Aquinas held that conscience, or conscientia was an imperfect process of judgment applied to activity because knowledge of the natural law (and all acts of natural virtue implicit therein) was obscured in most people by education and custom that promoted selfishness rather than fellow-feeling (Summa Theologiae, I–II, I). Aquinas also discussed conscience in relation to the virtue of prudence to explain why some people appear to be less "morally enlightened" than others, their weak will being incapable of adequately balancing their own needs with those of others.
Aquinas reasoned that acting contrary to conscience is an evil action but an errant conscience is only blameworthy if it is the result of culpable or vincible ignorance of factors that one has a duty to have knowledge of. Aquinas also argued that conscience should be educated to act towards real goods (from God) which encouraged human flourishing, rather than the apparent goods of sensory pleasures. In his Commentary on Aristotle's Nicomachean Ethics Aquinas claimed it was weak will that allowed a non-virtuous man to choose a principle allowing pleasure ahead of one requiring moral constraint.
Thomas A Kempis in the medieval contemplative classic The Imitation of Christ (ca 1418) stated that the glory of a good man is the witness of a good conscience. "Preserve a quiet conscience and you will always have joy. A quiet conscience can endure much, and remains joyful in all trouble, but an evil conscience is always fearful and uneasy." The anonymous medieval author of the Christian mystical work The Cloud of Unknowing similarly expressed the view that in profound and prolonged contemplation a soul dries up the "root and ground" of the sin that is always there, even after one's confession and however busy one is in holy things: "therefore, whoever would work at becoming a contemplative must first cleanse his [or her] conscience." The medieval Flemish mystic John of Ruysbroeck likewise held that true conscience has four aspects that are necessary to render a man just in the active and contemplative life: "a free spirit, attracting itself through love"; "an intellect enlightened by grace", "a delight yielding propension or inclination" and "an outflowing losing of oneself in the abyss of ... that eternal object which is the highest and chief blessedness ... those lofty amongst men, are absorbed in it, and immersed in a certain boundless thing."
#### Modern
Benedict de Spinoza in his Ethics, published after his death in 1677, argued that most people, even those that consider themselves to exercise free will, make moral decisions on the basis of imperfect sensory information, inadequate understanding of their mind and will, as well as emotions which are both outcomes of their contingent physical existence and forms of thought defective from being chiefly impelled by self-preservation. The solution, according to Spinoza, was to gradually increase the capacity of our reason to change the forms of thought produced by emotions and to fall in love with viewing problems requiring moral decision from the perspective of eternity. Thus, living a life of peaceful conscience means to Spinoza that reason is used to generate adequate ideas where the mind increasingly sees the world and its conflicts, our desires and passions sub specie aeternitatis, that is without reference to time. Hegel's obscure and mystical Philosophy of Mind held that the absolute right of freedom of conscience facilitates human understanding of an all-embracing unity, an absolute which was rational, real and true. Nevertheless, Hegel thought that a functioning State would always be tempted not to recognize conscience in its form of subjective knowledge, just as similar non-objective opinions are generally rejected in science. A similar idealist notion was expressed in the writings of Joseph Butler who argued that conscience is God-given, should always be obeyed, is intuitive, and should be considered the "constitutional monarch" and the "universal moral faculty": "conscience does not only offer itself to show us the way we should walk in, but it likewise carries its own authority with it." Butler advanced ethical speculation by referring to a duality of regulative principles in human nature: first, "self-love" (seeking individual happiness) and second, "benevolence" (compassion and seeking good for another) in conscience (also linked to the agape of situational ethics). Conscience tended to be more authoritative in questions of moral judgment, thought Butler, because it was more likely to be clear and certain (whereas calculations of self-interest tended to probable and changing conclusions). John Selden in his Table Talk expressed the view that an awake but excessively scrupulous or ill-trained conscience could hinder resolve and practical action; it being "like a horse that is not well wayed, he starts at every bird that flies out of the hedge".
As the sacred texts of ancient Hindu and Buddhist philosophy became available in German translations in the 18th and 19th centuries, they influenced philosophers such as Schopenhauer to hold that in a healthy mind only deeds oppress our conscience, not wishes and thoughts; "for it is only our deeds that hold us up to the mirror of our will"; the good conscience, thought Schopenhauer, we experience after every disinterested deed arises from direct recognition of our own inner being in the phenomenon of another, it affords us the verification "that our true self exists not only in our own person, this particular manifestation, but in everything that lives. By this the heart feels itself enlarged, as by egotism it is contracted."
Immanuel Kant, a central figure of the Age of Enlightenment, likewise claimed that two things filled his mind with ever new and increasing admiration and awe, the oftener and more steadily they were reflected on: "the starry heavens above me and the moral law within me ... the latter begins from my invisible self, my personality, and exhibits me in a world which has true infinity but which I recognise myself as existing in a universal and necessary (and not only, as in the first case, contingent) connection." The 'universal connection' referred to here is Kant's categorical imperative: "act only according to that maxim by which you can at the same time will that it should become a universal law." Kant considered critical conscience to be an internal court in which our thoughts accuse or excuse one another; he acknowledged that morally mature people do often describe contentment or peace in the soul after following conscience to perform a duty, but argued that for such acts to produce virtue their primary motivation should simply be duty, not expectation of any such bliss. Rousseau expressed a similar view that conscience somehow connected man to a greater metaphysical unity. John Plamenatz in his critical examination of Rousseau's work considered that conscience was there defined as the feeling that urges us, in spite of contrary passions, towards two harmonies: the one within our minds and between our passions, and the other within society and between its members; "the weakest can appeal to it in the strongest, and the appeal, though often unsuccessful, is always disturbing. However, corrupted by power or wealth we may be, either as possessors of them or as victims, there is something in us serving to remind us that this corruption is against nature."
Other philosophers expressed a more sceptical and pragmatic view of the operation of "conscience" in society.
John Locke in his Essays on the Law of Nature argued that the widespread fact of human conscience allowed a philosopher to infer the necessary existence of objective moral laws that occasionally might contradict those of the state. Locke highlighted the metaethics problem of whether accepting a statement like "follow your conscience" supports subjectivist or objectivist conceptions of conscience as a guide in concrete morality, or as a spontaneous revelation of eternal and immutable principles to the individual: "if conscience be a proof of innate principles, contraries may be innate principles; since some men with the same bent of conscience prosecute what others avoid." Thomas Hobbes likewise pragmatically noted that opinions formed on the basis of conscience with full and honest conviction, nevertheless should always be accepted with humility as potentially erroneous and not necessarily indicating absolute knowledge or truth. William Godwin expressed the view that conscience was a memorable consequence of the "perception by men of every creed when the descend into the scene of busy life" that they possess free will. Adam Smith considered that it was only by developing a critical conscience that we can ever see what relates to ourselves in its proper shape and dimensions; or that we can ever make any proper comparison between our own interests and those of other people. John Stuart Mill believed that idealism about the role of conscience in government should be tempered with a practical realisation that few men in society are capable of directing their minds or purposes towards distant or unobvious interests, of disinterested regard for others, and especially for what comes after them, for the idea of posterity, of their country, or of humanity, whether grounded on sympathy or on a conscientious feeling. Mill held that certain amount of conscience, and of disinterested public spirit, may fairly be calculated on in the citizens of any community ripe for representative government, but that "it would be ridiculous to expect such a degree of it, combined with such intellectual discernment, as would be proof against any plausible fallacy tending to make that which was for their class interest appear the dictate of justice and of the general good."
Josiah Royce (1855–1916) built on the transcendental idealism view of conscience, viewing it as the ideal of life which constitutes our moral personality, our plan of being ourself, of making common sense ethical decisions. But, he thought, this was only true insofar as our conscience also required loyalty to "a mysterious higher or deeper self".
In the modern Christian tradition this approach achieved expression with Dietrich Bonhoeffer who stated during his imprisonment by the Nazis in World War II that conscience for him was more than practical reason, indeed it came from a "depth which lies beyond a man's own will and his own reason and it makes itself heard as the call of human existence to unity with itself." For Bonhoeffer a guilty conscience arose as an indictment of the loss of this unity and as a warning against the loss of one's self; primarily, he thought, it is directed not towards a particular kind of doing but towards a particular mode of being. It protests against a doing which imperils the unity of this being with itself. Conscience for Bonhoeffer did not, like shame, embrace or pass judgment on the morality of the whole of its owner's life; it reacted only to certain definite actions: "it recalls what is long past and represents this disunion as something which is already accomplished and irreparable". The man with a conscience, he believed, fights a lonely battle against the "overwhelming forces of inescapable situations" which demand moral decisions despite the likelihood of adverse consequences. Simon Soloveychik has similarly claimed that the truth distributed in the world, as the statement about human dignity, as the affirmation of the line between good and evil, lives in people as conscience.
As Hannah Arendt pointed out, however, (following the utilitarian John Stuart Mill on this point): a bad conscience does not necessarily signify a bad character; in fact only those who affirm a commitment to applying moral standards will be troubled with remorse, guilt or shame by a bad conscience and their need to regain integrity and wholeness of the self.John Stuart Mill. "Utilitarianism" and "On Liberty" in Collected Works. University of Toronto Press. Toronto. 1969 Vols 10 and 18. Ch 3. pp. 228–29 and 263. Representing our soul or true self by analogy as our house, Arendt wrote that "conscience is the anticipation of the fellow who awaits you if and when you come home." Arendt believed that people who are unfamiliar with the process of silent critical reflection about what they say and do will not mind contradicting themselves by an immoral act or crime, since they can "count on its being forgotten the next moment;" bad people are not full of regrets. Arendt also wrote eloquently on the problem of languages distinguishing the word consciousness from conscience. One reason, she held, was that conscience, as we understand it in moral or legal matters, is supposedly always present within us, just like consciousness: "and this conscience is also supposed to tell us what to do and what to repent; before it became the lumen naturale or Kant's practical reason, it was the voice of God."
Albert Einstein, as a self-professed adherent of humanism and rationalism, likewise viewed an enlightened religious person as one whose conscience reflects that he "has, to the best of his ability, liberated himself from the fetters of his selfish desires and is preoccupied with thoughts, feelings and aspirations to which he clings because of their super-personal value."
Einstein often referred to the "inner voice" as a source of both moral and physical knowledge: "Quantum mechanics is very impressive. But an inner voice tells me that it is not the real thing. The theory produces a good deal but hardly brings one closer to the secrets of the Old One. I am at all events convinced that He does not play dice."
Simone Weil who fought for the French resistance (the Maquis) argued in her final book The Need for Roots: Prelude to a Declaration of Duties Towards Mankind that for society to become more just and protective of liberty, obligations should take precedence over rights in moral and political philosophy and a spiritual awakening should occur in the conscience of most citizens, so that social obligations are viewed as fundamentally having a transcendent origin and a beneficent impact on human character when fulfilled.Hellman, John. Simone Weil: An Introduction to Her Thought. Wilfrid Laurier, University Press, Waterloo, Ontario. 1982. Simone Weil also in that work provided a psychological explanation for the mental peace associated with a good conscience: "the liberty of men of goodwill, though limited in the sphere of action, is complete in that of conscience. For, having incorporated the rules into their own being, the prohibited possibilities no longer present themselves to the mind, and have not to be rejected."
Alternatives to such metaphysical and idealist opinions about conscience arose from realist and materialist perspectives such as those of Charles Darwin. Darwin suggested that "any animal whatever, endowed with well-marked social instincts, the parental and filial affections being here included, would inevitably acquire a moral sense or conscience, as soon as its intellectual powers had become as well, or as nearly as well developed, as in man." Émile Durkheim held that the soul and conscience were particular forms of an impersonal principle diffused in the relevant group and communicated by totemic ceremonies. A. J. Ayer was a more recent realist who held that the existence of conscience was an empirical question to be answered by sociological research into the moral habits of a given person or group of people, and what causes them to have precisely those habits and feelings. Such an inquiry, he believed, fell wholly within the scope of the existing social sciences. George Edward Moore bridged the idealistic and sociological views of 'critical' and 'traditional' conscience in stating that the idea of abstract 'rightness' and the various degrees of the specific emotion excited by it are what constitute, for many persons, the specifically 'moral sentiment' or conscience. For others, however, an action seems to be properly termed 'internally right', merely because they have previously regarded it as right, the idea of 'rightness' being present in some way to his or her mind, but not necessarily among his or her deliberately constructed motives.
The French philosopher Simone de Beauvoir in A Very Easy Death (Une mort très douce, 1964) reflects within her own conscience about her mother's attempts to develop such a moral sympathy and understanding of others.
Michael Walzer claimed that the growth of religious toleration in Western nations arose amongst other things, from the general recognition that private conscience signified some inner divine presence regardless of the religious faith professed and from the general respectability, piety, self-limitation, and sectarian discipline which marked most of the men who claimed the rights of conscience. Walzer also argued that attempts by courts to define conscience as a merely personal moral code or as sincere belief, risked encouraging an anarchy of moral egotisms, unless such a code and motive was necessarily tempered with shared moral knowledge: derived either from the connection of the individual to a universal spiritual order, or from the common principles and mutual engagements of unselfish people. Ronald Dworkin maintains that constitutional protection of freedom of conscience is central to democracy but creates personal duties to live up to it: "Freedom of conscience presupposes a personal responsibility of reflection, and it loses much of its meaning when that responsibility is ignored. A good life need not be an especially reflective one; most of the best lives are just lived rather than studied. But there are moments that cry out for self-assertion, when a passive bowing to fate or a mechanical decision out of deference or convenience is treachery, because it forfeits dignity for ease." Edward Conze stated it is important for individual and collective moral growth that we recognise the illusion of our conscience being wholly located in our body; indeed both our conscience and wisdom expand when we act in an unselfish way and conversely "repressed compassion results in an unconscious sense of guilt."
The philosopher Peter Singer considers that usually when we describe an action as conscientious in the critical sense we do so in order to deny either that the relevant agent was motivated by selfish desires, like greed or ambition, or that he acted on whim or impulse.
Moral anti-realists debate whether the moral facts necessary to activate conscience supervene on natural facts with a posteriori necessity; or arise a priori because moral facts have a primary intension and naturally identical worlds may be presumed morally identical. It has also been argued that there is a measure of moral luck in how circumstances create the obstacles which conscience must overcome to apply moral principles or human rights and that with the benefit of enforceable property rights and the rule of law, access to universal health care plus the absence of high adult and infant mortality from conditions such as malaria, tuberculosis, HIV/AIDS and famine, people in relatively prosperous developed countries have been spared pangs of conscience associated with the physical necessity to steal scraps of food, bribe tax inspectors or police officers, and commit murder in guerrilla wars against corrupt government forces or rebel armies. Roger Scruton has claimed that true understanding of conscience and its relationship with morality has been hampered by an "impetuous" belief that philosophical questions are solved through the analysis of language in an area where clarity threatens vested interests. Susan Sontag similarly argued that it was a symptom of psychological immaturity not to recognise that many morally immature people willingly experience a form of delight, in some an erotic breaking of taboo, when witnessing violence, suffering and pain being inflicted on others. Jonathan Glover wrote that most of us "do not spend our lives on endless landscape gardening of our self" and our conscience is likely shaped not so much by heroic struggles, as by choice of partner, friends and job, as well as where we choose to live. Garrett Hardin, in a famous article called "The Tragedy of the Commons", argues that any instance in which society appeals to an individual exploiting a commons to restrain himself or herself for the general good—by means of his or her conscience—merely sets up a system which, by selectively diverting societal power and physical resources to those lacking in conscience, while fostering guilt (including anxiety about his or her individual contribution to over-population) in people acting upon it, actually works toward the elimination of conscience from the race.Scott James Shackelford. 2008. "The Tragedy of the Common Heritage of Mankind". Retrieved 30 October 2009.
John Ralston Saul expressed the view in The Unconscious Civilization that in contemporary developed nations many people have acquiesced in turning over their sense of right and wrong, their critical conscience, to technical experts; willingly restricting their moral freedom of choice to limited consumer actions ruled by the ideology of the free market, while citizen participation in public affairs is limited to the isolated act of voting and private-interest lobbying turns even elected representatives against the public interest.
Some argue on religious or philosophical grounds that it is blameworthy to act against conscience, even if the judgement of conscience is likely to be erroneous (say because it is inadequately informed about the facts, or prevailing moral (humanist or religious), professional ethical, legal and human rights norms). Failure to acknowledge and accept that conscientious judgements can be seriously mistaken, may only promote situations where one's conscience is manipulated by others to provide unwarranted justifications for non-virtuous and selfish acts; indeed, insofar as it is appealed to as glorifying ideological content, and an associated extreme level of devotion, without adequate constraint of external, altruistic, normative justification, conscience may be considered morally blind and dangerous both to the individual concerned and humanity as a whole. Langston argues that philosophers of virtue ethics have unnecessarily neglected conscience for, once conscience is trained so that the principles and rules it applies are those one would want all others to live by, its practise cultivates and sustains the virtues; indeed, amongst people in what each society considers to be the highest state of moral development there is little disagreement about how to act. Emmanuel Levinas viewed conscience as a revelatory encountering of resistance to our selfish powers, developing morality by calling into question our naive sense of freedom of will to use such powers arbitrarily, or with violence, this process being more severe the more rigorously the goal of our self was to obtain control.
In other words, the welcoming of the Other, to Levinas, was the very essence of conscience properly conceived; it encouraged our ego to accept the fallibility of assuming things about other people, that selfish freedom of will "does not have the last word" and that realising this has a transcendent purpose: "I am not alone ... in conscience I have an experience that is not commensurate with any a priori [see a priori and a posteriori] framework-a conceptless experience."
## Conscientious acts and the law
In the late 13th and early 14th centuries, English litigants began to petition the Lord Chancellor of England for relief from unjust judgments. As Keeper of the King's Conscience, the Chancellor intervened to allow for "merciful exceptions" to the King's laws, "to ensure that the King's conscience was right before God". The Chancellor's office evolved into the Court of Chancery and the Chancellor's decisions evolved into the body of law known as equity.
English humanist lawyers in the 16th and 17th centuries interpreted conscience as a collection of universal principles given to man by god at creation to be applied by reason; this gradually reforming the medieval Roman law-based system with forms of action, written pleadings, use of juries and patterns of litigation such as Demurrer and Assumpsit that displayed an increased concern for elements of right and wrong on the actual facts. A conscience vote in a parliament allows legislators to vote without restrictions from any political party to which they may belong. In his trial in Jerusalem Nazi war criminal Adolf Eichmann claimed he was simply following legal orders under paragraph 48 of the German Military Code which provided: "punishability of an action or omission is not excused on the ground that the person considered his behaviour required by his conscience or the prescripts of his religion". The United Nations Universal Declaration on Human Rights (UDHR) which is part of international customary law specifically refers to conscience in Articles 1 and 18. Likewise, the United Nations International Covenant on Civil and Political Rights (ICCPR) mentions conscience in Article 18.1. It has been argued that these articles provide international legal obligations protecting conscientious objectors from service in the military.
John Rawls in his A Theory of Justice defines a conscientious objector as an individual prepared to undertake, in public (and often despite widespread condemnation), an action of civil disobedience to a legal rule justifying it (also in public) by reference to contrary foundational social virtues (such as justice as liberty or fairness) and the principles of morality and law derived from them. Rawls considered civil disobedience should be viewed as an appeal, warning or admonishment (showing general respect and fidelity to the rule of law by the non-violence and transparency of methods adopted) that a law breaches a community's fundamental virtue of justice. Objections to Rawls' theory include first, its inability to accommodate conscientious objections to the society's basic appreciation of justice or to emerging moral or ethical principles (such as respect for the rights of the natural environment) which are not yet part of it and second, the difficulty of predictably and consistently determining that a majority decision is just or unjust. Conscientious objection (also called conscientious refusal or evasion) to obeying a law, should not arise from unreasoning, naive "traditional conscience", for to do so merely encourages infantile abdication of responsibility to calibrate the law against moral or human rights norms and disrespect for democratic institutions. Instead it should be based on "critical conscience' – seriously thought out, conceptually mature, personal moral or religious beliefs held to be fundamentally incompatible (that is, not merely inconsistent on the basis of selfish desires, whim or impulse), for example, either with all laws requiring conscription for military service, or legal compulsion to fight for or financially support the State in a particular war. A famous example arose when Henry David Thoreau the author of Walden was willingly jailed for refusing to pay a tax because he profoundly disagreed with a government policy and was frustrated by the corruption and injustice of the democratic machinery of the state. A more recent case concerned Kimberly Rivera, a private in the US Army and mother of four children who, having served three months in Iraq War decided the conflict was immoral and sought refugee status in Canada in 2012 (see List of Iraq War resisters), but was deported and arrested in the US.
In the Second World War, Great Britain granted conscientious-objection status not just to complete pacifists, but to those who objected to fighting in that particular war; this was done partly out of genuine respect, but also to avoid the disgraceful and futile persecutions of conscientious objectors that occurred during the First World War.
Amnesty International organises campaigns to protect those arrested and or incarcerated as a prisoner of conscience because of their conscientious beliefs, particularly concerning intellectual, political and artistic freedom of expression and association. Aung San Suu Kyi of Burma, was the winner of the 2009 Amnesty International Ambassador of Conscience Award. In legislation, a conscience clause is a provision in a statute that excuses a health professional from complying with the law (for example legalising surgical or pharmaceutical abortion) if it is incompatible with religious or conscientious beliefs.
Expressed justifications for refusing to obey laws because of conscience vary. Many conscientious objectors are so for religious reasons—notably, members of the historic peace churches are pacifist by doctrine. Other objections can stem from a deep sense of responsibility toward humanity as a whole, or from the conviction that even acceptance of work under military orders acknowledges the principle of conscription that should be everywhere condemned before the world can ever become safe for real democracy. A conscientious objector, however, does not have a primary aim of changing the law. John Dewey considered that conscientious objectors were often the victims of "moral innocency" and inexpertness in moral training: "the moving force of events is always too much for conscience". The remedy was not to deplore the wickedness of those who manipulate world power, but to connect conscience with forces moving in another direction- to build institutions and social environments predicated on the rule of law, for example, "then will conscience itself have compulsive power instead of being forever the martyred and the coerced." As an example, Albert Einstein who had advocated conscientious objection during the First World War and had been a longterm supporter of War Resisters' International reasoned that "radical pacifism" could not be justified in the face of Nazi rearmament and advocated a world federalist organization with its own professional army.
Samuel Johnson pointed out that an appeal to conscience should not allow the law to bring unjust suffering upon another. Conscience, according to Johnson, was nothing more than a conviction felt by ourselves of something to be done or something to be avoided; in questions of simple unperplexed morality, conscience is very often a guide that may be trusted. But before conscience can conclusively determine what morally should be done, he thought that the state of the question should be thoroughly known. "No man's conscience", said Johnson "can tell him the right of another man ... it is a conscience very ill informed that violates the rights of one man, for the convenience of another."
Civil disobedience as nonviolent protest or civil resistance are also acts of conscience, but are designed by those who undertake them chiefly to change, by appealing to the majority and democratic processes, laws or government policies perceived to be incoherent with fundamental social virtues and principles (such as justice, equality or respect for intrinsic human dignity). Civil disobedience, in a properly functioning democracy, allows a minority who feel strongly that a law infringes their sense of justice (but have no capacity to obtain legislative amendments or a referendum on the issue) to make a potentially apathetic or uninformed majority take account of the intensity of opposing views. A notable example of civil resistance or satyagraha ("satya" in sanskrit means "truth and compassion", "agraha" means "firmness of will") involved Mahatma Gandhi making salt in India when that act was prohibited by a British statute, in order to create moral pressure for law reform. Rosa Parks similarly acted on conscience in 1955 in Montgomery, Alabama refusing a legal order to give up her seat to make room for a white passenger; her action (and the similar earlier act of 15-year-old Claudette Colvin) led to the Montgomery bus boycott. Rachel Corrie was a US citizen allegedly killed by a bulldozer operated by the Israel Defense Forces (IDF) while involved in direct action (based on the nonviolent principles of Martin Luther King Jr. and Mahatma Gandhi) to prevent demolition of the home of local Palestinian pharmacist Samir Nasrallah. Al Gore has argued "If you're a young person looking at the future of this planet and looking at what is being done right now, and not done, I believe we have reached the stage where it is time for civil disobedience to prevent the construction of new coal plants that do not have carbon capture and sequestration." In 2011, NASA climate scientist James E. Hansen, environmental leader Phil Radford and Professor Bill McKibben were arrested for opposing a tar sands oil pipelineBill McKibben. "The keystone pipeline revolt: why mass arrests are just the beginning". Rolling Stone. 28 September 2011. https://www.rollingstone.com/politics/news/the-keystone-pipeline-revolt-why-mass-arrests-are-just-the-beginning-20110928 (accessed 29 December 2012) and Canadian renewable energy professor Mark Jaccard was arrested for opposing mountain-top coal mining; in his book Storms of my Grandchildren Hansen calls for similar civil resistance on a global scale to help replace the 'business-as-usual' Kyoto Protocol cap and trade system, with a progressive carbon tax at emission source on the oil, gas and coal industries – revenue being paid as dividends to low carbon footprint families.James Hansen. Tell Barack Obama the Truth – The Whole Truth. accessed 10 December 2009.
Notable historical examples of conscientious noncompliance in a different professional context included the manipulation of the visa process in 1939 by Japanese Consul-General Chiune Sugihara in Kaunas (the temporary capital of Lithuania between Germany and the Soviet Union) and by Raoul Wallenberg in Hungary in 1944 to allow Jews to escape almost certain death. Ho Feng-Shan the Chinese Consul-General in Vienna in 1939, defied orders from the Chinese ambassador in Berlin to issue Jews with visas for Shanghai. John Rabe a German member of the Nazi Party likewise saved thousands of Chinese from massacre by the Japanese military at Nanjing. The White Rose German student movement against the Nazis declared in their 4th leaflet: "We will not be silent. We are your bad conscience. The White Rose will not leave you in peace!" Conscientious noncompliance may be the only practical option for citizens wishing to affirm the existence of an international moral order or 'core' historical rights (such as the right to life, right to a fair trial and freedom of opinion) in states where non-violent protest or civil disobedience are met with prolonged arbitrary detention, torture, forced disappearance, murder or persecution.
The controversial Milgram experiment into obedience by Stanley Milgram showed that many people lack the psychological resources to openly resist authority, even when they are directed to act callously and inhumanely against an innocent victim.
## World conscience
World conscience is the universalist idea that with ready global communication, all people on earth will no longer be morally estranged from one another, whether it be culturally, ethnically, or geographically; instead they will conceive ethics from the utopian point of view of the universe, eternity or infinity, rather than have their duties and obligations defined by forces arising solely within the restrictive boundaries of "blood and territory".
Often this derives from a spiritual or natural law perspective, that for world peace to be achieved, conscience, properly understood, should be generally considered as not necessarily linked (often destructively) to fundamentalist religious ideologies, but as an aspect of universal consciousness, access to which is the common heritage of humanity. Thinking predicated on the development of world conscience is common to members of the Global Ecovillage Network such as the Findhorn Foundation, international conservation organisations like Fauna and Flora International, as well as performers of world music such as Alan Stivell. Non-government organizations, particularly through their work in agenda-setting, policy-making and implementation of human rights-related policy, have been referred to as the conscience of the world
Edward O Wilson has developed the idea of consilience to encourage coherence of global moral and scientific knowledge supporting the premise that "only unified learning, universally shared, makes accurate foresight and wise choice possible". Thus, world conscience is a concept that overlaps with the Gaia hypothesis in advocating a balance of moral, legal, scientific and economic solutions to modern transnational problems such as global poverty and global warming, through strategies such as environmental ethics, climate ethics, natural conservation, ecology, cosmopolitanism, sustainability and sustainable development, biosequestration and legal protection of the biosphere and biodiversity.Edward Goldsmith. The Way. Shambhala, Boston. 1993. p. 64.Geoff Davies. Economia: New Economic Systems to Empower People and Support the Living World. ABC Books. Sydney. 2004. pp. 202–03. The NGO 350.org, for example, seeks to attract world conscience to the problems associated with elevation in atmospheric greenhouse gas concentrations.
The microcredit initiatives of Nobel Peace Prize winner Muhammad Yunus have been described as inspiring a "war on poverty that blends social conscience and business savvy".
The Green party politician Bob Brown (who was arrested by the Tasmanian state police for a conscientious act of civil disobedience during the Franklin Dam protest) expresses world conscience in these terms: "the universe, through us, is evolving towards experiencing, understanding and making choices about its future'; one example of policy outcomes from such thinking being a global tax (see Tobin tax) to alleviate global poverty and protect the biosphere, amounting to 1/10 of 1% placed on the worldwide speculative currency market. Such an approach sees world conscience best expressing itself through political reforms promoting democratically based globalisation or planetary democracy (for example internet voting for global governance organisations (see world government) based on the model of "one person, one vote, one value") which gradually will replace contemporary market-based globalisation.
The American cardiologist Bernard Lown and the Russian cardiologist Yevgeniy Chazov were motivated in conscience through studying the catastrophic public health consequences of nuclear war in establishing International Physicians for the Prevention of Nuclear War (IPPNW) which was awarded the Nobel Peace Prize in 1985 and continues to work to "heal an ailing planet".Worldwide expressions of conscience contributed to the decision of the French government to halt atmospheric nuclear tests at Mururoa in the Pacific in 1974 after 41 such explosions (although below-ground nuclear tests continued there into the 1990s).
A challenge to world conscience was provided by an influential 1968 article by Garrett Hardin that critically analyzed the dilemma in which multiple individuals, acting independently after rationally consulting self-interest (and, he claimed, the apparently low 'survival-of-the-fittest' value of conscience-led actions) ultimately destroy a shared limited resource, even though each acknowledges such an outcome is not in anyone's long-term interest. Hardin's conclusion that commons areas are practicably achievable only in conditions of low population density (and so their continuance requires state restriction on the freedom to breed), created controversy additionally through his direct deprecation of the role of conscience in achieving individual decisions, policies and laws that facilitate global justice and peace, as well as sustainability and sustainable development of world commons areas, for example including those officially designated such under United Nations treaties (see common heritage of humanity). Areas designated common heritage of humanity under international law include the Moon, Outer Space, deep sea bed, Antarctica, the world cultural and natural heritage (see World Heritage Convention) and the human genome. It will be a significant challenge for world conscience that as world oil, coal, mineral, timber, agricultural and water reserves are depleted, there will be increasing pressure to commercially exploit common heritage of mankind areas.
The philosopher Peter Singer has argued that the United Nations Millennium Development Goals represent the emergence of an ethics based not on national boundaries but on the idea of one world. Ninian Smart has similarly predicted that the increase in global travel and communication will gradually draw the world's religions towards a pluralistic and transcendental humanism characterized by an "open spirit" of empathy and compassion.
Noam Chomsky has argued that forces opposing the development of such a world conscience include free market ideologies that valorise corporate greed in nominal electoral democracies where advertising, shopping malls and indebtedness, shape citizens into apathetic consumers in relation to information and access necessary for democratic participation. John Passmore has argued that mystical considerations about the global expansion of all human consciousness, should take into account that if as a species we do become something much superior to what we are now, it will be as a consequence of conscience not only implanting a goal of moral perfectibility, but assisting us to remain periodically anxious, passionate and discontented, for these are necessary components of care and compassion. The Committee on Conscience of the US Holocaust Memorial Museum has targeted genocides such as those in Rwanda, Bosnia, Darfur, the Congo and Chechnya as challenges to the world's conscience. Oscar Arias Sanchez has criticised global arms industry spending as a failure of conscience by nation states: "When a country decides to invest in arms, rather than in education, housing, the environment, and health services for its people, it is depriving a whole generation of its right to prosperity and happiness. We have produced one firearm for every ten inhabitants of this planet, and yet we have not bothered to end hunger when such a feat is well within our reach. This is not a necessary or inevitable state of affairs. It is a deliberate choice" (see Campaign Against Arms Trade). US House of Representatives Speaker Nancy Pelosi, after meeting with the 14th Dalai Lama during the 2008 violent protests in Tibet and aftermath said: "The situation in Tibet is a challenge to the conscience of the world." Nelson Mandela, through his example and words, has been described as having shaped the conscience of the world. The Right Livelihood Award is awarded yearly in Sweden to those people, mostly strongly motivated by conscience, who have made exemplary practical contributions to resolving the great challenges facing our planet and its people. In 2009, for example, along with Catherine Hamlin (obstetric fistula and see fistula foundation)), David Suzuki (promoting awareness of climate change) and Alyn Ware (nuclear disarmament), René Ngongo shared the Right Livelihood Award "for his courage in confronting the forces that are destroying the Congo Basin's rainforests and building political support for their conservation and sustainable use".Mu Xuequan. "Alternative Nobel awards go to Congo, New Zealand, Australia" . www.chinaview.cn 2009-10-13 22:35:19. Retrieved 18 October 2009 Avaaz is one of the largest global on-line organizations launched in January 2007 to promote conscience-driven activism on issues such as climate change, human rights, animal rights, corruption, poverty, and conflict, thus "closing the gap between the world we have and the world most people everywhere want".
## Notable examples of modern acts based on conscience
In a notable contemporary act of conscience, Christian bushwalker Brenda Hean protested against the flooding of Lake Pedder despite threats and that ultimately led to her death. Another was the campaign by Ken Saro-Wiwa against oil extraction by multinational corporations in Nigeria that led to his execution. So too was the act by the Tank Man, or the Unknown Rebel photographed holding his shopping bag in the path of tanks during the protests at Beijing's Tiananmen Square on 5 June 1989. The actions of United Nations Secretary General Dag Hammarskjöld to try to achieve peace in the Congo despite the (eventuating) threat to his life were strongly motivated by conscience as is reflected in his diary, Vägmärken (Markings). Another example involved the actions of Warrant Officer Hugh Thompson, Jr to try to prevent the My Lai massacre in the Vietnam War. Evan Pederick voluntarily confessed and was convicted of the Sydney Hilton bombing stating that his conscience could not tolerate the guilt and that "I guess I was quite unique in the prison system in that I had to keep proving my guilt, whereas everyone else said they were innocent." Vasili Arkhipov was a Russian naval officer on out-of-radio-contact Soviet submarine B-59 being depth-charged by US warships during the Cuban Missile Crisis whose dissent when two other officers decided to launch a nuclear torpedo (unanimous agreement to launch was required) may have averted a nuclear war. In 1963 Buddhist monk Thich Quang Duc performed a famous act of self-immolation to protest against alleged persecution of his faith by the Vietnamese Ngo Dinh Diem regime.
Conscience played a major role in the actions by anaesthetist Stephen Bolsin to whistleblow (see list of whistleblowers) on incompetent paediatric cardiac surgeons at the Bristol Royal Infirmary. Jeffrey Wigand was motivated by conscience to expose the Big Tobacco scandal, revealing that executives of the companies knew that cigarettes were addictive and approved the addition of carcinogenic ingredients to the cigarettes. David Graham, a Food and Drug Administration employee, was motivated by conscience to whistleblow that the arthritis pain-reliever Vioxx increased the risk of cardiovascular deaths although the manufacturer suppressed this information. Rick Piltz, from the U.S. global warming Science Program, blew the whistle on a White House official who ignored majority scientific opinion to edit a climate change report ("Our Changing Planet") to reflect the Bush administration's view that the problem was unlikely to exist. Muntadhar al-Zaidi, an Iraqi journalist, was imprisoned and allegedly tortured for his act of conscience in throwing his shoes at George W. Bush. Mordechai Vanunu, an Israeli former nuclear technician, acted on conscience to reveal details of Israel's nuclear weapons program to the British press in 1986; was kidnapped by Israeli agents, transported to Israel, convicted of treason and spent 18 years in prison, including more than 11 years in solitary confinement.
At the awards ceremony for the 200 metres at the 1968 Summer Olympics in Mexico City John Carlos, Tommie Smith and Peter Norman ignored death threats and official warnings to take part in an anti-racism protest that destroyed their respective careers. W. Mark Felt an agent of the United States Federal Bureau of Investigation who retired in 1973 as the Bureau's Associate Director, acted on conscience to provide reporters Bob Woodward and Carl Bernstein with information that resulted in the Watergate scandal. Conscience was a major factor in US Public Health Service officer Peter Buxtun revealing the Tuskegee syphilis experiment to the public. The 2008 attack by the Israeli military on civilian areas of Palestinian Gaza was described as a "stain on the world's conscience". Conscience was a major factor in the refusal of Aung San Suu Kyi to leave Burma despite house arrest and persecution by the military dictatorship in that country.
Conscience was a factor in Peter Galbraith's criticism of fraud in the 2009 Afghanistan election despite it costing him his United Nations job. Conscience motivated Bunnatine Greenhouse to expose irregularities in the contracting of the Halliburton company for work in Iraq. Naji al-Ali a popular cartoon artist in the Arab world, loved for his defense of the ordinary people, and for his criticism of repression and despotism by both the Israeli military and Yasser Arafat's PLO, was murdered for refusing to compromise with his conscience. The journalist Anna Politkovskaya provided (prior to her murder) an example of conscience in her opposition to the Second Chechen War and then-Russian President Vladimir Putin. Conscience motivated the Russian human rights activist Natalia Estemirova, who was abducted and murdered in Grozny, Chechnya in 2009. The Death of Neda Agha-Soltan arose from conscience-driven protests against the 2009 Iranian presidential election. Muslim lawyer Shirin Ebadi (winner of the 2003 Nobel Peace Prize) has been described as the 'conscience of the Islamic Republic' for her work in protecting the human rights of women and children in Iran. The human rights lawyer Gao Zhisheng, often referred to as the 'conscience of China' and who had previously been arrested and allegedly tortured after calling for respect for human rights and for constitutional reform, was abducted by Chinese security agents in February 2009. 2010 Nobel Peace Prize winner Liu Xiaobo in his final statement before being sentenced by a closed Chinese court to over a decade in jail as a political prisoner of conscience stated: "For hatred is corrosive of a person’s wisdom and conscience; the mentality of enmity can poison a nation’s spirit." Sergei Magnitsky, a lawyer in Russia, was arrested, held without trial for almost a year and died in custody, as a result of exposing corruption. On 6 October 2001 Laura Whittle was a naval gunner on HMAS Adelaide (FFG 01) under orders to implement a new border protection policy when they encountered the SIEV-4 (Suspected Illegal Entry Vessel-4) refugee boat in choppy seas. After being ordered to fire warning shots from her 50 calibre machinegun to make the boat turn back she saw it beginning to break up and sink with a father on board holding out his young daughter that she might be saved (see Children Overboard Affair). Whittle jumped without a life vest 12 metres into the sea to help save the refugees from drowning thinking "this isn't right; this isn't how things should be." In February 2012 journalist Marie Colvin was deliberately targeted and killed by the Syrian Army in Homs during the Syrian uprising and Siege of Homs, after she decided to stay at the "epicentre of the storm" in order to "expose what is happening". In October 2012 the Taliban organised the attempted murder of Malala Yousafzai a teenage girl who had been campaigning, despite their threats, for female education in Pakistan. In December 2012 the 2012 Delhi gang rape case was said to have stirred the collective conscience of India to civil disobedience and public protest at the lack of legal action against rapists in that country (see Rape in India) In June 2013 Edward Snowden revealed details of a US National Security Agency internet and electronic communication PRISM (surveillance program) because of a conscience-felt obligation to the freedom of humanity greater than obedience to the laws that bound his employment.
## In literature, art, film, and music
The ancient epic of the Indian subcontinent, the Mahabharata of Vyasa, contains two pivotal moments of conscience. The first occurs when the warrior Arjuna being overcome with compassion against killing his opposing relatives in war, receives counsel (see Bhagavad-Gita) from Krishna about his spiritual duty ("work as though you are performing a sacrifice for the general good"). The second, at the end of the saga, is when king Yudhishthira having alone survived the moral tests of life, is offered eternal bliss, only to refuse it because a faithful dog is prevented from coming with him by purported divine rules and laws. The French author Montaigne (1533–1592) in one of the most celebrated of his essays ("On experience") expressed the benefits of living with a clear conscience: "Our duty is to compose our character, not to compose books, to win not battles and provinces, but order and tranquillity in our conduct. Our great and glorious masterpiece is to live properly". In his famous Japanese travel journal Oku no Hosomichi (Narrow Road to the Deep North) composed of mixed haiku poetry and prose, Matsuo Bashō (1644–94) in attempting to describe the eternal in this perishable world is often moved in conscience; for example by a thicket of summer grass being all that remains of the dreams and ambitions of ancient warriors. Chaucer's "Franklin's Tale" in The Canterbury Tales recounts how a young suitor releases a wife from a rash promise because of the respect in his conscience for the freedom to be truthful, gentle and generous.
The critic A. C. Bradley discusses the central problem of Shakespeare's tragic character Hamlet as one where conscience in the form of moral scruples deters the young Prince with his "great anxiety to do right" from obeying his father's hell-bound ghost and murdering the usurping King ("is't not perfect conscience to quit him with this arm?" (v.ii.67)).
Bradley develops a theory about Hamlet's moral agony relating to a conflict between "traditional" and "critical" conscience: "The conventional moral ideas of his time, which he shared with the Ghost, told him plainly that he ought to avenge his father; but a deeper conscience in him, which was in advance of his time, contended with these explicit conventional ideas. It is because this deeper conscience remains below the surface that he fails to recognise it, and fancies he is hindered by cowardice or sloth or passion or what not; but it emerges into light in that speech to Horatio. And it is just because he has this nobler moral nature in him that we admire and love him". The opening words of Shakespeare's Sonnet 94 ("They that have pow'r to hurt, and will do none") have been admired as a description of conscience. So has John Donne's commencement of his poem s:Goodfriday, 1613. Riding Westward: "Let man's soul be a sphere, and then, in this, Th' intelligence that moves, devotion is;"
Anton Chekhov in his plays The Seagull, Uncle Vanya and Three Sisters describes the tortured emotional states of doctors who at some point in their careers have turned their back on conscience. In his short stories, Chekhov also explored how people misunderstood the voice of a tortured conscience. A promiscuous student, for example, in The Fit describes it as a "dull pain, indefinite, vague; it was like anguish and the most acute fear and despair ... in his breast, under the heart" and the young doctor examining the misunderstood agony of compassion experienced by the factory owner's daughter in From a Case Book calls it an "unknown, mysterious power ... in fact close at hand and watching him." Characteristically, Chekhov's own conscience drove him on the long journey to Sakhalin to record and alleviate the harsh conditions of the prisoners at that remote outpost. As Irina Ratushinskaya writes in the introduction to that work: "Abandoning everything, he travelled to the distant island of Sakhalin, the most feared place of exile and forced labour in Russia at that time. One cannot help but wonder why? Simply, because the lot of the people there was a bitter one, because nobody really knew about the lives and deaths of the exiles, because he felt that they stood in greater need of help that anyone else. A strange reason, maybe, but not for a writer who was the epitome of all the best traditions of a Russian man of letters. Russian literature has always focused on questions of conscience and was, therefore, a powerful force in the moulding of public opinion."
E. H. Carr writes of Dostoevsky's character the young student Raskolnikov in the novel Crime and Punishment who decides to murder a 'vile and loathsome' old woman money lender on the principle of transcending conventional morals: "the sequel reveals to us not the pangs of a stricken conscience (which a less subtle writer would have given us) but the tragic and fruitless struggle of a powerful intellect to maintain a conviction which is incompatible with the essential nature of man." Hermann Hesse wrote his Siddhartha to describe how a young man in the time of the Buddha follows his conscience on a journey to discover a transcendent inner space where all things could be unified and simply understood, ending up discovering that personal truth through selfless service as a ferryman. J. R. R. Tolkien in his epic The Lord of the Rings describes how only the hobbit Frodo is pure enough in conscience to carry the ring of power through war-torn Middle-earth to destruction in the Cracks of Doom, Frodo determining at the end to journey without weapons, and being saved from failure by his earlier decision to spare the life of the creature Gollum. Conor Cruise O'Brien wrote that Albert Camus was the writer most representative of the Western consciousness and conscience in its relation to the non-Western world. Harper Lee's 1960 novel To Kill a Mockingbird portrays Atticus Finch (played by Gregory Peck in the classic film from the book (see To Kill a Mockingbird)) as a lawyer true to his conscience who sets an example to his children and community.
The Robert Bolt play A Man For All Seasons focuses on the conscience of Catholic lawyer Thomas More in his struggle with King Henry VIII ("the loyal subject is more bounden to be loyal to his conscience than to any other thing"). George Orwell wrote his novel Nineteen Eighty-Four on the isolated island of Jura, Scotland to describe how a man (Winston Smith) attempts to develop critical conscience in a totalitarian state which watches every action of the people and manipulates their thinking with a mixture of propaganda, endless war and thought control through language control (double think and newspeak) to the point where prisoners look up to and even love their torturers. In the Ministry of Love, Winston's torturer (O'Brien) states: "You are imagining that there is something called human nature which will be outraged by what we do and will turn against us. But we create human nature. Men are infinitely malleable".
A tapestry copy of Picasso's Guernica depicting a massacre of innocent women and children during the Spanish Civil War is displayed on the wall of the United Nations building in New York City, at the entrance to the Security Council room, demonstrably as a spur to the conscience of representatives from the nation states. Albert Tucker painted Man's Head to capture the moral disintegration, and lack of conscience, of a man convicted of kicking a dog to death.
The Impressionist painter Vincent van Gogh wrote in a letter to his brother Theo in 1878 that "one must never let the fire in one's soul die, for the time will inevitably come when it will be needed. And he who chooses poverty for himself and loves it possesses a great treasure and will hear the voice of his conscience address him every more clearly. He who hears that voice, which is God's greatest gift, in his innermost being and follows it, finds in it a friend at last, and he is never alone! ... That is what all great men have acknowledged in their works, all those who have thought a little more deeply and searched and worked and loved a little more than the rest, who have plumbed the depths of the sea of life."
The 1957 Ingmar Bergman film The Seventh Seal portrays the journey of a medieval knight (Max von Sydow) returning disillusioned from the crusades ("what is going to happen to those of us who want to believe, but aren't able to?") across a plague-ridden landscape, undertaking a game of chess with the personification of Death until he can perform one meaningful altruistic act of conscience (overturning the chess board to distract Death long enough for a family of jugglers to escape in their wagon). The 1942 Casablanca centers on the development of conscience in the cynical American Rick Blaine (Humphrey Bogart) in the face of oppression by the Nazis and the example of the resistance leader Victor Laszlo.The David Lean and Robert Bolt screenplay for Doctor Zhivago (an adaptation of Boris Pasternak's novel) focuses strongly on the conscience of a doctor-poet in the midst of the Russian Revolution (in the end "the walls of his heart were like paper").The 1982 Ridley Scott film Blade Runner focuses on the struggles of conscience between and within a bounty hunter (Rick Deckard (Harrison Ford)) and a renegade replicant android (Roy Batty (Rutger Hauer)) in a future society which refuses to accept that forms of artificial intelligence can have aspects of being such as conscience.
Johann Sebastian Bach wrote his last great choral composition the Mass in B minor (BWV 232) to express the alternating emotions of loneliness, despair, joy and rapture that arise as conscience reflects on a departed human life. Here JS Bach's use of counterpoint and contrapuntal settings, his dynamic discourse of melodically and rhythmically distinct voices seeking forgiveness of sins ("Qui tollis peccata mundi, miserere nobis") evokes a spiraling moral conversation of all humanity expressing his belief that "with devotional music, God is always present in his grace".
Ludwig van Beethoven's meditations on illness, conscience and mortality in the Late String Quartets led to his dedicating the third movement of String Quartet in A Minor (1825) Op. 132 (see String Quartet No. 15) as a "Hymn of Thanksgiving to God of a convalescent".Ludwig van Beethoven. The Late Quartets Vol II. String Quartet in A minor, Op. 132. Quartetto Italiano. Phillips Classics Productions 1996. John Lennon's work "Imagine" owes much of its popular appeal to its evocation of conscience against the atrocities created by war, religious fundamentalism and politics. The Beatles George Harrison-written track "The Inner Light" sets to Indian raga music a verse from the Tao Te Ching that "without going out of your door you can know the ways of heaven'. In the 1986 movie The Mission the guilty conscience and penance of the slave trader Mendoza is made more poignant by the haunting oboe music of Ennio Morricone ("On Earth as it is in Heaven") The song Sweet Lullaby by Deep Forest is based on a traditional Baegu lullaby from the Solomon Islands called "Rorogwela" in which a young orphan is comforted as an act of conscience by his older brother. The Dream Academy song 'Forest Fire' provided an early warning of the moral dangers of our 'black cloud' 'bringing down a different kind of weather ... letting the sunshine in, that's how the end begins."
The American Society of Journalists and Authors (ASJA) presents the Conscience-in-Media Award to journalists whom the society deems worthy of recognition for demonstrating "singular commitment to the highest principles of journalism at notable personal cost or sacrifice".
The Ambassador of Conscience Award, Amnesty International's most prestigious human rights award, takes its inspiration from a poem written by Irish Nobel prize-winning poet Seamus Heaney called "The Republic of Conscience". | https://en.wikipedia.org/wiki/Conscience |
A pump is a device that moves fluids (liquids or gases), or sometimes slurries, by mechanical action, typically converted from electrical energy into hydraulic or pneumatic energy.
Mechanical pumps serve in a wide range of applications such as pumping water from wells, aquarium filtering, pond filtering and aeration, in the car industry for water-cooling and fuel injection, in the energy industry for pumping oil and natural gas or for operating cooling towers and other components of heating, ventilation and air conditioning systems. In the medical industry, pumps are used for biochemical processes in developing and manufacturing medicine, and as artificial replacements for body parts, in particular the artificial heart and penile prosthesis.
When a pump contains two or more pump mechanisms with fluid being directed to flow through them in series, it is called a multi-stage pump. Terms such as two-stage or double-stage may be used to specifically describe the number of stages. A pump that does not fit this description is simply a single-stage pump in contrast.
In biology, many different types of chemical and biomechanical pumps have evolved; biomimicry is sometimes used in developing new types of mechanical pumps.
## Types
Mechanical pumps may be submerged in the fluid they are pumping or be placed external to the fluid.
Pumps can be classified by their method of displacement into electromagnetic pumps, positive-displacement pumps, impulse pumps, velocity pumps, gravity pumps, steam pumps and valveless pumps. There are three basic types of pumps: positive-displacement, centrifugal and axial-flow pumps. In centrifugal pumps the direction of flow of the fluid changes by ninety degrees as it flows over an impeller, while in axial flow pumps the direction of flow is unchanged.
### Electromagnetic pump
### Positive-displacement pumps
A positive-displacement pump makes a fluid move by trapping a fixed amount and forcing (displacing) that trapped volume into the discharge pipe.
Some positive-displacement pumps use an expanding cavity on the suction side and a decreasing cavity on the discharge side. Liquid flows into the pump as the cavity on the suction side expands and the liquid flows out of the discharge as the cavity collapses. The volume is constant through each cycle of operation.
#### Positive-displacement pump behavior and safety
Positive-displacement pumps, unlike centrifugal, can theoretically produce the same flow at a given rotational speed no matter what the discharge pressure. Thus, positive-displacement pumps are constant flow machines. However, a slight increase in internal leakage as the pressure increases prevents a truly constant flow rate.
A positive-displacement pump must not operate against a closed valve on the discharge side of the pump, because it has no shutoff head like centrifugal pumps. A positive-displacement pump operating against a closed discharge valve continues to produce flow and the pressure in the discharge line increases until the line bursts, the pump is severely damaged, or both.
A relief or safety valve on the discharge side of the positive-displacement pump is therefore necessary. The relief valve can be internal or external. The pump manufacturer normally has the option to supply internal relief or safety valves. The internal valve is usually used only as a safety precaution. An external relief valve in the discharge line, with a return line back to the suction line or supply tank, provides increased safety.
#### Positive-displacement types
A positive-displacement pump can be further classified according to the mechanism used to move the fluid:
- Rotary-type positive displacement: internal and external gear pump, screw pump, lobe pump, shuttle block, flexible vane and sliding vane, circumferential piston, flexible impeller, helical twisted roots (e.g. the Wendelkolben pump) and liquid-ring pumps
- Reciprocating-type positive displacement: piston pumps, plunger pumps and diaphragm pumps
- Linear-type positive displacement: rope pumps and chain pumps
##### Rotary positive-displacement pumps
These pumps move fluid using a rotating mechanism that creates a vacuum that captures and draws in the liquid.
Advantages: Rotary pumps are very efficient because they can handle highly viscous fluids with higher flow rates as viscosity increases.
Drawbacks: The nature of the pump requires very close clearances between the rotating pump and the outer edge, making it rotate at a slow, steady speed. If rotary pumps are operated at high speeds, the fluids cause erosion, which eventually causes enlarged clearances that liquid can pass through, which reduces efficiency.
Rotary positive-displacement pumps fall into five main types:
-
###### Gear pump
s – a simple type of rotary pump where the liquid is pushed around a pair of gears.
-
###### Screw pump
s – the shape of the internals of this pump is usually two screws turning against each other to pump the liquid
- Rotary vane pumps
- Hollow disc pumps (also known as eccentric disc pumps or hollow rotary disc pumps), similar to scroll compressors, these have an eccentric cylindrical rotor encased in a circular housing. As the rotor orbits, it traps fluid between the rotor and the casing, drawing the fluid through the pump. It is used for highly viscous fluids like petroleum-derived products, and it can also support high pressures of up to 290 psi.
-
###### Peristaltic pump
s have rollers which pinch a section of flexible tubing, forcing the liquid ahead as the rollers advance. Because they are very easy to keep clean, these are popular for dispensing food, medicine, and concrete.
##### Reciprocating positive-displacement pumps
Reciprocating pumps move the fluid using one or more oscillating pistons, plungers, or membranes (diaphragms), while valves restrict fluid motion to the desired direction. In order for suction to take place, the pump must first pull the plunger in an outward motion to decrease pressure in the chamber. Once the plunger pushes back, it will increase the chamber pressure and the inward pressure of the plunger will then open the discharge valve and release the fluid into the delivery pipe at constant flow rate and increased pressure.
Pumps in this category range from simplex, with one cylinder, to in some cases quad (four) cylinders, or more. Many reciprocating-type pumps are duplex (two) or triplex (three) cylinder. They can be either single-acting with suction during one direction of piston motion and discharge on the other, or double-acting with suction and discharge in both directions. The pumps can be powered manually, by air or steam, or by a belt driven by an engine. This type of pump was used extensively in the 19th century—in the early days of steam propulsion—as boiler feed water pumps. Now reciprocating pumps typically pump highly viscous fluids like concrete and heavy oils, and serve in special applications that demand low flow rates against high resistance. Reciprocating hand pumps were widely used to pump water from wells. Common bicycle pumps and foot pumps for inflation use reciprocating action.
These positive-displacement pumps have an expanding cavity on the suction side and a decreasing cavity on the discharge side. Liquid flows into the pumps as the cavity on the suction side expands and the liquid flows out of the discharge as the cavity collapses. The volume is constant given each cycle of operation and the pump's volumetric efficiency can be achieved through routine maintenance and inspection of its valves.
Typical reciprocating pumps are:
- Plunger pump – a reciprocating plunger pushes the fluid through one or two open valves, closed by suction on the way back.
-
###### Diaphragm pump
– similar to plunger pumps, where the plunger pressurizes hydraulic oil which is used to flex a diaphragm in the pumping cylinder. Diaphragm valves are used to pump hazardous and toxic fluids.
- Piston pump displacement pumps – usually simple devices for pumping small amounts of liquid or gel manually. The common hand soap dispenser is such a pump.
- Radial piston pumpa form of hydraulic pump where pistons extend in a radial direction.
- Vibratory pump or vibration pumpa particularly low-cost form of plunger pump, popular in low-cost espresso machines. The only moving part is a spring-loaded piston, the armature of a solenoid. Driven by half-wave rectified alternating current, the piston is forced forward while energized, and is retracted by the spring during the other half cycle. Due to their inefficiency, vibratory pumps typically cannot be operated for more than one minute without overheating, so are limited to intermittent duty.
##### Various positive-displacement pumps
The positive-displacement principle applies in these pumps:
- Rotary lobe pump
-
###### Progressing cavity pump
- Rotary gear pump
- Piston pump
- Diaphragm pump
- Screw pump
- Gear pump
- Hydraulic pump
- Rotary vane pump
- Peristaltic pump
-
###### Rope pump
- Flexible impeller pump
Gear pump
This is the simplest form of rotary positive-displacement pumps. It consists of two meshed gears that rotate in a closely fitted casing. The tooth spaces trap fluid and force it around the outer periphery. The fluid does not travel back on the meshed part, because the teeth mesh closely in the center. Gear pumps see wide use in car engine oil pumps and in various hydraulic power packs.
Screw pump
A screw pump is a more complicated type of rotary pump that uses two or three screws with opposing thread — e.g., one screw turns clockwise and the other counterclockwise. The screws are mounted on parallel shafts that often have gears that mesh so the shafts turn together and everything stays in place. In some cases the driven screw drives the secondary screw, without gears, often using the fluid to limit abrasion. The screws turn on the shafts and drive fluid through the pump. As with other forms of rotary pumps, the clearance between moving parts and the pump's casing is minimal.
Progressing cavity pump
Widely used for pumping difficult materials, such as sewage sludge contaminated with large particles, a progressing cavity pump consists of a helical rotor, about ten times as long as its width, and a stator, mainly made out of rubber. This can be visualized as a central core of diameter x with, typically, a curved spiral wound around of thickness half x, though in reality it is manufactured in a single lobe. This shaft fits inside a heavy-duty rubber sleeve or stator, of wall thickness also typically x. As the shaft rotates inside the stator, the rotor gradually forces fluid up the rubber cavity. Such pumps can develop very high pressure at low volumes at a rate of 90 PSI per stage on water for standard configurations.
###### Roots-type pump
Named after the Roots brothers who invented it, this lobe pump displaces the fluid trapped between two long helical rotors, each fitted into the other when perpendicular at 90°, rotating inside a triangular shaped sealing line configuration, both at the point of suction and at the point of discharge. This design produces a continuous flow with equal volume and no vortex. It can work at low pulsation rates, and offers gentle performance that some applications require.
## Applications
include:
- High capacity industrial air compressors.
- Roots superchargers on internal combustion engines.
- A brand of civil defense siren, the Federal Signal Corporation's Thunderbolt.
Peristaltic pump
A peristaltic pump is a type of positive-displacement pump. It contains fluid within a flexible tube fitted inside a circular pump casing (though linear peristaltic pumps have been made). A number of rollers, shoes, or wipers attached to a rotor compress the flexible tube. As the rotor turns, the part of the tube under compression closes (or occludes), forcing the fluid through the tube. Additionally, when the tube opens to its natural state after the passing of the cam it draws (restitution) fluid into the pump. This process is called peristalsis and is used in many biological systems such as the gastrointestinal tract.
###### Plunger pumps
Plunger pumps are reciprocating positive-displacement pumps.
These consist of a cylinder with a reciprocating plunger. The suction and discharge valves are mounted in the head of the cylinder. In the suction stroke, the plunger retracts and the suction valves open causing suction of fluid into the cylinder. In the forward stroke, the plunger pushes the liquid out of the discharge valve.
## Efficiency
and common problems: With only one cylinder in plunger pumps, the fluid flow varies between maximum flow when the plunger moves through the middle positions, and zero flow when the plunger is at the end positions. A lot of energy is wasted when the fluid is accelerated in the piping system. Vibration and water hammer may be a serious problem. In general, the problems are compensated for by using two or more cylinders not working in phase with each other. Centrifugal pumps are also susceptible to water hammer. Surge analysis, a specialized study, helps evaluate this risk in such systems.
###### Triplex-style plunger pump
Triplex plunger pumps use three plungers, which reduces the pulsation relative to single reciprocating plunger pumps. Adding a pulsation dampener on the pump outlet can further smooth the pump ripple, or ripple graph of a pump transducer. The dynamic relationship of the high-pressure fluid and plunger generally requires high-quality plunger seals. Plunger pumps with a larger number of plungers have the benefit of increased flow, or smoother flow without a pulsation damper. The increase in moving parts and crankshaft load is one drawback.
Car washes often use these triplex-style plunger pumps (perhaps without pulsation dampers). In 1968, William Bruggeman reduced the size of the triplex pump and increased the lifespan so that car washes could use equipment with smaller footprints. Durable high-pressure seals, low-pressure seals and oil seals, hardened crankshafts, hardened connecting rods, thick ceramic plungers and heavier duty ball and roller bearings improve reliability in triplex pumps. Triplex pumps now are in a myriad of markets across the world.
Triplex pumps with shorter lifetimes are commonplace to the home user. A person who uses a home pressure washer for 10 hours a year may be satisfied with a pump that lasts 100 hours between rebuilds. Industrial-grade or continuous duty triplex pumps on the other end of the quality spectrum may run for as much as 2,080 hours a year.
The oil and gas drilling industry uses massive semi-trailer-transported triplex pumps called mud pumps to pump drilling mud, which cools the drill bit and carries the cuttings back to the surface. Drillers use triplex or even quintuplex pumps to inject water and solvents deep into shale in the extraction process called fracking.
Diaphragm pump
Typically run on electricity compressed air, diaphragm pumps are relatively inexpensive and can perform a wide variety of duties, from pumping air into an aquarium, to liquids through a filter press. Double-diaphragm pumps can handle viscous fluids and abrasive materials with a gentle pumping process ideal for transporting shear-sensitive media.
Rope pump
### Impulse pump
Impulse pumps use pressure created by gas (usually air). In some impulse pumps the gas trapped in the liquid (usually water), is released and accumulated somewhere in the pump, creating a pressure that can push part of the liquid upwards.
Conventional impulse pumps include:
-
#### Hydraulic ram pump
s – kinetic energy of a low-head water supply is stored temporarily in an air-bubble hydraulic accumulator, then used to drive water to a higher head.
- Pulser pumps – run with natural resources, by kinetic energy only.
- Airlift pumps – run on air inserted into pipe, which pushes the water up when bubbles move upward
Instead of a gas accumulation and releasing cycle, the pressure can be created by burning of hydrocarbons. Such combustion driven pumps directly transmit the impulse from a combustion event through the actuation membrane to the pump fluid. In order to allow this direct transmission, the pump needs to be almost entirely made of an elastomer (e.g. silicone rubber). Hence, the combustion causes the membrane to expand and thereby pumps the fluid out of the adjacent pumping chamber. The first combustion-driven soft pump was developed by ETH Zurich.
Hydraulic ram pump
A hydraulic ram is a water pump powered by hydropower.
It takes in water at relatively low pressure and high flow-rate and outputs water at a higher hydraulic-head and lower flow-rate. The device uses the water hammer effect to develop pressure that lifts a portion of the input water that powers the pump to a point higher than where the water started.
The hydraulic ram is sometimes used in remote areas, where there is both a source of low-head hydropower, and a need for pumping water to a destination higher in elevation than the source. In this situation, the ram is often useful, since it requires no outside source of power other than the kinetic energy of flowing water.
### Velocity pumps
Rotodynamic pumps (or dynamic pumps) are a type of velocity pump in which kinetic energy is added to the fluid by increasing the flow velocity. This increase in energy is converted to a gain in potential energy (pressure) when the velocity is reduced prior to or as the flow exits the pump into the discharge pipe. This conversion of kinetic energy to pressure is explained by the First law of thermodynamics, or more specifically by Bernoulli's principle.
Dynamic pumps can be further subdivided according to the means in which the velocity gain is achieved.
These types of pumps have a number of characteristics:
1. Continuous energy
1. Conversion of added energy to increase in kinetic energy (increase in velocity)
1. Conversion of increased velocity (kinetic energy) to an increase in pressure head
A practical difference between dynamic and positive-displacement pumps is how they operate under closed valve conditions. Positive-displacement pumps physically displace fluid, so closing a valve downstream of a positive-displacement pump produces a continual pressure build up that can cause mechanical failure of pipeline or pump. Dynamic pumps differ in that they can be safely operated under closed valve conditions (for short periods of time).
#### Radial-flow pump
Such a pump is also referred to as a centrifugal pump. The fluid enters along the axis or center, is accelerated by the impeller and exits at right angles to the shaft (radially); an example is the centrifugal fan, which is commonly used to implement a vacuum cleaner. Another type of radial-flow pump is a vortex pump. The liquid in them moves in tangential direction around the working wheel. The conversion from the mechanical energy of motor into the potential energy of flow comes by means of multiple whirls, which are excited by the impeller in the working channel of the pump. Generally, a radial-flow pump operates at higher pressures and lower flow rates than an axial- or a mixed-flow pump.
#### Axial-flow pump
These are also referred to as all-fluid pumps. The fluid is pushed outward or inward to move fluid axially. They operate at much lower pressures and higher flow rates than radial-flow (centrifugal) pumps. Axial-flow pumps cannot be run up to speed without special precaution. If at a low flow rate, the total head rise and high torque associated with this pipe would mean that the starting torque would have to become a function of acceleration for the whole mass of liquid in the pipe system.
Mixed-flow pumps function as a compromise between radial and axial-flow pumps. The fluid experiences both radial acceleration and lift and exits the impeller somewhere between 0 and 90 degrees from the axial direction. As a consequence mixed-flow pumps operate at higher pressures than axial-flow pumps while delivering higher discharges than radial-flow pumps. The exit angle of the flow dictates the pressure head-discharge characteristic in relation to radial and mixed-flow.
#### Regenerative turbine pump
Also known as drag, friction, liquid-ring pump, peripheral, traction, turbulence, or vortex pumps, regenerative turbine pumps are a class of rotodynamic pump that operates at high head pressures, typically .
The pump has an impeller with a number of vanes or paddles which spins in a cavity. The suction port and pressure ports are located at the perimeter of the cavity and are isolated by a barrier called a stripper, which allows only the tip channel (fluid between the blades) to recirculate, and forces any fluid in the side channel (fluid in the cavity outside of the blades) through the pressure port. In a regenerative turbine pump, as fluid spirals repeatedly from a vane into the side channel and back to the next vane, kinetic energy is imparted to the periphery, thus pressure builds with each spiral, in a manner similar to a regenerative blower.
As regenerative turbine pumps cannot become vapor locked, they are commonly applied to volatile, hot, or cryogenic fluid transport. However, as tolerances are typically tight, they are vulnerable to solids or particles causing jamming or rapid wear. Efficiency is typically low, and pressure and power consumption typically decrease with flow. Additionally, pumping direction can be reversed by reversing direction of spin.
#### Side-channel pump
A side-channel pump has a suction disk, an impeller, and a discharge disk.
#### Eductor-jet pump
This uses a jet, often of steam, to create a low pressure. This low pressure sucks in fluid and propels it into a higher-pressure region.
### Gravity pumps
Gravity pumps include the syphon and Heron's fountain. The hydraulic ram is also sometimes called a gravity pump. In a gravity pump the fluid is lifted by gravitational force.
### Steam pump
Steam pumps have been for a long time mainly of historical interest. They include any type of pump powered by a steam engine and also pistonless pumps such as Thomas Savery's or the Pulsometer steam pump.
Recently there has been a resurgence of interest in low-power solar steam pumps for use in smallholder irrigation in developing countries. Previously small steam engines have not been viable because of escalating inefficiencies as vapour engines decrease in size. However the use of modern engineering materials coupled with alternative engine configurations has meant that these types of system are now a cost-effective opportunity.
### Valveless pumps
Valveless pumping assists in fluid transport in various biomedical and engineering systems. In a valveless pumping system, no valves (or physical occlusions) are present to regulate the flow direction. The fluid pumping efficiency of a valveless system, however, is not necessarily lower than that having valves. In fact, many fluid-dynamical systems in nature and engineering more or less rely upon valveless pumping to transport the working fluids therein. For instance, blood circulation in the cardiovascular system is maintained to some extent even when the heart's valves fail. Meanwhile, the embryonic vertebrate heart begins pumping blood long before the development of discernible chambers and valves. Similar to blood circulation in one direction, bird respiratory systems pump air in one direction in rigid lungs, but without any physiological valve. In microfluidics, valveless impedance pumps have been fabricated, and are expected to be particularly suitable for handling sensitive biofluids. Ink jet printers operating on the piezoelectric transducer principle also use valveless pumping. The pump chamber is emptied through the printing jet due to reduced flow impedance in that direction and refilled by capillary action.
## Pump repairs
Examining pump repair records and mean time between failures (MTBF) is of great importance to responsible and conscientious pump users. In view of that fact, the preface to the 2006 Pump User's Handbook alludes to "pump failure" statistics. For the sake of convenience, these failure statistics often are translated into MTBF (in this case, installed life before failure).
In early 2005, Gordon Buck, John Crane Inc.'s chief engineer for field operations in Baton Rouge, Louisiana, examined the repair records for a number of refinery and chemical plants to obtain meaningful reliability data for centrifugal pumps. A total of 15 operating plants having nearly 15,000 pumps were included in the survey. The smallest of these plants had about 100 pumps; several plants had over 2000. All facilities were located in the United States. In addition, considered as "new", others as "renewed" and still others as "established". Many of these plants—but not all—had an alliance arrangement with John Crane. In some cases, the alliance contract included having a John Crane Inc. technician or engineer on-site to coordinate various aspects of the program.
Not all plants are refineries, however, and different results occur elsewhere. In chemical plants, pumps have historically been "throw-away" items as chemical attack limits life. Things have improved in recent years, but the somewhat restricted space available in "old" DIN and ASME-standardized stuffing boxes places limits on the type of seal that fits. Unless the pump user upgrades the seal chamber, the pump only accommodates more compact and simple versions. Without this upgrading, lifetimes in chemical installations are generally around 50 to 60 percent of the refinery values.
Unscheduled maintenance is often one of the most significant costs of ownership, and failures of mechanical seals and bearings are among the major causes. Keep in mind the potential value of selecting pumps that cost more initially, but last much longer between repairs. The MTBF of a better pump may be one to four years longer than that of its non-upgraded counterpart. Consider that published average values of avoided pump failures range from US$2600 to US$12,000. This does not include lost opportunity costs. One pump fire occurs per 1000 failures. Having fewer pump failures means having fewer destructive pump fires.
As has been noted, a typical pump failure, based on actual year 2002 reports, costs US$5,000 on average. This includes costs for material, parts, labor and overhead. Extending a pump's MTBF from 12 to 18 months would save US$1,667 per year — which might be greater than the cost to upgrade the centrifugal pump's reliability.Submersible slurry pumps in high demand. Engineeringnews.co.za. Retrieved on 2011-05-25.
Applications
Pumps are used throughout society for a variety of purposes. Early applications includes the use of the windmill or watermill to pump water. Today, the pump is used for irrigation, water supply, gasoline supply, air conditioning systems, refrigeration (usually called a compressor), chemical movement, sewage movement, flood control, marine services, etc.
Because of the wide variety of applications, pumps have a plethora of shapes and sizes: from very large to very small, from handling gas to handling liquid, from high pressure to low pressure, and from high volume to low volume.
### Priming a pump
Typically, a liquid pump cannot simply draw air. The feed line of the pump and the internal body surrounding the pumping mechanism must first be filled with the liquid that requires pumping: An operator must introduce liquid into the system to initiate the pumping, known as priming the pump. Loss of prime is usually due to ingestion of air into the pump, or evaporation of the working fluid if the pump is used infrequently. Clearances and displacement ratios in pumps for liquids are insufficient for pumping compressible gas, so air or other gasses in the pump can not be evacuated by the pump's action alone. This is the case with most velocity (rotodynamic) pumps — for example, centrifugal pumps. For such pumps, the position of the pump and intake tubing should be lower than the suction point so it is primed by gravity; otherwise the pump should be manually filled with liquid or a secondary pump should be used until all air is removed from the suction line and the pump casing. Liquid ring pumps have a dedicated intake for the priming liquid separate from the intake of the fluid being pumped, as the fluid being pumped may be a gas or mix of gas, liquid, and solids. For these pumps the priming liquid intake must be supplied continuously (either by gravity or pressure), however the intake for the fluid being pumped is capable of drawing a vacuum equivalent to the boiling point of the priming liquid.
Positive–displacement pumps, however, tend to have sufficiently tight sealing between the moving parts and the casing or housing of the pump that they can be described as self-priming. Such pumps can also serve as priming pumps, so-called when they are used to fulfill that need for other pumps in lieu of action taken by a human operator.
### Pumps as public water supplies
One sort of pump once common worldwide was a hand-powered water pump, or 'pitcher pump'. It was commonly installed over community water wells in the days before piped water supplies.
In parts of the British Isles, it was often called the parish pump. Though such community pumps are no longer common, people still used the expression parish pump to describe a place or forum where matters of local interest are discussed.
Because water from pitcher pumps is drawn directly from the soil, it is more prone to contamination. If such water is not filtered and purified, consumption of it might lead to gastrointestinal or other water-borne diseases. A notorious case is the 1854 Broad Street cholera outbreak. At the time it was not known how cholera was transmitted, but physician John Snow suspected contaminated water and had the handle of the public pump he suspected removed; the outbreak then subsided.
Modern hand-operated community pumps are considered the most sustainable low-cost option for safe water supply in resource-poor settings, often in rural areas in developing countries. A hand pump opens access to deeper groundwater that is often not polluted and also improves the safety of a well by protecting the water source from contaminated buckets. Pumps such as the Afridev pump are designed to be cheap to build and install, and easy to maintain with simple parts. However, scarcity of spare parts for these type of pumps in some regions of Africa has diminished their utility for these areas.
### Sealing multiphase pumping applications
Multiphase pumping applications, also referred to as tri-phase, have grown due to increased oil drilling activity. In addition, the economics of multiphase production is attractive to upstream operations as it leads to simpler, smaller in-field installations, reduced equipment costs and improved production rates. In essence, the multiphase pump can accommodate all fluid stream properties with one piece of equipment, which has a smaller footprint. Often, two smaller multiphase pumps are installed in series rather than having just one massive pump.
#### Types and features of multiphase pumps
##### Helico-axial (centrifugal)
A rotodynamic pump with one single shaft that requires two mechanical seals, this pump uses an open-type axial impeller. It is often called a Poseidon pump, and can be described as a cross between an axial compressor and a centrifugal pump.
##### Twin-screw (positive-displacement)
The twin-screw pump is constructed of two inter-meshing screws that move the pumped fluid. Twin screw pumps are often used when pumping conditions contain high gas volume fractions and fluctuating inlet conditions. Four mechanical seals are required to seal the two shafts.
##### Progressive cavity (positive-displacement)
Progressive Cavity Pumps are well suited to pump sludge, slurries, viscous, and shear sensitive fluids. Progressive cavity pumps are single-screw types use in surface and downhole oil production. They serve a vast arrange of industries and applications ranging from Wastewater Treatment, Pulp and Paper, oil and gas, mining, and oil and gas.
##### Electric submersible (centrifugal)
These pumps are basically multistage centrifugal pumps and are widely used in oil well applications as a method for artificial lift. These pumps are usually specified when the pumped fluid is mainly liquid.
Buffer tank
A buffer tank is often installed upstream of the pump suction nozzle in case of a slug flow. The buffer tank breaks the energy of the liquid slug, smooths any fluctuations in the incoming flow and acts as a sand trap.
As the name indicates, multiphase pumps and their mechanical seals can encounter a large variation in service conditions such as changing process fluid composition, temperature variations, high and low operating pressures and exposure to abrasive/erosive media. The challenge is selecting the appropriate mechanical seal arrangement and support system to ensure maximized seal life and its overall effectiveness.John Crane Seal Sentinel – John Crane Increases Production Capabilities with Machine that Streamlines Four Machining Functions into One . Sealsentinel.com. Retrieved on 2011-05-25.
## Specifications
Pumps are commonly rated by horsepower, volumetric flow rate, outlet pressure in metres (or feet) of head, inlet suction in suction feet (or metres) of head.
The head can be simplified as the number of feet or metres the pump can raise or lower a column of water at atmospheric pressure.
From an initial design point of view, engineers often use a quantity termed the specific speed to identify the most suitable pump type for a particular combination of flow rate and head.
Net Positive Suction Head (NPSH) is crucial for pump performance. It has two key aspects:
1) NPSHr (Required): The Head required for the pump to operate without cavitation issues.
2) NPSHa (Available): The actual pressure provided by the system (e.g., from an overhead tank).
For optimal pump operation, NPSHa must always exceed NPSHr. This ensures the pump has enough pressure to prevent cavitation, a damaging condition.
## Pumping power
The power imparted into a fluid increases the energy of the fluid per unit volume. Thus the power relationship is between the conversion of the mechanical energy of the pump mechanism and the fluid elements within the pump. In general, this is governed by a series of simultaneous differential equations, known as the Navier–Stokes equations. However a more simple equation relating only the different energies in the fluid, known as Bernoulli's equation can be used. Hence the power, P, required by the pump:
$$
P = \frac{\Delta p Q}{\eta}
$$
where Δp is the change in total pressure between the inlet and outlet (in Pa), and Q, the volume flow-rate of the fluid is given in m3/s. The total pressure may have gravitational, static pressure and kinetic energy components; i.e. energy is distributed between change in the fluid's gravitational potential energy (going up or down hill), change in velocity, or change in static pressure. η is the pump efficiency, and may be given by the manufacturer's information, such as in the form of a pump curve, and is typically derived from either fluid dynamics simulation (i.e. solutions to the Navier–Stokes for the particular pump geometry), or by testing. The efficiency of the pump depends upon the pump's configuration and operating conditions (such as rotational speed, fluid density and viscosity etc.)
$$
\Delta p = \rho {(v_2^2 - v_1^2) \over 2} + \rho\Delta z g + {\Delta p_{\mathrm{static}}}
$$
For a typical "pumping" configuration, the work is imparted on the fluid, and is thus positive. For the fluid imparting the work on the pump (i.e. a turbine), the work is negative. Power required to drive the pump is determined by dividing the output power by the pump efficiency. Furthermore, this definition encompasses pumps with no moving parts, such as a siphon.
Efficiency
Pump efficiency is defined as the ratio of the power imparted on the fluid by the pump in relation to the power supplied to drive the pump. Its value is not fixed for a given pump, efficiency is a function of the discharge and therefore also operating head. For centrifugal pumps, the efficiency tends to increase with flow rate up to a point midway through the operating range (peak efficiency or Best Efficiency Point (BEP) ) and then declines as flow rates rise further. Pump performance data such as this is usually supplied by the manufacturer before pump selection. Pump efficiencies tend to decline over time due to wear (e.g. increasing clearances as impellers reduce in size).
When a system includes a centrifugal pump, an important design issue is matching the head loss-flow characteristic with the pump so that it operates at or close to the point of its maximum efficiency.
Pump efficiency is an important aspect and pumps should be regularly tested. Thermodynamic pump testing is one method.
## Minimum flow protection
Most large pumps have a minimum flow requirement below which the pump may be damaged by overheating, impeller wear, vibration, seal failure, drive shaft damage or poor performance. A minimum flow protection system ensures that the pump is not operated below the minimum flow rate. The system protects the pump even if it is shut-in or dead-headed, that is, if the discharge line is completely closed.
The simplest minimum flow system is a pipe running from the pump discharge line back to the suction line. This line is fitted with an orifice plate sized to allow the pump minimum flow to pass. The arrangement ensures that the minimum flow is maintained, although it is wasteful as it recycles fluid even when the flow through the pump exceeds the minimum flow.
A more sophisticated, but more costly, system (see diagram) comprises a flow measuring device (FE) in the pump discharge which provides a signal into a flow controller (FIC) which actuates a flow control valve (FCV) in the recycle line. If the measured flow exceeds the minimum flow then the FCV is closed. If the measured flow falls below the minimum flow the FCV opens to maintain the minimum flowrate.
As the fluids are recycled the kinetic energy of the pump increases the temperature of the fluid. For many pumps this added heat energy is dissipated through the pipework. However, for large industrial pumps, such as oil pipeline pumps, a recycle cooler is provided in the recycle line to cool the fluids to the normal suction temperature. Alternatively the recycled fluids may be returned to upstream of the export cooler in an oil refinery, oil terminal, or offshore installation.
## References
## Further reading
- Australian Pump Manufacturers' Association. Australian Pump Technical Handbook, 3rd edition. Canberra: Australian Pump Manufacturers' Association, 1987. .
- Hicks, Tyler G. and Theodore W. Edwards. Pump Application Engineering. McGraw-Hill Book Company.1971.
-
- Robbins, L. B. "Homemade Water Pressure Systems". Popular Science, February 1919, pages 83–84. Article about how a homeowner can easily build a pressurized home water system that does not use electricity.
Category:Ancient inventions | https://en.wikipedia.org/wiki/Pump |
Quickhull is a method of computing the convex hull of a finite set of points in n-dimensional space. It uses a divide and conquer approach similar to that of quicksort, from which its name derives. Its worst case time complexity for 2-dimensional and 3-dimensional space is
$$
O(n^2)
$$
, but when the input precision is restricted to
$$
O(\log n)
$$
bits, its worst case time complexity is conjectured to be
$$
O(n \log r)
$$
, where
$$
n
$$
is the number of input points and
$$
r
$$
is the number of processed points (up to
$$
n
$$
).
N-dimensional Quickhull was invented in 1996 by C. Bradford Barber, David P. Dobkin, and Hannu Huhdanpaa. It was an extension of Jonathan Scott Greenfield's 1990 planar Quickhull algorithm, although the 1996 authors did not know of his methods. Instead, Barber et al. describe it as a deterministic variant of Clarkson and Shor's 1989 algorithm.
## Algorithm
The 2-dimensional algorithm can be broken down into the following steps:
1. Find the points with minimum and maximum x coordinates, as these will always be part of the convex hull. If many points with the same minimum/maximum x exist, use the ones with the minimum/maximum y, respectively.
1. Use the line formed by the two points to divide the set into two subsets of points, which will be processed recursively. We next describe how to determine the part of the hull above the line; the part of the hull below the line can be determined similarly.
1. Determine the point above the line with the maximum distance from the line. This point forms a triangle with the two points on the line.
1. The points lying inside of that triangle cannot be part of the convex hull and can therefore be ignored in the next steps.
1. Recursively repeat the previous two steps on the two lines formed by the two new sides of the triangle.
1. Continue until no more points are left, the recursion has come to an end and the points selected constitute the convex hull.
The problem is more complex in the higher-dimensional case, as the hull is built from many facets; the data structure needs to account for that and record the line/plane/hyperplane (ridge) shared by neighboring facets too. For d dimensions:
1. Pick d + 1 points from the set that do not share a plane or a hyperplane. This forms an initial hull with facets Fs[].
1. For each F in Fs[], find all unassigned points that are "above" it; i.e., pointing away from the center of the hull, and assign them to an "outside" set F.O associated with F. The algorithm maintains the invariant that every point that has not been added to the hull but could potentially be a vertex of it is assigned to exactly one outside set.
1. For each F with a non-empty F.O:
1. Find the point p in F.O with the maximum distance from F and add it to the hull. Note that p will not necessarily be a vertex of the final hull, as it might be removed later.
1. Create a visible set V and initialize it to F. Extend V in all directions for neighboring facets Fv until no further facets are visible from p. Fv being visible from p means that p is above Fv.
1. The boundary of V then forms the set of horizon ridges H.
1. Let Fnew[] be the set of facets created from p and all ridges in H.
1. Unassign all points in the outside sets of facets in V. For each new facet in Fnew[], perform step (2) only considering these newly unassigned points to initialize its outside set. Note that every point that remains unassigned at the end of this process lies within the current hull.
1. Delete the now-internal facets in V from Fs[]. Add the new facets in Fnew[] to Fs[] and continue the iteration.
## Pseudocode for 2D set of points
Input = a set S of n points
Assume that there are at least 2 points in the input set S of points
function QuickHull(S) is
// Find convex hull from the set S of n points
Convex Hull := {}
Find left and right most points, say A & B, and add A & B to convex hull
Segment AB divides the remaining (n − 2) points into 2 groups S1 and S2
where S1 are points in S that are on the right side of the oriented line from A to B,
and S2 are points in S that are on the right side of the oriented line from B to A
FindHull(S1, A, B)
FindHull(S2, B, A)
Output := Convex Hull
end function
function FindHull(Sk, P, Q) is
// Find points on convex hull from the set Sk of points
// that are on the right side of the oriented line from P to Q
if Sk has no point then
return
From the given set of points in Sk, find farthest point, say C, from segment PQ
Add point C to convex hull at the location between P and Q
Three points P, Q, and C partition the remaining points of Sk into 3 subsets: S0, S1, and S2
where S0 are points inside triangle PCQ, S1 are points on the right side of the oriented
line from P to C, and S2 are points on the right side of the oriented line from C to Q.
FindHull(S1, P, C)
FindHull(S2, C, Q)
end function
A pseudocode specialized for the 3D case is available from Jordan Smith. It includes a similar "maximum point" strategy for choosing the starting hull. If these maximum points are degenerate, the whole point cloud is as well. | https://en.wikipedia.org/wiki/Quickhull |
In the mathematical field of graph theory, a transitive reduction of a directed graph is another directed graph with the same vertices and as few edges as possible, such that for all pairs of vertices , a (directed) path from to in exists if and only if such a path exists in the reduction. Transitive reductions were introduced by , who provided tight bounds on the computational complexity of constructing them.
More technically, the reduction is a directed graph that has the same reachability relation as . Equivalently, and its transitive reduction should have the same transitive closure as each other, and the transitive reduction of should have as few edges as possible among all graphs with that property.
The transitive reduction of a finite directed acyclic graph (a directed graph without directed cycles) is unique and is a subgraph of the given graph. However, uniqueness fails for graphs with (directed) cycles, and for infinite graphs not even existence is guaranteed.
The closely related concept of a minimum equivalent graph is a subgraph of that has the same reachability relation and as few edges as possible. The difference is that a transitive reduction does not have to be a subgraph of . For finite directed acyclic graphs, the minimum equivalent graph is the same as the transitive reduction. However, for graphs that may contain cycles, minimum equivalent graphs are NP-hard to construct, while transitive reductions can be constructed in polynomial time.
Transitive reduction can be defined for an abstract binary relation on a set, by interpreting the pairs of the relation as arcs in a directed graph.
## Classes of graphs
### In directed acyclic graphs
The transitive reduction of a finite directed graph G is a graph with the fewest possible edges that has the same reachability relation as the original graph. That is, if there is a path from a vertex x to a vertex y in graph G, there must also be a path from x to y in the transitive reduction of G, and vice versa. Specifically, if there is some path from x to y, and another from y to z, then there may be no path from x to z which does not include y. Transitivity for x, y, and z means that if x < y and y < z, then x < z. If for any path from y to z there is a path x to y, then there is a path x to z; however, it is not true that for any paths x to y and x to z that there is a path y to z, and therefore any edge between vertices x and z are excluded under a transitive reduction, as they represent walks which are not transitive. The following image displays drawings of graphs corresponding to a non-transitive binary relation (on the left) and its transitive reduction (on the right).
The transitive reduction of a finite directed acyclic graph G is unique, and consists of the edges of G that form the only path between their endpoints. In particular, it is always a spanning subgraph of the given graph. For this reason, the transitive reduction coincides with the minimum equivalent graph in this case.
In the mathematical theory of binary relations, any relation R on a set X may be thought of as a directed graph that has the set X as its vertex set and that has an arc xy for every ordered pair of elements that are related in R. In particular, this method lets partially ordered sets be reinterpreted as directed acyclic graphs, in which there is an arc xy in the graph whenever there is an order relation x < y between the given pair of elements of the partial order. When the transitive reduction operation is applied to a directed acyclic graph that has been constructed in this way, it generates the covering relation of the partial order, which is frequently given visual expression by means of a Hasse diagram.
Transitive reduction has been used on networks which can be represented as directed acyclic graphs (e.g. citation graphs or citation networks) to reveal structural differences between networks.
### In graphs with cycles
In a finite graph that has cycles, the transitive reduction may not be unique: there may be more than one graph on the same vertex set that has a minimum number of edges and has the same reachability relation as the given graph. Additionally, it may be the case that none of these minimum graphs is a subgraph of the given graph. Nevertheless, it is straightforward to characterize the minimum graphs with the same reachability relation as the given graph G. If G is an arbitrary directed graph, and H is a graph with the minimum possible number of edges having the same reachability relation as G, then H consists of
- A directed cycle for each strongly connected component of G, connecting together the vertices in this component
- An edge xy for each edge XY of the transitive reduction of the condensation of G, where X and Y are two strongly connected components of G that are connected by an edge in the condensation, x is any vertex in component X, and y is any vertex in component Y. The condensation of G is a directed acyclic graph that has a vertex for every strongly connected component of G and an edge for every two components that are connected by an edge in G. In particular, because it is acyclic, its transitive reduction can be defined as in the previous section.
The total number of edges in this type of transitive reduction is then equal to the number of edges in the transitive reduction of the condensation, plus the number of vertices in nontrivial strongly connected components (components with more than one vertex).
The edges of the transitive reduction that correspond to condensation edges can always be chosen to be a subgraph of the given graph G. However, the cycle within each strongly connected component can only be chosen to be a subgraph of G if that component has a Hamiltonian cycle, something that is not always true and is difficult to check. Because of this difficulty, it is NP-hard to find the smallest subgraph of a given graph G with the same reachability (its minimum equivalent graph).
### In infinite graphs
Aho et al. provide the following example to show that in infinite graphs, even when the graph is acyclic, a transitive reduction may not exist. Form a graph with a vertex for each real number, with an edge
$$
x\to y
$$
whenever
$$
x < y
$$
as real numbers. Then this graph is infinite, acyclic, and transitively closed. However, in any subgraph that has the same transitive closure, each remaining edge
$$
x\to y
$$
can be removed without changing the transitive closure, because there still must remain a path from
$$
x
$$
to
$$
y
$$
through any vertex between them. Therefore, among the subgraphs with the same transitive closure, none of these subgraphs is minimal: there is no transitive reduction.
## Computational complexity
As Aho et al. show, when the time complexity of graph algorithms is measured only as a function of the number n of vertices in the graph, and not as a function of the number of edges, transitive closure and transitive reduction of directed acyclic graphs have the same complexity. It had already been shown that transitive closure and multiplication of Boolean matrices of size n × n had the same complexity as each other, so this result put transitive reduction into the same class. The best exact algorithms for matrix multiplication, as of 2023, take time O(n2.371552), and this gives the fastest known worst-case time bound for transitive reduction in dense graphs, by applying it to matrices over the integers and looking at the nonzero entries in the result.
### Computing the reduction using the closure
To prove that transitive reduction is as easy as transitive closure, Aho et al. rely on the already-known equivalence with Boolean matrix multiplication. They let A be the adjacency matrix of the given directed acyclic graph, and B be the adjacency matrix of its transitive closure (computed using any standard transitive closure algorithm). Then an edge uv belongs to the transitive reduction if and only if there is a nonzero entry in row u and column v of matrix A, and there is a zero entry in the same position of the matrix product AB. In this construction, the nonzero elements of the matrix AB represent pairs of vertices connected by paths of length two or more.
### Computing the closure using the reduction
To prove that transitive reduction is as hard as transitive closure, Aho et al. construct from a given directed acyclic graph G another graph H, in which each vertex of G is replaced by a path of three vertices, and each edge of G corresponds to an edge in H connecting the corresponding middle vertices of these paths. In addition, in the graph H, Aho et al. add an edge from every path start to every path end. In the transitive reduction of H, there is an edge from the path start for u to the path end for v, if and only if edge uv does not belong to the transitive closure of G. Therefore, if the transitive reduction of H can be computed efficiently, the transitive closure of G can be read off directly from it.
### Computing the reduction in sparse graphs
When measured both in terms of the number n of vertices and the number m of edges in a directed acyclic graph, transitive reductions can also be found in time O(nm), a bound that may be faster than the matrix multiplication methods for sparse graphs. To do so, apply a linear time longest path algorithm in the given directed acyclic graph, for each possible choice of starting vertex. From the computed longest paths, keep only those of length one (single edge); in other words, keep those edges (u,v) for which there exists no other path from u to v. This O(nm) time bound matches the complexity of constructing transitive closures by using depth-first search or breadth first search to find the vertices reachable from every choice of starting vertex, so again with these assumptions transitive closures and transitive reductions can be found in the same amount of time.
### Output-sensitive
For a graph with n vertices, m edges, and r edges in the transitive reduction, it is possible to find the transitive reduction using an output-sensitive algorithm in an amount of time that depends on r in place of m. The algorithm is:
- For each vertex v, in the reverse of a topological order of the input graph:
- Initialize a set of vertices reachable from v, initially the singleton set {v}.
- For each edge vw, in topological order by w, test whether w is in the reachable set of v, and if not:
- Output edge vw as part of the transitive reduction.
- Replace the set of vertices reachable from v by its union with the reachable set of w.
The ordering of the edges in the inner loop can be obtained by using two passes of counting sort or another stable sorting algorithm to sort the edges, first by the topological numbering of their end vertex, and secondly by their starting vertex. If the sets are represented as bit arrays, each set union operation can be performed in time O(n), or faster using bitwise operations. The number of these set operations is proportional to the number of output edges, leading to the overall time bound of O(nr). The reachable sets obtained during the algorithm describe the transitive closure of the input.
If the graph is given together with a partition of its vertices into k chains (pairwise-reachable subsets), this time can be further reduced to O(kr), by representing each reachable set concisely as a union of suffixes of chains.
## Notes
## References
- .
- .
-
- .
- .
- .
- .
## External links
-
Category:Set theory
Category:Graph theory
Category:Graph algorithms
de:Transitive_Hülle_(Relation)#Transitive_Reduktion | https://en.wikipedia.org/wiki/Transitive_reduction |
In computer science, a queap is a priority queue data structure. The data structure allows insertions and deletions of arbitrary elements, as well as retrieval of the highest-priority element. Each deletion takes amortized time logarithmic in the number of items that have been in the structure for a longer time than the removed item. Insertions take constant amortized time.
The data structure consists of a doubly linked list and a 2–4 tree data structure, each modified to keep track of its minimum-priority element. The basic operation of the structure is to keep newly inserted elements in the doubly linked list, until a deletion would remove one of the list items, at which point they are all moved into the 2–4 tree. The 2–4 tree stores its elements in insertion order, rather than the more conventional priority-sorted order.
Both the data structure and its name were devised by John Iacono and Stefan Langerman.
## Description
A queap is a priority queue that inserts elements in O(1) amortized time, and removes the minimum element in O(log(k + 2)) if there are k items that have been in the heap for a longer time than the element to be extracted. The queap has a property called the queueish property: the time to search for element x is O(lg q(x)) where q(x) is equal to n − 1 − w(x) and w(x) is the number of distinct items that has been accessed by operations such as searching, inserting, or deleting. q(x) is defined as how many elements have not been accessed since x's last access. Indeed, the queueish property is the complement of the splay tree working set property: the time to search for element x is O(lg w(x)).
A queap can be represented by two data structures: a doubly linked list and a modified version of 2–4 tree. The doubly linked list, L, is used for a series of insert and locate-min operations. The queap keeps a pointer to the minimum element stored in the list. To add element x to list l, the element x is added to the end of the list and a bit variable in element x is set to one. This operation is done to determine if the element is either in the list or in a 2–4 tree.
A 2–4 tree is used when a delete operation occurs. If the item x is already in tree T, the item is removed using the 2–4 tree delete operation. Otherwise, the item x is in list L (done by checking if the bit variable is set). All the elements stored in list L are then added to the 2–4 tree, setting the bit variable of each element to zero. x is then removed from T.
A queap uses only the 2–4 tree structure properties, not a search tree. The modified 2–4 tree structure is as follows. Suppose list L has the following set of elements:
$$
x_1, x_2, x_3, \dots , x_k
$$
. When the deletion operation is invoked, the set of elements stored in L is then added to the leaves of the 2–4 tree in that order, proceeded by a dummy leaf containing an infinite key. Each internal node of T has a pointer
$$
h_v
$$
, which points to the smallest item in subtree v. Each internal node on path P from the root to
$$
x_0
$$
has a pointer
$$
c_v
$$
, which points to the smallest key in
$$
T - T_v - \{r\}
$$
. The
$$
h_v
$$
pointers of each internal node on path P are ignored. The queap has a pointer to
$$
c_{x_0}
$$
, which points to the smallest element in T.
An application of queaps includes a unique set of high priority events and extraction of the highest priority event for processing.
## Operations
Let minL be a pointer that points to the minimum element in the doubly linked list L,
$$
c_{x_0}
$$
be the minimum element stored in the 2–4 tree, T, k be the number of elements stored in T, and n be the total number of elements stored in queap Q. The operations are as follows:
New(Q): Initializes a new empty queap.
Initialize an empty doubly linked list L and 2–4 tree T. Set k and n to zero.
Insert(Q, x): Add the element x to queap Q.
Insert the element x in list L. Set the bit in element x to one to demonstrate that the element is in the list L. Update the minL pointer if x is the smallest element in the list. Increment n by 1.
Minimum(Q): Retrieve a pointer to the smallest element from queap Q.
If key(minL) < key(
$$
c_{x_0}
$$
), return minL. Otherwise return
$$
c_{x_0}
$$
.
Delete(Q, x): Remove element x from queap Q.
If the bit of the element x is set to one, the element is stored in list L. Add all the elements from L to T, setting the bit of each element to zero. Each element is added to the parent of the right most child of T using the insert operation of the 2–4 tree. L becomes empty. Update
$$
h_v
$$
pointers for all the nodes v whose children are new/modified, and repeat the process with the next parent until the parent is equal to the root. Walk from the root to node
$$
x_0
$$
, and update the
$$
c_v
$$
values. Set k equal to n.
If the bit of the element x is set to zero, x is a leaf of T. Delete x using the 2–4 tree delete operation. Starting from node x, walk in T to node
$$
x_0
$$
, updating
$$
h_v
$$
and
$$
c_v
$$
pointers. Decrement n and k by 1.
DeleteMin(Q): Delete and return the smallest element from queap Q.
Invoke the Minimum(Q) operation. The operation returns min. Invoke the Delete(Q, min) operation. Return min.
CleanUp(Q): Delete all the elements in list L and tree T.
Starting from the first element in list L, traverse the list, deleting each node.
Starting from the root of the tree T, traverse the tree using the post-order traversal algorithm, deleting each node in the tree.
## Analysis
The running time is analyzed using the amortized analysis. The potential function for queap Q will be
$$
\phi(Q)=c|L|
$$
where
$$
Q=(T, L)
$$
.
Insert(Q, x): The cost of the operation is O(1). The size of list L grows by one, the potential increases by some constant c.
Minimum(Q): The operation does not alter the data structure so the amortized cost is equal to its actual cost, O(1).
Delete(Q, x): There are two cases.
### Case 1
If x is in tree T, then the amortized cost is not modified. The delete operation is O(1) amortized 2–4 tree. Since x was removed from the tree,
$$
h_v
$$
and
$$
c_v
$$
pointers may need updating. At most, there will be
$$
O(lgq(x))
$$
updates.
### Case 2
If x is in list L, then all the elements from L are inserted in T. This has a cost of
$$
a|L|
$$
of some constant a, amortized over the 2–4 tree. After inserting and updating the
$$
h_v
$$
and
$$
c_v
$$
pointers, the total time spent is bounded by
$$
2a|L|
$$
. The second operation is to delete x from T, and to walk on the path from x to
$$
x_0
$$
, correcting
$$
h_v
$$
and
$$
c_v
$$
values. The time is spent at most
$$
2a|L| + O(lgq(x))
$$
. If
$$
c > 2a
$$
, then the amortized cost will be
$$
O(lgq(x))
$$
.
Delete(Q, x): is the addition of the amortized cost of Minimum(Q) and Delete(Q, x), which is
$$
O(lgq(x))
$$
.
## Code example
A small Java implementation of a queap:
```java
public class Queap
{
public int n, k;
public List<Element> l; // Element is a generic data type.
public QueapTree t; // a 2-4 tree, modified for Queap purpose
public Element minL;
private Queap() {
n = 0;
k = 0;
l = new LinkedList<Element>();
t = new QueapTree();
}
public static Queap New() {
return new Queap();
}
public static void Insert(Queap Q, Element x) {
if (Q.n == 0)
Q.minL = x;
Q.l.add(x);
x.inList = true;
if (x.compareTo(Q.minL) < 0)
Q.minL = x;
}
public static Element Minimum(Queap Q) {
// t is a 2-4 tree and x0, cv are tree nodes.
if (Q.minL.compareTo(Q.t.x0.cv.key) < 0)
return Q.minL;
return Q.t.x0.cv.key;
}
public static void Delete(Queap Q, QueapNode x) {
Q.t.deleteLeaf(x);
--Q.n;
--Q.k;
}
public static void Delete(Queap Q, Element x) {
QueapNode n;
if (x.inList) {
// set inList of all the elements in the list to false
n = Q.t.insertList(Q.l, x);
Q.k = Q.n;
Delete(Q, n);
}
else if ((n = Q.t.x0.cv).key == x)
Delete(Q, n);
}
public static Element DeleteMin(Queap Q) {
Element min = Minimum(Q);
Delete(Q, min);
return min;
}
}
``` | https://en.wikipedia.org/wiki/Queap |
A domain-specific language (DSL) is a computer language specialized to a particular application domain. This is in contrast to a general-purpose language (GPL), which is broadly applicable across domains. There are a wide variety of DSLs, ranging from widely used languages for common domains, such as HTML for web pages, down to languages used by only one or a few pieces of software, such as MUSH soft code. DSLs can be further subdivided by the kind of language, and include domain-specific markup languages, domain-specific modeling languages (more generally, specification languages), and domain-specific programming languages. Special-purpose computer languages have always existed in the computer age, but the term "domain-specific language" has become more popular due to the rise of domain-specific modeling. Simpler DSLs, particularly ones used by a single application, are sometimes informally called mini-languages.
The line between general-purpose languages and domain-specific languages is not always sharp, as a language may have specialized features for a particular domain but be applicable more broadly, or conversely may in principle be capable of broad application but in practice used primarily for a specific domain. For example, Perl was originally developed as a text-processing and glue language, for the same domain as AWK and shell scripts, but was mostly used as a general-purpose programming language later on. By contrast, PostScript is a Turing-complete language, and in principle can be used for any task, but in practice is narrowly used as a page description language.
## Use
The design and use of appropriate DSLs is a key part of domain engineering, by using a language suitable to the domain at hand – this may consist of using an existing DSL or GPL, or developing a new DSL. Language-oriented programming considers the creation of special-purpose languages for expressing problems as standard part of the problem-solving process. Creating a domain-specific language (with software to support it), rather than reusing an existing language, can be worthwhile if the language allows a particular type of problem or solution to be expressed more clearly than an existing language would allow and the type of problem in question reappears sufficiently often. Pragmatically, a DSL may be specialized to a particular problem domain, a particular problem representation technique, a particular solution technique, or other aspects of a domain.
## Overview
A domain-specific language is created specifically to solve problems in a particular domain and is not intended to be able to solve problems outside of it (although that may be technically possible). In contrast, general-purpose languages are created to solve problems in many domains. The domain can also be a business area. Some examples of business areas include:
- life insurance policies (developed internally by a large insurance enterprise)
- combat simulation
- salary calculation
- billing
A domain-specific language is somewhere between a tiny programming language and a scripting language, and is often used in a way analogous to a programming library. The boundaries between these concepts are quite blurry, much like the boundary between scripting languages and general-purpose languages.
### In design and implementation
Domain-specific languages are languages (or often, declared syntaxes or grammars) with very specific goals in design and implementation. A domain-specific language can be one of a visual diagramming language, such as those created by the Generic Eclipse Modeling System, programmatic abstractions, such as the Eclipse Modeling Framework, or textual languages. For instance, the command line utility grep has a regular expression syntax which matches patterns in lines of text. The sed utility defines a syntax for matching and replacing regular expressions. Often, these tiny languages can be used together inside a shell to perform more complex programming tasks.
The line between domain-specific languages and scripting languages is somewhat blurred, but domain-specific languages often lack low-level functions for filesystem access, interprocess control, and other functions that characterize full-featured programming languages, scripting or otherwise. Many domain-specific languages do not compile to byte-code or executable code, but to various kinds of media objects: GraphViz exports to PostScript, GIF, JPEG, etc., where Csound compiles to audio files, and a ray-tracing domain-specific language like POV compiles to graphics files.
### Data definition languages
A data definition language like SQL presents an interesting case: it can be deemed a domain-specific language because it is specific to a specific domain (in SQL's case, accessing and managing relational databases), and is often called from another application, but SQL has more keywords and functions than many scripting languages, and is often thought of as a language in its own right, perhaps because of the prevalence of database manipulation in programming and the amount of mastery required to be an expert in the language.
Further blurring this line, many domain-specific languages have exposed APIs, and can be accessed from other programming languages without breaking the flow of execution or calling a separate process, and can thus operate as programming libraries.
### Programming tools
Some domain-specific languages expand over time to include full-featured programming tools, which further complicates the question of whether a language is domain-specific or not. A good example is the functional language XSLT, specifically designed for transforming one XML graph into another, which has been extended since its inception to allow (particularly in its 2.0 version) for various forms of filesystem interaction, string and date manipulation, and data typing.
In model-driven engineering, many examples of domain-specific languages may be found like OCL, a language for decorating models with assertions or QVT, a domain-specific transformation language. However, languages like UML are typically general-purpose modeling languages.
To summarize, an analogy might be useful: a Very Little Language is like a knife, which can be used in thousands of different ways, from cutting food to cutting down trees. A domain-specific language is like an electric drill: it is a powerful tool with a wide variety of uses, but a specific context, namely, putting holes in things. A General Purpose Language is a complete workbench, with a variety of tools intended for performing a variety of tasks. Domain-specific languages should be used by programmers who, looking at their current workbench, realize they need a better drill and find that a particular domain-specific language provides exactly that.
## Domain-specific language topics
### External and Embedded Domain Specific Languages
DSLs implemented via an independent interpreter or compiler are known as External Domain Specific Languages. Well known examples include TeX or AWK. A separate category known as Embedded (or Internal) Domain Specific Languages are typically implemented within a host language as a library and tend to be limited to the syntax of the host language, though this depends on host language capabilities.
### Usage patterns
There are several usage patterns for domain-specific languages:
- Processing with standalone tools, invoked via direct user operation, often on the command line or from a Makefile (e.g., grep for regular expression matching, sed, lex, yacc, the GraphViz toolset, etc.)
- Domain-specific languages which are implemented using programming language macro systems, and which are converted or expanded into a host general purpose language at compile-time or realtime
- As embedded domain-specific language (eDSL) also known as an internal domain-specific language, is a DSL that is implemented as a library in a "host" programming language. The embedded domain-specific language leverages the syntax, semantics and runtime environment (sequencing, conditionals, iteration, functions, etc.) and adds domain-specific primitives that allow programmers to use the "host" programming language to create programs that generate code in the "target" programming language. Multiple eDSLs can easily be combined into a single program and the facilities of the host language can be used to extend an existing eDSL. Other possible advantages using an eDSL are improved type safety and better IDE tooling. eDSL examples: SQLAlchemy "Core" an SQL eDSL in Python, jOOQ an SQL eDSL in Java, LINQ's "method syntax" an SQL eDSL in C# and kotlinx.html an HTML eDSL in Kotlin.
- Domain-specific languages which are called (at runtime) from programs written in general purpose languages like C or Perl, to perform a specific function, often returning the results of operation to the "host" programming language for further processing; generally, an interpreter or virtual machine for the domain-specific language is embedded into the host application (e.g. format strings, a regular expression engine)
- Domain-specific languages which are embedded into user applications (e.g., macro languages within spreadsheets) and which are (1) used to execute code that is written by users of the application, (2) dynamically generated by the application, or (3) both.
Many domain-specific languages can be used in more than one way. DSL code embedded in a host language may have special syntax support, such as regexes in sed, AWK, Perl or JavaScript, or may be passed as strings.
### Design goals
Adopting a domain-specific language approach to software engineering involves both risks and opportunities. The well-designed domain-specific language manages to find the proper balance between these.
Domain-specific languages have important design goals that contrast with those of general-purpose languages:
- Domain-specific languages are less comprehensive.
- Domain-specific languages are much more expressive in their domain.
- Domain-specific languages should exhibit minimal redundancy.
### Idioms
In programming, idioms are methods imposed by programmers to handle common development tasks, e.g.:
- Ensure data is saved before the window is closed.
- Edit code whenever command-line parameters change because they affect program behavior.
General purpose programming languages rarely support such idioms, but domain-specific languages can describe them, e.g.:
- A script can automatically save data.
- A domain-specific language can parameterize command line input.
## Examples
Examples of domain-specific programming languages include HTML, Logo for pencil-like drawing, Verilog and VHDL hardware description languages, MATLAB and GNU Octave for matrix programming, Mathematica, Maple and Maxima for symbolic mathematics, Specification and Description Language for reactive and distributed systems, spreadsheet formulas and macros, SQL for relational database queries, YACC grammars for creating parsers, regular expressions for specifying lexers, the Generic Eclipse Modeling System for creating diagramming languages, Csound for sound and music synthesis, and the input languages of GraphViz and GrGen, software packages used for graph layout and graph rewriting, Hashicorp Configuration Language used for Terraform and other Hashicorp tools, Puppet also has its own configuration language.
### GameMaker Language
The GML scripting language used by GameMaker Studio is a domain-specific language targeted at novice programmers to easily be able to learn programming. While the language serves as a blend of multiple languages including Delphi, C++, and BASIC. Most of functions in that language after compiling in fact calls runtime functions written in language specific for targeted platform, so their final implementation is not visible to user. The language primarily serves to make it easy for anyone to pick up the language and develop a game, and thanks to GM runtime which handles main game loop and keeps implementation of called functions, few lines of code is required for simplest game, instead of thousands.
### ColdFusion Markup Language
ColdFusion's associated scripting language is another example of a domain-specific language for data-driven websites.
This scripting language is used to weave together languages and services such as Java, .NET, C++, SMS, email, email servers, http, ftp, exchange, directory services, and file systems for use in websites.
The ColdFusion Markup Language (CFML) includes a set of tags that can be used in ColdFusion pages to interact with data
sources, manipulate data, and display output. CFML tag syntax is similar to HTML element syntax.
### FilterMeister
FilterMeister is a programming environment, with a programming language that is based on C, for the specific purpose of creating Photoshop-compatible image processing filter plug-ins; FilterMeister runs as a Photoshop plug-in itself and it can load and execute scripts or compile and export them as independent plug-ins.
Although the FilterMeister language reproduces a significant portion of the C language and function library, it contains only those features which can be used within the context of Photoshop plug-ins and adds a number of specific features only useful in this specific domain.
### MediaWiki templates
The Template feature of MediaWiki is an embedded domain-specific language whose fundamental purpose is to support the creation of page templates and the transclusion (inclusion by reference) of MediaWiki pages into other MediaWiki pages.
### Software engineering uses
There has been much interest in domain-specific languages to improve the productivity and quality of software engineering. Domain-specific language could possibly provide a robust set of tools for efficient software engineering. Such tools are beginning to make their way into the development of critical software systems.
The Software Cost Reduction Toolkit is an example of this. The toolkit is a suite of utilities including a specification editor to create a requirements specification, a dependency graph browser to display variable dependencies, a consistency checker to catch missing cases in well-formed formulas in the specification, a model checker and a theorem prover to check program properties against the specification, and an invariant generator that automatically constructs invariants based on the requirements.
A newer development is language-oriented programming, an integrated software engineering methodology based mainly on creating, optimizing, and using domain-specific languages.
### Metacompilers
Complementing language-oriented programming, as well as all other forms of domain-specific languages, are the class of compiler writing tools called metacompilers. A metacompiler is not only useful for generating parsers and code generators for domain-specific languages, but a metacompiler itself compiles a domain-specific metalanguage specifically designed for the domain of metaprogramming.
Besides parsing domain-specific languages, metacompilers are useful for generating a wide range of software engineering and analysis tools. The meta-compiler methodology is often found in program transformation systems.
Metacompilers that played a significant role in both computer science and the computer industry include Meta-II, and its descendant TreeMeta.
### Unreal Engine before version 4 and other games
Unreal and Unreal Tournament unveiled a language called UnrealScript. This allowed for rapid development of modifications compared to the competitor Quake (using the Id Tech 2 engine). The Id Tech engine used standard C code meaning C had to be learned and properly applied, while UnrealScript was optimized for ease of use and efficiency. Similarly, more recent games have introduced their own specific languages for development. One more common example is Lua for scripting.
### Rules engines for policy automation
Various business rules engines have been developed for automating policy and business rules used in both government and private industry. ILOG, Oracle Policy Automation, DTRules, Drools and others provide support for DSLs aimed to support various problem domains. DTRules goes so far as to define an interface for the use of multiple DSLs within a rule set.
The purpose of business rules engines is to define a representation of business logic in as human-readable fashion as possible. This allows both subject-matter experts and developers to work with and understand the same representation of the business logic. Most rules engines provide both an approach to simplifying the control structures for business logic (for example, using declarative rules or decision tables) coupled with alternatives to programming syntax in favor of DSLs.
### Statistical modelling languages
Statistical modelers have developed domain-specific languages such as R (an implementation of the S language),
Bugs, Jags, and Stan. These languages provide a syntax for describing a Bayesian model and generate a method for solving it using simulation.
### Generate model and services to multiple programming Languages
Generate object handling and services based on an Interface Description Language for a domain-specific language such as JavaScript for web applications, HTML for documentation, C++ for high-performance code, etc. This is done by cross-language frameworks such as Apache Thrift or Google Protocol Buffers.
### Gherkin
Gherkin is a language designed to define test cases to check the behavior of software, without specifying how that behavior is implemented. It is meant to be read and used by non-technical users using a natural language syntax and a line-oriented design. The tests defined with Gherkin must then be implemented in a general programming language. Then, the steps in a Gherkin program acts as a syntax for method invocation accessible to non-developers.
### Other examples
Other prominent examples of domain-specific languages include:
- Game Description Language
- OpenGL Shading Language
- Gradle
- ActionScript
## Advantages and disadvantages
Some of the advantages:
- Domain-specific languages allow solutions to be expressed in the idiom and at the level of abstraction of the problem domain. The idea is that domain experts themselves may understand, validate, modify, and often even develop domain-specific language programs. However, this is seldom the case.
- Domain-specific languages allow validation at the domain level. As long as the language constructs are safe any sentence written with them can be considered safe.
- Domain-specific languages can help to shift the development of business information systems from traditional software developers to the typically larger group of domain-experts who (despite having less technical expertise) have a deeper knowledge of the domain.
- Domain-specific languages are easier to learn, given their limited scope.
Some of the disadvantages:
- Cost of learning a new language
- Limited applicability
- Cost of designing, implementing, and maintaining a domain-specific language as well as the tools required to develop with it (IDE)
- Finding, setting, and maintaining proper scope.
- Difficulty of balancing trade-offs between domain-specificity and general-purpose programming language constructs.
- Potential loss of processor efficiency compared with hand-coded software.
- Proliferation of similar non-standard domain-specific languages, for example, a DSL used within one insurance company versus a DSL used within another insurance company.
- Non-technical domain experts can find it hard to write or modify DSL programs by themselves.
- Increased difficulty of integrating the DSL with other components of the IT system (as compared to integrating with a general-purpose language).
- Low supply of experts in a particular DSL tends to raise labor costs.
- Harder to find code examples.
## Tools for designing domain-specific languages
- JetBrains MPS is a tool for designing domain-specific languages. It uses projectional editing which allows overcoming the limits of language parsers and building DSL editors, such as ones with tables and diagrams. It implements language-oriented programming. MPS combines an environment for language definition, a language workbench, and an Integrated Development Environment (IDE) for such languages.
- MontiCore is a language workbench for the efficient development of domain-specific languages. It processes an extended grammar format that defines the DSL and generates Java components for processing the DSL documents.
- Xtext is an open-source software framework for developing programming languages and domain-specific languages (DSLs). Unlike standard parser generators, Xtext generates not only a parser but also a class model for the abstract syntax tree. In addition, it provides a fully featured, customizable Eclipse-based IDE. The project was archived in April 2023.
- Racket is a cross-platform language toolchain including native code, JIT and JavaScript compiler, IDE (in addition to supporting Emacs, Vim, VSCode and others) and command line tools designed to accommodate creating both domain-specific and general purpose languages. | https://en.wikipedia.org/wiki/Domain-specific_language |
Category:Articles with example pseudocode
Category:Debian
Category:Electoral systems
Category:Monotonic Condorcet methods
Category:Single-winner electoral systems
The Schulze method (), also known as the beatpath method, is a single winner ranked-choice voting rule developed by Markus Schulze. The Schulze method is a Condorcet completion method, which means it will elect a majority-preferred candidate if one exists. In other words, if most people rank A above B, A will defeat B (whenever this is possible). Schulze's method breaks cyclic ties by using indirect victories. The idea is that if Alice beats Bob, and Bob beats Charlie, then Alice (indirectly) beats Charlie; this kind of indirect win is called a "beatpath".
For proportional representation, a single transferable vote (STV) variant known as Schulze STV also exists. The Schulze method is used by several organizations including Debian, Ubuntu, Gentoo, Pirate Party political parties and many others. It was also used by Wikimedia prior to their adoption of score voting.
## Description of the method
Schulze's method uses ranked ballots with equal ratings allowed. There are two common (equivalent) descriptions of Schulze's method.
### Beatpath explanation
The idea behind Schulze's method is that if Alice defeats Bob, and Bob beats Charlie, then Alice "indirectly" defeats Charlie. These chained sequences of "beats" are called 'beatpaths'.
Every beatpath is assigned a particular strength. The strength of a single-step beatpath from Alice to Bob is just the number of voters who rank Alice over Bob. For a longer beatpath, consisting of multiple beats, a beatpath is as strong as its weakest link (i.e. the beat with the smallest number of winning votes).
We say Alice has a "beatpath-win" over Bob if her strongest beatpath to Bob is stronger than all of Bob's strongest beatpaths to Alice. The winner is the candidate who has a beatpath-win over every other candidate.
Markus Schulze proved that this definition of a beatpath-win is transitive: in other words, if Alice has a beatpath-win over Bob, and Bob has a beatpath-win over Charlie, Alice has a beatpath-win over Charlie. As a result, the Schulze method is a Condorcet method, providing a full extension of the majority rule to any set of ballots.
### Iterative description
The Schulze winner can also be constructed iteratively, using a defeat-dropping method:
1. Draw a directed graph with all the candidates as nodes; label the edges with the number of votes supporting the winner.
1. If there is more than one candidate left:
1. Check if any candidates are tied (and if so, break the ties by random ballot).
1. Eliminate all candidates outside the majority-preferred set.
1. Delete the edge closest to being tied.
The winner is the only candidate left at the end of the procedure.
## Example
In the following example 45 voters rank 5 candidates.
Number of votersOrder of preference5ACBED5ADECB8BEDAC3CABED7CAEBD2CBADE7DCEBA8EBADC
The pairwise preferences have to be computed first. For example, when comparing and pairwise, there are voters who prefer to , and voters who prefer to . So
$$
d[A, B] = 20
$$
and
$$
d[B, A] = 25
$$
. The full set of pairwise preferences is:
+ Matrix of pairwise preferences 20 26 30 22 25 16 33 18 19 29 17 24 15 12 28 14 23 27 21 31
The cells for d[X, Y] have a light green background if d[X, Y] > d[Y, X], otherwise the background is light red. There is no undisputed winner by only looking at the pairwise differences here.
Now the strongest paths have to be identified. To help visualize the strongest paths, the set of pairwise preferences is depicted in the diagram on the right in the form of a directed graph. An arrow from the node representing a candidate X to the one representing a candidate Y is labelled with d[X, Y]. To avoid cluttering the diagram, an arrow has only been drawn from X to Y when d[X, Y] > d[Y, X] (i.e. the table cells with light green background), omitting the one in the opposite direction (the table cells with light red background).
One example of computing the strongest path strength is p[B, D] = 33: the strongest path from B to D is the direct path (B, D) which has strength 33. But when computing p[A, C], the strongest path from A to C is not the direct path (A, C) of strength 26, rather the strongest path is the indirect path (A, D, C) which has strength min(30, 28) = 28. The strength of a path is the strength of its weakest link.
For each pair of candidates X and Y, the following table shows the strongest path from candidate X to candidate Y in red, with the weakest link underlined.
+ Strongest paths A B C D E A A-(30)-D-(28)-C-(29)-B A-(30)-D-(28)-C A-(30)-D A-(30)-D-(28)-C-(24)-E A B B-(25)-A B-(33)-D-(28)-C B-(33)-D B-(33)-D-(28)-C-(24)-E B C C-(29)-B-(25)-A C-(29)-B C-(29)-B-(33)-D C-(24)-E C D D-(28)-C-(29)-B-(25)-A D-(28)-C-(29)-B D-(28)-C D-(28)-C-(24)-E D E E-(31)-D-(28)-C-(29)-B-(25)-A E-(31)-D-(28)-C-(29)-B E-(31)-D-(28)-C E-(31)-D E A B C D E
+Strengths of the strongest paths 28 28 30 24 25 28 33 24 25 29 29 24 25 28 28 24 25 28 28 31
Now the output of the Schulze method can be determined. For example, when comparing and , since
$$
(28 =) p[A,B] > p[B,A] (= 25)
$$
, for the Schulze method candidate is better than candidate . Another example is that
$$
(31 =) p[E,D] > p[D,E] (= 24)
$$
, so candidate E is better than candidate D. Continuing in this way, the result is that the Schulze ranking is
$$
E > A > C > B > D
$$
, and wins. In other words, wins since
$$
p[E,X] \ge p[X,E]
$$
for every other candidate X.
## Implementation
The only difficult step in implementing the Schulze method is computing the strongest path strengths. However, this is a well-known problem in graph theory sometimes called the widest path problem. One simple way to compute the strengths, therefore, is a variant of the Floyd–Warshall algorithm. The following pseudocode illustrates the algorithm.
```text
1. Input: d[i,j], the number of voters who prefer candidate i to candidate j.
1. Output: p[i,j], the strength of the strongest path from candidate i to candidate j.
for i from 1 to C
for j from 1 to C
if i ≠ j then
if d[i,j] > d[j,i] then
p[i,j] := d[i,j]
else
p[i,j] := 0
for i from 1 to C
for j from 1 to C
if i ≠ j then
for k from 1 to C
if i ≠ k and j ≠ k then
p[j,k] := max (p[j,k], min (p[j,i], p[i,k]))
```
This algorithm is efficient and has running time O(C3) where C is the number of candidates.
## Ties and alternative implementations
When allowing users to have ties in their preferences, the outcome of the Schulze method naturally depends on how these ties are interpreted in defining d[*,*]. Two natural choices are that d[A, B] represents either the number of voters who strictly prefer A to B (A>B), or the margin of (voters with A>B) minus (voters with B>A). But no matter how the ds are defined, the Schulze ranking has no cycles, and assuming the ds are unique it has no ties.
Although ties in the Schulze ranking are unlikely, they are possible. Schulze's original paper recommended breaking ties by random ballot.
There is another alternative way to demonstrate the winner of the Schulze method. This method is equivalent to the others described here, but the presentation is optimized for the significance of steps being visually apparent as a human goes through it, not for computation.
1. Make the results table, called the "matrix of pairwise preferences", such as used above in the example. Then, every positive number is a pairwise win for the candidate on that row (and marked green), ties are zeroes, and losses are negative (marked red). Order the candidates by how long they last in elimination.
1. If there is a candidate with no red on their line, they win.
1. Otherwise, draw a square box around the Schwartz set in the upper left corner. It can be described as the minimal "winner's circle" of candidates who do not lose to anyone outside the circle. Note that to the right of the box there is no red, which means it is a winner's circle, and note that within the box there is no reordering possible that would produce a smaller winner's circle.
1. Cut away every part of the table outside the box.
1. If there is still no candidate with no red on their line, something needs to be compromised on; every candidate lost some race, and the loss we tolerate the best is the one where the loser obtained the most votes. So, take the red cell with the highest number (if going by margins, the least negative), make it green—or any color other than red—and go back step 2.
Here is a margins table made from the above example. Note the change of order used for demonstration purposes.
+ Initial results table E A C B D E 1 −3 9 17 A−1 7 −5 15 C3 −7 13 −11 B−9 5 −13 21 D−17 −15 11 −21
The first drop (A's loss to E by 1 vote) does not help shrink the Schwartz set.
+ First drop E A C B D E 1 −3 9 17 A−1 7 −5 15 C3 −7 13 −11 B−9 5 −13 21 D−17 −15 11 −21
So we get straight to the second drop (E's loss to C by 3 votes), and that shows us the winner, E, with its clear row.
+ Second drop, final E A C B D E 1 −3 9 17 A−1 7 −5 15 C3 −7 13 −11 B−9 5 −13 21 D−17 −15 11 −21
This method can also be used to calculate a result, if the table is remade in such a way that one can conveniently and reliably rearrange the order of the candidates on both the row and the column, with the same order used on both at all times.
## Satisfied and failed criteria
### Satisfied criteria
The Schulze method satisfies the following criteria:
- Monotonicity criterion
- Majority criterion
- Majority loser criterion
- Condorcet criterion
- Condorcet loser criterion
- Smith criterion
- Independence of Smith-dominated alternatives
- Mutual majority criterion
- Independence of clones
- Reversal symmetry
- Mono-append
- Mono-add-plump
- Resolvability criterion
- Polynomial runtime
- prudence
- MinMax sets
- Woodall's plurality criterion if winning votes are used for d[X,Y]
- Symmetric-completion if margins are used for d[X,Y]
### Failed criteria
Since the Schulze method satisfies the Condorcet criterion, it automatically fails the following criteria:
- Participation
- Consistency
- Invulnerability to burying
- Later-no-harm
Likewise, since the Schulze method is not a dictatorship and is a ranked voting system (not rated), Arrow's Theorem implies it fails independence of irrelevant alternatives, meaning it can be vulnerable to the spoiler effect in some rare circumstances. The Schulze method also fails Peyton Young's criterion of Local Independence of Irrelevant Alternatives.
### Comparison table
The following table compares the Schulze method with other single-winner election methods:
### Difference from ranked pairs
Ranked pairs is another Condorcet method which is very similar to Schulze's rule, and typically produces the same outcome. There are slight differences, however. The main difference between the beatpath method and ranked pairs is that Schulze retains behavior closer to minimax. Say that the minimax score of a set X of candidates is the strength of the strongest pairwise win of a candidate A ∉ X against a candidate B ∈ X. Then the Schulze method, but not ranked pairs, guarantees the winner is always a candidate of the set with minimum minimax score. This is the sense in which the Schulze method minimizes the largest majority that has to be reversed when determining the winner.
On the other hand, Ranked Pairs minimizes the largest majority that has to be reversed to determine the order of finish. In other words, when Ranked Pairs and the Schulze method produce different orders of finish, for the majorities on which the two orders of finish disagree, the Schulze order reverses a larger majority than the Ranked Pairs order.
## History
The Schulze method was developed by Markus Schulze in 1997. It was first discussed in public mailing lists in 1997–1998 and in 2000. In 2011, Schulze published the method in the academic journal Social Choice and Welfare.
## Usage
### Government
The Schulze method is used by the city of Silla, Spain for all referendums. It is also used by the cities of Turin and San Donà di Piave in Italy and by the London Borough of Southwark through their use of the WeGovNow platform, which in turn uses the LiquidFeedback decision tool.
### Political parties
Schulze was adopted by the Pirate Party of Sweden (2009), and the Pirate Party of Germany (2010). The Boise, Idaho chapter of the Democratic Socialists of America in February chose this method for their first special election held in March 2018.
- Five Star Movement of Campobasso, Fondi, Monte Compatri, Montemurlo, Pescara, and San Cesareo
- Pirate Parties of Australia, Austria, Belgium, Brazil, Germany, Iceland, Italy, the Netherlands, Sweden, Switzerland, and the United States
- SustainableUnion
- Volt Europe
### Student government and associations
- AEGEE – European Students' Forum
- Club der Ehemaligen der Deutschen SchülerAkademien e. V.
- Associated Student Government at École normale supérieure de Paris
- Flemish Society of Engineering Students Leuven
- Graduate Student Organization at the State University of New York: Computer Science (GSOCS)
- Hillegass Parker House
- Kingman Hall
- Associated Students of Minerva Schools at KGI
- Associated Student Government at Northwestern University
- Associated Student Government at University of Freiburg
- Associated Student Government at the Computer Sciences Department of the University of Kaiserslautern-Landau
### Organizations
It is used by the Institute of Electrical and Electronics Engineers, by the Association for Computing Machinery, and by USENIX through their use of the HotCRP decision tool.
Organizations which currently use the Schulze method include:
- Annodex Association
- (BVKJ)
- BoardGameGeek
- Cloud Foundry Foundation
- County Highpointers
- Dapr
- Debian
- EuroBillTracker
- European Democratic Education Community (EUDEC)
- FFmpeg
- Free Geek
- Free Hardware Foundation of Italy
- Gentoo Foundation
- GNU Privacy Guard (GnuPG)
- Haskell
- Homebrew
- Internet Corporation for Assigned Names and Numbers (ICANN) (until 2023)
- Kanawha Valley Scrabble Club
- KDE e.V.
- Knight Foundation
- Kubernetes
- Kumoricon
- League of Professional System Administrators (LOPSA)
- LiquidFeedback
- Madisonium
- Metalab
- MTV
- Neo
- Noisebridge
- OpenEmbedded
- Open Neural Network Exchange
- OpenStack
- OpenSwitch
- RLLMUK
- Squeak
- Students for Free Culture
- Sugar Labs
- Sverok
- TopCoder
- Ubuntu
- Vidya Gaem Awards
- Wikimedia (2008)
- Wikipedia in French, Hebrew, Hungarian, Russian, and Persian.
## Generalizations
In 2008, Camps et. al devised a method that, while ranking candidates in the same order of finish as Schulze, also provides ratings indicating the candidates' relative strength of victory.
## Notes
## External links
-
- The Schulze Method by Hubert Bray
- Spieltheorie by Bernhard Nebel
- Accurate Democracy by Rob Loring
- Christoph Börgers (2009), Mathematics of Social Choice: Voting, Compensation, and Division, SIAM,
- Nicolaus Tideman (2006), Collective Decisions and Voting: The Potential for Public Choice, Burlington: Ashgate,
- preftools by the Public Software Group
- Arizonans for Condorcet Ranked Voting
- Condorcet PHP Command line application and PHP library, supporting multiple Condorcet methods, including Schulze.
- Implementation in Java
- Implementation in Ruby
- Implementation in Python 2
- Implementation in Python 3 | https://en.wikipedia.org/wiki/Schulze_method |
In probability theory, Kolmogorov's three-series theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions. Kolmogorov's three-series theorem, combined with Kronecker's lemma, can be used to give a relatively easy proof of the strong law of large numbers.
## Statement of the theorem
Let
$$
(X_n)_{n \in \mathbb{N}}
$$
be independent random variables. The random series
$$
\sum_{n=1}^\infty X_n
$$
converges almost surely in
$$
\mathbb{R}
$$
if the following conditions hold for some
$$
A > 0
$$
, and only if the following conditions hold for any
$$
A > 0
$$
:
## Proof
### Sufficiency of conditions ("if")
Condition (i) and Borel–Cantelli give that
$$
X_n = Y_n
$$
for
$$
n
$$
large, almost surely. Hence
$$
\textstyle\sum_{n=1}^{\infty}X_n
$$
converges if and only if
$$
\textstyle\sum_{n=1}^{\infty}Y_n
$$
converges. Conditions (ii)-(iii) and Kolmogorov's two-series theorem give the almost sure convergence of
$$
\textstyle\sum_{n=1}^{\infty}Y_n
$$
.
### Necessity of conditions ("only if")
Suppose that
$$
\textstyle\sum_{n=1}^{\infty}X_n
$$
converges almost surely.
Without condition (i), by Borel–Cantelli there would exist some
$$
A > 0
$$
such that
$$
\{|X_n| \ge A\}
$$
for infinitely many
$$
n
$$
, almost surely. But then the series would diverge. Therefore, we must have condition (i).
We see that condition (iii) implies condition (ii): Kolmogorov's two-series theorem along with condition (i) applied to the case
$$
A = 1
$$
gives the convergence of
$$
\textstyle\sum_{n=1}^{\infty}(Y_n - \mathbb{E}[Y_n])
$$
. So given the convergence of
$$
\textstyle\sum_{n=1}^{\infty}Y_n
$$
, we have
$$
\textstyle\sum_{n=1}^{\infty}\mathbb{E}[Y_n]
$$
converges, so condition (ii) is implied.
Thus, it only remains to demonstrate the necessity of condition (iii), and we will have obtained the full result. It is equivalent to check condition (iii) for the series
$$
\textstyle\sum_{n=1}^{\infty}Z_n = \textstyle\sum_{n=1}^{\infty}(Y_n - Y'_n)
$$
where for each
$$
n
$$
,
$$
Y_n
$$
and
$$
Y'_n
$$
are IID—that is, to employ the assumption that
$$
\mathbb{E}[Y_n] = 0
$$
, since
$$
Z_n
$$
is a sequence of random variables bounded by 2, converging almost surely, and with
$$
\mathrm{var}(Z_n) = 2\mathrm{var}(Y_n)
$$
. So we wish to check that if
$$
\textstyle\sum_{n=1}^{\infty}Z_n
$$
converges, then
$$
\textstyle\sum_{n=1}^{\infty}\mathrm{var}(Z_n)
$$
converges as well. This is a special case of a more general result from martingale theory with summands equal to the increments of a martingale sequence and the same conditions (
$$
\mathbb{E}[Z_n] = 0
$$
; the series of the variances is converging; and the summands are bounded).M. Loève, "Probability theory", Princeton Univ. Press (1963) pp. Sect. 16.3
## Example
As an illustration of the theorem, consider the example of the harmonic series with random signs:
$$
\sum_{n=1}^\infty \pm \frac{1}{n}.
$$
Here, "
$$
\pm
$$
" means that each term
$$
1/n
$$
is taken with a random sign that is either
$$
1
$$
or
$$
-1
$$
with respective probabilities
$$
1/2,\ 1/2
$$
, and all random signs are chosen independently. Let
$$
X_n
$$
in the theorem denote a random variable that takes the values
$$
1/n
$$
and
$$
-1/n
$$
with equal probabilities. With
$$
A=2
$$
the summands of the first two series are identically zero and var(Yn)=
$$
n^{-2}
$$
. The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely. On the other hand, the analogous series of (for example) square root reciprocals with random signs, namely
$$
\sum_{n=1}^\infty \pm \frac{1}{\sqrt{n}},
$$
diverges almost surely, since condition (3) in the theorem is not satisfied for any A. Note that this is different from the behavior of the analogous series with alternating signs,
$$
\sum_{n=1}^\infty (-1)^n/\sqrt{n}
$$
, which does converge.
## Notes
Category:Series (mathematics)
Category:Theorems in probability theory | https://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem |
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field
$$
\mathbb F_p
$$
with
$$
p
$$
elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers.
## History
The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in
$$
\mathbb F_p
$$
. In 2000 Peralta's method was generalized for cubic equations.
## Statement of problem
Let
$$
p
$$
be an odd prime number. Consider the polynomial
$$
f(x) = a_0 + a_1 x + \cdots + a_n x^n
$$
over the field
$$
\mathbb F_p\simeq \mathbb Z/p\mathbb Z
$$
of remainders modulo
$$
p
$$
. The algorithm should find all
$$
\lambda
$$
in
$$
\mathbb F_p
$$
such that
$$
f(\lambda)= 0
$$
in
$$
\mathbb F_p
$$
.
## Algorithm
### Randomization
Let
$$
f(x) = (x-\lambda_1)(x-\lambda_2)\cdots(x-\lambda_n)
$$
. Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial
$$
f_z(x)=f(x-z) = (x-\lambda_1 - z)(x-\lambda_2 - z) \cdots (x-\lambda_n-z)
$$
where
$$
z
$$
is some element of
$$
\mathbb F_p
$$
. If one can represent this polynomial as the product
$$
f_z(x)=p_0(x)p_1(x)
$$
then in terms of the initial polynomial it means that
$$
f(x) =p_0(x+z)p_1(x+z)
$$
, which provides needed factorization of
$$
f(x)
$$
.
### Classification of elements
Due to Euler's criterion, for every monomial
$$
(x-\lambda)
$$
exactly one of following properties holds:
1. The monomial is equal to
$$
x
$$
if
$$
\lambda = 0
$$
,
1. The monomial divides
$$
g_0(x)=(x^{(p-1)/2}-1)
$$
if
$$
\lambda
$$
is quadratic residue modulo
$$
p
$$
,
1. The monomial divides
$$
g_1(x)=(x^{(p-1)/2}+1)
$$
if
$$
\lambda
$$
is quadratic non-residual modulo
$$
p
$$
.
Thus if
$$
f_z(x)
$$
is not divisible by
$$
x
$$
, which may be checked separately, then
$$
f_z(x)
$$
is equal to the product of greatest common divisors
$$
\gcd(f_z(x);g_0(x))
$$
and
$$
\gcd(f_z(x);g_1(x))
$$
.
### Berlekamp's method
The property above leads to the following algorithm:
1. Explicitly calculate coefficients of
$$
f_z(x) = f(x-z)
$$
,
1. Calculate remainders of
$$
x,x^2, x^{2^2},x^{2^3}, x^{2^4}, \ldots, x^{2^{\lfloor \log_2 p \rfloor}}
$$
modulo
$$
f_z(x)
$$
by squaring the current polynomial and taking remainder modulo
$$
f_z(x)
$$
,
1. Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of
$$
x^{(p-1)/2}
$$
modulo
$$
f_z(x)
$$
,
1. If
$$
x^{(p-1)/2} \not \equiv \pm 1 \pmod{f_z(x)}
$$
then
$$
\gcd
$$
mentioned below provide a non-trivial factorization of
$$
f_z(x)
$$
,
1. Otherwise all roots of
$$
f_z(x)
$$
are either residues or non-residues simultaneously and one has to choose another
$$
z
$$
.
If
$$
f(x)
$$
is divisible by some non-linear primitive polynomial
$$
g(x)
$$
over
$$
\mathbb F_p
$$
then when calculating
$$
\gcd
$$
with
$$
g_0(x)
$$
and
$$
g_1(x)
$$
one will obtain a non-trivial factorization of
$$
f_z(x)/g_z(x)
$$
, thus algorithm allows to find all roots of arbitrary polynomials over
$$
\mathbb F_p
$$
.
### Modular square root
Consider equation
$$
x^2 \equiv a \pmod{p}
$$
having elements
$$
\beta
$$
and
$$
-\beta
$$
as its roots. Solution of this equation is equivalent to factorization of polynomial
$$
f(x) = x^2-a=(x-\beta)(x+\beta)
$$
over
$$
\mathbb F_p
$$
. In this particular case problem it is sufficient to calculate only
$$
\gcd(f_z(x); g_0(x))
$$
. For this polynomial exactly one of the following properties will hold:
1. GCD is equal to
$$
1
$$
which means that
$$
z+\beta
$$
and
$$
z-\beta
$$
are both quadratic non-residues,
1. GCD is equal to
$$
f_z(x)
$$
which means that both numbers are quadratic residues,
1. GCD is equal to
$$
(x-t)
$$
which means that exactly one of these numbers is quadratic residue.
In the third case GCD is equal to either
$$
(x-z-\beta)
$$
or
$$
(x-z+\beta)
$$
. It allows to write the solution as
$$
\beta = (t - z) \pmod{p}
$$
.
### Example
Assume we need to solve the equation
$$
x^2 \equiv 5\pmod{11}
$$
. For this we need to factorize
$$
f(x)=x^2-5=(x-\beta)(x+\beta)
$$
. Consider some possible values of
$$
z
$$
:
1. Let
$$
z=3
$$
. Then
$$
f_z(x) = (x-3)^2 - 5 = x^2 - 6x + 4
$$
, thus
$$
\gcd(x^2 - 6x + 4 ; x^5 - 1) = 1
$$
. Both numbers
$$
3 \pm \beta
$$
are quadratic non-residues, so we need to take some other
$$
z
$$
.
1. Let
$$
z=2
$$
. Then
$$
f_z(x) = (x-2)^2 - 5 = x^2 - 4x - 1
$$
, thus
$$
\gcd( x^2 - 4x - 1 ; x^5 - 1)\equiv x - 9 \pmod{11}
$$
. From this follows
$$
x - 9 = x - 2 - \beta
$$
, so
$$
\beta \equiv 7 \pmod{11}
$$
and
$$
-\beta \equiv -7 \equiv 4 \pmod{11}
$$
.
A manual check shows that, indeed,
$$
7^2 \equiv 49 \equiv 5\pmod{11}
$$
and
$$
4^2\equiv 16 \equiv 5\pmod{11}
$$
.
## Correctness proof
The algorithm finds factorization of
$$
f_z(x)
$$
in all cases except for ones when all numbers
$$
z+\lambda_1, z+\lambda_2, \ldots, z+\lambda_n
$$
are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when
$$
\lambda_1, \ldots, \lambda_n
$$
are all residues or non-residues simultaneously (that is, when
$$
z=0
$$
would fail) may be estimated as
$$
2^{-k}
$$
where
$$
k
$$
is the number of distinct values in
$$
\lambda_1, \ldots, \lambda_n
$$
. In this way even for the worst case of
$$
k=1
$$
and
$$
f(x)=(x-\lambda)^n
$$
, the probability of error may be estimated as
$$
1/2
$$
and for modular square root case error probability is at most
$$
1/4
$$
.
## Complexity
Let a polynomial have degree
$$
n
$$
. We derive the algorithm's complexity as follows:
1. Due to the binomial theorem
$$
(x-z)^k = \sum\limits_{i=0}^k \binom{k}{i} (-z)^{k-i}x^i
$$
, we may transition from
$$
f(x)
$$
to
$$
f(x-z)
$$
in
$$
O(n^2)
$$
time.
1. Polynomial multiplication and taking remainder of one polynomial modulo another one may be done in
$$
O(n^2)
$$
, thus calculation of
$$
x^{2^k} \bmod f_z(x)
$$
is done in
$$
O(n^2 \log p)
$$
.
1. Binary exponentiation works in
$$
O(n^2 \log p)
$$
.
1. Taking the
$$
\gcd
$$
of two polynomials via Euclidean algorithm works in
$$
O(n^2)
$$
.
Thus the whole procedure may be done in
$$
O(n^2 \log p)
$$
. Using the fast Fourier transform and Half-GCD algorithm, the algorithm's complexity may be improved to
$$
O(n \log n \log pn)
$$
. For the modular square root case, the degree is
$$
n = 2
$$
, thus the whole complexity of algorithm in such case is bounded by
$$
O(\log p)
$$
per iteration.
## References
Category:Algorithms
Category:Algebra
Category:Number theoretic algorithms
Category:Polynomials | https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Rabin_algorithm |
In physics, the Bardeen–Cooper–Schrieffer (BCS) theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes superconductivity as a microscopic effect caused by a condensation of Cooper pairs. The theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.
It was proposed by Bardeen, Cooper, and Schrieffer in 1957; they received the Nobel Prize in Physics for this theory in 1972.
## History
Rapid progress in the understanding of superconductivity gained momentum in the mid-1950s. It began with the 1948 paper, "On the Problem of the Molecular Theory of Superconductivity", where Fritz London proposed that the phenomenological London equations may be consequences of the coherence of a quantum state. In 1953, Brian Pippard, motivated by penetration experiments, proposed that this would modify the London equations via a new scale parameter called the coherence length. John Bardeen then argued in the 1955 paper, "Theory of the Meissner Effect in Superconductors", that such a modification naturally occurs in a theory with an energy gap. The key ingredient was Leon Cooper's calculation of the bound states of electrons subject to an attractive force in his 1956 paper, "Bound Electron Pairs in a Degenerate Fermi Gas".
In 1957 Bardeen and Cooper assembled these ingredients and constructed such a theory, the BCS theory, with Robert Schrieffer. The theory was first published in April 1957 in the letter, "Microscopic theory of superconductivity". The demonstration that the phase transition is second order, that it reproduces the Meissner effect and the calculations of specific heats and penetration depths appeared in the December 1957 article, "Theory of superconductivity". They received the Nobel Prize in Physics in 1972 for this theory.
In 1986, high-temperature superconductivity was discovered in La-Ba-Cu-O, at temperatures up to 30 K. Following experiments determined more materials with transition temperatures up to about 130 K, considerably above the previous limit of about 30 K. It is experimentally very well known that the transition temperature strongly depends on pressure. In general, it is believed that BCS theory alone cannot explain this phenomenon and that other effects are in play. These effects are still not yet fully understood; it is possible that they even control superconductivity at low temperatures for some materials.
## Overview
At sufficiently low temperatures, electrons near the Fermi surface become unstable against the formation of Cooper pairs. Cooper showed such binding will occur in the presence of an attractive potential, no matter how weak. In conventional superconductors, an attraction is generally attributed to an electron-lattice interaction. The BCS theory, however, requires only that the potential be attractive, regardless of its origin. In the BCS framework, superconductivity is a macroscopic effect which results from the condensation of Cooper pairs. These have some bosonic properties, and bosons, at sufficiently low temperature, can form a large Bose–Einstein condensate. Superconductivity was simultaneously explained by Nikolay Bogolyubov, by means of the Bogoliubov transformations.
In many superconductors, the attractive interaction between electrons (necessary for pairing) is brought about indirectly by the interaction between the electrons and the vibrating crystal lattice (the phonons). Roughly speaking the picture is the following:
An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density. The two electrons then become correlated. Because there are a lot of such electron pairs in a superconductor, these pairs overlap very strongly and form a highly collective condensate. In this "condensed" state, the breaking of one pair will change the energy of the entire condensate - not just a single electron, or a single pair. Thus, the energy required to break any single pair is related to the energy required to break all of the pairs (or more than just two electrons). Because the pairing increases this energy barrier, kicks from oscillating atoms in the conductor (which are small at sufficiently low temperatures) are not enough to affect the condensate as a whole, or any individual "member pair" within the condensate. Thus the electrons stay paired together and resist all kicks, and the electron flow as a whole (the current through the superconductor) will not experience resistance. Thus, the collective behavior of the condensate is a crucial ingredient necessary for superconductivity.
### Details
BCS theory starts from the assumption that there is some attraction between electrons, which can overcome the Coulomb repulsion. In most materials (in low temperature superconductors), this attraction is brought about indirectly by the coupling of electrons to the crystal lattice (as explained above). However, the results of BCS theory do not depend on the origin of the attractive interaction. For instance, Cooper pairs have been observed in ultracold gases of fermions where a homogeneous magnetic field has been tuned to their Feshbach resonance. The original results of BCS (discussed below) described an s-wave superconducting state, which is the rule among low-temperature superconductors but is not realized in many unconventional superconductors such as the d-wave high-temperature superconductors.
Extensions of BCS theory exist to describe these other cases, although they are insufficient to completely describe the observed features of high-temperature superconductivity.
BCS is able to give an approximation for the quantum-mechanical many-body state of the system of (attractively interacting) electrons inside the metal. This state is now known as the BCS state. In the normal state of a metal, electrons move independently, whereas in the BCS state, they are bound into Cooper pairs by the attractive interaction. The BCS formalism is based on the reduced potential for the electrons' attraction. Within this potential, a variational ansatz for the wave function is proposed. This ansatz was later shown to be exact in the dense limit of pairs. Note that the continuous crossover between the dilute and dense regimes of attracting pairs of fermions is still an open problem, which now attracts a lot of attention within the field of ultracold gases.
### Underlying evidence
The hyperphysics website pages at Georgia State University summarize some key background to BCS theory as follows:
- Evidence of a band gap at the Fermi level (described as "a key piece in the puzzle")
the existence of a critical temperature and critical magnetic field implied a band gap, and suggested a phase transition, but single electrons are forbidden from condensing to the same energy level by the Pauli exclusion principle. The site comments that "a drastic change in conductivity demanded a drastic change in electron behavior". Conceivably, pairs of electrons might perhaps act like bosons instead, which are bound by different condensate rules and do not have the same limitation.
- Isotope effect on the critical temperature, suggesting lattice interactions
The Debye frequency of phonons in a lattice is proportional to the inverse of the square root of the mass of lattice ions. It was shown that the superconducting transition temperature of mercury indeed showed the same dependence, by substituting the most abundant natural mercury isotope, 202Hg, with a different isotope, 198Hg.
- An exponential rise in heat capacity near the critical temperature for some superconductors
An exponential increase in heat capacity near the critical temperature also suggests an energy bandgap for the superconducting material. As superconducting vanadium is warmed toward its critical temperature, its heat capacity increases greatly in a very few degrees; this suggests an energy gap being bridged by thermal energy.
- The lessening of the measured energy gap towards the critical temperature
This suggests a type of situation where some kind of binding energy exists but it is gradually weakened as the temperature increases toward the critical temperature. A binding energy suggests two or more particles or other entities that are bound together in the superconducting state. This helped to support the idea of bound particles – specifically electron pairs – and together with the above helped to paint a general picture of paired electrons and their lattice interactions.
## Implications
BCS derived several important theoretical predictions that are independent of the details of the interaction, since the quantitative predictions mentioned below hold for any sufficiently weak attraction between the electrons and this last condition is fulfilled for many low temperature superconductors - the so-called weak-coupling case. These have been confirmed in numerous experiments:
- The electrons are bound into Cooper pairs, and these pairs are correlated due to the Pauli exclusion principle for the electrons, from which they are constructed. Therefore, in order to break a pair, one has to change energies of all other pairs. This means there is an energy gap for single-particle excitation, unlike in the normal metal (where the state of an electron can be changed by adding an arbitrarily small amount of energy). This energy gap is highest at low temperatures but vanishes at the transition temperature when superconductivity ceases to exist. The BCS theory gives an expression that shows how the gap grows with the strength of the attractive interaction and the (normal phase) single particle density of states at the Fermi level. Furthermore, it describes how the density of states is changed on entering the superconducting state, where there are no electronic states any more at the Fermi level. The energy gap is most directly observed in tunneling experiments and in reflection of microwaves from superconductors.
- BCS theory predicts the dependence of the value of the energy gap Δ at temperature T on the critical temperature Tc. The ratio between the value of the energy gap at zero temperature and the value of the superconducting transition temperature (expressed in energy units) takes the universal value
$$
\Delta(T=0) = 1.764 \, k_{\rm B}T_{\rm c},
$$
independent of material. Near the critical temperature the relation asymptotes to
$$
\Delta(T \to T_{\rm c})\approx 3.06 \, k_{\rm B}T_{\rm c}\sqrt{1-(T/T_{\rm c})}
$$
which is of the form suggested the previous year by M. J. Buckingham based on the fact that the superconducting phase transition is second order, that the superconducting phase has a mass gap and on Blevins, Gordy and Fairbank's experimental results the previous year on the absorption of millimeter waves by superconducting tin.
- Due to the energy gap, the specific heat of the superconductor is suppressed strongly (exponentially) at low temperatures, there being no thermal excitations left. However, before reaching the transition temperature, the specific heat of the superconductor becomes even higher than that of the normal conductor (measured immediately above the transition) and the ratio of these two values is found to be universally given by 2.5.
- BCS theory correctly predicts the Meissner effect, i.e. the expulsion of a magnetic field from the superconductor and the variation of the penetration depth (the extent of the screening currents flowing below the metal's surface) with temperature.
- It also describes the variation of the critical magnetic field (above which the superconductor can no longer expel the field but becomes normal conducting) with temperature. BCS theory relates the value of the critical field at zero temperature to the value of the transition temperature and the density of states at the Fermi level.
- In its simplest form, BCS gives the superconducting transition temperature Tc in terms of the electron-phonon coupling potential V and the Debye cutoff energy ED:
$$
k_{\rm B}\,T_{\rm c} = 1.134E_{\rm D}\,{e^{-1/N(0)\,V}},
$$
where N(0) is the electronic density of states at the Fermi level. For more details, see Cooper pairs.
- The BCS theory reproduces the isotope effect, which is the experimental observation that for a given superconducting material, the critical temperature is inversely proportional to the square-root of the mass of the isotope used in the material. The isotope effect was reported by two groups on 24 March 1950, who discovered it independently working with different mercury isotopes, although a few days before publication they learned of each other's results at the ONR conference in Atlanta. The two groups are Emanuel Maxwell, and C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt. The choice of isotope ordinarily has little effect on the electrical properties of a material, but does affect the frequency of lattice vibrations. This effect suggests that superconductivity is related to vibrations of the lattice. This is incorporated into BCS theory, where lattice vibrations yield the binding energy of electrons in a Cooper pair.
- Little–Parks experiment - One of the first indications to the importance of the Cooper-pairing principle. | https://en.wikipedia.org/wiki/BCS_theory |
In mathematics, an Apollonian gasket, Apollonian net, or Apollonian circle packing is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga.
## Construction
The construction of the Apollonian gasket starts with three circles
$$
C_1
$$
,
$$
C_2
$$
, and
$$
C_3
$$
(black in the figure), that are each tangent to the other two, but that do not have a single point of triple tangency. These circles may be of different sizes to each other, and it is allowed for two to be inside the third, or for all three to be outside each other. As Apollonius discovered, there exist two more circles
$$
C_4
$$
and
$$
C_5
$$
(red) that are tangent to all three of the original circles – these are called Apollonian circles. These five circles are separated from each other by six curved triangular regions, each bounded by the arcs from three pairwise-tangent circles. The construction continues by adding six more circles, one in each of these six curved triangles, tangent to its three sides. These in turn create 18 more curved triangles, and the construction continues by again filling these with tangent circles, ad infinitum.
Continued stage by stage in this way, the construction adds
$$
2\cdot 3^n
$$
new circles at stage
$$
n
$$
, giving a total of
$$
3^{n+1}+2
$$
circles after
$$
n
$$
stages. In the limit, this set of circles is an Apollonian gasket. In it, each pair of tangent circles has an infinite Pappus chain of circles tangent to both circles in the pair.
The size of each new circle is determined by Descartes' theorem, which states that, for any four mutually tangent circles, the radii
$$
r_i
$$
of the circles obeys the equation
$$
\left(\frac1{r_1}+\frac1{r_2}+\frac1{r_3}+\frac1{r_4}\right)^2=2\left(\frac1{r_1^2}+\frac1{r_2^2}+\frac1{r_3^2}+\frac1{r_4^2}\right).
$$
This equation may have a solution with a negative radius; this means that one of the circles (the one with negative radius) surrounds the other three.
One or two of the initial circles of this construction, or the circles resulting from this construction, can degenerate to a straight line, which can be thought of as a circle with infinite radius. When there are two lines, they must be parallel, and are considered to be tangent at a point at infinity. When the gasket includes two lines on the
$$
x
$$
-axis and one unit above it, and a circle of unit diameter tangent to both lines centered on the
$$
y
$$
-axis, then the circles that are tangent to the
$$
x
$$
-axis are the Ford circles, important in number theory.
The Apollonian gasket has a Hausdorff dimension of about 1.3056867, which has been extended to at least 128 decimal places. Because it has a well-defined fractional dimension, even though it is not precisely self-similar, it can be thought of as a fractal.
## Symmetries
The Möbius transformations of the plane preserve the shapes and tangencies of circles, and therefore preserve the structure of an Apollonian gasket.
Any two triples of mutually tangent circles in an Apollonian gasket may be mapped into each other by a Möbius transformation, and any two Apollonian gaskets may be mapped into each other by a Möbius transformation. In particular, for any two tangent circles in any Apollonian gasket, an inversion in a circle centered at the point of tangency (a special case of a Möbius transformation) will transform these two circles into two parallel lines, and transform the rest of the gasket into the special form of a gasket between two parallel lines. Compositions of these inversions can be used to transform any two points of tangency into each other. Möbius transformations are also isometries of the hyperbolic plane, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry.
The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group.
For Euclidean symmetry transformations rather than Möbius transformations, in general, the Apollonian gasket will inherit the symmetries of its generating set of three circles. However, some triples of circles can generate Apollonian gaskets with higher symmetry than the initial triple; this happens when the same gasket has a different and more-symmetric set of generating circles. Particularly symmetric cases include the Apollonian gasket between two parallel lines (with infinite dihedral symmetry), the Apollonian gasket generated by three congruent circles in an equilateral triangle (with the symmetry of the triangle), and the Apollonian gasket generated by two circles of radius 1 surrounded by a circle of radius 2 (with two lines of reflective symmetry).
## Integral Apollonian circle packings
If any four mutually tangent circles in an Apollonian gasket all have integer curvature (the inverse of their radius) then all circles in the gasket will have integer curvature.
Since the equation relating curvatures in an Apollonian gasket, integral or not, is
$$
a^2 + b^2 + c^2 + d^2 = 2ab + 2 a c + 2 a d + 2 bc+2bd+2cd,\,
$$
it follows that one may move from one quadruple of curvatures to another by Vieta jumping, just as when finding a new Markov number.
The first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket – all other curvatures can be derived from these three.
+ Integral Apollonian gaskets Beginning curvatures Symmetry −1, 2, 2, 3, 3 D2 −2, 3, 6, 7, 7 D1 −3, 4, 12, 13, 13 D1 −3, 5, 8, 8, 12 D1 −4, 5, 20, 21, 21 D1 −4, 8, 9, 9, 17 D1 −5, 6, 30, 31, 31 D1 −5, 7, 18, 18, 22 D1 −6, 7, 42, 43, 43 D1 −6, 10, 15, 19, 19 D1 −6, 11, 14, 15, 23 C1 −7, 8, 56, 57, 57 D1 −7, 9, 32, 32, 36 D1 −7, 12, 17, 20, 24 C1 −8, 9, 72, 73, 73 D1 −8, 12, 25, 25, 33 D1 −8, 13, 21, 24, 28 C1 −9, 10, 90, 91, 91 D1 −9, 11, 50, 50, 54 D1 −9, 14, 26, 27, 35 C1 −9, 18, 19, 22, 34 C1 −10, 11, 110, 111, 111 D1 −10, 14, 35, 39, 39 D1 −10, 18, 23, 27, 35 C1 −11, 12, 132, 133, 133 D1 −11, 13, 72, 72, 76 D1 −11, 16, 36, 37, 45 C1 −11, 21, 24, 28, 40 C1 −12, 13, 156, 157, 157 D1 −12, 16, 49, 49, 57 D1 −12, 17, 41, 44, 48 C1 −12, 21, 28, 37, 37 D1 −12, 21, 29, 32, 44 C1 −12, 25, 25, 28, 48 D1 −13, 14, 182, 183, 183 D1 −13, 15, 98, 98, 102 D1 −13, 18, 47, 50, 54 C1 −13, 23, 30, 38, 42 C1 −14, 15, 210, 211, 211 D1 −14, 18, 63, 67, 67 D1 −14, 19, 54, 55, 63 C1 −14, 22, 39, 43, 51 C1 −14, 27, 31, 34, 54 C1 −15, 16, 240, 241, 241 D1 −15, 17, 128, 128, 132 D1 −15, 24, 40, 49, 49 D1 −15, 24, 41, 44, 56 C1 −15, 28, 33, 40, 52 C1 −15, 32, 32, 33, 65 D1
### Enumerating integral Apollonian circle packings
The curvatures
$$
(a, b, c, d)
$$
are a root quadruple (the smallest in some integral circle packing) if
$$
a < 0 \leq b \leq c \leq d
$$
. They are primitive when
$$
\gcd(a, b, c, d)=1
$$
. Defining a new set of variables
$$
(x, d_1, d_2, m)
$$
by the matrix equation
$$
\begin{bmatrix}
a \\ b \\ c \\ d
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & 0\\
-1 & 1 & 0 & 0\\
-1 & 0 & 1 & 0\\
-1 & 1 & 1 &-2
\end{bmatrix}
\begin{bmatrix}
x \\ d_1 \\ d_2 \\ m
\end{bmatrix}
$$
gives a system where
$$
(a, b, c, d)
$$
satisfies the Descartes equation precisely when
$$
x^2+m^2=d_1 d_2
$$
. Furthermore,
$$
(a, b, c, d)
$$
is primitive precisely when
$$
\gcd(x, d_1, d_2)=1
$$
, and
$$
(a, b, c, d)
$$
is a root quadruple precisely when
$$
x<0\leq 2m\leq d_1\leq d_2
$$
.
This relationship can be used to find all the primitive root quadruples with a given negative bend
$$
x
$$
. It follows from
$$
2m\leq d_1
$$
and
$$
2m\leq d_2
$$
that
$$
4m^2\leq d_1d_2
$$
, and hence that
$$
3m^2\leq d_1d_2-m^2=x^2
$$
. Therefore, any root quadruple will satisfy
$$
0\leq m \leq |x|/\sqrt{3}
$$
. By iterating over all the possible values of
$$
m
$$
,
$$
d_1
$$
, and
$$
d_2
$$
one can find all the primitive root quadruples. The following Python code demonstrates this algorithm, producing the primitive root quadruples listed above.
```python
import math
def get_primitive_bends(n: int) -> tuple[int, int, int, int]:
if n == 0:
yield 0, 0, 1, 1
return
for m in range(math.ceil(n / math.sqrt(3))):
s = m**2 + n**2
for d1 in range(max(2 * m, 1), math.floor(math.sqrt(s)) + 1):
d2, remainder = divmod(s, d1)
if remainder == 0 and math.gcd(n, d1, d2) == 1:
yield -n, d1 + n, d2 + n, d1 + d2 + n - 2 * m
for n in range(15):
for bends in get_primitive_bends(n):
print(bends)
```
#### The Local-Global Conjecture
The curvatures appearing in a primitive integral Apollonian circle packing must belong to a set of six or eight possible residues classes modulo 24, and theoretical results and numerical evidence supported that any sufficiently large integer from these residue classes would also be present as a curvature within the packing. This conjecture, known as the local-global conjecture, was proved to be false in 2023.
### Symmetry of integral Apollonian circle packings
There are multiple types of dihedral symmetry that can occur with a gasket depending on the curvature of the circles.
#### No symmetry
If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group C1; the gasket described by curvatures (−10, 18, 23, 27) is an example.
#### D1 symmetry
Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have D1 symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry.
#### D2 symmetry
If two different curvatures are repeated within the first five, the gasket will have D2 symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a two-fold rotational symmetry of 180°. The gasket described by curvatures (−1, 2, 2, 3) is the only Apollonian gasket (up to a scaling factor) to possess D2 symmetry.
#### D3 symmetry
There are no integer gaskets with D3 symmetry.
If the three circles with smallest positive curvature have the same curvature, the gasket will have D3 symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with three-fold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2 − 3. As this ratio is not rational, no integral Apollonian circle packings possess this D3 symmetry, although many packings come close.
#### Almost-D3 symmetry
The figure at left is an integral Apollonian gasket that appears to have D3 symmetry. The same figure is displayed at right, with labels indicating the curvatures of the interior circles, illustrating that the gasket actually possesses only the D1 symmetry common to many other integral Apollonian gaskets.
The following table lists more of these almost-D3 integral Apollonian gaskets. The sequence has some interesting properties, and the table lists a factorization of the curvatures, along with the multiplier needed to go from the previous set to the current one. The absolute values of the curvatures of the "a" disks obey the recurrence relation , from which it follows that the multiplier converges to + 2 ≈ 3.732050807.
+ Integral Apollonian gaskets with near-D3 symmetry Curvature Factors Multiplier a b c d a b d a b c d −1 2 2 3 1×1 1×2 1×3 −4 8 9 9 2×2 2×4 3×3 4.000000000 4.000000000 4.500000000 3.000000000 −15 32 32 33 3×5 4×8 3×11 3.750000000 4.000000000 3.555555556 3.666666667 −56 120 121 121 8×7 8×15 11×11 3.733333333 3.750000000 3.781250000 3.666666667 −209 450 450 451 11×19 15×30 11×41 3.732142857 3.750000000 3.719008264 3.727272727 −780 1680 1681 1681 30×26 30×56 41×41 3.732057416 3.733333333 3.735555556 3.727272727 −2911 6272 6272 6273 41×71 56×112 41×153 3.732051282 3.733333333 3.731112433 3.731707317 −10864 23408 23409 23409 112×97 112×209 153×153 3.732050842 3.732142857 3.732302296 3.731707317 −40545 87362 87362 87363 153×265 209×418 153×571 3.732050810 3.732142857 3.731983425 3.732026144
### Sequential curvatures
For any integer n > 0, there exists an Apollonian gasket defined by the following curvatures: (−n, n + 1, n(n + 1), n(n + 1) + 1). For example, the gaskets defined by (−2, 3, 6, 7), (−3, 4, 12, 13), (−8, 9, 72, 73), and (−9, 10, 90, 91) all follow this pattern. Because every interior circle that is defined by n + 1 can become the bounding circle (defined by −n) in another gasket, these gaskets can be nested. This is demonstrated in the figure at right, which contains these sequential gaskets with n running from 2 through 20.
## History
Although the Apollonian gasket is named for Apollonius of Perga -- because of its construction's dependence on the solution to the problem of Apollonius -- the earliest description of the gasket is from 1706 by Leibniz in a letter to Des Bosses.
The first modern definition of the Apollonian gasket is given by Kasner and Supnick. | https://en.wikipedia.org/wiki/Apollonian_gasket |
Pulse-coupled networks or pulse-coupled neural networks (PCNNs) are neural models proposed by modeling a cat's visual cortex, and developed for high-performance biomimetic image processing.
In 1989, Eckhorn introduced a neural model to emulate the mechanism of cat's visual cortex. The Eckhorn model provided a simple and effective tool for studying small mammal’s visual cortex, and was soon recognized as having significant application potential in image processing.
In 1994, Johnson adapted the Eckhorn model to an image processing algorithm, calling this algorithm a pulse-coupled neural network.
The basic property of the Eckhorn's linking-field model (LFM) is the coupling term. LFM is a modulation of the primary input by a biased offset factor driven by the linking input. These drive a threshold variable that decays from an initial high value. When the threshold drops below zero it is reset to a high value and the process starts over. This is different than the standard integrate-and-fire neural model, which accumulates the input until it passes an upper limit and effectively "shorts out" to cause the pulse.
LFM uses this difference to sustain pulse bursts, something the standard model does not do on a single neuron level. It is valuable to understand, however, that a detailed analysis of the standard model must include a shunting term, due to the floating voltages level in the dendritic compartment(s), and in turn this causes an elegant multiple modulation effect that enables a true higher-order network (HON).
A PCNN is a two-dimensional neural network. Each neuron in the network corresponds to one pixel in an input image, receiving its corresponding pixel's color information (e.g. intensity) as an external stimulus. Each neuron also connects with its neighboring neurons, receiving local stimuli from them. The external and local stimuli are combined in an internal activation system, which accumulates the stimuli until it exceeds a dynamic threshold, resulting in a pulse output. Through iterative computation, PCNN neurons produce temporal series of pulse outputs. The temporal series of pulse outputs contain information of input images and can be used for various image processing applications, such as image segmentation and feature generation. Compared with conventional image processing means, PCNNs have several significant merits, including robustness against noise, independence of geometric variations in input patterns, capability of bridging minor intensity variations in input patterns, etc.
A simplified PCNN called a spiking cortical model was developed in 2009.
## Applications
PCNNs are useful for image processing, as discussed in a book by Thomas Lindblad and Jason M. Kinser.
PCNNs have been used in a variety of image processing applications, including: image segmentation, pattern recognition, feature generation, face extraction, motion detection, region growing, image denoising and image enhancement
Multidimensional pulse image processing of chemical structure data using PCNN has been discussed by Kinser, et al.
They have also been applied to an all pairs shortest path problem.
## References
Category:Artificial neural networks
Category:Image processing | https://en.wikipedia.org/wiki/Pulse-coupled_networks |
A Colonel Blotto game is a type of two-person constant-sum game in which the players (officers) are tasked to simultaneously distribute limited resources over several objects (battlefields). In the classic version of the game, the player devoting the most resources to a battlefield wins that battlefield, and the gain (or payoff) is equal to the total number of battlefields won.
The game was first proposed by Émile Borel in 1921. In 1938 Borel and Ville published a particular optimal strategy (the "disk" solution). The game was studied after the Second World War by scholars in Operation Research, and became a classic in game theory. Gross and Wagner's 1950 research memorandum states Borel's optimal strategy, and coined the fictitious Colonel Blotto and Enemy names. For three battlefields or more, the space of pure strategies is multi-dimensional (two dimensions for three battlefields) and a mixed strategy is thus a probability distribution over a continuous set. The game is a rare example of a non trivial game of that kind where optimal strategies can be explicitly found.
In addition to military strategy applications, the Colonel Blotto game has applications to political strategy (resource allocations across political battlefields), network defense, R&D patent races, and strategic hiring decisions. Consider two sports teams with must spend budget caps (or two Economics departments with use-or-lose grants) are pursuing the same set of candidates, and must decide between many modest offers or aggressive pursuit of a subset of candidates.
## Example
As an example Blotto game, consider the game in which two players each write down three positive integers in non-decreasing order and such that they add up to a pre-specified number S. Subsequently, the two players show each other their writings, and compare corresponding numbers. The player who has two numbers higher than the corresponding ones of the opponent wins the game.
For S = 6 only three choices of numbers are possible: (2, 2, 2), (1, 2, 3) and (1, 1, 4). It is easy to see that:
Any triplet against itself is a draw
(1, 1, 4) against (1, 2, 3) is a draw
(1, 2, 3) against (2, 2, 2) is a draw
(2, 2, 2) beats (1, 1, 4)
It follows that the optimum strategy is (2, 2, 2) as it does not do worse than breaking even against any other strategy while beating one other strategy. There are however several Nash equilibria. If both players choose the strategy (2, 2, 2) or (1, 2, 3), then none of them can beat the other one by changing strategies, so every such strategy pair is a Nash equilibrium.
For larger S the game becomes progressively more difficult to analyze. For S = 12, it can be shown that (2, 4, 6) represents the optimal strategy, while for S > 12, deterministic strategies fail to be optimal. For S = 13, choosing (3, 5, 5), (3, 3, 7) and (1, 5, 7) with probability 1/3 each can be shown to be the optimal probabilistic strategy.
Borel's game is similar to the above example for very large S, but the players are not limited to round integers. They thus have an infinite number of available pure strategies, indeed a continuum.
This concept is also implemented in a story of Sun Bin (田忌赛马) when watching a chariot race with three different races running concurrently. In the races each party had the option to have one chariot team in each race, and each chose to use a strategy of 1, 2, 3 (with 3 being the fastest chariot and 1 being the slowest) to deploy their chariots between the three races creating close wins in each race and few sure outcomes on the winners. When asked how to win Sun Bin advised the chariot owner to change his deployment to that of 2, 3, 1. Though he would be sure to lose the race against the fastest chariots (the 3 chariots); he would win each of the other races, with his 3 chariot easily beating the 2 chariots and his 2 chariot beating the 1 chariots.
## The case of two battlefields
In the simpler case of two battlefields, Macdonell and Mastronardi 2015 provide the first complete characterization of all Nash equilibria to the canonical simplest version of the Colonel Blotto game. This solution, which includes a graphical algorithm for characterizing all the Nash equilibrium strategies, includes previously unidentified Nash equilibrium strategies as well as helps identify what behaviors should never be expected by rational players. Nash equilibrium strategies in this version of the game are a set of bivariate probability distributions: distributions over a set of possible resource allocations for each player, often referred to as Mixed Nash Equilibria (such as can be found in Paper-Rock-Scissors or Matching Pennies as much simpler examples).
Macdonell and Mastronardi 2015 solution, proof, and graphical algorithm for identifying Nash equilibria strategies also pertains to generalized versions of the game such as when Colonel Blotto have differing valuations of the battlefields, when their resources have differing effectiveness on the two battlefields (e.g. one battlefield includes a water landing and Colonel Blotto's resources are Marines instead of Soldiers), and provides insights into versions of the game with three or more battlefields.
Consider two players (Colonel Blotto and Enemy), two battlefields both of equal value, both players know each other's total level of resources prior to allocation, and they then must make a simultaneous allocation decision. It is often assumed Colonel Blotto is the more-resourced officer (his level of resource can be defined to be 1), and Enemy has a fraction of resources less than 1. The Nash equilibrium allocation strategies and payoffs depend on that resource level relationship.
## Application
This game is commonly used as a metaphor for electoral competition, with two political parties devoting money or resources to attract the support of a fixed number of voters. Each voter is a "battlefield" that can be won by one or the other party. The same game also finds application in auction theory where bidders must make simultaneous bids.
Several variations on the original game have been solved by Jean-François Laslier, Brian Roberson, and Dmitriy Kvasov. | https://en.wikipedia.org/wiki/Blotto_game |
In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal lens, a new Gaussian beam is produced. The electric and magnetic field amplitude profiles along a circular Gaussian beam of a given wavelength and polarization are determined by two parameters: the waist , which is a measure of the width of the beam at its narrowest point, and the position relative to the waist.
Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approximation to a real-world beam for cases where lenses or mirrors in the beam are significantly larger than the spot size w(z) of the beam.
Fundamentally, the Gaussian is a solution of the paraxial Helmholtz equation, the wave equation for an electromagnetic field. Although there exist other solutions, the Gaussian families of solutions are useful for problems involving compact beams.
## Mathematical form
The equations below assume a beam with a circular cross-section at all values of ; this can be seen by noting that a single transverse dimension, , appears. Beams with elliptical cross-sections, or with waists at different positions in for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of and of the location for the two transverse dimensions and .
The Gaussian beam is a transverse electromagnetic (TEM) mode. The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation. Assuming polarization in the direction and propagation in the direction, the electric field in phasor (complex) notation is given by:
$$
{\mathbf E(r,z)} = E_0 \, \hat{\mathbf x} \, \frac{w_0}{w(z)}
\exp \left( \frac{-r^2}{w(z)^2}\right )
\exp \left(\! -i \left(kz +k \frac{r^2}{2R(z)} - \psi(z) \right) \!\right)
$$
where
- is the radial distance from the center axis of the beam,
- is the axial distance from the beam's focus (or "waist"),
- is the imaginary unit,
- is the wave number (in radians per meter) for a free-space wavelength , and is the index of refraction of the medium in which the beam propagates,
- , the electric field amplitude at the origin (, ),
- is the radius at which the field amplitudes fall to of their axial values (i.e., where the intensity values fall to of their axial values), at the plane along the beam,
- is the waist radius,
- is the radius of curvature of the beam's wavefronts at , and
- is the
### Gouy phase
at , an extra phase term beyond that attributable to the phase velocity of light.
The physical electric field is obtained from the phasor field amplitude given above by taking the real part of the amplitude times a time factor:
$$
\mathbf E_\text{phys}(r,z,t) = \operatorname{Re}(\mathbf E(r,z) \cdot e^{i\omega t}),
$$
where
$$
\omega
$$
is the angular frequency of the light and is time. The time factor involves an arbitrary sign convention, as discussed at .
Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above form is valid in most practical cases, where .
The corresponding intensity (or irradiance) distribution is given by
$$
I(r,z) = { |E(r,z)|^2 \over 2 \eta } = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( \frac{-2r^2}{w(z)^2}\right),
$$
where the constant is the wave impedance of the medium in which the beam is propagating. For free space, ≈ 377 Ω. is the intensity at the center of the beam at its waist.
If is the total power of the beam,
$$
I_0 = {2P_0 \over \pi w_0^2}.
$$
### Evolving beam width
At a position along the beam (measured from the focus), the spot size parameter is given by a hyperbolic relation:
$$
w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 },
$$
where
$$
z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}
$$
is called the Rayleigh range as further discussed below, and
$$
n
$$
is the refractive index of the medium.
The radius of the beam , at any position along the beam, is related to the full width at half maximum (FWHM) of the intensity distribution at that position according to:
$$
w(z)={\frac {\text{FWHM}(z)}{\sqrt {2\ln2}}}.
$$
### Wavefront curvature
The wavefronts have zero curvature (radius = ∞) at the waist. Wavefront curvature increases away from the waist, with the maximum rate of change occurring at the Rayleigh distance, . Beyond the Rayleigh distance, , the curvature again decreases in magnitude, approaching zero as . The curvature is often expressed in terms of its reciprocal, , the radius of curvature; for a fundamental Gaussian beam the curvature at position is given by:
$$
\frac{1}{R(z)} = \frac{z} {z^2 + z_\mathrm{R}^2} ,
$$
so the radius of curvature is
$$
R(z) = z \left[{ 1+ {\left( \frac{z_\mathrm{R}}{z} \right)}^2 } \right].
$$
Being the reciprocal of the curvature, the radius of curvature reverses sign and is infinite at the beam waist where the curvature goes through zero.
### Elliptical and astigmatic beams
Many laser beams have an elliptical cross-section. Also common are beams with waist positions which are different for the two transverse dimensions, called astigmatic beams. These beams can be dealt with using the above two evolution equations, but with distinct values of each parameter for and and distinct definitions of the point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range contributed by each dimension.
An elliptical beam will invert its ellipticity ratio as it propagates from the far field to the waist. The dimension which was the larger far from the waist, will be the smaller near the waist.
### Gaussian as a decomposition into modes
Arbitrary solutions of the paraxial Helmholtz equation can be decomposed as the sum of Hermite–Gaussian modes (whose amplitude profiles are separable in and using Cartesian coordinates), Laguerre–Gaussian modes (whose amplitude profiles are separable in and using cylindrical coordinates) or similarly as combinations of Ince–Gaussian modes (whose amplitude profiles are separable in and using elliptical coordinates).probably first considered by Goubau and Schwering (1961). At any point along the beam these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different Gouy phase which is why the net transverse profile due to a superposition of modes evolves in , whereas the propagation of any single Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.
Although there are other modal decompositions, Gaussians are useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is not operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.
## Beam parameters
The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength (in the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.
### Beam waist
The shape of a Gaussian beam of a given wavelength is governed solely by one parameter, the beam waist . This is a measure of the beam size at the point of its focus ( in the above equations) where the beam width (as defined above) is the smallest (and likewise where the intensity on-axis () is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range and asymptotic beam divergence , as detailed below.
### Rayleigh range and confocal parameter
The Rayleigh distance or Rayleigh range is determined given a Gaussian beam's waist size:
$$
z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}.
$$
Here is the wavelength of the light, is the index of refraction. At a distance from the waist equal to the Rayleigh range , the width of the beam is larger than it is at the focus where , the beam waist. That also implies that the on-axis () intensity there is one half of the peak intensity (at ). That point along the beam also happens to be where the wavefront curvature () is greatest.
The distance between the two points is called the confocal parameter or depth of focus of the beam.
### Beam divergence
Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where . That is where the intensity has dropped to of its on-axis value. Now, for the parameter increases linearly with . This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose ) and the beam axis () defines the divergence of the beam:
$$
\theta = \lim_{z\to\infty} \arctan\left(\frac{w(z)}{z}\right).
$$
In the paraxial case, as we have been considering, (in radians) is then approximately
$$
\theta = \frac{\lambda}{\pi n w_0}
$$
where is the refractive index of the medium the beam propagates through, and is the free-space wavelength. The total angular spread of the diverging beam, or apex angle of the above-described cone, is then given by
$$
\Theta = 2 \theta\, .
$$
That cone then contains 86% of the Gaussian beam's total power.
Because the divergence is inversely proportional to the spot size, for a given wavelength , a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to minimize the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section () at the waist (and thus a large diameter where it is launched, since is never less than ). This relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes Fraunhofer diffraction. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case.
Since the Gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the axis of the beam. From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about .
Laser beam quality is quantified by the beam parameter product (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size . The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as ("M squared"). The for a Gaussian beam is one. All real laser beams have values greater than one, although very high quality beams can have values very close to one.
The numerical aperture of a Gaussian beam is defined to be , where is the index of refraction of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by
$$
z_\mathrm{R} = \frac{n w_0}{\mathrm{NA}} .
$$
Gouy phase
The Gouy phase is a phase shift gradually acquired by a beam around the focal region. At position the Gouy phase of a fundamental Gaussian beam is given by
$$
\psi(z) = \arctan \left( \frac{z}{z_\mathrm{R}} \right).
$$
The Gouy phase results in an increase in the apparent wavelength near the waist (). Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large numerical aperture, in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the wave equation is satisfied at every position.
The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor. With dependence, the Gouy phase changes from to , while with dependence it changes from to along the axis.
For a fundamental Gaussian beam, the Gouy phase results in a net phase discrepancy with respect to the speed of light amounting to radians (thus a phase reversal) as one moves from the far field on one side of the waist to the far field on the other side. This phase variation is not observable in most experiments. It is, however, of theoretical importance and takes on a greater range for higher-order Gaussian modes.
## Power and intensity
### Power through an aperture
With a beam centered on an aperture, the power passing through a circle of radius in the transverse plane at position is
$$
P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right],
$$
where
$$
P_0 = \frac{ 1 }{ 2 } \pi I_0 w_0^2
$$
is the total power transmitted by the beam.
For a circle of radius , the fraction of power transmitted through the circle is
$$
\frac{P(z)}{P_0} = 1 - e^{-2} \approx 0.865.
$$
Similarly, about 90% of the beam's power will flow through a circle of radius , 95% through a circle of radius , and 99% through a circle of radius .
### Peak intensity
The peak intensity at an axial distance from the beam waist can be calculated as the limit of the enclosed power within a circle of radius , divided by the area of the circle as the circle shrinks:
$$
I(0,z) = \lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2} .
$$
The limit can be evaluated using L'Hôpital's rule:
$$
I(0,z)
= \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)}
= {2P_0 \over \pi w^2(z)} .
$$
## Complex beam parameter
The spot size and curvature of a Gaussian beam as a function of along the beam can also be encoded in the complex beam parameter Garg, pp. 165–168. given by:
$$
q(z) = z + iz_\mathrm{R} .
$$
The reciprocal of contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:
$$
{1 \over q(z)} = {1 \over R(z)} - i {\lambda \over n \pi w^2(z)} .
$$
The complex beam parameter simplifies the mathematical analysis of Gaussian beam propagation, and especially in the analysis of optical resonator cavities using ray transfer matrices.
Then using this form, the earlier equation for the electric (or magnetic) field is greatly simplified. If we call the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the and directions) then it can be separated in and according to:
$$
u(x,y,z) = u_x(x,z)\, u_y(y,z) ,
$$
where
$$
\begin{align}
u_x(x,z) &= \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right), \\
u_y(y,z) &= \frac{1}{\sqrt{{q}_y(z)}} \exp\left(-i k \frac{y^2}{2 {q}_y(z)}\right),
\end{align}
$$
where and are the complex beam parameters in the and directions.
For the common case of a circular beam profile, and , which yields
$$
u(r,z) = \frac{1}{q(z)}\exp\left( -i k\frac{r^2}{2 q(z)}\right) .
$$
## Beam optics
When a gaussian beam propagates through a thin lens, the outgoing beam is also a (different) gaussian beam, provided that the beam travels along the cylindrical symmetry axis of the lens, and that the lens is larger than the width of the beam. The focal length of the lens
$$
f
$$
, the beam waist radius
$$
w_0
$$
, and beam waist position
$$
z_0
$$
of the incoming beam can be used to determine the beam waist radius
$$
w_0'
$$
and position
$$
z_0'
$$
of the outgoing beam.
### Lens equation
As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the phase that is added to each point
$$
(x,y)
$$
of the gaussian beam as it travels through the lens. An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.
The exact solution to the above problem is expressed simply in terms of the magnification
$$
M
$$
$$
\begin{align}
w_0' &= Mw_0\\[1.2ex]
(z_0'-f) &= M^2(z_0-f).
\end{align}
$$
The magnification, which depends on
$$
w_0
$$
and
$$
z_0
$$
, is given by
$$
M = \frac{M_r}{\sqrt{1+r^2}}
$$
where
$$
r = \frac{z_R}{z_0-f}, \quad M_r = \left|\frac{f}{z_0-f}\right|.
$$
An equivalent expression for the beam position
$$
z_0'
$$
is
$$
\frac{1}{z_0+\frac{z_R^2}{(z_0-f)}}+\frac{1}{z_0'} = \frac{1}{f}.
$$
This last expression makes clear that the ray optics thin lens equation is recovered in the limit that
$$
\left|\left(\tfrac{z_R}{z_0}\right)\left(\tfrac{z_R}{z_0-f}\right)\right|\ll 1
$$
. It can also be noted that if
$$
\left|z_0+\frac{z_R^2}{z_0-f}\right|\gg f
$$
then the incoming beam is "well collimated" so that
$$
z_0'\approx f
$$
.
### Beam focusing
In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot. Mathematically, this implies minimization of the magnification
$$
M
$$
. If the beam size is constrained by the size of available optics, this is typically best achieved by sending the largest possible collimated beam through a small focal length lens, i.e. by maximizing
$$
z_R
$$
and minimizing
$$
f
$$
. In this situation, it is justifiable to make the approximation
$$
z_R^2/(z_0-f)^2\gg 1
$$
, implying that
$$
M\approx f/z_R
$$
and yielding the result
$$
w_0'\approx fw_0/z_R
$$
. This result is often presented in the form
$$
\begin{align}
2w_0' &\approx \frac{4}{\pi}\lambda F_\# \\[1.2ex]
z_0' &\approx f
\end{align}
$$
where
$$
F_\# = \frac{f}{2w_0},
$$
which is found after assuming that the medium has index of refraction
$$
n\approx 1
$$
and substituting
$$
z_R=\pi w_0^2/\lambda
$$
. The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters
$$
2w_0'
$$
and
$$
2w_0
$$
, rather than the waist radii
$$
w_0'
$$
and
$$
w_0
$$
.
## Wave equation
As a special case of electromagnetic radiation, Gaussian beams (and the higher-order Gaussian modes detailed below) are solutions to the wave equation for an electromagnetic field in free space or in a homogeneous dielectric medium, obtained by combining Maxwell's equations for the curl of and the curl of , resulting in:
$$
\nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2},
$$
where is the speed of light in the medium, and could either refer to the electric or magnetic field vector, as any specific solution for either determines the other. The Gaussian beam solution is valid only in the paraxial approximation, that is, where wave propagation is limited to directions within a small angle of an axis. Without loss of generality let us take that direction to be the direction in which case the solution can generally be written in terms of which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the wavenumber in the direction:
$$
U(x, y, z, t) = u(x, y, z) e^{-i(kz-\omega t)} \, \hat{\mathbf x} \, .
$$
Using this form along with the paraxial approximation, can then be essentially neglected. Since solutions of the electromagnetic wave equation only hold for polarizations which are orthogonal to the direction of propagation (), we have without loss of generality considered the polarization to be in the direction so that we now solve a scalar equation for .
Substituting this solution into the wave equation above yields the paraxial approximation to the scalar wave equation:
$$
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2ik \frac{\partial u}{\partial z}.
$$
Writing the wave equations in the light-cone coordinates returns this equation without utilizing any approximation. Gaussian beams of any beam waist satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at in terms of the complex beam parameter as defined above. There are many other solutions. As solutions to a linear system, any combination of solutions (using addition or multiplication by a constant) is also a solution. The fundamental Gaussian happens to be the one that minimizes the product of minimum spot size and far-field divergence, as noted above. In seeking paraxial solutions, and in particular ones that would describe laser radiation that is not in the fundamental Gaussian mode, we will look for families of solutions with gradually increasing products of their divergences and minimum spot sizes. Two important orthogonal decompositions of this sort are the Hermite–Gaussian or
### Laguerre-Gaussian modes
, corresponding to rectangular and circular symmetry respectively, as detailed in the next section. With both of these, the fundamental Gaussian beam we have been considering is the lowest order mode.
## Higher-order modes
### Hermite-Gaussian modes
It is possible to decompose a coherent paraxial beam using the orthogonal set of so-called Hermite-Gaussian modes, any of which are given by the product of a factor in and a factor in . Such a solution is possible due to the separability in and in the paraxial Helmholtz equation as written in Cartesian coordinates. Thus given a mode of order referring to the and directions, the electric field amplitude at may be given by:
$$
E(x,y,z) = u_l(x,z) \, u_m(y,z) \, \exp(-ikz),
$$
where the factors for the and dependence are each given by:
$$
u_J(x,z) = \left(\frac{\sqrt{2/\pi}}{ 2^J \, J! \; w_0}\right)^{\!\!1/2} \!\! \left( \frac{{q}_0}{{q}(z)}\right)^{\!\!1/2} \!\! \left(- \frac{{q}^\ast(z)}{{q}(z)}\right)^{\!\! J/2} \!\! H_J\!\left(\frac{\sqrt{2}x}{w(z)}\right) \, \exp \left(\! -i \frac{k x^2}{2 {q}(z)}\right) ,
$$
where we have employed the complex beam parameter (as defined above) for a beam of waist at from the focus. In this form, the first factor is just a normalizing constant to make the set of orthonormal. The second factor is an additional normalization dependent on which compensates for the expansion of the spatial extent of the mode according to (due to the last two factors). It also contains part of the Gouy phase. The third factor is a pure phase which enhances the Gouy phase shift for higher orders .
The final two factors account for the spatial variation over (or ). The fourth factor is the Hermite polynomial of order ("physicists' form", i.e. ), while the fifth accounts for the Gaussian amplitude fall-off , although this isn't obvious using the complex in the exponent. Expansion of that exponential also produces a phase factor in which accounts for the wavefront curvature () at along the beam.
Hermite-Gaussian modes are typically designated "TEMlm"; the fundamental Gaussian beam may thus be referred to as TEM00 (where TEM is transverse electro-magnetic). Multiplying and to get the 2-D mode profile, and removing the normalization so that the leading factor is just called , we can write the mode in the more accessible form:
$$
\begin{align}
E_{l, m}(x, y, z) ={}
& E_0 \frac{w_0}{w(z)}\, H_l \!\Bigg(\frac{\sqrt{2} \,x}{w(z)}\Bigg)\, H_m \!\Bigg(\frac{\sqrt{2} \,y}{w(z)}\Bigg) \times {} \exp \left( {-\frac{x^2+y^2}{w^2(z)}} \right) \exp \left( {-i\frac{k(x^2 + y^2)}{2R(z)}} \right) \times {} \exp \big(i \psi(z)\big) \exp(-ikz).
\end{align}
$$
In this form, the parameter , as before, determines the family of modes, in particular scaling the spatial extent of the fundamental mode's waist and all other mode patterns at . Given that , and have the same definitions as for the fundamental Gaussian beam described above. It can be seen that with we obtain the fundamental Gaussian beam described earlier (since ). The only specific difference in the and profiles at any are due to the Hermite polynomial factors for the order numbers and . However, there is a change in the evolution of the modes' Gouy phase over :
$$
\psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right),
$$
where the combined order of the mode is defined as . While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by radians over all of (and only by radians between ), this is increased by the factor for the higher order modes.
Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.
Laguerre-Gaussian modes
Beam profiles which are circularly symmetric (or lasers with cavities that are cylindrically symmetric) are often best solved using the Laguerre-Gaussian modal decomposition. These functions are written in cylindrical coordinates using generalized Laguerre polynomials. Each transverse mode is again labelled using two integers, in this case the radial index and the azimuthal index which can be positive or negative (or zero):
$$
\begin{align}
u(r, \phi, z) ={}
&C^{LG}_{lp}\frac{1}{w(z)}\left(\frac{r \sqrt{2}}{w(z)}\right)^{\! |l|} \exp\! \left(\! -\frac{r^2}{w^2(z)}\right)L_p^{|l|} \! \left(\frac{2r^2}{w^2(z)}\right) \times {} \\
&\exp \! \left(\! - i k \frac{r^2}{2 R(z)}\right) \exp(-i l \phi) \, \exp(i \psi(z)) ,
\end{align}
$$
where are the generalized Laguerre polynomials. is a required normalization constant:
$$
C^{LG}_{lp} = \sqrt{\frac{2 p!}{\pi(p+|l|)!}} \Rightarrow \int_0^{2\pi}d\phi\int_0^\infty dr\; r \,|u(r,\phi,z)|^2=1,
$$
.
and have the same definitions as above. As with the higher-order Hermite-Gaussian modes the magnitude of the Laguerre-Gaussian modes' Gouy phase shift is exaggerated by the factor :
$$
\psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right) ,
$$
where in this case the combined mode number . As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in but now multiplied by a Laguerre polynomial. The effect of the rotational mode number , in addition to affecting the Laguerre polynomial, is mainly contained in the phase factor , in which the beam profile is advanced (or retarded) by complete phases in one rotation around the beam (in ). This is an example of an optical vortex of topological charge , and can be associated with the orbital angular momentum of light in that mode.
### Ince-Gaussian modes
In elliptic coordinates, one can write the higher-order modes using Ince polynomials. The even and odd Ince-Gaussian modes are given by
$$
u_\varepsilon \left( \xi ,\eta ,z\right) = \frac{w_{0}}{w\left(
z\right) }\mathrm{C}_{p}^{m}\left( i\xi ,\varepsilon \right) \mathrm{C}
_{p}^{m}\left( \eta ,\varepsilon \right) \exp \left[ -ik\frac{r^{2}}{
2q\left( z\right) }-\left( p+1\right) \zeta\left( z\right) \right] ,
$$
where and are the radial and angular elliptic coordinates defined by
$$
\begin{align}
x &= \sqrt{\varepsilon /2}\;w(z) \cosh \xi \cos \eta ,\\
y &= \sqrt{\varepsilon /2}\;w(z) \sinh \xi \sin \eta .
\end{align}
$$
are the even Ince polynomials of order and degree where is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for and respectively.
### Hypergeometric-Gaussian modes
There is another important class of paraxial wave modes in cylindrical coordinates in which the complex amplitude is proportional to a confluent hypergeometric function.
These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. Their intensity profiles are characterized by a single brilliant ring; like Laguerre–Gaussian modes, their intensities fall to zero at the center (on the optical axis) except for the fundamental (0,0) mode. A mode's complex amplitude can be written in terms of the normalized (dimensionless) radial coordinate and the normalized longitudinal coordinate as follows:
$$
\begin{align}
u_{\mathsf{p}m}(\rho, \phi, \Zeta) {}={}
&\sqrt{\frac{2^{\mathsf{p} + |m| + 1}}{\pi\Gamma(\mathsf{p} + |m| + 1)}}\; \frac{\Gamma\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}{\Gamma(|m| + 1)}\, i^{|m|+1} \times{} \\
&\Zeta^{\frac{\mathsf{p}}{2}}\, (\Zeta + i)^{-\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}\, \rho^{|m|} \times{} \\
&\exp\left(-\frac{i\rho^2}{\Zeta + i}\right)\, e^{im\phi}\, {}_1F_1 \left(-\frac{\mathsf{p}}{2}, |m| + 1; \frac{\rho^2}{\Zeta(\Zeta + i)}\right)
\end{align}
$$
where the rotational index is an integer, and
$$
{\mathsf p}\ge-|m|
$$
is real-valued, is the gamma function and is a confluent hypergeometric function.
Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre–Gaussian modes.
The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field profile, HyGG modes have a very simple profile at the beam waist ():
$$
u(\rho, \phi, 0) \propto \rho^{\mathsf{p} + |m|}e^{-\rho^2 + im\phi}.
$$ | https://en.wikipedia.org/wiki/Gaussian_beam |
In the theory of integrable systems, a peakon ("peaked soliton") is a soliton with discontinuous first derivative; the wave profile is shaped like the graph of the function
$$
e^{-|x|}
$$
. Some examples of non-linear partial differential equations with (multi-)peakon solutions are the Camassa–Holm shallow water wave equation, the Degasperis–Procesi equation and the Fornberg–Whitham equation.
Since peakon solutions are only piecewise differentiable, they must be interpreted in a suitable weak sense.
The concept was introduced in 1993 by Camassa and Holm in the short but much cited paper where they derived their shallow water equation.
## A family of equations with peakon solutions
The primary example of a PDE which supports peakon solutions is
$$
u_t - u_{xxt} + (b+1) u u_x = b u_x u_{xx} + u u_{xxx}, \,
$$
where
$$
u(x,t)
$$
is the unknown function, and b is a parameter.
In terms of the auxiliary function
$$
m(x,t)
$$
defined by the relation
$$
m = u-u_{xx}
$$
, the equation takes the simpler form
$$
m_t + m_x u + b m u_x = 0. \,
$$
This equation is integrable for exactly two values of b, namely b = 2 (the Camassa–Holm equation) and b = 3 (the Degasperis–Procesi equation).
## Single peakon solution
The PDE above admits the travelling wave solution
$$
u(x,t) = c \, e^{-|x-ct|}
$$
,
which is a peaked solitary wave with amplitude c and speed c.
This solution is called a (single) peakon solution,
or simply a peakon.
If c is negative, the wave moves to the left with the peak pointing downwards,
and then it is sometimes called an antipeakon.
It is not immediately obvious in what sense the peakon solution satisfies the PDE.
Since the derivative ux has a jump discontinuity at the peak,
the second derivative uxx must be taken in the sense of distributions and will contain a Dirac delta function;
in fact,
$$
m = u - u_{xx} = c \, \delta(x-ct)
$$
.
Now the product
$$
m u_x
$$
occurring in the PDE seems to be undefined, since the distribution m is supported at the very point where the derivative ux is undefined. An ad hoc interpretation is to take the value of ux at that point to equal the average of its left and right limits (zero, in this case). A more satisfactory way to make sense of the solution is to invert the relationship between u and m by writing
$$
m = (G/2) * u
$$
, where
$$
G(x) = \exp(-|x|)
$$
, and use this to rewrite the PDE as a (nonlocal) hyperbolic conservation law:
$$
\partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \left(\frac{b u^2}{2} + \frac{(3-b) u_x^2}{2} \right) \right] = 0.
$$
(The star denotes convolution with respect to x.)
In this formulation the function u can simply be interpreted as a weak solution in the usual sense.
## Multipeakon solutions
Multipeakon solutions are formed by taking a linear combination of several peakons, each with its own time-dependent amplitude and position. (This is a very simple structure compared to the multisoliton solutions of most other integrable PDEs, like the Korteweg–de Vries equation for instance.)
The n-peakon solution thus takes the form
$$
u(x,t) = \sum_{i=1}^n m_i(t) \, e^{-|x-x_i(t)|},
$$
where the 2n functions
$$
x_i(t)
$$
and
$$
m_i(t)
$$
must be chosen suitably in order for u to satisfy the PDE.
For the "b-family" above it turns out that this ansatz indeed gives a solution, provided that the system of ODEs
$$
\dot{x}_k = \sum_{i=1}^n m_i e^{-|x_k-x_i|},
\qquad
\dot{m}_k = (b-1) \sum_{i=1}^n m_k m_i \sgn(x_k-x_i) e^{-|x_k-x_i|}
\qquad
(k = 1,\dots,n)
$$
is satisfied. (Here sgn denotes the sign function.)
Note that the right-hand side of the equation for
$$
x_k
$$
is obtained by substituting
$$
x=x_k
$$
in the formula for u.
Similarly, the equation for
$$
m_k
$$
can be expressed in terms of
$$
u_x
$$
, if one interprets the derivative of
$$
\exp(-|x|)
$$
at x = 0 as being zero.
This gives the following convenient shorthand notation for the system:
$$
\dot{x}_k = u(x_k),
\qquad
\dot{m}_k = -(b-1) m_k u_x(x_k)
\qquad
(k = 1,\dots,n).
$$
The first equation provides some useful intuition about peakon dynamics: the velocity of each peakon equals the elevation of the wave at that point.
## Explicit solution formulas
In the integrable cases b = 2 and b = 3, the system of ODEs describing the peakon dynamics can be solved explicitly for arbitrary n in terms of elementary functions, using inverse spectral techniques. For example, the solution for n = 3 in the Camassa–Holm case b = 2 is given by
$$
\begin{align}
x_1(t) &= \log\frac{(\lambda_1-\lambda_2)^2 (\lambda_1-\lambda_3)^2 (\lambda_2-\lambda_3)^2 a_1 a_2 a_3}{\sum_{j<k} \lambda_j^2 \lambda_k^2 (\lambda_j-\lambda_k)^2 a_j a_k}
\\
x_2(t) &= \log\frac{\sum_{j<k} (\lambda_j-\lambda_k)^2 a_j a_k}{\lambda_1^2 a_1 + \lambda_2^2 a_2 + \lambda_3^2 a_3}
\\
x_3(t) &= \log(a_1+a_2+a_3)
\\
m_1(t) &= \frac{\sum_{j<k} \lambda_j^2 \lambda_k^2 (\lambda_j-\lambda_k)^2 a_j a_k}{\lambda_1 \lambda_2 \lambda_3 \sum_{j<k} \lambda_j \lambda_k (\lambda_j-\lambda_k)^2 a_j a_k}
\\
m_2(t) &= \frac{ \left( \lambda_1^2 a_1 + \lambda_2^2 a_2 + \lambda_3^2 a_3 \right) \sum_{j<k} (\lambda_j-\lambda_k)^2 a_j a_k}{ \left( \lambda_1 a_1 + \lambda_2 a_2 + \lambda_3 a_3 \right) \sum_{j<k} \lambda_j \lambda_k (\lambda_j-\lambda_k)^2 a_j a_k}
\\
m_3(t) &= \frac{a_1+a_2+a_3}{\lambda_1 a_1 + \lambda_2 a_2 + \lambda_3 a_3}
\end{align}
$$
where
$$
a_k(t) = a_k(0) e^{t/\lambda_k}
$$
, and where the 2n constants
$$
a_k(0)
$$
and
$$
\lambda_k
$$
are determined from initial conditions. The general solution for arbitrary n can be expressed in terms of symmetric functions of
$$
a_k
$$
and
$$
\lambda_k
$$
. The general n-peakon solution in the Degasperis–Procesi case b = 3 is similar in flavour, although the detailed structure is more complicated.
## Notes
## References
-
-
-
-
-
Category:Solitons | https://en.wikipedia.org/wiki/Peakon |
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H
$$
f \colon V(G) \to V(H)
$$
such that any two vertices u and v of G are adjacent in G if and only if
$$
f(u)
$$
and
$$
f(v)
$$
are adjacent in H. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection.
If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as
$$
G\simeq H
$$
. In the case when the isomorphism is a mapping of a graph onto itself, i.e., when G and H are one and the same graph, the isomorphism is called an automorphism of G.
Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each other is called an isomorphism class of graphs. The question of whether graph isomorphism can be determined in polynomial time is a major unsolved problem in computer science, known as the graph isomorphism problem.
The two graphs shown below are isomorphic, despite their different looking drawings.
Graph G Graph H An isomorphismbetween G and Hf(a) = 1
f(b) = 6
f(c) = 8
f(d) = 3
f(g) = 5
f(h) = 2
f(i) = 4
f(j) = 7
## Variations
In the above definition, graphs are understood to be undirected non-labeled non-weighted graphs. However, the notion of isomorphism may be applied to all other variants of the notion of graph, by adding the requirements to preserve the corresponding additional elements of structure: arc directions, edge weights, etc., with the following exception.
### Isomorphism of labeled graphs
For labeled graphs, two definitions of isomorphism are in use.
Under one definition, an isomorphism is a vertex bijection which is both edge-preserving and label-preserving.
Under another definition, an isomorphism is an edge-preserving vertex bijection which preserves equivalence classes of labels, i.e., vertices with equivalent (e.g., the same) labels are mapped onto the vertices with equivalent labels and vice versa; same with edge labels.
For example, the
$$
K_2
$$
graph with the two vertices labelled with 1 and 2 has a single automorphism under the first definition, but under the second definition there are two auto-morphisms.
The second definition is assumed in certain situations when graphs are endowed with unique labels commonly taken from the integer range 1,...,n, where n is the number of the vertices of the graph, used only to uniquely identify the vertices. In such cases two labeled graphs are sometimes said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic (otherwise the definition of isomorphism would be trivial).
## Motivation
The formal notion of "isomorphism", e.g., of "graph isomorphism", captures the informal notion that some objects have "the same structure" if one ignores individual distinctions of "atomic" components of objects in question. Whenever individuality of "atomic" components (vertices and edges, for graphs) is important for correct representation of whatever is modeled by graphs, the model is refined by imposing additional restrictions on the structure, and other mathematical objects are used: digraphs, labeled graphs, colored graphs, rooted trees and so on. The isomorphism relation may also be defined for all these generalizations of graphs: the isomorphism bijection must preserve the elements of structure which define the object type in question: arcs, labels, vertex/edge colors, the root of the rooted tree, etc.
The notion of "graph isomorphism" allows us to distinguish graph properties inherent to the structures of graphs themselves from properties associated with graph representations: graph drawings, data structures for graphs, graph labelings, etc. For example, if a graph has exactly one cycle, then all graphs in its isomorphism class also have exactly one cycle. On the other hand, in the common case when the vertices of a graph are (represented by) the integers 1, 2,... N, then the expression
$$
\sum_{v \in V(G)} v\cdot\text{deg }v
$$
may be different for two isomorphic graphs.
## Whitney theorem
The Whitney graph isomorphism theorem, shown by Hassler Whitney, states that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K3, the complete graph on three vertices, and the complete bipartite graph K1,3, which are not isomorphic but both have K3 as their line graph. The Whitney graph theorem can be extended to hypergraphs.
## Recognition of graph isomorphism
While graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the graph isomorphism problem.
Its practical applications include primarily cheminformatics, mathematical chemistry (identification of chemical compounds), and electronic design automation (verification of equivalence of various representations of the design of an electronic circuit).
The graph isomorphism problem is one of few standard problems in computational complexity theory belonging to NP, but not known to belong to either of its well-known (and, if P ≠ NP, disjoint) subsets: P and NP-complete. It is one of only two, out of 12 total, problems listed in whose complexity remains unresolved, the other being integer factorization. It is however known that if the problem is NP-complete then the polynomial hierarchy collapses to a finite level.
In November 2015, László Babai, a mathematician and computer scientist at the University of Chicago, claimed to have proven that the graph isomorphism problem is solvable in quasi-polynomial time. He published preliminary versions of these results in the proceedings of the 2016 Symposium on Theory of Computing, and of the 2018 International Congress of Mathematicians. In January 2017, Babai briefly retracted the quasi-polynomiality claim and stated a sub-exponential time complexity bound instead. He restored the original claim five days later. as of 2024, the full journal version of Babai's paper has not yet been published.
Its generalization, the subgraph isomorphism problem, is known to be NP-complete.
The main areas of research for the problem are design of fast algorithms and theoretical investigations of its computational complexity, both for the general problem and for special classes of graphs.
The Weisfeiler Leman graph isomorphism test can be used to heuristically test for graph isomorphism. If the test fails the two input graphs are guaranteed to be non-isomorphic. If the test succeeds the graphs may or may not be isomorphic. There are generalizations of the test algorithm that are guaranteed to detect isomorphisms, however their run time is exponential.
Another well-known algorithm for graph isomorphism is the vf2 algorithm, developed by Cordella et al. in 2001. The vf2 algorithm is a depth-first search algorithm that tries to build an isomorphism between two graphs incrementally. It uses a set of feasibility rules to prune the search space, allowing it to efficiently handle graphs with thousands of nodes. The vf2 algorithm has been widely used in various applications, such as pattern recognition, computer vision, and bioinformatics. While it has a worst-case exponential time complexity, it performs well in practice for many types of graphs. | https://en.wikipedia.org/wiki/Graph_isomorphism |
Pulse-density modulation (PDM) is a form of modulation used to represent an analog signal with a binary signal. In a PDM signal, specific amplitude values are not encoded into codewords of pulses of different weight as they would be in pulse-code modulation (PCM); rather, the relative density of the pulses corresponds to the analog signal's amplitude. The output of a 1-bit DAC is the same as the PDM encoding of the signal.
## Description
In a pulse-density modulation bitstream, a 1 corresponds to a pulse of positive polarity (+A), and a 0 corresponds to a pulse of negative polarity (−A). Mathematically, this can be represented as
$$
x[n] = -A (-1)^{a[n]},
$$
where x[n] is the bipolar bitstream (either −A or +A), and a[n] is the corresponding binary bitstream (either 0 or 1).
A run consisting of all 1s would correspond to the maximum (positive) amplitude value, all 0s would correspond to the minimum (negative) amplitude value, and alternating 1s and 0s would correspond to a zero amplitude value. The continuous amplitude waveform is recovered by low-pass filtering the bipolar PDM bitstream.
## Examples
A single period of the trigonometric sine function, sampled 100 times and represented as a PDM bitstream, is:
0101011011110111111111111111111111011111101101101010100100100000010000000000000000000001000010010101
Two periods of a higher frequency sine wave would appear as:
0101101111111111111101101010010000000000000100010011011101111111111111011010100100000000000000100101
In pulse-density modulation, a high density of 1s occurs at the peaks of the sine wave, while a low density of 1s occurs at the troughs of the sine wave.
## Analog-to-digital conversion
A PDM bitstream is encoded from an analog signal through the process of a 1-bit delta-sigma modulation. This process uses a one-bit quantizer that produces either a 1 or 0 depending on the amplitude of the analog signal. A 1 or 0 corresponds to a signal that is all the way up or all the way down, respectively. Because in the real world, analog signals are rarely all the way in one direction, there is a quantization error, the difference between the 1 or 0 and the actual amplitude it represents. This error is fed back negatively in the ΔΣ process loop. In this way, every error successively influences every other quantization measurement and its error. This has the effect of averaging out the quantization error.
## Digital-to-analog conversion
The process of decoding a PDM signal into an analog one is simple: one only has to pass the PDM signal through a low-pass filter. This works because the function of a low-pass filter is essentially to average the signal. The average amplitude of pulses is measured by the density of those pulses over time, thus a low-pass filter is the only step required in the decoding process.
## Relationship to PWM
Pulse-width modulation (PWM) is a special case of PDM where the switching frequency is fixed and all the pulses corresponding to one sample are contiguous in the digital signal. The method for demodulation to an analogue signal remains the same, but the representation of a 50% signal with a resolution of 8-bits, a PWM waveform will turn on for 128 clock cycles and then off for the remaining 128 cycles. With PDM and the same clock rate the signal would alternate between on and off every other cycle. The average obtained by a low-pass filter is 50% of the maximum signal level for both waveforms, but the PDM signal switches more often. For 100% or 0% level, they are the same, with the signal permanently on or off respectively.
## Relationship to biology
Notably, one of the ways animal nervous systems represent sensory and other information is through rate coding whereby the magnitude of the signal is related to the rate of firing of the sensory neuron. In direct analogy, each neural event – called an action potential – represents one bit (pulse), with the rate of firing of the neuron representing the pulse density.
## Algorithm
The following digital model of pulse-density modulation can be obtained from a digital model of a 1st-order 1-bit delta-sigma modulator. Consider a signal
$$
x[n]
$$
in the discrete time domain as the input to a first-order delta-sigma modulator, with
$$
y[n]
$$
the output. In the discrete frequency domain, where the Z-transform has been applied to the amplitude time-series
$$
x[n]
$$
to yield
$$
X(z)
$$
, the output
$$
Y(z)
$$
of the delta-sigma modulator's operation is represented by
$$
Y(z) = X(z) + E(z) \left(1 - z^{-1}\right),
$$
where
$$
E(z)
$$
is the frequency-domain quantization error of the delta-sigma modulator.
Rearranging terms, we obtain
$$
Y(z) = E(z) + \left[X(z) - Y(z) z^{-1}\right] \left(\frac{1}{1 - z^{-1}}\right).
$$
The factor
$$
1 - z^{-1}
$$
represents a high-pass filter, so it is clear that
$$
E(z)
$$
contributes less to the output
$$
Y(z)
$$
at low frequencies and more at high frequencies. This demonstrates the noise shaping effect of the delta-sigma modulator: the quantization noise is "pushed" out of the low frequencies up into the high-frequency range.
Using the inverse Z-transform, we may convert this into a difference equation relating the input of the delta-sigma modulator to its output in the discrete time domain,
$$
y[n] = x[n] + e[n] - e[n-1].
$$
There are two additional constraints to consider: first, at each step the output sample
$$
y[n]
$$
is chosen so as to minimize the "running" quantization error
$$
e[n].
$$
Second,
$$
y[n]
$$
is represented as a single bit, meaning it can take on only two values. We choose
$$
y[n] = \pm 1
$$
for convenience, allowing us to write
$$
\begin{align} y[n] &= \sgn\big(x[n] - e[n-1]\big) \\
&= \begin{cases} +1 & x[n] > e[n-1] \\
-1 & x[n] < e[n-1] \end{cases} \\
&= (x[n] - e[n-1]) + e[n]. \\
\end{align}
$$
Rearranging to solve for
$$
e[n]
$$
yields:
$$
e[n] = y[n] - \big(x[n] - e[n-1]\big) = \sgn\big(x[n] - e[n-1]\big) - \big(x[n] - e[n-1]\big).
$$
This, finally, gives a formula for the output sample
$$
y[n]
$$
in terms of the input sample
$$
x[n]
$$
. The quantization error of each sample is fed back into the input for the following sample.
The following pseudo-code implements this algorithm to convert a pulse-code modulation signal into a PDM signal:
// Encode samples into pulse-density modulation
// using a first-order sigma-delta modulator
function pdm(real[0..s] x, real qe = 0) // initial running error is zero
var int[0..s] y
for n from 0 to s do
qe := qe + x[n]
if qe > 0 then
y[n] := 1
else
y[n] := −1
qe := qe - y[n]
return y, qe // return output and running error
## Applications
PDM is the encoding used in Sony's Super Audio CD (SACD) format, under the name Direct Stream Digital.
PDM is also the output of some MEMS microphones.
Some systems transmit PDM stereo audio over a single data wire. The rising edge of the master clock indicates a bit from the left channel, while the falling edge of the master clock indicates a bit from the right channel.Maxim Integrated. "PDM Input Class D Audio Power Amplifier" (PDF). 2013. Figure 1 on p. 5; and the "Digital Audio Interface" section on p. 13. | https://en.wikipedia.org/wiki/Pulse-density_modulation |
In engineering, a requirement is a condition that must be satisfied for the output of a work effort to be acceptable. It is an explicit, objective, clear and often quantitative description of a condition to be satisfied by a material, design, product, or service.
A specification or spec is a set of requirements that is typically used by developers in the design stage of product development and by testers in their verification process.
With iterative and incremental development such as agile software development, requirements are developed in parallel with design and implementation. With the waterfall model, requirements are completed before design or implementation start.
Requirements are used in many engineering fields including engineering design, system engineering, software engineering, enterprise engineering, product development, and process optimization.
Requirement is a relatively broad concept that can describe any necessary or desired function, attribute, capability, characteristic, or quality of a system for it to have value and utility to a customer, organization, user, or other stakeholder.
## Origins of term
The term requirement has been in use in the software engineering community since at least the 1960s.
According to the Guide to the Business Analysis Body of Knowledge® version 2 from IIBA (BABOK), a requirement is:
1. A condition or capability needed by a stakeholder to solve a problem or achieve an objective.
1. A condition or capability that must be met or possessed by a solution or solution component to satisfy a contract, standard, specification, or other formally imposed documents.
1. A documented representation of a condition or capability as in (1) or (2).
This definition is based on IEEE 610.12-1990: IEEE Standard Glossary of Software Engineering Terminology.
## Product versus process requirements
Requirements can be said to relate to two fields:
- Product requirements prescribe properties of a system or product.
- Process requirements prescribe activities to be performed by the developing organization. For instance, process requirements could specify the methodologies that must be followed, and constraints that the organization must obey.
Product and process requirements are closely linked; a product requirement could be said to specify the automation required to support a process requirement while a process requirement could be said to specify the activities required to support a product requirement. For example, a maximum development cost requirement (a process requirement) may be imposed to help achieve a maximum sales price requirement (a product requirement); a requirement that the product be maintainable (a product requirement) often is addressed by imposing requirements to follow particular development styles (e.g., object-oriented programming), style-guides, or a review/inspection process (process requirements).
## Types of requirements
Requirements are typically classified into types produced at different stages in a development progression, with the taxonomy depending on the overall model being used. For example, the following scheme was devised by the International Institute of Business Analysis in their Business Analysis Body of Knowledge (see also FURPS and Types of requirements).
Architectural requirements
Architectural requirements explain what has to be done by identifying the necessary integration of system structure and system behavior, i.e., system architecture of a system.
In software engineering, they are called architecturally significant requirements, which is defined as those requirements that have a measurable impact on a software system’s architecture.
Business requirements
High-level statements of the goals, objectives, or needs of an organization. They usually describe opportunities that an organization wants to realise or problems that they want to solve. Often stated in a business case.
User (stakeholder) requirements
Mid-level statements of the needs of a particular stakeholder or group of stakeholders. They usually describe how someone wants to interact with the intended solution. Often acting as a mid-point between the high-level business requirements and more detailed solution requirements.
Functional (solution) requirements
Usually detailed statements of capabilities, behavior, and information that the solution will need. Examples include formatting text, calculating a number, modulating a signal. They are also sometimes known as capabilities.
Quality-of-service (non-functional) requirements
Usually detailed statements of the conditions under which the solution must remain effective, qualities that the solution must have, or constraints within which it must operate. Examples include: reliability, testability, maintainability, availability. They are also known as characteristics, constraints or the ilities.
Implementation (transition) requirements
Usually, detailed statements of capabilities or behavior required only to enable the transition from the current state of the enterprise to the desired future state, but that will thereafter no longer be required. Examples include recruitment, role changes, education, migration of data from one system to another.
Regulatory requirements
Requirements defined by laws (Federal, State, Municipal, or Regional), contracts (terms and conditions), or policies (company, departmental, or project-level).
## Characteristics of good requirements
The characteristics of good requirements are variously stated by different writers, with each writer generally emphasizing the characteristics most appropriate to their general discussion or the specific technology domain being addressed. However, the following characteristics are generally acknowledged.
Characteristic Explanation Unitary (Cohesive) The requirement addresses one and only one thing. Complete The requirement is fully stated in one place with no missing information. Consistent The requirement does not contradict any other requirement and is fully consistent with all authoritative external documentation. Non-Conjugated (Atomic) The requirement is atomic, i.e., it does not contain conjunctions. E.g., "The postal code field must validate American and Canadian postal codes" should be written as two separate requirements: (1) "The postal code field must validate American postal codes" and (2) "The postal code field must validate Canadian postal codes". Traceable The requirement meets all or part of a business need as stated by stakeholders and authoritatively documented. Current The requirement has not been made obsolete by the passage of time. Unambiguous The requirement is concisely stated without recourse to technical jargon, acronyms (unless defined elsewhere in the Requirements document), or other esoteric verbiage. It expresses objective facts, not subjective opinions. It is subject to one and only one interpretation. Vague subjects, adjectives, prepositions, verbs and subjective phrases are avoided. Negative statements and compound statements are avoided. Specify Importance Many requirements represent a stakeholder-defined characteristic the absence of which will result in a major or even fatal deficiency. Others represent features that may be implemented if time and budget permits. The requirement must specify a level of importance. Verifiable The implementation of the requirement can be determined through basic possible methods: inspection, demonstration, test (instrumented) or analysis (to include validated modeling & simulation).
There are many more attributes to consider that contribute to the quality of requirements. If requirements are subject to rules of data integrity (for example) then accuracy/correctness and validity/authorization are also worthy attributes. Traceability confirms that the requirement set satisfies the need (no more - and no less than what is required).
To the above some add Externally Observable, that is, the requirement specifies a characteristic of the product that is externally observable or experienced by the user. Such advocates argue that requirements that specify internal architecture, design, implementation, or testing decisions are probably constraints, and should be clearly articulated in the Constraints section of the Requirements document. The contrasting view is that this perspective fails on two points. First, the perspective does not recognize that the user experience may be supported by requirements not perceivable by the user. For example, a requirement to present geocoded information to the user may be supported by a requirement for an interface with an external third party business partner. The interface will be imperceptible to the user, though the presentation of information obtained through the interface certainly would not. Second, a constraint limits design alternatives, whereas a requirement specifies design characteristics. To continue the example, a requirement selecting a web service interface is different from a constraint limiting design alternatives to methods compatible with a Single Sign-On architecture.
### Verification
All requirements should be verifiable. The most common method is by test. If this is not the case, another verification method should be used instead (e.g. analysis, demonstration, inspection, or review of design).
Certain requirements, by their very structure, are not verifiable. These include requirements that say the system must never or always exhibit a particular property. Proper testing of these requirements would require an infinite testing cycle. Such requirements must be rewritten to be verifiable. As stated above all requirements must be verifiable.
Non-functional requirements, which are unverifiable at the software level, must still be kept as a documentation of customer intent. However, they may be traced to process requirements that are determined to be a practical way of meeting them. For example, a non-functional requirement to be free from backdoors may be satisfied by replacing it with a process requirement to use pair programming. Other non-functional requirements will trace to other system components and be verified at that level. For example, system reliability is often verified by analysis at the system level. Avionics software with its complicated safety requirements must follow the DO-178B development process.
Activities that lead to the derivation of the system or software requirements. Requirements engineering may involve a feasibility study or a conceptual analysis phase of the project and requirements elicitation (gathering, understanding, reviewing, and articulating the needs of the stakeholders) and requirements analysis, analysis (checking for consistency and completeness), specification (documenting the requirements) and validation (making sure the specified requirements are correct).
Requirements are prone to issues of ambiguity, incompleteness, and inconsistency. Techniques such as rigorous inspection have been shown to help deal with these issues. Ambiguities, incompleteness, and inconsistencies that can be resolved in the requirements phase typically cost orders of magnitude less to correct than when these same issues are found in later stages of product development. Requirements analysis strives to address these issues.
There is an engineering trade off to consider between requirements which are too vague, and those which are so detailed that they
- take a long time to produce - sometimes to the point of being obsolete once completed
- limit the implementation options available
- are costly to produce
Agile approaches evolved as a way of overcoming these problems, by baselining requirements at a high-level, and elaborating detail on a just-in-time or last responsible moment basis.
## Documenting requirements
Requirements are usually written as a means for communication between the different stakeholders. This means that the requirements should be easy to understand both for normal users and for developers. One common way to document a requirement is stating what the system must do. Example: 'The contractor must deliver the product no later than xyz date.' Other methods include use cases and user stories.
## Changes in requirements
Requirements generally change with time. Once defined and approved, requirements should fall under change control. For many projects, requirements are altered before the system is complete. This is partly due to the complexity of computer software and the fact that users don't know what they want before they see it. This characteristic of requirements has led to requirements management studies and practices.
## Issues
### Competing standards
There are several competing views of what requirements are and how they should be managed and used. Two leading bodies in the industry are the IEEE and the IIBA. Both of these groups have different but similar definitions of what a requirement is.
### Disputes regarding the necessity and effects of software requirements
Many projects have succeeded with little or no agreement on requirements. Some evidence furthermore indicates that specifying requirements can decrease creativity and design performance Requirements hinder creativity and design because designers become overly preoccupied with provided information. More generally, some research suggests that software requirements are an illusion created by misrepresenting design decisions as requirements in situations where no real requirements are evident.
Meanwhile, most agile software development methodologies question the need for rigorously describing software requirements upfront, which they consider a moving target. Instead, extreme programming for example describes requirements informally using user stories (short summaries fitting on an index card explaining one aspect of what the system should do), and considers it the developer's duty to directly ask the customer for clarification. Agile methodologies attempt to capture requirements in a series of automated acceptance tests.
### Requirements creep
Scope creep may occur from requirements moving over time. In Requirements management the alteration of requirements is allowed but if not adequately tracked or preceding steps (business goals then user requirements) are not throttled by additional oversight or handled as a cost and potential program failure, then requirements changes are easy and likely to happen. It is easy for requirement changes to occur faster than developers are able to produce work, and the effort to go backwards as a result.
### Multiple requirements taxonomies
There are multiple taxonomies for requirements depending on which framework one is operating under. (For example, the stated standards of IEEE, vice IIBA or U.S. DoD approaches). Differing language and processes in different venues or casual speech can cause confusion and deviation from desired process.
### Process corruptions
A process being run by humans is subject to human flaws in governance, where convenience or desires or politics may lead to exceptions or outright subversion of the process and deviations from the textbook way the process is supposed to proceed. Examples include:
- Process with no rigor gets no respect - If exceptions or changes are common, such as the organization running it having little independence or power or not being reliable and transparent in records, it may lead to the overall process being ignored.
- New players wanting a do-over - e.g., The natural tendency of new people to want to change their predecessor's work to demonstrate their power or claims of value, such as a new CEO wanting to change the previous CEO's planning, including business goals, of something (such as a software solution) already in development, or a newly created office objects to current development of a project because they did not exist when user requirements were crafted, so they begin an effort to backtrack and re-baseline the project.
- Coloring outside the lines - e.g., Users wanting more control do not just input things that meet the requirements management definition of "user requirement" or priority level, but insert design details or favored vendor characteristic as user requirements or everything their office says as the highest possible priority.
- Showing up late - e.g., Doing little or no effort in requirements elicitation prior to development. This may be due to thinking they will get the same benefit regardless of individual participation, or that there is no point if they can just insert demands at the testing stage and next spin, or the preference to be always right by waiting for post-work critique.
Within the U.S. Department of Defense process, some historical examples of requirements issues are
- the M-2 Bradley issues of casual requirements movement portrayed in Pentagon Wars;
- the F-16 growth from lightweight fighter concept of the Fighter mafia, attributed to F-15 program attempting to sabotage competition or individual offices putting in local desires eroding the concept of being lightweight and low cost.
- enthusiasm ca. 1998 for 'Net-Ready' led to its mandate as Key Performance Parameter from the Net-Ready office, outside the office defining requirements process and not consistent to that office's previously defined process, their definition of what a KPP was, or that some efforts might not be appropriate or able to define what constituted 'Net-Ready'. | https://en.wikipedia.org/wiki/Requirement |
In cartography, a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane.
Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography.
All projections of a sphere on a plane necessarily distort the surface in some way. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections.
More generally, projections are considered in several fields of pure mathematics, including differential geometry, projective geometry, and manifolds. However, the term "map projection" refers specifically to a cartographic projection.
Despite the name's literal meaning, projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate. Rather, any mathematical function that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection. Few projections in practical use are perspective.
Most of this article assumes that the surface to be mapped is that of a sphere. The Earth and other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes. The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid.
The most well-known map projection is the Mercator projection. This map projection has the property of being conformal. However, it has been criticized throughout the 20th century for enlarging regions further from the equator. To contrast, equal-area projections such as the Sinusoidal projection and the Gall–Peters projection show the correct sizes of countries relative to each other, but distort angles. The National Geographic Society and most atlases favor map projections that compromise between area and angular distortion, such as the Robinson projection and the Winkel tripel projection.
## Metric properties of maps
Many properties can be measured on the Earth's surface independently of its geography:
- Area
- Shape
- Direction
- Bearing
- Distance
Map projections can be constructed to preserve some of these properties at the expense of others. Because the Earth's curved surface is not isometric to a plane, preservation of shapes inevitably requires a variable scale and, consequently, non-proportional presentation of areas. Similarly, an area-preserving projection can not be conformal, resulting in shapes and bearings distorted in most places of the map. Each projection preserves, compromises, or approximates basic metric properties in different ways. The purpose of the map determines which projection should form the base for the map. Because maps have many different purposes, a diversity of projections have been created to suit those purposes.
Another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information; their collection depends on the chosen datum (model) of the Earth. Different datums assign slightly different coordinates to the same location, so in large scale maps, such as those from national mapping systems, it is important to match the datum to the projection. The slight differences in coordinate assignation between different datums is not a concern for world maps or those of large regions, where such differences are reduced to imperceptibility.
### Distortion
Carl Friedrich Gauss's Theorema Egregium proved that a sphere's surface cannot be represented on a plane without distortion. The same applies to other reference surfaces used as models for the Earth, such as oblate spheroids, ellipsoids, and geoids. Since any map projection is a representation of one of those surfaces on a plane, all map projections distort.
The classical way of showing the distortion inherent in a projection is to use Tissot's indicatrix. For a given point, using the scale factor h along the meridian, the scale factor k along the parallel, and the angle θ′ between them, Nicolas Tissot described how to construct an ellipse that illustrates the amount and orientation of the components of distortion. By spacing the ellipses regularly along the meridians and parallels, the network of indicatrices shows how distortion varies across the map.
#### Other distortion metrics
Many other ways have been described of showing the distortion in projections. Like Tissot's indicatrix, the Goldberg-Gott indicatrix is based on infinitesimals, and depicts flexion and skewness (bending and lopsidedness) distortions.
Rather than the original (enlarged) infinitesimal circle as in Tissot's indicatrix, some visual methods project finite shapes that span a part of the map.
For example, a small circle of fixed radius (e.g., 15 degrees angular radius). Sometimes spherical triangles are used.
In the first half of the 20th century, projecting a human head onto different projections was common to show how distortion varies across one projection as compared to another.
In dynamic media, shapes of familiar coastlines and boundaries can be dragged across an interactive map to show how the projection distorts sizes and shapes according to position on the map.
Another way to visualize local distortion is through grayscale or color gradations whose shade represents the magnitude of the angular deformation or areal inflation. Sometimes both are shown simultaneously by blending two colors to create a bivariate map.
To measure distortion globally across areas instead of at just a single point necessarily involves choosing priorities to reach a compromise. Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result. Many have been described.
## Design and construction
The creation of a map projection involves two steps:
1. Selection of a model for the shape of the Earth or planetary body (usually choosing between a sphere or ellipsoid). Because the Earth's actual shape is irregular, information is lost in this step.
1. Transformation of geographic coordinates (longitude and latitude) to Cartesian (x,y) or polar (r, θ) plane coordinates. In large-scale maps, Cartesian coordinates normally have a simple relation to eastings and northings defined as a grid superimposed on the projection. In small-scale maps, eastings and northings are not meaningful, and grids are not superimposed.
Some of the simplest map projections are literal projections, as obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface. Although most projections are not defined in this way, picturing the light source-globe model can be helpful in understanding the basic concept of a map projection.
### Choosing a projection surface
A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface. The cylinder, cone and the plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to distort the image. (To compare, one cannot flatten an orange peel without tearing and warping it.)
One way of describing a projection is first to project from the Earth's surface to a developable surface such as a cylinder or cone, and then to unroll the surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without further distortion.
### Aspect of the projection
Once a choice is made between projecting onto a cylinder, cone, or plane, the aspect of the shape must be specified. The aspect describes how the developable surface is placed relative to the globe: it may be normal (such that the surface's axis of symmetry coincides with the Earth's axis), transverse (at right angles to the Earth's axis) or oblique (any angle in between).
### Notable lines
The developable surface may also be either tangent or secant to the sphere or ellipsoid. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Moving the developable surface away from contact with the globe never preserves or optimizes metric properties, so that possibility is not discussed further here.
Tangent and secant lines (standard lines) are represented undistorted. If these lines are a parallel of latitude, as in conical projections, it is called a standard parallel. The central meridian is the meridian to which the globe is rotated before projecting. The central meridian (usually written λ) and a parallel of origin (usually written φ) are often used to define the origin of the map projection.
### Scale
A globe is the only way to represent the Earth with constant scale throughout the entire map in all directions. A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines.
Some possible properties are:
- The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a conformal map.
- Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect.
- Combination of the above: the scale depends on latitude only, not on longitude or direction. This applies for the Mercator projection in normal aspect.
- Scale is constant along all straight lines radiating from a particular geographic location. This is the defining characteristic of an equidistant projection such as the azimuthal equidistant projection. There are also projections (Maurer's two-point equidistant projection, Close) where true distances from two points are preserved.
### Choosing a model for the shape of the body
Projection construction is also affected by how the shape of the Earth or planetary body is approximated. In the following section on projection categories, the earth is taken as a sphere in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate ellipsoid. Whether spherical or ellipsoidal, the principles discussed hold without loss of generality.
Selecting a model for a shape of the Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct topographic maps and for other large- and medium-scale maps that need to accurately depict the land surface. Auxiliary latitudes are often employed in projecting the ellipsoid.
A third model is the geoid, a more complex and accurate representation of Earth's shape coincident with what mean sea level would be if there were no winds, tides, or land. Compared to the best fitting ellipsoid, a geoidal model would change the characterization of important properties such as distance, conformality and equivalence. Therefore, in geoidal projections that preserve such properties, the mapped graticule would deviate from a mapped ellipsoid's graticule. Normally the geoid is not used as an Earth model for projections, however, because Earth's shape is very regular, with the undulation of the geoid amounting to less than 100 m from the ellipsoidal model out of the 6.3 million m Earth radius. For irregular planetary bodies such as asteroids, however, sometimes models analogous to the geoid are used to project maps from.
Other regular solids are sometimes used as generalizations for smaller bodies' geoidal equivalent. For example, Io is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities. Haumea's shape is a Jacobi ellipsoid, with its major axis twice as long as its minor and with its middle axis one and half times as long as its minor.
See map projection of the triaxial ellipsoid for further information.
## Classification
One way to classify map projections is based on the type of surface onto which the globe is projected. In this scheme, the projection process is described as placing a hypothetical projection surface the size of the desired study area in contact with part of the Earth, transferring features of the Earth's surface onto the projection surface, then unraveling and scaling the projection surface into a flat map. The most common projection surfaces are cylindrical (e.g., Mercator), conic (e.g., Albers), and planar (e.g., stereographic). Many mathematical projections, however, do not neatly fit into any of these three projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic.
Another way to classify projections is according to properties of the model they preserve. Some of the more common categories are:
- Preserving direction (azimuthal or zenithal), a trait possible only from one or two points to every other point
- Preserving shape locally (conformal or orthomorphic)
- Preserving area (equal-area or equiareal or equivalent or authalic)
- Preserving distance (equidistant), a trait possible only between one or two points and every other point
- Preserving shortest route, a trait preserved only by the gnomonic projection
Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.
## Projections by surface
The three developable surfaces (plane, cylinder, cone) provide useful models for understanding, describing, and developing map projections. However, these models are limited in two fundamental ways. For one thing, most world projections in use do not fall into any of those categories. For another thing, even most projections that do fall into those categories are not naturally attainable through physical projection. As L. P. Lee notes,
Lee's objection refers to the way the terms cylindrical, conic, and planar (azimuthal) have been abstracted in the field of map projections. If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such a cylindrical projection (for example) is one which:
1. Is rectangular;
1. Has straight vertical meridians, spaced evenly;
1. Has straight parallels symmetrically placed about the equator;
1. Has parallels constrained to where they fall when light shines through the globe onto the cylinder, with the light source someplace along the line formed by the intersection of the prime meridian with the equator, and the center of the sphere.
(If you rotate the globe before projecting then the parallels and meridians will not necessarily still be straight lines. Rotations are normally ignored for the purpose of classification.)
Where the light source emanates along the line described in this last constraint is what yields the differences between the various "natural" cylindrical projections. But the term cylindrical as used in the field of map projections relaxes the last constraint entirely. Instead the parallels can be placed according to any algorithm the designer has decided suits the needs of the map. The famous Mercator projection is one in which the placement of parallels does not arise by projection; instead parallels are placed how they need to be in order to satisfy the property that a course of constant bearing is always plotted as a straight line.
### Cylindrical
#### Normal cylindrical
A normal cylindrical projection is any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines.
The mapping of meridians to vertical lines can be visualized by imagining a cylinder whose axis coincides with the Earth's axis of rotation. This cylinder is wrapped around the Earth, projected onto, and then unrolled.
By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ):
- North-south stretching equals east-west stretching (sec φ): The east-west scale matches the north-south scale: conformal cylindrical or Mercator; this distorts areas excessively in high latitudes.
- North-south stretching grows with latitude faster than east-west stretching (sec φ): The cylindric perspective (or central cylindrical) projection; unsuitable because distortion is even worse than in the Mercator projection.
- North-south stretching grows with latitude, but less quickly than the east-west stretching: such as the Miller cylindrical projection (sec φ).
- North-south distances neither stretched nor compressed (1): equirectangular projection or "plate carrée".
- North-south compression equals the cosine of the latitude (the reciprocal of east-west stretching): equal-area cylindrical. This projection has many named specializations differing only in the scaling constant, such as the Gall–Peters or Gall orthographic (undistorted at the 45° parallels), Behrmann (undistorted at the 30° parallels), and Lambert cylindrical equal-area (undistorted at the equator). Since this projection scales north-south distances by the reciprocal of east-west stretching, it preserves area at the expense of shapes.
In the first case (Mercator), the east-west scale always equals the north-south scale. In the second case (central cylindrical), the north-south scale exceeds the east-west scale everywhere away from the equator. Each remaining case has a pair of secant lines—a pair of identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale.
Normal cylindrical projections map the whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width.
#### Transverse cylindrical
A transverse cylindrical projection is a cylindrical projection that in the tangent case uses a great circle along a meridian as contact line for the cylinder.
See: transverse Mercator.
#### Oblique cylindrical
An oblique cylindrical projection aligns with a great circle, but not the equator and not a meridian.
### Pseudocylindrical
Pseudocylindrical projections represent the central meridian as a straight line segment. Other meridians are longer than the central meridian and bow outward, away from the central meridian. Pseudocylindrical projections map parallels as straight lines. Along parallels, each point from the surface is mapped at a distance from the central meridian that is proportional to its difference in longitude from the central meridian. Therefore, meridians are equally spaced along a given parallel. On a pseudocylindrical map, any point further from the equator than some other point has a higher latitude than the other point, preserving north-south relationships. This trait is useful when illustrating phenomena that depend on latitude, such as climate. Examples of pseudocylindrical projections include:
- Sinusoidal, which was the first pseudocylindrical projection developed. On the map, as in reality, the length of each parallel is proportional to the cosine of the latitude. The area of any region is true.
- Collignon projection, which in its most common forms represents each meridian as two straight line segments, one from each pole to the equator.
- Tobler hyperelliptical
- Mollweide
- Goode homolosine
- Eckert IV
- Eckert VI
- Kavrayskiy VII
### Hybrid
The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas.
### Conic
The term "conic projection" is used to refer to any projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex.
When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as secant lines where the cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe. The resulting conic map has low distortion in scale, shape, and area near those standard parallels. Distances along the parallels to the north of both standard parallels or to the south of both standard parallels are stretched; distances along parallels between the standard parallels are compressed. When a single standard parallel is used, distances along all other parallels are stretched.
Conic projections that are commonly used are:
-
### Equidistant
conic, which keeps parallels evenly spaced along the meridians to preserve a constant distance scale along each meridian, typically the same or similar scale as along the standard parallels.
- Albers conic, which adjusts the north-south distance between non-standard parallels to compensate for the east-west stretching or compression, giving an equal-area map.
- Lambert conformal conic, which adjusts the north-south distance between non-standard parallels to equal the east-west stretching, giving a conformal map.
### Pseudoconic
- Bonne, an equal-area projection on which most meridians and parallels appear as curved lines. It has a configurable standard parallel along which there is no distortion.
- Werner cordiform, upon which distances are correct from one pole, as well as along all parallels.
- American polyconic and other projections in the polyconic projection class.
### Azimuthal (projections onto a plane)
Azimuthal projections have the property that directions from a central point are preserved and therefore great circles through the central point are represented by straight lines on the map. These projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function r(d) of the true distance d, independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map.
The mapping of radial lines can be visualized by imagining a plane tangent to the Earth, with the central point as tangent point.
The radial scale is r′(d) and the transverse scale r(d)/(R sin ) where R is the radius of the Earth.
Some azimuthal projections are true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a point of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane:
- The gnomonic projection displays great circles as straight lines. Can be constructed by using a point of perspective at the center of the Earth. r(d) = c tan ; so that even just a hemisphere is already infinite in extent.
- The orthographic projection maps each point on the Earth to the closest point on the plane. Can be constructed from a point of perspective an infinite distance from the tangent point; r(d) = c sin . Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the Moon, approximate this perspective.
- Near-sided perspective projection, which simulates the view from space at a finite distance and therefore shows less than a full hemisphere, such as used in The Blue Marble 2012).
- The General Perspective projection can be constructed by using a point of perspective outside the Earth. Photographs of Earth (such as those from the International Space Station) give this perspective. It is a generalization of near-sided perspective projection, allowing tilt.
- The stereographic projection, which is conformal, can be constructed by using the tangent point's antipode as the point of perspective. r(d) = c tan ; the scale is c/(2R cos ). Can display nearly the entire sphere's surface on a finite circle. The sphere's full surface requires an infinite map.
Other azimuthal projections are not true perspective projections:
- Azimuthal equidistant: r(d) = cd; it is used by amateur radio operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the Earth (; for the case where the tangent point is the North Pole, see the flag of the United Nations)
- Lambert azimuthal equal-area. Distance from the tangent point on the map is proportional to straight-line distance through the Earth: r(d) = c sin
- Logarithmic azimuthal is constructed so that each point's distance from the center of the map is the logarithm of its distance from the tangent point on the Earth. r(d) = c ln ); locations closer than at a distance equal to the constant d0 are not shown.
### Polyhedral
Polyhedral map projections use a polyhedron to subdivide the globe into faces, and then projects each face to the globe. The most well-known polyhedral map projection is Buckminster Fuller's Dymaxion map.
## Projections by preservation of a metric property
### Conformal
Conformal, or orthomorphic, map projections preserve angles locally, implying that they map infinitesimal circles of constant size anywhere on the Earth to infinitesimal circles of varying sizes on the map. In contrast, mappings that are not conformal distort most such small circles into ellipses of distortion. An important consequence of conformality
is that relative angles at each point of the map are correct, and the local scale (although varying throughout the map) in every direction around any one point is constant. These are some conformal projections:
- Mercator: Rhumb lines are represented by straight segments
- Transverse Mercator
- Stereographic: Any circle of a sphere, great and small, maps to a circle or straight line.
- Roussilhe
- Lambert conformal conic
- Peirce quincuncial projection
- Adams hemisphere-in-a-square projection
- Guyou hemisphere-in-a-square projection
### Equal-area
Equal-area maps preserve area measure, generally distorting shapes in order to do so. Equal-area maps are also called equivalent or authalic. These are some projections that preserve area:
- Albers conic
- Boggs eumorphic
- Bonne
- Bottomley
- Collignon
- Cylindrical equal-area
- Eckert II, IV and VI
- Equal Earth
- Gall orthographic (also known as Gall–Peters, or Peters, projection)
- Goode's homolosine
- Hammer
- Hobo–Dyer
- Lambert azimuthal equal-area
- Lambert cylindrical equal-area
- Mollweide
- Sinusoidal
- Strebe 1995
- Snyder's equal-area polyhedral projection, used for geodesic grids.
- Tobler hyperelliptical
- Werner
Equidistant
If the length of the line segment connecting two projected points on the plane is proportional to the geodesic (shortest surface) distance between the two unprojected points on the globe, then we say that distance has been preserved between those two points. An equidistant projection preserves distances from one or two special points to all other points. The special point or points may get stretched into a line or curve segment when projected. In that case, the point on the line or curve segment closest to the point being measured to must be used to measure the distance.
- Plate carrée: Distances from the two poles are preserved, in equatorial aspect.
- Azimuthal equidistant: Distances from the center and edge are preserved.
- Equidistant conic: Distances from the two poles are preserved, in equatorial aspect.
- Werner cordiform Distances from the North Pole are preserved, in equatorial aspect.
- Two-point equidistant: Two "control points" are arbitrarily chosen by the map maker; distances from each control point are preserved.
### Gnomonic
Great circles are displayed as straight lines:
- Gnomonic projection
### Retroazimuthal
Direction to a fixed location B (the bearing at the starting location A of the shortest route) corresponds to the direction on the map from A to B:
- Littrow—the only conformal retroazimuthal projection
- Hammer retroazimuthal—also preserves distance from the central point
- Craig retroazimuthal aka Mecca or Qibla—also has vertical meridians
### Compromise projections
Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things look right. Most of these types of projections distort shape in the polar regions more than at the equator. These are some compromise projections:
- Robinson
- van der Grinten
- Miller cylindrical
- Winkel Tripel
- Buckminster Fuller's Dymaxion
- B. J. S. Cahill's Butterfly Map
- Kavrayskiy VII projection
- Wagner VI projection
- Chamberlin trimetric
- Oronce Finé's cordiform
- AuthaGraph projection
## Suitability of projections for application
The mathematics of projection do not permit any particular map projection to be best for everything. Something will always be distorted. Thus, many projections exist to serve the many uses of maps and their vast range of scales.
Modern national mapping systems typically employ a transverse Mercator or close variant for large-scale maps in order to preserve conformality and low variation in scale over small areas. For smaller-scale maps, such as those spanning continents or the entire world, many projections are in common use according to their fitness for the purpose, such as Winkel tripel, Robinson and Mollweide. Reference maps of the world often appear on compromise projections. Due to distortions inherent in any map of the world, the choice of projection becomes largely one of aesthetics.
Thematic maps normally require an equal area projection so that phenomena per unit area are shown in correct proportion.
However, representing area ratios correctly necessarily distorts shapes more than many maps that are not equal-area.
The Mercator projection, developed for navigational purposes, has often been used in world maps where other projections would have been more appropriate.Robinson, Arthur Howard. (1960). Elements of Cartography, second edition. New York: John Wiley and Sons. p. 82. This problem has long been recognized even outside professional circles. For example, a 1943 New York Times editorial states:
A controversy in the 1980s over the Peters map motivated the American Cartographic Association (now the Cartography and Geographic Information Society) to produce a series of booklets (including Which Map Is Best) designed to educate the public about map projections and distortion in maps. In 1989 and 1990, after some internal debate, seven North American geographic organizations adopted a resolution recommending against using any rectangular projection (including Mercator and Gall–Peters) for reference maps of the world. | https://en.wikipedia.org/wiki/Map_projection |
Project management is the process of supervising the work of a team to achieve all project goals within the given constraints. This information is usually described in project documentation, created at the beginning of the development process. The primary constraints are scope, time and budget. The secondary challenge is to optimize the allocation of necessary inputs and apply them to meet predefined objectives.
The objective of project management is to produce a complete project which complies with the client's objectives. In many cases, the objective of project management is also to shape or reform the client's brief to feasibly address the client's objectives. Once the client's objectives are established, they should influence all decisions made by other people involved in the project– for example, project managers, designers, contractors and subcontractors. Ill-defined or too tightly prescribed project management objectives are detrimental to the decisionmaking process.
A project is a temporary and unique endeavor designed to produce a product, service or result with a defined beginning and end (usually time-constrained, often constrained by funding or staffing) undertaken to meet unique goals and objectives, typically to bring about beneficial change or added value. The temporary nature of projects stands in contrast with business as usual (or operations), which are repetitive, permanent or semi-permanent functional activities to produce products or services. In practice, the management of such distinct production approaches requires the development of distinct technical skills and management strategies.
## History
Until 2001, civil engineering projects were generally managed by creative architects, engineers, and master builders themselves, for example, Vitruvius (first century BC), Christopher Wren (1632–1723), Thomas Telford (1757–1834), and Isambard Kingdom Brunel (1806–1859). In the 1950s, organizations started to apply project-management tools and techniques more systematically to complex engineering projects.
As a discipline, project management developed from several fields of application including civil construction, engineering, and heavy defense activity. Two forefathers of project management are Henry Gantt, called the father of planning and control techniques, who is famous for his use of the Gantt chart as a project management tool (alternatively Harmonogram first proposed by Karol Adamiecki); and Henri Fayol for his creation of the five management functions that form the foundation of the body of knowledge associated with project and program management. Both Gantt and Fayol were students of Frederick Winslow Taylor's theories of scientific management. His work is the forerunner to modern project management tools including work breakdown structure (WBS) and resource allocation.
The 1950s marked the beginning of the modern project management era, where core engineering fields came together to work as one. Project management became recognized as a distinct discipline arising from the management discipline with the engineering model. In the United States, prior to the 1950s, projects were managed on an ad-hoc basis, using mostly Gantt charts and informal techniques and tools. At that time, two mathematical project-scheduling models were developed. The critical path method (CPM) was developed as a joint venture between DuPont Corporation and Remington Rand Corporation for managing plant maintenance projects. The program evaluation and review technique (PERT), was developed by the U.S. Navy Special Projects Office in conjunction with the Lockheed Corporation and Booz Allen Hamilton as part of the Polaris missile submarine program.
PERT and CPM are very similar in their approach but still present some differences. CPM is used for projects that assume deterministic activity times; the times at which each activity will be carried out are known. PERT, on the other hand, allows for stochastic activity times; the times at which each activity will be carried out are uncertain or varied. Because of this core difference, CPM and PERT are used in different contexts. These mathematical techniques quickly spread into many private enterprises.
At the same time, as project-scheduling models were being developed, technology for project cost estimating, cost management and engineering economics was evolving, with pioneering work by Hans Lang and others. In 1956, the American Association of Cost Engineers (now AACE International; the Association for the Advancement of Cost Engineering) was formed by early practitioners of project management and the associated specialties of planning and scheduling, cost estimating, and project control. AACE continued its pioneering work and in 2006, released the first integrated process for portfolio, program, and project management (total cost management framework).
In 1969, the Project Management Institute (PMI) was formed in the USA. PMI publishes the original version of A Guide to the Project Management Body of Knowledge (PMBOK Guide) in 1996 with William Duncan as its primary author, which describes project management practices that are common to "most projects, most of the time."
## Project management types
Project management methods can be applied to any project. It is often tailored to a specific type of project based on project size, nature, industry or sector. For example, the construction industry, which focuses on the delivery of things like buildings, roads, and bridges, has developed its own specialized form of project management that it refers to as construction project management and in which project managers can become trained and certified. The information technology industry has also evolved to develop its own form of project management that is referred to as IT project management and which specializes in the delivery of technical assets and services that are required to pass through various lifecycle phases such as planning, design, development, testing, and deployment. Biotechnology project management focuses on the intricacies of biotechnology research and development. Localization project management includes application of many standard project management practices to translation works even though many consider this type of management to be a very different discipline. For example, project managers have a key role in improving the translation even when they do not speak the language of the translation, because they know the study objectives well to make informed decisions. Similarly, research study management can also apply a project manage approach. There is public project management that covers all public works by the government, which can be carried out by the government agencies or contracted out to contractors. Another classification of project management is based on the hard (physical) or soft (non-physical) type.
Common among all the project management types is that they focus on three important goals: time, quality, and cost. Successful projects are completed on schedule, within budget, and according to previously agreed quality standards i.e. meeting the Iron Triangle or Triple Constraint in order for projects to be considered a success or failure.
For each type of project management, project managers develop and utilize repeatable templates that are specific to the industry they're dealing with. This allows project plans to become very thorough and highly repeatable, with the specific intent to increase quality, lower delivery costs, and lower time to deliver project results.
## Approaches of project management
A 2017 study suggested that the success of any project depends on how well four key aspects are aligned with the contextual dynamics affecting the project, these are referred to as the four P's:
- Plan: The planning and forecasting activities.
- Process: The overall approach to all activities and project governance.
- People: Including dynamics of how they collaborate and communicate.
- Power: Lines of authority, decision-makers, organograms, policies for implementation and the like.
There are a number of approaches to organizing and completing project activities, including phased, lean, iterative, and incremental. There are also several extensions to project planning, for example, based on outcomes (product-based) or activities (process-based).
Regardless of the methodology employed, careful consideration must be given to the overall project objectives, timeline, and cost, as well as the roles and responsibilities of all participants and stakeholders.
### Benefits realization management
Benefits realization management (BRM) enhances normal project management techniques through a focus on outcomes (benefits) of a project rather than products or outputs and then measuring the degree to which that is happening to keep a project on track. This can help to reduce the risk of a completed project being a failure by delivering agreed upon requirements (outputs) i.e. project success but failing to deliver the benefits (outcomes) of those requirements i.e. product success. Note that good requirements management will ensure these benefits are captured as requirements of the project and their achievement monitored throughout the project.
In addition, BRM practices aim to ensure the strategic alignment between project outcomes and business strategies. The effectiveness of these practices is supported by recent research evidencing BRM practices influencing project success from a strategic perspective across different countries and industries. These wider effects are called the strategic impact.
An example of delivering a project to requirements might be agreeing to deliver a computer system that will process staff data and manage payroll, holiday, and staff personnel records in shorter times with reduced errors. Under BRM, the agreement might be to achieve a specified reduction in staff hours and errors required to process and maintain staff data after the system installation when compared without the system.
### Critical path method
Critical path method (CPM) is an algorithm for determining the schedule for project activities. It is the traditional process used for predictive-based project planning. The CPM method evaluates the sequence of activities, the work effort required, the inter-dependencies, and the resulting float time per line sequence to determine the required project duration. Thus, by definition, the critical path is the pathway of tasks on the network diagram that has no extra time available (or very little extra time)."
### Critical chain project management
Critical chain project management (CCPM) is an application of the theory of constraints (TOC) to planning and managing projects and is designed to deal with the uncertainties inherent in managing projects, while taking into consideration the limited availability of resources (physical, human skills, as well as management & support capacity) needed to execute projects.
The goal is to increase the flow of projects in an organization (throughput). Applying the first three of the five focusing steps of TOC, the system constraint for all projects, as well as the resources, are identified. To exploit the constraint, tasks on the critical chain are given priority over all other activities.
### Earned value management
Earned value management (EVM) extends project management with techniques to improve project monitoring. It illustrates project progress towards completion in terms of work and value (cost). Earned Schedule is an extension to the theory and practice of EVM.
### Iterative and incremental project management
In critical studies of project management, it has been noted that phased approaches are not well suited for projects which are large-scale and multi-company, with undefined, ambiguous, or fast-changing requirements, or those with high degrees of risk, dependency, and fast-changing technologies. The cone of uncertainty explains some of this as the planning made on the initial phase of the project suffers from a high degree of uncertainty. This becomes especially true as software development is often the realization of a new or novel product.
These complexities are better handled with a more exploratory or iterative and incremental approach. Several models of iterative and incremental project management have evolved, including agile project management, dynamic systems development method, extreme project management, and Innovation Engineering®.
### Lean project management
Lean project management uses the principles from lean manufacturing to focus on delivering value with less waste and reduced time.
### Project lifecycle
There are five phases to a project lifecycle; known as process groups. Each process group represents a series of inter-related processes to manage the work through a series of distinct steps to be completed. This type of project approach is often referred to as "traditional" or "waterfall". The five process groups are:
1.
### Initiating
1.
### Planning
1.
### Executing
1. Monitoring and Controlling
1.
### Closing
Some industries may use variations of these project stages and rename them to better suit the organization. For example, when working on a brick-and-mortar design and construction, projects will typically progress through stages like pre-planning, conceptual design, schematic design, design development, construction drawings (or contract documents), and construction administration.
While the phased approach works well for small, well-defined projects, it often results in challenge or failure on larger projects, or those that are more complex or have more ambiguities, issues, and risks - see the parodying 'six phases of a big project'.
### Process-based management
The incorporation of process-based management has been driven by the use of maturity models such as the OPM3 and the CMMI (capability maturity model integration; see Image:Capability Maturity Model.jpg
### Project production management
Project production management is the application of operations management to the delivery of capital projects. The Project production management framework is based on a project as a production system view, in which a project transforms inputs (raw materials, information, labor, plant & machinery) into outputs (goods and services).
### Product-based planning
Product-based planning is a structured approach to project management, based on identifying all of the products (project deliverables) that contribute to achieving the project objectives. As such, it defines a successful project as output-oriented rather than activity- or task-oriented. The most common implementation of this approach is PRINCE2.
## Process groups
Traditionally (depending on what project management methodology is being used), project management includes a number of elements: four to five project management process groups, and a control system. Regardless of the methodology or terminology used, the same basic project management processes or stages of development will be used. Major process groups generally include:
- Initiation
- Planning
- Production or execution
-
### Monitoring and controlling
- Closing
In project environments with a significant exploratory element (e.g., research and development), these stages may be supplemented with decision points (go/no go decisions) at which the project's continuation is debated and decided. An example is the Phase–gate model.
Project management relies on a wide variety of meetings to coordinate actions. For instance, there is the kick-off meeting, which broadly involves stakeholders at the project's initiation. Project meetings or project committees enable the project team to define and monitor action plans. Steering committees are used to transition between phases and resolve issues. Project portfolio and program reviews are conducted in organizations running parallel projects. Lessons learned meetings are held to consolidate learnings. All these meetings employ techniques found in meeting science, particularly to define the objective, participant list, and facilitation methods.
Initiating
The initiating processes determine the nature and scope of the project. If this stage is not performed well, it is unlikely that the project will be successful in meeting the business' needs. The key project controls needed here are an understanding of the business environment and making sure that all necessary controls are incorporated into the project. Any deficiencies should be reported and a recommendation should be made to fix them.
The initiating stage should include a plan that encompasses the following areas. These areas can be recorded in a series of documents called Project Initiation documents.
Project Initiation documents are a series of planned documents used to create an order for the duration of the project.
These tend to include:
- project proposal (idea behind project, overall goal, duration)
- project scope (project direction and track)
- product breakdown structure (PBS) (a hierarchy of deliverables/outcomes and components thereof)
- work breakdown structure (WBS) (a hierarchy of the work to be done, down to daily tasks)
- responsibility assignment matrix (RACI - Responsible, Accountable, Consulted, Informed) (roles and responsibilities aligned to deliverables / outcomes)
- tentative project schedule (milestones, important dates, deadlines)
- analysis of business needs and requirements against measurable goals
- review of the current operations
- financial analysis of the costs and benefits, including a budget
- stakeholder analysis, including users and support personnel for the project
- project charter including costs, tasks, deliverables, and schedules
- SWOT analysis: strengths, weaknesses, opportunities, and threats to the business
Planning
After the initiation stage, the project is planned to an appropriate level of detail (see an example of a flowchart). The main purpose is to plan time, cost, and resources adequately to estimate the work needed and to effectively manage risk during project execution. As with the Initiation process group, a failure to adequately plan greatly reduces the project's chances of successfully accomplishing its goals.
Project planning generally consists of
- determining the project management methodology to follow (e.g. whether the plan will be defined wholly upfront, iteratively, or in rolling waves);
- developing the scope statement;
- selecting the planning team;
- identifying deliverables and creating the product and work breakdown structures;
- identifying the activities needed to complete those deliverables and networking the activities in their logical sequence;
- estimating the resource requirements for the activities;
- estimating time and cost for activities;
- developing the schedule;
- developing the budget;
- risk planning;
- developing quality assurance measures;
- gaining formal approval to begin work.
Additional processes, such as planning for communications and for scope management, identifying roles and responsibilities, determining what to purchase for the project, and holding a kick-off meeting are also generally advisable.
For new product development projects, conceptual design of the operation of the final product may be performed concurrent with the project planning activities and may help to inform the planning team when identifying deliverables and planning activities.
Executing
While executing we must know what are the planned terms that need to be executed.
The execution/implementation phase ensures that the project management plan's deliverables are executed accordingly. This phase involves proper allocation, coordination, and management of human resources and any other resources such as materials and budgets. The output of this phase is the project deliverables.
### Project documentation
Documenting everything within a project is key to being successful. To maintain budget, scope, effectiveness and pace a project must have physical documents pertaining to each specific task. With correct documentation, it is easy to see whether or not a project's requirement has been met. To go along with that, documentation provides information regarding what has already been completed for that project. Documentation throughout a project provides a paper trail for anyone who needs to go back and reference the work in the past.
In most cases, documentation is the most successful way to monitor and control the specific phases of a project. With the correct documentation, a project's success can be tracked and observed as the project goes on. If performed correctly, documentation can be the backbone of a project's success
Monitoring and controlling
Monitoring and controlling consist of those processes performed to observe project execution so that potential problems can be identified in a timely manner and corrective action can be taken, when necessary, to control the execution of the project. The key benefit is that project performance is observed and measured regularly to identify variances from the project management plan.
Monitoring and controlling include:
- Measuring the ongoing project activities ('where we are');
- Monitoring the project variables (cost, effort, scope, etc.) against the project management plan and the project performance baseline (where we should be);
- Identifying corrective actions to address issues and risks properly (How can we get on track again);
- Influencing the factors that could circumvent integrated change control so only approved changes are implemented.
Two main mechanisms support monitoring and controlling in projects. On the one hand, contracts offer a set of rules and incentives often supported by potential penalties and sanctions. On the other hand, scholars in business and management have paid attention to the role of integrators (also called project barons) to achieve a project's objectives. In turn, recent research in project management has questioned the type of interplay between contracts and integrators. Some have argued that these two monitoring mechanisms operate as substitutes as one type of organization would decrease the advantages of using the other one.
In multi-phase projects, the monitoring and control process also provides feedback between project phases, to implement corrective or preventive actions to bring the project into compliance with the project management plan.
Project maintenance is an ongoing process, and it includes:
- Continuing support of end-users
- Correction of errors
- Updates to the product over time
In this stage, auditors should pay attention to how effectively and quickly user problems are resolved.
Over the course of any construction project, the work scope may change. Change is a normal and expected part of the construction process. Changes can be the result of necessary design modifications, differing site conditions, material availability, contractor-requested changes, value engineering, and impacts from third parties, to name a few. Beyond executing the change in the field, the change normally needs to be documented to show what was actually constructed. This is referred to as change management. Hence, the owner usually requires a final record to show all changes or, more specifically, any change that modifies the tangible portions of the finished work. The record is made on the contract documents – usually, but not necessarily limited to, the design drawings. The end product of this effort is what the industry terms as-built drawings, or more simply, "as built." The requirement for providing them is a norm in construction contracts. Construction document management is a highly important task undertaken with the aid of an online or desktop software system or maintained through physical documentation. The increasing legality pertaining to the construction industry's maintenance of correct documentation has caused an increase in the need for document management systems.
When changes are introduced to the project, the viability of the project has to be re-assessed. It is important not to lose sight of the initial goals and targets of the projects. When the changes accumulate, the forecasted result may not justify the original proposed investment in the project. Successful project management identifies these components, and tracks and monitors progress, so as to stay within time and budget frames already outlined at the commencement of the project. Exact methods were suggested to identify the most informative monitoring points along the project life-cycle regarding its progress and expected duration.
Closing
Closing includes the formal acceptance of the project and the ending thereof. Administrative activities include the archiving of the files and documenting lessons learned.
This phase consists of:
- Contract closure: Complete and settle each contract (including the resolution of any open items) and close each contract applicable to the project or project phase.
- Project close: Finalize all activities across all of the process groups to formally close the project or a project phase
Also included in this phase is the post implementation review. This is a vital phase of the project for the project team to learn from experiences and apply to future projects. Normally a post implementation review consists of looking at things that went well and analyzing things that went badly on the project to come up with lessons learned.
### Project control and project control systems
Project control (also known as Cost Engineering)
should be established as an independent function in project management. It implements verification and controlling functions during the processing of a project to reinforce the defined performance and formal goals. The tasks of project control are also:
- the creation of infrastructure for the supply of the right information and its update
- the establishment of a way to communicate disparities in project parameters
- the development of project information technology based on an intranet or the determination of a project key performance indicator system (KPI)
- divergence analyses and generation of proposals for potential project regulations
- the establishment of methods to accomplish an appropriate project structure, project workflow organization, project control, and governance
- creation of transparency among the project parameters
Fulfillment and implementation of these tasks can be achieved by applying specific methods and instruments of project control. The following methods of project control can be applied:
- investment analysis
- cost–benefit analysis
- value benefit analysis
- expert surveys
- simulation calculations
- risk-profile analysis
- surcharge calculations
- milestone trend analysis
- cost trend analysis
- target/actual comparison
Project control is that element of a project that keeps it on track, on time, and within budget. Project control begins early in the project with planning and ends late in the project with post-implementation review, having a thorough involvement of each step in the process. Projects may be audited or reviewed while the project is in progress. Formal audits are generally risk or compliance-based and management will direct the objectives of the audit. An examination may include a comparison of approved project management processes with how the project is actually being managed. Each project should be assessed for the appropriate level of control needed: too much control is too time-consuming, too little control is very risky. If project control is not implemented correctly, the cost to the business should be clarified in terms of errors and fixes.
Control systems are needed for cost, risk, quality, communication, time, change, procurement, and human resources. In addition, auditors should consider how important the projects are to the financial statements, how reliant the stakeholders are on controls, and how many controls exist. Auditors should review the development process and procedures for how they are implemented. The process of development and the quality of the final product may also be assessed if needed or requested. A business may want the auditing firm to be involved throughout the process to catch problems earlier on so that they can be fixed more easily. An auditor can serve as a controls consultant as part of the development team or as an independent auditor as part of an audit.
Businesses sometimes use formal systems development processes. This help assure systems are developed successfully. A formal process is more effective in creating strong controls, and auditors should review this process to confirm that it is well designed and is followed in practice. A good formal systems development plan outlines:
- A strategy to align development with the organization's broader objectives
- Standards for new systems
- Project management policies for timing and budgeting
- Procedures describing the process
- Evaluation of quality of change
## Characteristics of projects
There are five important characteristics of a project:
(i) It should always have specific start and end dates.
(ii) They are performed and completed by a group of people.
(iii) The output is the delivery of a unique product or service.
(iv) They are temporary in nature.
(v) It is progressively elaborated.
Examples are: designing a new car or writing a book.
### Project complexity
Complexity and its nature play an important role in the area of project management. Despite having a number of debates on this subject matter, studies suggest a lack of definition and reasonable understanding of complexity in relation to the management of complex projects.
Project complexity is the property of a project which makes it difficult to understand, foresee, and keep under control its overall behavior, even when given reasonably complete information about the project system.
The identification of complex projects is specifically important to multi-project engineering environments.
As it is considered that project complexity and project performance are closely related, it is important to define and measure the complexity of the project for project management to be effective.
Complexity can be:
- Structural complexity (also known as detail complexity, or complicatedness), i.e. consisting of many varied interrelated parts. It is typically expressed in terms of size, variety, and interdependence of project components, and described by technological and organizational factors.
- Dynamic complexity refers to phenomena, characteristics, and manifestations such as ambiguity, uncertainty, propagation, emergence, and chaos.
Based on the Cynefin framework, complex projects can be classified as:
- Simple (or clear, obvious, known) projects, systems, or contexts. These are characterized by known knowns, stability, and clear cause-and-effect relationships. They can be solved with standard operating procedures and best practices.
- Complicated: characterized by known unknowns. A complicated system is the sum of its parts. In principle, it can be deconstructed into smaller simpler components. While difficult, complicated problems are theoretically solvable with additional resources, specialized expertise, analytical, reductionist, simplification, decomposition techniques, scenario planning, and following good practices.
- Complex are characterized by unknown unknowns, and emergence. Patterns could be uncovered, but they are not obvious. A complex system can be described by Euclid's statement that the whole is more than the sum of its parts.
- Really complex projects, a.k.a. very complex, or chaotic: characterized by unknowables. No patterns are discernible in really complex projects. Causes and effects are unclear even in retrospect. Paraphrasing Aristotle, a really complex system is different from the sum of its parts.
By applying the discovery in measuring work complexity described in Requisite Organization and Stratified Systems Theory, Elliott Jaques classifies projects and project work (stages, tasks) into seven basic levels of project complexity based on such criteria as time-span of discretion and complexity of a project's output:
- Level 1 Project – improve the direct output of an activity (quantity, quality, time) within a business process with a targeted completion time up to 3 months.
- Level 2 Project – develop and improve compliance to a business process with a targeted completion time of 3 months to 1 year.
- Level 3 Project – develop, change, and improve a business process with a targeted completion time of 1 to 2 years.
- Level 4 Project – develop, change, and improve a functional system with a targeted completion time of 2 to 5 years.
- Level 5 Project – develop, change, and improve a group of functional systems/business functions with a targeted completion time of 5 to 10 years.
- Level 6 Project – develop, change, and improve a whole single value chain of a company with targeted completion time from 10 to 20 years.
- Level 7 Project – develop, change, and improve multiple value chains of a company with target completion time from 20 to 50 years.
Benefits from measuring Project Complexity are to improve project people feasibility by matching the level of a project's complexity with an effective targeted completion time, with the respective capability level of the project manager and of the project members.
### Positive, appropriate (requisite), and negative complexity
Similarly with the Law of requisite variety and The law of requisite complexity, project complexity is sometimes required in order for the project to reach its objectives, and sometimes it has beneficial outcomes. Based on the effects of complexity, Stefan Morcov proposed its classification as Positive, Appropriate, or Negative.
- Positive complexity is the complexity that adds value to the project, and whose contribution to project success outweighs the associated negative consequences.
- Appropriate (or requisite) complexity is the complexity that is needed for the project to reach its objectives, or whose contribution to project success balances the negative effects, or the cost of mitigation outweighs negative manifestations.
- Negative complexity is the complexity that hinders project success.
## Project managers
A project manager is a professional in the field of project management. Project managers are in charge of the people in a project. People are the key to any successful project. Without the correct people in the right place and at the right time a project cannot be successful. Project managers can have the responsibility of the planning, execution, controlling, and closing of any project typically relating to the construction industry, engineering, architecture, computing, and telecommunications. Many other fields of production engineering, design engineering, and heavy industrial have project managers.
A project manager needs to understand the order of execution of a project to schedule the project correctly as well as the time necessary to accomplish each individual task within the project. A project manager is the person accountable for accomplishing the stated project objectives on behalf of the client. Project Managers tend to have multiple years' experience in their field. A project manager is required to know the project in and out while supervising the workers along with the project. Typically in most construction, engineering, architecture, and industrial projects, a project manager has another manager working alongside of them who is typically responsible for the execution of task on a daily basis. This position in some cases is known as a superintendent. A superintendent and project manager work hand in hand in completing daily project tasks. Key project management responsibilities include creating clear and attainable project objectives, building the project requirements, and managing the triple constraint (now including more constraints and calling it competing constraints) for projects, which is cost, time, quality and scope for the first three but about three additional ones in current project management. A typical project is composed of a team of workers who work under the project manager to complete the assignment within the time and budget targets. A project manager normally reports directly to someone of higher stature on the completion and success of the project.
A project manager is often a client representative and has to determine and implement the exact needs of the client, based on knowledge of the firm they are representing. The ability to adapt to the various internal procedures of the contracting party, and to form close links with the nominated representatives, is essential in ensuring that the key issues of cost, time, quality and above all, client satisfaction, can be realized.
A complete project manager, a term first coined by Robert J. Graham in his simulation, has been expanded upon by Randall L. Englund and Alfonso Bucero. They describe a complete project manager as a person who embraces multiple disciplines, such as leadership, influence, negotiations, politics, change and conflict management, and humor. These are all "soft" people skills that enable project leaders to be more effective and achieve optimized, consistent results.
## Multilevel success framework and criteria - project success vs. project performance
There is a tendency to confuse the project success with project management success. They are two different things. "Project success" has 2 perspectives:
- the perspective of the process, i.e. delivering efficient outputs; typically called project management performance or project efficiency.
- the perspective of the result, i.e. delivering beneficial outcomes; typically called project performance (sometimes just project success).
Project management success criteria are different from project success criteria. The project management is said to be successful if the given project is completed within the agreed upon time, met the agreed upon scope and within the agreed upon budget. Subsequent to the triple constraints, multiple constraints have been considered to ensure project success. However, the triple or multiple constraints indicate only the efficiency measures of the project, which are indeed the project management success criteria during the project lifecycle.
The priori criteria leave out the more important after-completion results of the project which comprise four levels i.e. the output (product) success, outcome (benefits) success and impact (strategic) success during the product lifecycle. These posterior success criteria indicate the effectiveness measures of the project product, service or result, after the project completion and handover. This overarching multilevel success framework of projects, programs and portfolios has been developed by Paul Bannerman in 2008. In other words, a project is said to be successful, when it succeeds in achieving the expected business case which needs to be clearly identified and defined during the project inception and selection before starting the development phase. This multilevel success framework conforms to the theory of project as a transformation depicted as the input-process / activity-output-outcome-impact in order to generate whatever value intended. Emanuel Camilleri in 2011 classifies all the critical success and failure factors into groups and matches each of them with the multilevel success criteria in order to deliver business value.
An example of a performance indicator used in relation to project management is the "backlog of commissioned projects" or "project backlog".
## Risk management
The United States Department of Defense states that "Cost, Schedule, Performance, and Risk" are the four elements through which Department of Defense acquisition professionals make trade-offs and track program status. There are also international standards. Risk management applies proactive identification (see tools) of future problems and understanding of their consequences allowing predictive decisions about projects. ERM system plays a role in overall risk management.
## Work breakdown structure and other breakdown structures
The work breakdown structure (WBS) is a tree structure that shows a subdivision of the activities required to achieve an objective – for example a portfolio, program, project, and contract. The WBS may be hardware-, product-, service-, or process-oriented (see an example in a NASA reporting structure (2001)). Beside WBS for project scope management, there are organizational breakdown structure (chart), cost breakdown structure and risk breakdown structure.
A WBS can be developed by starting with the end objective and successively subdividing it into manageable components in terms of size, duration, and responsibility (e.g., systems, subsystems, components, tasks, sub-tasks, and work packages), which include all steps necessary to achieve the objective.
The work breakdown structure provides a common framework for the natural development of the overall planning and control of a contract and is the basis for dividing work into definable increments from which the statement of work can be developed and technical, schedule, cost, and labor hour reporting can be established.
The work breakdown structure can be displayed in two forms, as a table with subdivision of tasks or as an organizational chart whose lowest nodes are referred to as "work packages".
It is an essential element in assessing the quality of a plan, and an initial element used during the planning of the project. For example, a WBS is used when the project is scheduled, so that the use of work packages can be recorded and tracked.
Similarly to work breakdown structure (WBS), other decomposition techniques and tools are: organization breakdown structure (OBS), product breakdown structure (PBS), cost breakdown structure (CBS), risk breakdown structure (RBS), and resource breakdown structure (ResBS).
## International standards
There are several project management standards, including:
- The ISO standards ISO 9000, a family of standards for quality management systems, and the ISO 10006:2003, for Quality management systems and guidelines for quality management in projects.
- ISO 21500:2012 – Guidance on project management. This is the first International Standard related to project management published by ISO. Other standards in the 21500 family include 21503:2017 Guidance on programme management; 21504:2015 Guidance on portfolio management; 21505:2017 Guidance on governance; 21506:2018 Vocabulary; 21508:2018 Earned value management in project and programme management; and 21511:2018 Work breakdown structures for project and programme management.
- ISO 21502:2020 Project, programme and portfolio management — Guidance on project management
- ISO 21503:2022 Project, programme and portfolio management — Guidance on programme management
- ISO 21504:2015 Project, programme and portfolio management – Guidance on portfolio management
- ISO 21505:2017 Project, programme and portfolio management - Guidance on governance
- ISO 31000:2009 – Risk management.
- ISO/IEC/IEEE 16326:2009 – Systems and Software Engineering—Life Cycle Processes—Project Management
- Individual Competence Baseline (ICB) from the International Project Management Association (IPMA).
- Capability Maturity Model (CMM) from the Software Engineering Institute.
- GAPPS, Global Alliance for Project Performance Standards – an open source standard describing COMPETENCIES for project and program managers.
- HERMES method, Swiss general project management method, selected for use in Luxembourg and international organizations.
- The logical framework approach (LFA), which is popular in international development organizations.
- PMBOK Guide from the Project Management Institute (PMI).
- PRINCE2 from AXELOS.
- PM2: The Project Management methodology developed by the [European Commission].
- Procedures for Project Formulation and Management (PPFM) by the Indian Ministry of Defence
- Team Software Process (TSP) from the Software Engineering Institute.
- Total Cost Management Framework, AACE International's Methodology for Integrated Portfolio, Program and Project Management.
- V-Model, an original systems development method.
## Program management and project networks
Some projects, either identical or different, can be managed as program management.
Programs are collections of projects that support a common objective and set of goals. While individual projects have clearly defined and specific scope and timeline, a program's objectives and duration are defined with a lower level of granularity.
Besides programs and portfolios, additional structures that combine their different characteristics are: project networks, mega-projects, or mega-programs.
A project network is a temporary project formed of several different distinct evolving phases, crossing organizational lines.
Mega-projects and mega-programs are defined as exceptional in terms of size, cost, public and political attention, and competencies required.
## Project portfolio management
An increasing number of organizations are using what is referred to as project portfolio management (PPM) as a means of selecting the right projects and then using project management techniques as the means for delivering the outcomes in the form of benefits to the performing public, private or not-for-profit organization.
Portfolios are collections of similar projects. Portfolio management supports efficiencies of scale, increasing success rates, and reducing project risks, by applying similar standardized techniques to all projects in the portfolio, by a group of project management professionals sharing common tools and knowledge.
Organizations often create project management offices as an organizational structure to support project portfolio management in a structured way.
Thus, PPM is usually performed by a dedicated team of managers organized within an enterprise project management office (PMO), usually based within the organization, and headed by a PMO director or chief project officer. In cases where strategic initiatives of an organization form the bulk of the PPM, the head of the PPM is sometimes titled as the chief initiative officer.
## Project management software
Project management software is software used to help plan, organize, and manage resource pools, develop resource estimates and implement plans. Depending on the sophistication of the software, functionality may include estimation and planning, scheduling, cost control and budget management, resource allocation, collaboration software, communication, decision-making, workflow, risk, quality, documentation, and/or administration systems.Kendrick, Tom (2013). The Project Management Tool Kit: 100 Tips and Techniques for Getting the Job Done Right, Third Edition. AMACOM Books.
## Virtual project management
Virtual program management (VPM) is management of a project done by a virtual team, though it rarely may refer to a project implementing a virtual environment It is noted that managing a virtual project is fundamentally different from managing traditional projects, combining concerns of remote work and global collaboration (culture, time zones, language). | https://en.wikipedia.org/wiki/Project_management |
EEPROM or E2PROM (electrically erasable programmable read-only memory) is a type of non-volatile memory. It is used in computers, usually integrated in microcontrollers such as smart cards and remote keyless systems, or as a separate chip device, to store relatively small amounts of data by allowing individual bytes to be erased and reprogrammed.
EEPROMs are organized as arrays of floating-gate transistors. EEPROMs can be programmed and erased in-circuit, by applying special programming signals. Originally, EEPROMs were limited to single-byte operations, which made them slower, but modern EEPROMs allow multi-byte page operations. An EEPROM has a limited life for erasing and reprogramming, reaching a million operations in modern EEPROMs. In an EEPROM that is frequently reprogrammed, the life of the EEPROM is an important design consideration.
Flash memory is a type of EEPROM designed for high speed and high density, at the expense of large erase blocks (typically 512 bytes or larger) and limited number of write cycles (often 10,000). There is no clear boundary dividing the two, but the term "EEPROM" is generally used to describe non-volatile memory with small erase blocks (as small as one byte) and a long lifetime (typically 1,000,000 cycles). Many past microcontrollers included both (flash memory for the firmware and a small EEPROM for parameters), though the trend with modern microcontrollers is to emulate EEPROM using flash.
As of 2020, flash memory costs much less than byte-programmable EEPROM and is the dominant memory type wherever a system requires a significant amount of non-volatile solid-state storage. EEPROMs, however, are still used on applications that only require small amounts of storage, like in serial presence detect.
## History
### Early attempts
In the early 1970s, some studies, inventions, and development for electrically re-programmable non-volatile memories were performed by various companies and organizations.
In 1971, early research was presented at the 3rd Conference on Solid State Devices, Tokyo in Japan by Yasuo Tarui, Yutaka Hayashi, and Kiyoko Nagai at Electrotechnical Laboratory; a Japanese national research institute.
They fabricated an electrically re-programmable non-volatile memory in 1972, and continued this study for more than 10 years. However this early memory depended on capacitors to work, which modern EEPROM lacks.
In 1972 IBM patented an electrically re-programmable non-volatile memory invention. Later that year, an avalanche injection type MOS was patented by Fujio Masuoka, the inventor of flash memory, at Toshiba and IBM patented another later that year.
In 1974, NEC patented a electrically erasable carrier injection device. The next year, NEC applied for the trademark "EEPROM®" with the Japan Patent Office. The trademark was granted in 1978.
The theoretical basis of these devices is avalanche hot-carrier injection. In general, programmable memories, including EPROM, of early 1970s had reliability and endurance problems such as the data retention periods and the number of erase/write cycles.
Most of the major semiconductor manufactures, such as
Toshiba,
Sanyo (later, ON Semiconductor),
IBM,
Intel,
NEC (later, Renesas Electronics),
Philips (later, NXP Semiconductors),
Siemens (later, Infineon Technologies),
Honeywell (later, Atmel),
Texas Instruments,
studied, invented, and manufactured some electrically re-programmable non-volatile devices until 1977.
### Modern EEPROM
The first EEPROM that used Fowler-Nordheim tunnelling to erase data was invented by Bernward and patented by Siemens in 1974. In February 1977, Israeli-American Eliyahou Harari at Hughes Aircraft Company patented in the US a modern EEPROM technology, based on Fowler-Nordheim tunnelling through a thin silicon dioxide layer between the floating-gate and the wafer.
Hughes went on to produce this new EEPROM devices.
In May 1977, some important research result was disclosed by Fairchild and Siemens. They used SONOS (polysilicon-oxynitride-nitride-oxide-silicon) structure with thickness of silicon dioxide less than 30 Å, and SIMOS (stacked-gate injection MOS) structure, respectively, for using Fowler-Nordheim tunnelling hot-carrier injection.
Around 1976 to 1978, Intel's team, including George Perlegos, made some inventions to improve this tunneling E2PROM technology.
In 1978, they developed a 16K (2K word × 8) bit Intel 2816 chip with a thin silicon dioxide layer, which was less than 200 Å.
In 1980, this structure was publicly introduced as FLOTOX; floating gate tunnel oxide.
The FLOTOX structure improved reliability of erase/write cycles per byte up to 10,000 times.
But this device required additional 2022V VPP bias voltage supply for byte erase, except for 5V read operations.
In 1981, Perlegos and 2 other members left Intel to form Seeq Technology,
which used on-device charge pumps to supply the high voltages necessary for programming E2PROMs.
In 1984, Perlogos left Seeq Technology to found Atmel, then Seeq Technology was acquired by Atmel.
Electrically alterable read-only memory (EAROM) is a type of EEPROM that can be modified one or a few bits at a time. Writing is a very slow process and again needs higher voltage (usually around 12 V) than is used for read access. EAROMs are intended for applications that require infrequent and only partial rewriting.
## Theoretical basis of FLOTOX structure
As is described in former section, old EEPROMs are based on avalanche breakdown-based hot-carrier injection with high reverse breakdown voltage. But FLOTOX theoretical basis is Fowler–Nordheim tunneling hot-carrier injection through a thin silicon dioxide layer between the floating gate and the wafer. In other words, it uses a tunnel junction.
Theoretical basis of the physical phenomenon itself is the same as today's flash memory. But each FLOTOX structure is in conjunction with another read-control transistor because the floating gate itself is just programming and erasing one data bit.
Intel's FLOTOX device structure improved EEPROM reliability, in other words, the endurance of the write and erase cycles, and the data retention period. A material of study for single-event effect about FLOTOX is available.
Today, an academic explanation of the FLOTOX device structure can be found in several sources.
## Today's EEPROM structure
Nowadays, EEPROM is used for embedded microcontrollers as well as standard EEPROM products.
EEPROM still requires a 2-transistor structure per bit to erase a dedicated byte in the memory, while flash memory has 1 transistor per bit to erase a region of the memory.
## Security protections
Because EEPROM technology is used for some security gadgets, such as credit cards, SIM cards, key-less entry, etc., some devices have security protection mechanisms, such as copy-protection.
## Electrical interface
EEPROM devices use a serial or parallel interface for data input/output.
### Serial bus devices
The common serial interfaces are SPI, I²C, Microwire, UNI/O, and 1-Wire. These use from 1 to 4 device pins and allow devices to use packages with 8 pins or less.
A typical EEPROM serial protocol consists of three phases: OP-code phase, address phase and data phase. The OP-code is usually the first 8 bits input to the serial input pin of the EEPROM device (or with most I²C devices, is implicit); followed by 8 to 24 bits of addressing, depending on the depth of the device, then the read or write data.
Each EEPROM device typically has its own set of OP-code instructions mapped to different functions. Common operations on SPI EEPROM devices are:
- Write enable (WRENAL)
- Write disable (WRDI)
- Read status register (RDSR)
- Write status register (WRSR)
- Read data (READ)
- Write data (WRITE)
Other operations supported by some EEPROM devices are:
- Program
- Sector erase
- Chip erase commands
### Parallel bus devices
Parallel EEPROM devices typically have an 8-bit data bus and an address bus wide enough to cover the complete memory. Most devices have chip select and write protect pins. Some microcontrollers also have integrated parallel EEPROM.
Operation of a parallel EEPROM is simple and fast when compared to serial EEPROM, but these devices are larger due to the higher pin count (28 pins or more) and have been decreasing in popularity in favor of serial EEPROM or flash.
### Other devices
EEPROM memory is used to enable features in other types of products that are not strictly memory products. Products such as real-time clocks, digital potentiometers, digital temperature sensors, among others, may have small amounts of EEPROM to store calibration information or other data that needs to be available in the event of power loss.
It was also used on video game cartridges to save game progress and configurations, before the usage of external and internal flash memories.
## Failure modes
There are two limitations of stored information: endurance and data retention.
During rewrites, the gate oxide in the floating-gate transistors gradually accumulates trapped electrons. The electric field of the trapped electrons adds to the electrons in the floating gate, lowering the window between threshold voltages for zeros vs ones. After sufficient number of rewrite cycles, the difference becomes too small to be recognizable, the cell is stuck in programmed state, and endurance failure occurs. The manufacturers usually specify the maximum number of rewrites being 1 million or more.
During storage, the electrons injected into the floating gate may drift through the insulator, especially at increased temperature, and cause charge loss, reverting the cell into erased state. The manufacturers usually guarantee data retention of 10 years or more.
## Related types
Flash memory is a later form of EEPROM. In the industry, there is a convention to reserve the term EEPROM to byte-wise erasable memories compared to block-wise erasable flash memories. EEPROM occupies more die area than flash memory for the same capacity, because each cell usually needs a read, a write, and an erase transistor, while flash memory erase circuits are shared by large blocks of cells (often 512×8).
Newer non-volatile memory technologies such as FeRAM and MRAM are slowly replacing EEPROMs in some applications, but are expected to remain a small fraction of the EEPROM market for the foreseeable future.
### Comparison with EPROM and EEPROM/flash
The difference between EPROM and EEPROM lies in the way that the memory programs and erases. EEPROM can be programmed and erased electrically using field electron emission (more commonly known in the industry as "Fowler–Nordheim tunneling").
EPROMs can't be erased electrically and are programmed by hot-carrier injection onto the floating gate. Erase is by an ultraviolet light source, although in practice many EPROMs are encapsulated in plastic that is opaque to UV light, making them "one-time programmable".
Most NOR flash memory is a hybrid style—programming is through hot-carrier injection and erase is through Fowler–Nordheim tunneling.
Type Inject electrons onto gate(mostly interpreted as bit=0) Duration Remove electrons from gate(mostly interpreted as bit=1) Duration/mode EEPROM field electron emission 0.1—5 ms, bytewise field electron emission 0.1—5 ms, blockwise NOR flash memory hot-carrier injection 0.01—1 ms field electron emission 0.01—1 ms, blockwise EPROM hot-carrier injection 3—50 ms, bytewise ultraviolet light <400nm 5—30 minutes, whole chip | https://en.wikipedia.org/wiki/EEPROM |
A GLR parser (generalized left-to-right rightmost derivation parser) is an extension of an LR parser algorithm to handle non-deterministic and ambiguous grammars. The theoretical foundation was provided in a 1974 paper by Bernard Lang (along with other general context-free parsers such as GLL). It describes a systematic way to produce such algorithms, and provides uniform results regarding correctness proofs, complexity with respect to grammar classes, and optimization techniques. The first actual implementation of GLR was described in a 1984 paper by Masaru Tomita, it has also been referred to as a "parallel parser". Tomita presented five stages in his original work, though in practice it is the second stage that is recognized as the GLR parser.
Though the algorithm has evolved since its original forms, the principles have remained intact. As shown by an earlier publication, Lang was primarily interested in more easily used and more flexible parsers for extensible programming languages. Tomita's goal was to parse natural language text thoroughly and efficiently. Standard LR parsers cannot accommodate the nondeterministic and ambiguous nature of natural language, and the GLR algorithm can.
## Algorithm
Briefly, the GLR algorithm works in a manner similar to the LR parser algorithm, except that, given a particular grammar, a GLR parser will process all possible interpretations of a given input in a breadth-first search. On the front-end, a GLR parser generator converts an input grammar into parser tables, in a manner similar to an LR generator. However, where LR parse tables allow for only one state transition (given a state and an input token), GLR parse tables allow for multiple transitions. In effect, GLR allows for shift/reduce and reduce/reduce conflicts.
When a conflicting transition is encountered, the parse stack is forked into two or more parallel parse stacks, where the state corresponding to each possible transition is at the top. Then, the next input token is read and used to determine the next transition(s) for each of the "top" states – and further forking can occur. If any given top state and input token do not result in at least one transition, then that "path" through the parse tables is invalid and can be discarded.
A crucial optimization known as a graph-structured stack allows sharing of common prefixes and suffixes of these stacks, which constrains the overall search space and memory usage required to parse input text. The complex structures that arise from this improvement make the search graph a directed acyclic graph (with additional restrictions on the "depths" of various nodes), rather than a tree.
## Advantages
Recognition using the GLR algorithm has the same worst-case time complexity as the CYK algorithm and Earley algorithm: O(n3). However, GLR carries two additional advantages:
- The time required to run the algorithm is proportional to the degree of nondeterminism in the grammar: on deterministic grammars the GLR algorithm runs in O(n) time (this is not true of the Earley and CYK algorithms, but the original Earley algorithms can be modified to ensure it)
- The GLR algorithm is "online" – that is, it consumes the input tokens in a specific order and performs as much work as possible after consuming each token (also true for Earley).
In practice, the grammars of most programming languages are deterministic or "nearly deterministic", meaning that any nondeterminism is usually resolved within a small (though possibly unbounded) number of tokens. Compared to other algorithms capable of handling the full class of context-free grammars (such as Earley parser or CYK algorithm), the GLR algorithm gives better performance on these "nearly deterministic" grammars, because only a single stack will be active during the majority of the parsing process.
GLR can be combined with the LALR(1) algorithm, in a hybrid parser, allowing still higher performance. | https://en.wikipedia.org/wiki/GLR_parser |
In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted true and false) which is intended to represent the two truth values of logic and Boolean algebra. It is named after George Boole, who first defined an algebraic system of logic in the mid 19th century. The Boolean data type is primarily associated with conditional statements, which allow different actions by changing control flow depending on whether a programmer-specified Boolean condition evaluates to true or false. It is a special case of a more general logical data type—logic does not always need to be Boolean (see probabilistic logic).
## Generalities
In programming languages with a built-in Boolean data type, such as Pascal, C, Python or
### Java
, the comparison operators such as `>` and `≠` are usually defined to return a Boolean value. Conditional and iterative commands may be defined to test Boolean-valued expressions.
Languages with no explicit Boolean data type, like C90 and Lisp, may still represent truth values by some other data type. Common Lisp uses an empty list for false, and any other value for true. The C programming language uses an integer type, where relational expressions like `i > j` and logical expressions connected by `&&` and `||` are defined to have value 1 if true and 0 if false, whereas the test parts of `if`, `while`, `for`, etc., treat any non-zero value as true. Indeed, a Boolean variable may be regarded (and implemented) as a numerical variable with one binary digit (bit), or as a bit string of length one, which can store only two values. The implementation of Booleans in computers are most likely represented as a full word, rather than a bit; this is usually due to the ways computers transfer blocks of information.
Most programming languages, even those with no explicit Boolean type, have support for Boolean algebraic operations such as conjunction (`AND`, `&`, `*`), disjunction (`OR`, `|`, `+`), equivalence (`EQV`, `=`, `==`), exclusive or/non-equivalence (`XOR`, `NEQV`, `^`, `!=`, `¬`), and negation (`NOT`, `~`, `!`, `¬`).
In some languages, like Ruby, Smalltalk, and Alice the true and false values belong to separate classes, e.g., `True` and `False`, respectively, so there is no one Boolean type.
In
### SQL
, which uses a three-valued logic for explicit comparisons because of its special treatment of Nulls, the Boolean data type (introduced in SQL:1999) is also defined to include more than two truth values, so that SQL Booleans can store all logical values resulting from the evaluation of predicates in SQL. A column of Boolean type can be restricted to just `TRUE` and `FALSE` though.
## Language-specific implementations
### ALGOL and the built-in BOOLEAN type
One of the earliest programming languages to provide an explicit `BOOLEAN` data type is ALGOL 60 (1960) with values true and false and logical operators denoted by symbols '
$$
\wedge
$$
' (and), '
$$
\vee
$$
' (or), '
$$
\supset
$$
' (implies), '
$$
\equiv
$$
' (equivalence), and '
$$
\neg
$$
' (not). Due to input device and character set limits on many computers of the time, however, most compilers used alternative representations for many of the operators, such as `AND` or `'AND'`.
This approach with `BOOLEAN` as a built-in (either primitive or otherwise predefined) data type was adopted by many later programming languages, such as Simula 67 (1967), ALGOL 68 (1970), Pascal (1970), Ada (1980), Java (1995), and C# (2000), among others.
### C, C++, D, Objective-C, AWK
Initial implementations of the language C (1972) provided no Boolean type, and to this day Boolean values are commonly represented by integers (`int`s) in C programs. The comparison operators (`>`, `==`, etc.) are defined to return a signed integer (`int`) result, either 0 (for false) or 1 (for true). Logical operators (`&&`, `||`, `!`, etc.) and condition-testing statements (`if`, `while`) assume that zero (and hence a NULL pointer or a null string terminator '\0' also) is false and all other values are true.
After enumerated types (`enum`s) were added to the American National Standards Institute version of C, ANSI C (1989), many C programmers got used to defining their own Boolean types as such, for readability reasons. However, enumerated types are equivalent to integers according to the language standards; so the effective identity between Booleans and integers is still valid for C programs.
Standard C (since C99) provides a Boolean type, called `_Bool`. Since C23, the Boolean is now a core data type called `bool`, with values `true` and `false` (previously these was provided by macros from the header `stdbool.h`, which is now obsolete). The language guarantees that any two true values will compare equal (which was impossible to achieve before the introduction of the type). Boolean values still behave as integers, can be stored in integer variables, and used anywhere integers would be valid, including in indexing, arithmetic, parsing, and formatting. This approach (Boolean values are just integers) has been retained in all later versions of C. Note, that this does not mean that any integer value can be stored in a Boolean variable.
C++ has had the Boolean data type `bool` since C++98, but with automatic conversions from scalar and pointer values that are very similar to those of C. This approach was adopted also by many later languages, especially by some scripting languages such as AWK.
The D programming language has a proper Boolean data type `bool`. The `bool` type is a byte-sized type that can only hold the value true or false.
The only operators that can accept operands of type bool are: &, |, ^, &=, |=, ^=, !, &&, || and ?:.
A `bool` value can be implicitly converted to any integral type, with false becoming 0 and true becoming 1.
The numeric literals 0 and 1 can be implicitly converted to the bool values false and true, respectively. Casting an expression to `bool` means testing for 0 or !=0 for arithmetic types, and null or !=null for pointers or references.
Objective-C also has a separate Boolean data type `BOOL`, with possible values being `YES` or `NO`, equivalents of true and false respectively. Also, in Objective-C compilers that support C99, C's `_Bool` type can be used, since Objective-C is a superset of C.
### Forth
Forth (programming language) has no Boolean type, it uses regular integers: value 0 (all bits low) represents false, and -1 (all bits high) represents true. This allows the language to define only one set of logical operators, instead of one for mathematical calculations and one for conditions.
### Fortran
The first version of FORTRAN (1957) and its successor FORTRAN II (1958) have no logical values or operations; even the conditional `IF` statement takes an arithmetic expression and branches to one of three locations according to its sign; see arithmetic IF. FORTRAN IV (1962), however, follows the ALGOL 60 example by providing a Boolean data type (`LOGICAL`), truth literals (`.TRUE.` and `.FALSE.`), logical `IF` statement, Boolean-valued numeric comparison operators (`.EQ.`, `.GT.`, etc.), and logical operators (`.NOT.`, `.AND.`, `.OR.`, `.EQV.`, and `.NEQV.`). In `FORMAT` statements, a specific format descriptor ('`L`') is provided for the parsing or formatting of logical values. Fortran 90 added alternative comparison operators `<`, `<=`, `==`, `/=`, `>`, and `>=`.
Java
In Java, the value of the `boolean` data type can only be either `true` or `false`.
### Lisp and Scheme
The language Lisp (1958) never had a built-in Boolean data type. Instead, conditional constructs like `cond` assume that the logical value false is represented by the empty list `()`, which is defined to be the same as the special atom `nil` or `NIL`; whereas any other s-expression is interpreted as true. For convenience, most modern dialects of Lisp predefine the atom `t` to have value `t`, so that `t` can be used as a mnemonic notation for true.
This approach (any value can be used as a Boolean value) was retained in most Lisp dialects (Common Lisp, Scheme, Emacs Lisp), and similar models were adopted by many scripting languages, even ones having a distinct Boolean type or Boolean values; although which values are interpreted as false and which are true vary from language to language. In Scheme, for example, the false value is an atom distinct from the empty list, so the latter is interpreted as true. Common Lisp, on the other hand, also provides the dedicated `boolean` type, derived as a specialization of the symbol.
### Pascal, Ada, and Haskell
The language Pascal (1970) popularized the concept of programmer-defined enumerated types, previously available with different nomenclature in COBOL, FACT and JOVIAL. A built-in `Boolean` data type was then provided as a predefined enumerated type with values `FALSE` and `TRUE`. By definition, all comparisons, logical operations, and conditional statements applied to and/or yielded `Boolean` values. Otherwise, the `Boolean` type had all the facilities which were available for enumerated types in general, such as ordering and use as indices. In contrast, converting between `Boolean`s and integers (or any other types) still required explicit tests or function calls, as in ALGOL 60. This approach (Boolean is an enumerated type) was adopted by most later languages which had enumerated types, such as Modula, Ada, and Haskell.
### Perl and Lua
Perl has no Boolean data type. Instead, any value can behave as Boolean in Boolean context (condition of `if` or `while` statement, argument of `&&` or `||`, etc.). The number `0`, the strings `"0"` and `""`, the empty list `()`, and the special value `undef` evaluate to false. All else evaluates to true.
Lua has a Boolean data type, but non-Boolean values can also behave as Booleans. The non-value `nil` evaluates to false, whereas every other data type value evaluates to true. This includes the empty string `""` and the number `0`, which are very often considered `false` in other languages.
### PL/I
PL/I has no Boolean data type. Instead, comparison operators generate BIT(1) values; '0'B represents false and '1'B represents true. The operands of, e.g., `&`, `|`, `¬`, are converted to bit strings and the operations are performed on each bit. The element-expression of an `IF` statement is true if any bit is 1.
### Python and Ruby
Python, from version 2.3 forward, has a `bool` type which is a subclass of `int`, the standard integer type. It has two possible values: `True` and `False`, which are special versions of 1 and 0 respectively and behave as such in arithmetic contexts. Also, a numeric value of zero (integer or fractional), the null value (`None`), the empty string, and empty containers (lists, sets, etc.) are considered Boolean false; all other values are considered Boolean true by default. Classes can define how their instances are treated in a Boolean context through the special method `__nonzero__` (Python 2) or `__bool__` (Python 3). For containers, `__len__` (the special method for determining the length of containers) is used if the explicit Boolean conversion method is not defined.
In Ruby, in contrast, only `nil` (Ruby's null value) and a special `false` object are false; all else (including the integer 0 and empty arrays) is true.
### Rexx
Rexx has no Boolean data type. Instead, comparison operators generate 0 or 1; 0 represents false and 1 represents true. The operands of, e.g., `&`, `|`, `¬`, must be 0 or 1.
SQL
Booleans appear in SQL when a condition is needed, such as clause, in form of predicate which is produced by using operators such as comparison operators, operator, etc. However, apart from and , these operators can also yield a third state, called , when comparison with is made.
The SQL92 standard introduced and operators which evaluate a predicate, which predated the introduction of Boolean type in SQL:1999.
The SQL:1999 standard introduced a data type as an optional feature (T031). When restricted by a constraint, a SQL behaves like Booleans in other languages, which can store only and values. However, if it is nullable, which is the default like all other SQL data types, it can have the special null value also. Although the SQL standard defines three literals for the type – and it also says that the and "may be used interchangeably to mean exactly the same thing".ISO/IEC 9075-2:2011 §4.5 This has caused some controversy because the identification subjects to the equality comparison rules for NULL. More precisely is not but . As of 2012 few major SQL systems implement the T031 feature. Firebird and PostgreSQL are notable exceptions, although PostgreSQL implements no literal; can be used instead.
The treatment of Boolean values differs between SQL systems.
For example, in Microsoft SQL Server, Boolean value is not supported at all, neither as a standalone data type nor representable as an integer. It shows the error message "An expression of non-Boolean type specified in a context where a condition is expected" if a column is directly used in the clause, e.g. , while a statement such as yields a syntax error. The data type, which can only store integers 0 and 1 apart from , is commonly used as a workaround to store Boolean values, but workarounds need to be used such as to convert between the integer and Boolean expression.
Microsoft Access, which uses the Access Database Engine (ACE/JET), also does not have a Boolean data type.
Similar to MS SQL Server, it uses a data type. In Access it is known as a Yes/No data type which can have two values; Yes (True) or No (False).
The BIT data type in Access can also be represented numerically:
True is −1 and False is 0.
This differs from MS SQL Server in two ways, even though both are Microsoft products:
1. Access represents as −1, while it is 1 in SQL Server
1. Access does not support the Null tri-state, supported by SQL Server
PostgreSQL has a distinct type as in the standard, which allows predicates to be stored directly into a column, and allows using a column directly as a predicate in a clause.
In MySQL, is treated as an alias of ;
is the same as integer 1 and is the same as integer 0.
Any non-zero integer is true in conditions.
### Tableau
Tableau Software has a BOOLEAN data type. The literal of a Boolean value is `True` or `False`.
The Tableau `INT()` function converts a Boolean to a number, returning 1 for True and 0 for False.
### Tcl
Tcl has no separate Boolean type. Like in C, the integers 0 (false) and 1 (true—in fact any nonzero integer) are used.
Examples of coding:
The above will show since the expression evaluates to 1.
The above will render an error, as variable cannot be evaluated as 0 or 1.
## Truthy | https://en.wikipedia.org/wiki/Boolean_data_type |
C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date".
It became quite popular after ranking #1 in the Top 10
## Algorithm
s in Data Mining pre-eminent paper published by Springer LNCS in 2008.
Algorithm
C4.5 builds decision trees from a set of training data in the same way as ID3, using the concept of information entropy. The training data is a set
$$
S = {s_1, s_2, ...}
$$
of already classified samples. Each sample
$$
s_i
$$
consists of a p-dimensional vector
$$
(x_{1,i}, x_{2,i}, ...,x_{p,i})
$$
, where the
$$
x_j
$$
represent attribute values or features of the sample, as well as the class in which
$$
s_i
$$
falls.
At each node of the tree, C4.5 chooses the attribute of the data that most effectively splits its set of samples into subsets enriched in one class or the other. The splitting criterion is the normalized information gain (difference in entropy). The attribute with the highest normalized information gain is chosen to make the decision. The C4.5 algorithm then recurses on the partitioned sublists.
This algorithm has a few base cases.
- All the samples in the list belong to the same class. When this happens, it simply creates a leaf node for the decision tree saying to choose that class.
- None of the features provide any information gain. In this case, C4.5 creates a decision node higher up the tree using the expected value of the class.
- Instance of previously unseen class encountered. Again, C4.5 creates a decision node higher up the tree using the expected value.
### Pseudocode
In pseudocode, the general algorithm for building decision trees is:
1. Check for the above base cases.
1. For each attribute a, find the normalized information gain ratio from splitting on a.
1. Let a_best be the attribute with the highest normalized information gain.
1. Create a decision node that splits on a_best.
1. Recurse on the sublists obtained by splitting on a_best, and add those nodes as children of node.
## Implementations
J48 is an open source Java implementation of the C4.5 algorithm in the Weka data mining tool.
## Improvements from ID3 algorithm
C4.5 made a number of improvements to ID3. Some of these are:
- Handling both continuous and discrete attributes - In order to handle continuous attributes, C4.5 creates a threshold and then splits the list into those whose attribute value is above the threshold and those that are less than or equal to it.
- Handling training data with missing attribute values - C4.5 allows attribute values to be marked as ? for missing. Missing attribute values are simply not used in gain and entropy calculations.
- Handling attributes with differing costs.
- Pruning trees after creation - C4.5 goes back through the tree once it's been created and attempts to remove branches that do not help by replacing them with leaf nodes.
## Improvements in C5.0/See5 algorithm
Quinlan went on to create C5.0 and See5 (C5.0 for Unix/Linux, See5 for Windows) which he markets commercially. C5.0 offers a number of improvements on C4.5. Some of these are:M. Kuhn and K. Johnson, Applied Predictive Modeling, Springer 2013
- Speed - C5.0 is significantly faster than C4.5 (several orders of magnitude)
- Memory usage - C5.0 is more memory efficient than C4.5
- Smaller decision trees - C5.0 gets similar results to C4.5 with considerably smaller decision trees.
- Support for boosting - Boosting improves the trees and gives them more accuracy.
- Weighting - C5.0 allows you to weight different cases and misclassification types.
- Winnowing - a C5.0 option automatically winnows the attributes to remove those that may be unhelpful.
Source for a single-threaded Linux version of C5.0 is available under the GNU General Public License (GPL). | https://en.wikipedia.org/wiki/C4.5_algorithm |
Surfing is a surface water sport in which an individual, a surfer (or two in tandem surfing), uses a board to ride on the forward section, or face, of a moving wave of water, which usually carries the surfer towards the shore. Waves suitable for surfing are primarily found on ocean shores, but can also be found as standing waves in the open ocean, in lakes, in rivers in the form of a tidal bore, or wave pools.
The term surfing refers to a person riding a wave using a board, regardless of the stance. There are several types of boards. The Moche of
### Peru
would often surf on reed craft, while the native peoples of the Pacific surfed waves on alaia, paipo, and other such water craft. Ancient cultures often surfed on their belly and knees, while the modern-day definition of surfing most often refers to a surfer riding a wave standing on a surfboard; this is also referred to as stand-up surfing.
Another prominent form of surfing is body boarding, where a surfer rides the wave on a bodyboard, either lying on their belly, drop knee (one foot and one knee on the board), or sometimes even standing up on a body board. Other types of surfing include knee boarding, surf matting (riding inflatable mats) and using foils. Body surfing, in which the wave is caught and ridden using the surfer's own body rather than a board, is very common and is considered by some surfers to be the purest form of surfing. The closest form of body surfing using a board is a handboard which normally has one strap over it to fit on one hand. Surfers who body board, body surf, or handboard feel more drag as they move through the water than stand up surfers do. This holds body surfers into a more turbulent part of the wave (often completely submerged by whitewater). In contrast, surfers who instead ride a hydrofoil feel substantially less drag and may ride unbroken waves in the open ocean.
Three major subdivisions within stand-up surfing are stand-up paddling, long boarding and short boarding with several major differences including the board design and length, the riding style and the kind of wave that is ridden.
In tow-in surfing (most often, but not exclusively, associated with big wave surfing), a motorized water vehicle such as a personal watercraft, tows the surfer into the wave front, helping the surfer match a large wave's speed, which is generally a higher speed than a self-propelled surfer can produce. Surfing-related sports such as paddle boarding and sea kayaking that are self-propelled by hand paddles do not require waves, and other derivative sports such as kite surfing and windsurfing rely primarily on wind for power, yet all of these platforms may also be used to ride waves. Recently with the use of V-drive boats, wakesurfing, in which one surfs on the wake of a boat, has emerged. As of 2023, the Guinness Book of World Records recognized a wave ride by Sebastian Steudtner in Nazaré, Portugal, as the largest wave ever surfed.
During the winter season in the northern hemisphere, the North Shore of Oahu, the third-largest island of Hawaii, is known for having some of the best waves in the world. Surfers from around the world flock to breaks like Backdoor, Waimea Bay, and Pipeline. However, there are still many popular surf spots around the world: Teahupo'o, located off the coast of Tahiti; Mavericks,
### California
, United States; Cloudbreak, Tavarua Island, Fiji; Superbank, Gold Coast, Australia.
In 2016, surfing was added by the International Olympic Committee (IOC) as an Olympic sport to begin at the 2020 Summer Olympics in Japan.
The first gold medalists of the Tokyo 2020 surfing men and women's competitions were, respectively, the Brazilian Ítalo Ferreira and the American from Hawaii, Carissa Moore.
## Origins and history
Peru
About three to five thousand years ago, cultures in ancient Peru fished in kayak-like watercraft (mochica) made of reeds that the fishermen surfed back to shore. The Moche culture used the caballito de totora (little horse of totora), with archaeological evidence showing its use around 200 CE. An early description of the Inca surfing in Callao was documented by Jesuit missionary José de Acosta in his 1590 publication Historia natural y moral de las Indias, writing:
### Polynesia
In Polynesian culture, surfing was an important activity. Modern surfing as we know it today is thought to have originated in Hawaii. The history of surfing dates to in Polynesia, where Polynesians began to make their way to the Hawaiian Islands from Tahiti and the Marquesas Islands. They brought many of their customs with them including playing in the surf on Paipo (belly/body) boards. It was in Hawaii that the art of standing and surfing upright on boards was invented.
Various European explorers witnessed surfing in Polynesia. Surfing may have been observed by British explorers at Tahiti in 1767. Samuel Wallis and the crew members of were the first Britons to visit the island in June of that year. Another candidate is the botanist Joseph Banks who was part of the first voyage of James Cook on , arriving on Tahiti on 10 April 1769. Lieutenant James King was the first person to write about the art of surfing on Hawaii, when he was completing the journals of Captain James Cook (upon Cook's death in 1779).
In Herman Melville's 1849 novel Mardi, based on his experiences in Polynesia earlier that decade, the narrator describes the "Rare Sport at Ohonoo" (title of chap. 90): “For this sport, a surf-board is indispensable: some five feet in length; the width of a man's body; convex on both sides; highly polished; and rounded at the ends. It is held in high estimation; invariably oiled after use; and hung up conspicuously in the dwelling of the owner.” When Mark Twain visited Hawaii in 1866 he wrote, "In one place, we came upon a large company of naked natives of both sexes and all ages, amusing themselves with the national pastime of surf-bathing."
References to surf riding on planks and single canoe hulls are also verified for pre-contact Samoa, where surfing was called fa'ase'e or se'egalu (see Augustin Krämer, The Samoa Islands), and Tonga, far pre-dating the practice of surfing by Hawaiians and eastern Polynesians by over a thousand years.
### West Africa
West Africans (e.g., Ghana, Ivory Coast, Liberia, Senegal) and western Central Africans (e.g., Cameroon) independently developed the skill of surfing. Amid the 1640s CE, Michael Hemmersam provided an account of surfing in the Gold Coast: “the parents ‘tie their children to boards and throw them into the water.’” In 1679 CE, Barbot provided an account of surfing among Elmina children in Ghana: “children at Elmina learned “to swim, on bits of boards, or small bundles of rushes, fasten’d under their stomachs, which is a good diversion to the spectators.” James Alexander provided an account of surfing in Accra, Ghana in 1834 CE: “From the beach, meanwhile, might be seen boys swimming into the sea, with light boards under their stomachs. They waited for a surf; and came rolling like a cloud on top of it. But I was told that sharks occasionally dart in behind the rocks and ‘yam’ them.” Thomas Hutchinson provided an account of surfing in southern Cameroon in 1861: “Fishermen rode small dugouts ‘no more than six feet in length, fourteen to sixteen inches in width, and from four to six inches in depth.’”
California
In July 1885, three teenage Hawaiian princes took a break from their boarding school, St. Matthew's Hall in San Mateo, and came to cool off in Santa Cruz, California. There, David Kawānanakoa, Edward Keliʻiahonui and Jonah Kūhiō Kalanianaʻole surfed the mouth of the San Lorenzo River on custom-shaped redwood boards, according to surf historians Kim Stoner and Geoff Dunn. In 1890, the pioneer in agricultural education John Wrightson reputedly became the first British surfer when instructed by two Hawaiian students at his college.
George Freeth (1883–1919), of English and Native Hawaiian descent, is generally credited as the person who had done more than anyone else to renew interest in surfing at Waikiki in the early twentieth century after the sport had declined in popularity in Hawaii during the latter half of the nineteenth century.
In 1907, the eclectic interests of land developer Abbot Kinney (founder of Venice of America, now Venice, California) helped bring Freeth to California. Freeth had sought the help of the Hawaii Promotion Committee (HPC) in Honolulu to sponsor him on a trip to California to give surfing exhibitions. The HPC arranged through their contacts in Los Angeles to secure a contract for Freeth to perform at Venice of America in July, 1907. Later that year, land baron Henry E. Huntington brought surfing to Redondo Beach. Looking for a way to entice visitors to his own budding resort community south of Venice where he had heavily invested in real estate, he hired Freeth as a lifeguard and to give surfing exhibitions in front of the Hotel Redondo. Another native Hawaiian, Duke Kahanamoku, spread surfing to both the U.S. and Australia, riding the waves after displaying the swimming prowess that won him Olympic gold medals in 1912 and 1920.
Mary Ann Hawkins, inspired by Duke Kahanamoku's surfing during the late 1920s, developed a lifelong passion for surfing. In 1935, her family relocated to Santa Monica, providing her with opportunities to further immerse herself in surfing and paddleboarding. On September 12, 1936, Hawkins achieved a historic milestone by winning California’s first women’s paddleboard race at the Santa Monica Breakwater. She continued to dominate the sport, winning numerous competitions, including the women’s half-mile paddleboard race and the Venice Breakwater event in 1938, both held on the same day.
Hawkins was also a pioneer in tandem surfing, a discipline that highlights synchronized surfing between two individuals on a single board. She gained further recognition in 1939 when she performed exhibition paddleboarding and tandem surfing displays at various Southern California beaches, inspiring a new generation of women surfers.
In January 1939, Hawkins was appointed head of the women’s auxiliary group of the Santa Monica Paddle Club and rose to vice president by January 1940. Her surfing peers frequently lauded her achievements, with "Whitey" Harrison describing her as "the best tandem rider." Throughout her career, Hawkins exemplified grace and athleticism, leaving an indelible mark on the history of women’s surfing and paddleboarding.
In 1975, a professional tour started. That year Margo Oberg became the first female professional surfer.
## Surf waves
Swell is generated when the wind blows consistently over a large space of open water, called the wind's fetch. The size of a swell is determined by the strength of the wind, and the length of its fetch and duration. Because of these factors, the surf tends to be larger and more prevalent on coastlines exposed to large expanses of ocean traversed by intense low pressure systems.
Local wind conditions affect wave quality since the surface of a wave can become choppy in blustery conditions. Ideal conditions include a light to moderate "offshore" wind, because it blows into the front of the wave, making it a "barrel" or "tube" wave. Waves are left-handed and right-handed depending upon the breaking formation of the wave.
Waves are generally recognized by the surfaces over which they break. For example, there are beach breaks, reef breaks and point breaks.
The most important influence on wave shape is the topography of the seabed directly behind and immediately beneath the breaking wave. Each break is different since each location's underwater topography is unique. At beach breaks, sandbanks change shape from week to week. Surf forecasting is aided by advances in information technology. Mathematical modeling graphically depicts the size and direction of swells around the globe.
Swell regularity varies across the globe and throughout the year. During winter, heavy swells are generated in the mid-latitudes, when the North and South polar fronts shift toward the Equator. The predominantly Westerly winds generate swells that advance Eastward, so waves tend to be largest on West coasts during winter months. However, an endless train of mid-latitude cyclones cause the isobars to become undulated, redirecting swells at regular intervals toward the tropics.
East coasts also receive heavy winter swells when low-pressure cells form in the sub-tropics, where slow moving highs inhibit their movement. These lows produce a shorter fetch than polar fronts, however, they can still generate heavy swells since their slower movement increases the duration of a particular wind direction. The variables of fetch and duration both influence how long wind acts over a wave as it travels since a wave reaching the end of a fetch behaves as if the wind died.
During summer, heavy swells are generated when cyclones form in the tropics. Tropical cyclones form over warm seas, so their occurrence is influenced by El Niño and La Niña cycles. Their movements are unpredictable.
Surf travel and some surf camps offer surfers access to remote, tropical locations, where tradewinds ensure offshore conditions. Since winter swells are generated by mid-latitude cyclones, their regularity coincides with the passage of these lows. Swells arrive in pulses, each lasting for a couple of days, with a few days between each swell.
The availability of free model data from the NOAA has allowed the creation of several surf forecasting websites.
### Tube shape and speed
Tube shape is defined by length to width ratio. A perfectly cylindrical vortex has a ratio of 1:1. Other forms include:
- Square: <1:1
- Round: 1–2:1
- Almond: >2:1
Peel or peeling off as a descriptive term for the quality of a break has been defined as "a fast, clean, evenly falling curl line, perfect for surfing, and usually found at pointbreaks."
Tube speed is the rate of advance of the break along the length of the wave, and is the speed at which the surfer must move along the wave to keep up with the advance of the tube.
Tube speed can be described using the peel angle and wave celerity. Peel angle is the angle between the wave front and the horizontal projection of the point of break over time, which in a regular break is most easily represented by the line of white water left after the break. A break that closes out, or breaks all at once along its length, leaves white water parallel to the wave front, and has a peel angle of 0°. This is unsurfable as it would require infinite speed to progress along the face fast enough to keep up with the break. A break which advances along the wave face more slowly will leave a line of new white water at an angle to the line of the wave face.
$$
V_s = \frac {c}{sin \alpha}
$$
Where:
$$
V_s =
$$
velocity of surfer along the wave face
$$
c =
$$
wave celerity (velocity in direction of propagation)
$$
\alpha =
$$
peel angle
In most cases a peel angle less than 25° is too fast to surf.
- Fast: 30°
- Medium: 45°
- Slow: 60°
+
### Wave intensity
table Fast Medium Slow Square The Cobra Teahupoo Shark Island Round Speedies, Gnaraloo Banzai Pipeline Almond Lagundri Bay, Superbank Jeffreys Bay, Bells Beach Angourie Point
Wave intensity
The type of break depends on shoaling rate. Breaking waves can be classified as four basic types: spilling (ξb<0.4), plunging (0.4<ξb<2), collapsing (ξb>2) and surging (ξb>2), and which type occurs depends on the slope of the bottom.
Waves suitable for surfing break as spilling or plunging types, and when they also have a suitable peel angle, their value for surfing is enhanced. Other factors such as wave height and period, and wind strength and direction can also influence steepness and intensity of the break, but the major influence on the type and shape of breaking waves is determined by the slope of the seabed before the break. The breaker type index and Iribarren number allow classification of breaker type as a function of wave steepness and seabed slope.
### Artificial reefs
The value of good surf in attracting surf tourism has prompted the construction of artificial reefs and sand bars. Artificial surfing reefs can be built with durable sandbags or concrete, and resemble a submerged breakwater. These artificial reefs not only provide a surfing location, but also dissipate wave energy and shelter the coastline from erosion. Ships such as Seli 1 that have accidentally stranded on sandy bottoms, can create sandbanks that give rise to good waves.
An artificial reef known as Chevron Reef was constructed in El Segundo, California in hopes of creating a new surfing area. However, the reef failed to produce any quality waves and was removed in 2008. In Kovalam, South West India, an artificial reef has successfully provided the local community with a quality lefthander, stabilized coastal soil erosion, and provided good habitat for marine life. ASR Ltd., a New Zealand-based company, constructed the Kovalam reef and is working on another reef in Boscombe, England.
### Artificial waves
Even with artificial reefs in place, a tourist's vacation time may coincide with a "flat spell", when no waves are available. Completely artificial wave pools aim to solve that problem by controlling all the elements that go into creating perfect surf, however there are only a handful of wave pools that can simulate good surfing waves, owing primarily to construction and operation costs and potential liability. Most wave pools generate waves that are too small and lack the power necessary to surf. The Seagaia Ocean Dome, located in Miyazaki, Japan, was an example of a surfable wave pool. Able to generate waves with up to faces, the specialized pump held water in 20 vertical tanks positioned along the back edge of the pool. This allowed the waves to be directed as they approach the artificial sea floor. Lefts, Rights, and A-frames could be directed from this pump design providing for rippable surf and barrel rides. The Ocean Dome cost about $2 billion to build and was expensive to maintain. The Ocean Dome was closed in 2007. In England, construction is nearing completion on the Wave, situated near Bristol, which will enable people unable to get to the coast to enjoy the waves in a controlled environment, set in the heart of nature.
There are two main types of artificial waves that exist today. One being artificial or stationary waves which simulate a moving, breaking wave by pumping a layer of water against a smooth structure mimicking the shape of a breaking wave. Because of the velocity of the rushing water, the wave and the surfer can remain stationary while the water rushes by under the surfboard. Artificial waves of this kind provide the opportunity to try surfing and learn its basics in a moderately small and controlled environment near or far from locations with natural surf.
## Maneuvers
Standup surfing begins when the surfer paddles toward shore in an attempt to match the speed of the wave (the same applies whether the surfer is standup paddling, bodysurfing, boogie-boarding or using some other type of watercraft, such as a waveski or kayak). Once the wave begins to carry the surfer forward, the surfer stands up and proceeds to ride the wave. The basic idea is to position the surfboard so it is just ahead of the breaking part (whitewash) of the wave, in the so-called 'pocket'. It is difficult for beginners to catch the wave at all.
Surfers' skills are tested by their ability to control their board in difficult conditions, riding challenging waves, and executing maneuvers such as strong turns and cutbacks (turning board back to the breaking wave) and carving (a series of strong back-to-back maneuvers). More advanced skills include the floater (riding on top of the breaking curl of the wave), and off the lip (banking off crest of the breaking wave). A newer addition to surfing is the progression of the air, whereby a surfer propels off the wave entirely up into the air and then successfully lands the board back on the wave.
The tube ride is considered to be the ultimate maneuver in surfing. As a wave breaks, if the conditions are ideal, the wave will break in an orderly line from the middle to the shoulder, enabling the experienced surfer to position themselves inside the wave as it is breaking. This is known as a tube ride. Viewed from the shore, the tube rider may disappear from view as the wave breaks over the rider's head. The longer the surfer remains in the tube, the more successful the ride. This is referred to as getting tubed, barrelled, shacked or pitted. Some of the world's best-known waves for tube riding include Pipeline on the North Shore of Oahu, Teahupoo in Tahiti and G-Land in Java. Other names for the tube include "the barrel", and "the pit".
Hanging ten and hanging five are moves usually specific to longboarding. Hanging Ten refers to having both feet on the front end of the board with all of the surfer's toes off the edge, also known as nose-riding. Hanging Five is having just one foot near the front, with five toes off the edge.
Cutback: Generating speed down the line and then turning back to reverse direction.
Snap: Quickly turning along the face or top of the wave, almost as if snapping the board back towards the wave. Typically done on steeper waves.
Blowtail: Pushing the tail of the board out of the back of the wave so that the fins leave the water.
Floater: Suspending the board atop the wave. Very popular on small waves.
Top-Turn: Turn off the top of the wave. Sometimes used to generate speed and sometimes to shoot spray.
Bottom Turn: A turn at the bottom or mid-face of the wave, this maneuver is used to set up other maneuvers such as the top turn, cutback and even aerials.
Airs/Aerials: These maneuvers have been becoming more and more prevalent in the sport in both competition and free surfing. An air is when the surfer can achieve enough speed and approach a certain type of section of a wave that is supposed to act as a ramp and launch the surfer above the lip line of the wave, “catching air”, and landing either in the transition of the wave or the whitewash when hitting a close-out section.
Airs can either be straight airs or rotational airs. Straight airs have minimal rotation if any, but definitely no more rotation than 90 degrees. Rotational airs require a rotation of 90 degrees or more depending on the level of the surfer.
Types of rotations:
- 180 degrees – called an air reverse, this is when the surfer spins enough to land backwards, then reverts to their original positional with the help of the fins. This rotation can either be done frontside or backside and can spin right or left.
- 360 degrees – this is a full rotation air or “full rotor” where the surfer lands where they started or more, as long as they do not land backwards. When this is achieved front side on a wave spinning the opposite of an air reverse is called an alley-oop.
- 540 degrees – the surfer does a full rotation plus another 180 degrees and can be inverted or spinning straight, few surfers have been able to land this air.
- Backflip – usually done with a double grab, this hard to land air is made for elite-level surfers.
- Rodeo flip – usually done backside, it is a backflip with a 180 rotation, and is actually easier than a straight backflip.
- Grabs – a surfer can help land an aerial maneuver by grabbing the surfboard, keeping them attached to the board and keeping the board under their feet. Common types of grabs include:
- Indy – a grab on the surfers inside rail going frontside, outside rail going backside with their backhand.
- Slob – a grab on the surfers inside rail going frontside, outside rail going backside with their front hand.
- Lien – A grab on the surfers outside rail frontside, inside rail going backside with their front hand.
- Stalefish – A grab on the surfers outside rail frontside, inside rail backside with their backhand.
- Double grab – A grab on the surfers inside and outside rail, the inside rail with the backhand and the outside rail with the front hand.
### Terms
The Glossary of surfing includes some of the extensive vocabulary used to describe various aspects of the sport of surfing as described in literature on the subject. In some cases terms have spread to a wider cultural use. These terms were originally coined by people who were directly involved in the sport of surfing.
## Learning
Many popular surfing destinations have surf schools and surf camps that offer lessons. Surf camps for beginners and intermediates are multi-day lessons that focus on surfing fundamentals. They are designed to take new surfers and help them become proficient riders. All-inclusive surf camps offer overnight accommodations, meals, lessons and surfboards. Most surf lessons begin with instruction and a safety briefing on land, followed by instructors helping students into waves on longboards or "softboards". The softboard is considered the ideal surfboard for learning, due to the fact it is safer, and has more paddling speed and stability than shorter boards. Funboards are also a popular shape for beginners as they combine the volume and stability of the longboard with the manageable size of a smaller surfboard. New and inexperienced surfers typically learn to catch waves on softboards around the funboard size. Due to the softness of the surfboard the chance of getting injured is substantially minimized.
It is possible to learn to surf without an instructor, but the process is usually safer and quicker with a surf instructor. Typical surfing instruction is best-performed one-on-one, but can also be done in a group setting. Post-COVID, there's been a shift towards online and land-based surf coaching and training. Online surf coaching is allowing surfers to learn at their own pace and convenience from anywhere. Land-based training, such as skateboard simulations, offers a way to practice maneuvers repeatedly, refining techniques with the guidance of professional coaches either in person or remotely using video analysis apps. The most popular surf locations offer perfect surfing conditions for beginners, as well as challenging breaks for advanced students. The ideal conditions for learning would be small waves that crumble and break softly, as opposed to the steep, fast-peeling waves desired by more experienced surfers. When available, a sandy seabed is generally safer.
Surfing can be broken into several skills: paddling strength, positioning to catch the wave, timing, and balance. Paddling out requires strength, but also the mastery of techniques to break through oncoming waves (duck diving, eskimo roll also known as turtle roll). Take-off positioning requires experience at predicting the wave set and where it will break. The surfer must pop up quickly as soon as the wave starts pushing the board forward. Preferred positioning on the wave is determined by experience at reading wave features including where the wave is breaking. Balance plays a crucial role in standing on a surfboard. Thus, balance training exercises are good preparation. Practicing with a balance board, longboard (skateboard), surfskate or swing board helps novices master the art of surfing. However, it's important to note that these land-based training methods have faced criticism within the surf coaching community. Concerns include the potential for developing poor surfing style and habits, such as excessive wiggling, due to training on flat surfaces which do not accurately mimic the dynamic nature of ocean waves.
To address these limitations, training in a skate bowl is recommended. Skate bowls can offer a more realistic simulation of the centrifugal forces experienced while surfing. This type of training helps in developing better control and style by replicating the curved, wave-like shapes and motions surfers encounter in the water. Integrating skate bowl training can provide a more comprehensive preparation for the surfing experience, balancing the benefits of basic balance training with the nuances of wave dynamics.
The repetitive cycle of paddling, popping up, and balancing requires stamina and physical strength. Having a proper warm-up routine can help prevent injuries.
## Equipment
Surfing can be done on various equipment, including surfboards, longboards, stand up paddle boards (SUPs), bodyboards, wave skis, skimboards, kneeboards, surf mats and macca's trays. Surfboards were originally made of solid wood and were large and heavy (often up to long and having a mass of ). Lighter balsa wood surfboards (first made in the late 1940s and early 1950s) were a significant improvement, not only in portability, but also in increasing maneuverability.
Most modern surfboards are made of fiberglass foam (PU), with one or more wooden strips or "stringers", fiberglass cloth, and polyester resin (PE). An emerging board material is epoxy resin and Expanded Polystyrene foam (EPS) which is stronger and lighter than traditional PU/PE construction. Even newer designs incorporate materials such as carbon fiber and variable-flex composites in conjunction with fiberglass and epoxy or polyester resins. Since epoxy/EPS surfboards are generally lighter, they will float better than a traditional PU/PE board of similar size, shape and thickness. This makes them easier to paddle and faster in the water. However, a common complaint of EPS boards is that they do not provide as much feedback as a traditional PU/PE board. For this reason, many advanced surfers prefer that their surfboards be made from traditional materials.
Other equipment includes a leash (to stop the board from drifting away after a wipeout and to prevent it from hitting other surfers), surf wax, traction pads (to keep a surfer's feet from slipping off the deck of the board), and fins (also known as skegs) which can either be permanently attached (glassed-on) or interchangeable. Sportswear designed or particularly suitable for surfing may be sold as boardwear (the term is also used in snowboarding). In warmer climates, swimsuits, surf trunks or boardshorts are worn, and occasionally rash guards; in cold water, surfers can opt to wear wetsuits, boots, hoods, and gloves to protect them against lower water temperatures. A newer introduction is a rash vest with a thin layer of titanium to provide maximum warmth without compromising mobility. In recent years, there have been advancements in technology that have allowed surfers to pursue even bigger waves with added elements of safety. Big wave surfers are now experimenting with inflatable vests or colored dye packs to help decrease their odds of drowning.
There are many different surfboard sizes, shapes, and designs in use today. Modern longboards, generally in length, are reminiscent of the earliest surfboards, but now benefit from modern innovations in surfboard shaping and fin design. Competitive longboard surfers need to be competent at traditional walking manoeuvres, as well as the short-radius turns normally associated with shortboard surfing. The modern shortboard began life in the late 1960s and has evolved into today's common thruster style, defined by its three fins, usually around in length. The thruster was invented by Australian shaper Simon Anderson.
Midsize boards, often called funboards, provide more maneuverability than a longboard, with more flotation than a shortboard. While many surfers find that funboards live up to their name, providing the best of both surfing modes, others are critical.
"It is the happy medium of mediocrity," writes Steven Kotler. "Funboard riders either have nothing left to prove or lack the skills to prove anything."
There are also various niche styles, such as the Egg, a longboard-style short board targeted at people who want to ride a shortboard but need more paddle power. The Fish, a board that is typically shorter, flatter, and wider than a normal shortboard, often with a split tail (known as a swallow tail). The Fish often has two or four fins and is specifically designed for surfing smaller waves. For big waves, there is the Gun, a long, thick board with a pointed nose and tail (known as a pintail) specifically designed for big waves.
## The physics of surfing
The physics of surfing involves the physical oceanographic properties of wave creation in the surf zone, the characteristics of the surfboard, and the surfer's interaction with the water and the board.
### Wave formation
Ocean waves are defined as a collection of dislocated water parcels that undergo a cycle of being forced past their normal position and being restored back to their normal position. Wind causes ripples and eddies to form waves that gradually gain speed and distance (fetch). Waves increase in energy and speed and then become longer and stronger. The fully-developed sea has the strongest wave action that experiences storms lasting 10-hours and creates wave heights in the open ocean.
The waves created in the open ocean are classified as deep-water waves. Deep-water waves have no bottom interaction and the orbits of these water molecules are circular; their wavelength is short relative to water depth and the velocity decays before reaching the bottom of the water basin. Deep water waves are waves in water depths greater than half their wavelengths. Wind forces waves to break in the deep sea.
Deep-water waves travel to shore and become shallow-water waves when the water depth is less than half of their wavelength, and the wave motion becomes constrained by the bottom, causing the orbit paths to be flattened to ellipses. The bottom exerts a frictional drag on the bottom of the wave, which decreases the celerity (or the speed of the waveform), and causes refraction. Slowing the wave forces it to shorten which increases the height and steepness, and the top (crest) falls because the velocity of the top of the wave becomes greater than the velocity of the bottom of the wave where the drag occurs.
The surf zone is the place of convergence of multiple waves types creating complex wave patterns. A wave suitable for surfing results from maximum speeds of . This speed is relative because local onshore winds can cause waves to break. In the surf zone, shallow water waves are carried by global winds to the beach and interact with local winds to make surfing waves.
Different onshore and off-shore wind patterns in the surf zone create different types of waves. Onshore winds cause random wave breaking patterns and are more suitable for experienced surfers. Light offshore winds create smoother waves, while strong direct offshore winds cause plunging or large barrel waves. Barrel waves are large because the water depth is small when the wave breaks. Thus, the breaker intensity (or force) increases, and the wave speed and height increase. Off-shore winds produce non-surfable conditions by flattening a weak swell. Weak swell is made from surface gravity forces and has long wavelengths.
### Wave conditions for surfing
Surfing waves can be analyzed using the following parameters: breaking wave height, wave peel angle (α), wave breaking intensity, and wave section length. The breaking wave height has two measurements, the relative heights estimated by surfers and the exact measurements done by physical oceanographers. Measurements done by surfers were 1.36 to 2.58 times higher than the measurements done by scientists. The scientifically concluded wave heights that are physically possible to surf are .
The wave peel angle is one of the main constituents of a potential surfing wave. Wave peel angle measures the distance between the peel-line and the line tangent to the breaking crest line. This angle controls the speed of the wave crest. The speed of the wave is an addition of the propagation velocity vector (Vw) and peel velocity vector (Vp), which results in the overall velocity of the wave (Vs).
Wave breaking intensity measures the force of the wave as it breaks, spills, or plunges (a plunging wave is termed by surfers as a "barrel wave"). Wave section length is the distance between two breaking crests in a wave set. Wave section length can be hard to measure because local winds, non-linear wave interactions, island sheltering, and swell interactions can cause multifarious wave configurations in the surf zone.
The parameters breaking wave height, wave peel angle (α), and wave breaking intensity, and wave section length are important because they are standardized by past oceanographers who researched surfing; these parameters have been used to create a guide that matches the type of wave formed and the skill level of surfer.
+ Table 1: Wave type and surfer skill level Skill level Peel angle (degrees) Wave height (meters) Section speed (meters/second) Section length (meters) General locations of waves Beginner 60-70 2.5 10 25 Low Gradient Breaks; Atlantic Beach, Florida Intermediate 55 2.5 20 40 Bells Beach; Australia Competent 40-50 3 20 40-60 Kirra Point; Burleigh Heads Top Amateur 30 3 20 60 Bingin Beach; Padang Padang Beach Top World Surfer >27 3 20 60 Banzai Pipeline; Shark Island
Table 1 shows a relationship of smaller peel angles correlating with a higher skill level of the surfer. Smaller wave peel angles increase the velocities of waves. A surfer must know how to react and paddle quickly to match the speed of the wave to catch it. Therefore, more experience is required to catch low peel angle waves. More experienced surfers can handle longer section lengths, increased velocities, and higher wave heights. Different locations offer different types of surfing conditions for each skill level.
### Surf breaks
A surf break is an area with an obstruction or an object that causes a wave to break. Surf breaks entail multiple scale phenomena. Wave section creation has microscale factors of peel angle and wave breaking intensity. The micro-scale components influence wave height and variations on wave crests. The mesoscale components of surf breaks are the ramp, platform, wedge, or ledge that may be present at a surf break. Macro-scale processes are the global winds that initially produce offshore waves. Types of surf breaks are headlands (point break), beach break, river/estuary entrance bar, reef breaks, and ledge breaks.
#### Headland (point break)
A headland or point break interacts with the water by causing refraction around the point or headland. The point absorbs the high-frequency waves and long-period waves persist, which are easier to surf. Examples of locations that have headland or point break-induced surf breaks are Dunedin (New Zealand), Raglan (New Zealand), Malibu (California), Rincon (California), and Kirra (Australia).
#### Beach break
A beach break is an area of open coastline where the waves break over a sand-bottom. They are the most common, yet also the most volatile of surf breaks. Wave breaks happen successively at beach breaks, as in there are multiple peaks to surf at a single beach break location. Example locations are Tairua and Aramoana Beach (New Zealand) and the Gold Coast (Australia).
#### River or estuary entrance bar
A river or estuary entrance bar creates waves from the ebb-tidal delta, sediment outflow, and tidal currents. An ideal estuary entrance bar exists in Whangamata Bar, New Zealand.
#### Reef break
A reef break is conducive to surfing because large waves consistently break over the reef. The reef is usually made of coral, and because of this, many injuries occur while surfing reef breaks. However, the waves that are produced by reef breaks are some of the best in the world. Famous reef breaks are present in Padang Padang (Indonesia), Pipeline (Hawaii), Uluwatu (Bali), and Teahupo'o (Tahiti).
#### Ledge break
A ledge break is formed by steep rocks ledges that make intense waves because the waves travel through deeper water then abruptly reach shallower water at the ledge. Shark Island, Australia is a location with a ledge break. Ledge breaks create difficult surfing conditions, sometimes only allowing body surfing as the only feasible way to confront the waves.
### Jetties and their impacts on wave formation in the surf zone
Jetties are added to bodies of water to regulate erosion, preserve navigation channels, and make harbors. Jetties are classified into four different types and have two main controlling variables: the type of delta and the size of the jetty.
#### Type 1 jetty
The first classification is a type 1 jetty. This type of jetty is significantly longer than the surf zone width and the waves break at the shore end of the jetty. The effect of a Type 1 jetty is sediment accumulation in a wedge formation on the jetty. These waves are large and increase in size as they pass over the sediment wedge formation. An example of a Type 1 jetty is Mission Beach, San Diego, California. This 1000-meter jetty was installed in 1950 at the mouth of Mission Bay. The surf waves happen north of the jetty, are longer waves, and are powerful. The bathymetry of the sea bottom in Mission Bay has a wedge shape formation that causes the waves to refract as they become closer to the jetty. The waves converge constructively after they refract and increase the sizes of the waves.
#### Type 2 jetty
A type 2 jetty occurs in an ebb-tidal delta, a delta transitioning between high and low tide. This area has shallow water, refraction, and distinctive seabed shapes that create large wave heights.
An example of a type 2 jetty is called "The Poles" in Atlantic Beach, Florida. Atlantic Beach is known to have flat waves, with exceptions during major storms. However, "The Poles" has larger than normal waves due to a 500-meter jetty that was installed on the south side of St. Johns. This jetty was built to make a deep channel in the river. It formed a delta at "The Poles". This is a special area because the jetty increases wave size for surfing when comparing pre-conditions and post-conditions of the southern St. Johns River mouth area.
The wave size at "The Poles" depends on the direction of the incoming water. When easterly waters (from 55°) interact with the jetty, they create waves larger than southern waters (from 100°). When southern waves (from 100°) move toward "The Poles", one of the waves breaks north of the southern jetty and the other breaks south of the jetty. This does not allow for merging to make larger waves. Easterly waves, from 55°, converge north of the jetty and unite to make bigger waves.
#### Type 3 jetty
A type 3 jetty is in an ebb-tidal area with an unchanging seabed that has naturally created waves. Examples of a Type 3 jetty occurs in “Southside” Tamarack, Carlsbad, California.
#### Type 4 jetty
A type 4 jetty is one that no longer functions nor traps sediment. The waves are created from reefs in the surf zone. A type 4 jetty can be found in Tamarack, Carlsbad, California.
###
### Rip currents
Rip currents are fast, narrow currents that are caused by onshore transport within the surf zone and the successive return of the water seaward. The wedge bathymetry makes a convenient and consistent rip current of 5–10 meters that brings the surfers to the “take-off point” then out to the beach.
Oceanographers have two theories on rip current formation. The wave interaction model assumes that two edges of waves interact, create differing wave heights, and cause longshore transport of nearshore currents. The Boundary Interaction Model assumes that the topography of the sea bottom causes nearshore circulation and longshore transport; the result of both models is a rip current.
Rip currents can be extremely strong and narrow as they extend out of the surf zone into deeper water, reaching speeds from and up to , which is faster than any human can swim. The water in the jet is sediment rich, bubble rich, and moves rapidly. The rip head of the rip current has long shore movement. Rip currents are common on beaches with mild slopes that experience sizeable and frequent oceanic swell.
### On the surfboard
A longer surfboard of causes more friction with the water; therefore, it will be slower than a smaller and lighter board with a length of . Longer boards are good for beginners who need help balancing. Smaller boards are good for more experienced surfers who want to have more control and maneuverability.
When practicing the sport of surfing, the surfer paddles out past the wave break to wait for a wave. When a surfable wave arrives, the surfer must paddle extremely fast to match the velocity of the wave so the wave can accelerate him or her.
When the surfer is at wave speed, the surfer must quickly pop up, stay low, and stay toward the front of the wave to become stable and prevent falling as the wave steepens. The acceleration is less toward the front than toward the back. The physics behind the surfing of the wave involves the horizontal acceleration force (F·sinθ) and the vertical force (F·cosθ=mg). Therefore, the surfer should lean forward to gain speed, and lean on the back foot to brake. Also, to increase the length of the ride of the wave, the surfer should travel parallel to the wave crest.
## Dangers
### Drowning
Surfing, like all water sports, carries the inherent risk of drowning. Although the board assists a surfer in staying buoyant, it can become separated from the user. A leash, attached to the ankle or knee, can keep a board from being swept away, but does not keep a rider on the board or above water. In some cases, possibly including the drowning of professional surfer Mark Foo, the leash can be a cause of drowning by snagging on a reef or other object and holding the surfer underwater. By keeping the surfboard close to the surfer during a wipeout, a leash also increases the chances that the board may strike the rider, which could knock them unconscious and lead to drowning (especially with a hard surfboard instead of a soft surfboard). A fallen rider's board can become trapped in larger waves, and if the rider is attached by a leash, they can be dragged for long distances underwater. Surfers should be careful to remain in smaller surf until they have acquired the advanced skills and experience necessary to handle bigger waves and more challenging conditions. Even world-class surfers have drowned in extremely challenging conditions.
### Collisions
Under the wrong set of conditions, anything that a surfer's body can come in contact with is a potential hazard, including sand bars, rocks, small ice, reefs, surfboards, and other surfers. Collisions with these objects can sometimes cause injuries such as cuts and scrapes and in rare instances, death.
A large number of injuries, up to 66%, are caused by collision with a surfboard (nose or fins). Fins can cause deep lacerations and cuts, as well as bruising. While these injuries can be minor, they can open the skin to infection from the sea; groups like Surfers Against Sewage campaign for cleaner waters to reduce the risk of infections. Local bugs and diseases can be risk factors when surfing around the globe.
Falling off a surfboard or colliding with others is commonly referred to as a wipeout.
### Marine life
Sea life can sometimes cause injuries (Bethany Hamilton) and even fatalities. Animals such as sharks, stingrays, Weever fish, seals and jellyfish can sometimes present a danger. Warmer-water surfers often do the "stingray shuffle" as they walk out through the shallows, shuffling their feet in the sand to scare away stingrays that may be resting on the bottom.
Rip currents
Rip currents are water channels that flow away from the shore. Under the wrong circumstances these currents can endanger both experienced and inexperienced surfers. Since a rip current appears to be an area of flat water, tired or inexperienced swimmers or surfers may enter one and be carried out beyond the breaking waves. Although many rip currents are much smaller, the largest rip currents have a width of . The flow of water moving out towards the sea in a rip will be stronger than most swimmers, making swimming back to shore difficult, however, by paddling parallel to the shore, a surfer can easily exit a rip current. Alternatively, some surfers actually ride on a rip current because it is a fast and effortless way to get out beyond the zone of breaking waves.
### Seabed
The seabed can pose a risk for surfers. If a surfer falls while riding a wave, the wave tosses and tumbles the surfer around, often in a downwards direction. At reef breaks and beach breaks, surfers have been seriously injured and even killed, because of a violent collision with the sea bed, the water above which can sometimes be very shallow, especially at beach breaks or reef breaks during low tide. Cyclops, Western Australia, for example, is one of the biggest and thickest reef breaks in the world, with waves measuring up to high, but the reef below is only about below the surface of the water.
### Microorganisms
A January 2018 study by the University of Exeter called the "Beach Bum Survey" found surfers and bodyboarders to be three times as likely as non-surfers to harbor antibiotic-resistant E. coli and four times as likely to harbor other bacteria capable of easily becoming antibiotic resistant. The researchers attributed this to the fact that surfers swallow roughly ten times as much seawater as swimmers.
### Ear damage
Surfers sometimes use ear protection such as ear plugs to avoid surfer's ear, inflammation of the ear or other damage. Surfer's ear is where the bone near the ear canal grows after repeated exposure to cold water, making the ear canal narrower. The narrowed canal makes it harder for water to drain from the ear. This can result in pain, infection and sometimes ringing of the ear.
Ear plugs designed for surfers, swimmers and other water athletes are primarily made to keep water out of the ear, thereby letting a protective pocket of air stay inside the ear canal. They can also block cold air, dirt and bacteria. Many designs are made to let sound through, and either float and/or have a leash in case the plug accidentally gets bumped out.
### Surf rash
Surf rash appears in many different ways on the skin, commonly as a painful red bumpy patch located on the surfer's chest or inner legs. A rash guard will lessen the incidence of surf rash caused by abrasion or sunburn. Healing ointments such as petroleum jelly can be used to treat irritated skin.
### Spinal cord
Surfer's myelopathy is a rare spinal cord injury causing paralysis of the lower extremities, caused by hyperextension of the back. This is due to one of the main blood vessels of the spine becoming kinked, depriving the spinal cord of oxygen. In some cases the paralysis is permanent. Although any activity where the back is arched can cause this condition (i.e. yoga, pilates, etc.), this rare phenomenon has most often been seen in those surfing for the first time. According to DPT Sergio Florian, some recommendations for preventing myelopathy is proper warm up, limiting the session length and sitting on the board while waiting for waves, rather than lying.
## Surfers and surf culture
Surfers represent a diverse culture based on riding the waves. Some people practice surfing as a recreational activity, while others make it the central focus of their lives. Surfing culture in the US is most dominant in Hawaii and California, because these two states offer the best surfing conditions. However, waves can be found wherever there is coastline, and a tight-knit yet far-reaching subculture of surfers has emerged throughout America. Some historical markers of the culture included the woodie, the station wagon used to carry surfers' boards, as well as boardshorts, the long swim shorts typically worn while surfing. Surfers also wear wetsuits in colder regions and when the seasons cool the air and water.
During the 1960s, as surfing caught on in California, its popularity spread through American pop culture. Several teen movies, starting with the Gidget series in 1959, transformed surfing into a dream life for American youth. Later movies, including Beach Party (1963), Ride the Wild Surf (1964), and Beach Blanket Bingo (1965) promoted the California dream of sun and surf. Surf culture also fueled the early records of the Beach Boys.
The sport is also a significant part of Australia's eastern coast sub-cultural life, especially in New South Wales, where the weather and water conditions are most favourable for surfing.
The sport of surfing now represents a multibillion-dollar industry, especially in clothing and fashion markets. Founded in 1964, the International Surfing Association (ISA) is the oldest foundation associated with surfing formed to better improve surfing and recognized by the International Olympic Committee as the leading authority on surfing. National and international surf competitions began in 1964. In addition, The World Surf League (WSL) was established in 1976 and promotes various championship tours, hosting top competitors in some of the best surf spots around the globe. A small number of people make a career out of surfing by receiving corporate sponsorships and performing for photographers and videographers in far-flung destinations; they are typically referred to as freesurfers. Sixty-six surfers on a long surfboard set a record in Huntington Beach, California for most people on a surfboard at one time. Dale Webster consecutively surfed for 14,641 days, making it his main life focus.
As of 2023, the Guinness Book of World Records recognized a 26.2 m (86 ft) wave ride by Sebastian Steudtner at Nazaré, Portugal as the largest wave ever surfed.
When the waves were flat, surfers persevered with sidewalk surfing, which is now called skateboarding. Sidewalk surfing has a similar feel to surfing and requires only a paved road or sidewalk. To create the feel of the wave, surfers even sneaked into empty backyard swimming pools to ride in, known as pool skating. Eventually, surfing made its way to the slopes with the invention of the Snurfer, later credited as the first snowboard. Many other board sports have been invented over the years, but all can trace their heritage back to surfing.
Many surfers claim to have a spiritual connection with the ocean, describing surfing, the surfing experience, both in and out of the water, as a type of spiritual experience or a religion. Recent academic studies and practitioner testimonies have demonstrated the mental health and well-being benefits of surfing which has spurred the development of para surfing and surf therapy. | https://en.wikipedia.org/wiki/Surfing |
The structured program theorem, also called the Böhm–Jacopini theorem, is a result in programming language theory. It states that a class of control-flow graphs (historically called flowcharts in this context) can compute any computable function if it combines subprograms in only three specific ways (control structures). These are
1. Executing one subprogram, and then another subprogram (sequence)
1. Executing one of two subprograms according to the value of a boolean expression (selection)
1. Repeatedly executing a subprogram as long as a boolean expression is true (iteration)
The structured chart subject to these constraints, particularly the loop constraint implying a single exit (as described later in this article), may however use additional variables in the form of bits (stored in an extra integer variable in the original proof) in order to keep track of information that the original program represents by the program location. The construction was based on Böhm's programming language P′′.
The theorem forms the basis of structured programming, a programming paradigm which eschews goto commands and exclusively uses subroutines, sequences, selection and iteration.
## Origin and variants
The theorem is typically credited to a 1966 paper by Corrado Böhm and . David Harel wrote in 1980 that the Böhm–Jacopini paper enjoyed "universal popularity", particularly with proponents of structured programming. Harel also noted that "due to its rather technical style [the 1966 Böhm–Jacopini paper] is apparently more often cited than read in detail" and, after reviewing a large number of papers published up to 1980, Harel argued that the contents of the Böhm–Jacopini proof were usually misrepresented as a folk theorem that essentially contains a simpler result, a result which itself can be traced to the inception of modern computing theory in the papers of von Neumann and Kleene.
Harel also writes that the more generic name was proposed by H.D. Mills as "The Structure Theorem" in the early 1970s.
### Single-while-loop, folk version of the theorem
This version of the theorem replaces all the original program's control flow with a single global `while` loop that simulates a program counter going over all possible labels (flowchart boxes) in the original non-structured program. Harel traced the origin of this folk theorem to two papers marking the beginning of computing. One is the 1946 description of the von Neumann architecture, which explains how a program counter operates in terms of a while loop. Harel notes that the single loop used by the folk version of the structured programming theorem basically just provides operational semantics for the execution of a flowchart on a von Neumann computer. Another, even older source that Harel traced the folk version of the theorem is Stephen Kleene's normal form theorem from 1936.
Donald Knuth criticized this form of the proof, which results in pseudocode like the one below, by pointing out that the structure of the original program is completely lost in this transformation. Similarly, Bruce Ian Mills wrote about this approach that "The spirit of block structure is a style, not a language. By simulating a Von Neumann machine, we can produce the behavior of any spaghetti code within the confines of a block-structured language. This does not prevent it from being spaghetti."
```pascal
p := 1
while p > 0 do
if p = 1 then
perform step 1 from the flowchart
p := resulting successor step number of step 1 from the flowchart (0 if no successor)
end if
if p = 2 then
perform step 2 from the flowchart
p := resulting successor step number of step 2 from the flowchart (0 if no successor)
end if
...
if p = n then
perform step n from the flowchart
p := resulting successor step number of step n from the flowchart (0 if no successor)
end if
end while
```
### Böhm and Jacopini's proof
The proof in Böhm and Jacopini's paper proceeds by induction on the structure of the flow chart. Because it employed pattern matching in graphs, the proof of Böhm and Jacopini's was not really practical as a program transformation algorithm, and thus opened the door for additional research in this direction.
### Reversible version
The Reversible Structured Program Theorem is an important concept in the field of reversible computing. It posits that any computation achievable by a reversible program can also be accomplished through a reversible program using only a structured combination of control flow constructs such as sequences, selections, and iterations. Any computation achievable by a traditional, irreversible program can also be accomplished through a reversible program, but with the additional constraint that each step must be reversible and some extra output. Furthermore, any reversible unstructured program can also be accomplished through a structured reversible program with only one iteration without any extra output. This theorem lays the foundational principles for constructing reversible algorithms within a structured programming framework.
For the Structured Program Theorem, both local and global methods of proof are known. However, for its reversible version, while a global method of proof is recognized, a local approach similar to that undertaken by Böhm and Jacopini is not yet known. This distinction is an example that underscores the challenges and nuances in establishing the foundations of reversible computing compared to traditional computing paradigms.
## Implications and refinements
The Böhm–Jacopini proof did not settle the question of whether to adopt structured programming for software development, partly because the construction was more likely to obscure a program than to improve it. On the contrary, it signalled the beginning of the debate. Edsger Dijkstra's famous letter, "Go To Statement Considered Harmful," followed in 1968.
Some academics took a purist approach to the Böhm–Jacopini result and argued that even instructions like `break` and `return` from the middle of loops are bad practice as they are not needed in the Böhm–Jacopini proof, and thus they advocated that all loops should have a single exit point. This purist approach is embodied in the Pascal programming language (designed in 1968–1969), which up to the mid-1990s was the preferred tool for teaching introductory programming classes in academia.
Edward Yourdon notes that in the 1970s there was even philosophical opposition to transforming unstructured programs into structured ones by automated means, based on the argument that one needed to think in structured programming fashion from the get go. The pragmatic counterpoint was that such transformations benefited a large body of existing programs. Among the first proposals for an automated transformation was a 1971 paper by Edward Ashcroft and Zohar Manna.
The direct application of the Böhm–Jacopini theorem may result in additional local variables being introduced in the structured chart, and may also result in some code duplication. The latter issue is called the loop and a half problem in this context. Pascal is affected by both of these problems and according to empirical studies cited by Eric S. Roberts, student programmers had difficulty formulating correct solutions in Pascal for several simple problems, including writing a function for searching an element in an array. A 1980 study by Henry Shapiro cited by Roberts found that using only the Pascal-provided control structures, the correct solution was given by only 20% of the subjects, while no subject wrote incorrect code for this problem if allowed to write a return from the middle of a loop.
In 1973, S. Rao Kosaraju proved that it's possible to avoid adding additional variables in structured programming, as long as arbitrary-depth, multi-level breaks from loops are allowed.KOSARAJU, S. RAO. "Analysis of structured programs," Proc. Fifth Annual ACM Syrup.
Theory of Computing, (May 1973), 240-252; also cited by Furthermore, Kosaraju proved that a strict hierarchy of programs exists, nowadays called the Kosaraju hierarchy, in that for every integer n, there exists a program containing a multi-level break of depth n that cannot be rewritten as program with multi-level breaks of depth less than n (without introducing additional variables). Kosaraju cites the multi-level break construct to the BLISS programming language. The multi-level breaks, in the form a `leave label` keyword were actually introduced in the BLISS-11 version of that language; the original BLISS only had single-level breaks. The BLISS family of languages didn't provide an unrestricted goto. The Java programming language would later follow this approach as well.
A simpler result from Kosaraju's paper is that a program is reducible to a structured program (without adding variables) if and only if it does not contain a loop with two distinct exits. Reducibility was defined by Kosaraju, loosely speaking, as computing the same function and using the same "primitive actions" and predicates as the original program, but possibly using different control flow structures. (This is a narrower notion of reducibility than what Böhm–Jacopini used.) Inspired by this result, in section VI of his highly-cited paper that introduced the notion of cyclomatic complexity, Thomas J. McCabe described an analogue of Kuratowski's theorem for the control-flow graphs (CFG) of non-structured programs, which is to say, the minimal subgraphs that make the CFG of a program non-structured. These subgraphs have a very good description in natural language. They are:
1. branching out of a loop (other than from the loop cycle test)
1. branching into a loop
1. branching into a decision (i.e. into an if "branch")
1. branching out of a decision
McCabe actually found that these four graphs are not independent when appearing as subgraphs, meaning that a necessary and sufficient condition for a program to be non-structured is for its CFG to have as subgraph one of any subset of three of these four graphs. He also found that if a non-structured program contains one of these four sub-graphs, it must contain another distinct one from the set of four. This latter result helps explain how the control flow of non-structured program becomes entangled in what is popularly called "spaghetti code". McCabe also devised a numerical measure that, given an arbitrary program, quantifies how far off it is from the ideal of being a structured program; McCabe called his measure essential complexity.
McCabe's characterization of the forbidden graphs for structured programming can be considered incomplete, at least if the Dijkstra's D structures are considered the building blocks.
Up to 1990 there were quite a few proposed methods for eliminating gotos from existing programs, while preserving most of their structure. The various approaches to this problem also proposed several notions of equivalence, which are stricter than simply Turing equivalence, in order to avoid output like the folk theorem discussed above. The strictness of the chosen notion of equivalence dictates the minimal set of control flow structures needed. The 1988 JACM paper by Lyle Ramshaw surveys the field up to that point, as well proposing its own method. Ramshaw's algorithm was used for example in some Java decompilers because the Java virtual machine code has branch instructions with targets expressed as offsets, but the high-level Java language only has multi-level `break` and `continue` statements. Ammarguellat (1992) proposed a transformation method that goes back to enforcing single-exit.
## Application to Cobol
In the 1980s IBM researcher Harlan Mills oversaw the development of the COBOL Structuring Facility, which applied a structuring algorithm to COBOL code. Mills's transformation involved the following steps for each procedure.
1. Identify the basic blocks in the procedure.
1. Assign a unique label to each block's entry path, and label each block's exit paths with the labels of the entry paths they connect to. Use 0 for return from the procedure and 1 for the procedure's entry path.
1. Break the procedure into its basic blocks.
1. For each block that is the destination of only one exit path, reconnect that block to that exit path.
1. Declare a new variable in the procedure (called L for reference).
1. On each remaining unconnected exit path, add a statement that sets L to the label value on that path.
1. Combine the resulting programs into a selection statement that executes the program with the entry path label indicated by L
1. Construct a loop that executes this selection statement as long as L is not 0.
1. Construct a sequence that initializes L to 1 and executes the loop.
This construction can be improved by converting some cases of the selection statement into subprocedures. | https://en.wikipedia.org/wiki/Structured_program_theorem |
Agroecosystems are the ecosystems supporting the food production systems in farms and gardens. As the name implies, at the core of an agroecosystem lies the human activity of agriculture. As such they are the basic unit of study in Agroecology, and Regenerative Agriculture using ecological approaches.
Like other ecosystems, agroecosystems form partially closed systems in which animals, plants, microbes, and other living organisms and their environment are interdependent and regularly interact. They are somewhat arbitrarily defined as a spatially and functionally coherent unit of agricultural activity.
An agroecosystem can be seen as not restricted to the immediate site of agricultural activity (e.g. the farm). That is, it includes the region that is impacted by this activity, usually by changes to the complexity of species assemblages and energy flows, as well as to the net nutrient balance. Agroecosystems, particularly those managed intensively, are characterized as having simpler species composition, energy and nutrient flows than "natural" ecosystems. Likewise, agroecosystems are often associated with elevated nutrient input, much of which exits the farm leading to eutrophication of connected ecosystems not directly engaged in agriculture.
## Utilization
Forest gardens are probably the world's oldest and most resilient agroecosystem.
Some major organizations are hailing farming within agroecosystems as the way forward for mainstream agriculture. Current farming methods have resulted in over-stretched water resources, high levels of erosion and reduced soil fertility. According to a report by the International Water Management Institute and the United Nations Environment Programme, there is not enough water to continue farming using current practices; therefore how critical water, land, and ecosystem resources are used to boost crop yields must be reconsidered. The report suggested assigning value to ecosystems, recognizing environmental and livelihood tradeoffs, and balancing the rights of a variety of users and interests, as well addressing inequities that sometimes result when such measures are adopted, such as the reallocation of water from poor to rich, the clearing of land to make way for
more productive farmland, or the preservation of a wetland system that limits fishing rights.
One of the major efforts of disciplines such as agroecology is to promote management styles that blur the distinction between agroecosystems and "natural" ecosystems, both by decreasing the impact of agriculture (increasing the biological and trophic complexity of the agricultural system as well as decreasing the nutrient inputs/outflow) and by increasing awareness that "downstream" effects extend agroecosystems beyond the boundaries of the farm (e.g. the Corn Belt agroecosystem includes the hypoxic zone in the Gulf of Mexico). In the first case, polyculture or buffer strips for wildlife habitat can restore some complexity to a cropping system, while organic farming can reduce nutrient inputs. Efforts of the second type are most common at the watershed scale. An example is the National Association of Conservation Districts' Lake Mendota Watershed Project, which seeks to reduce runoff from the agricultural lands feeding into the lake with the aim of reducing algal blooms. | https://en.wikipedia.org/wiki/Agroecosystem |
Supersonic speed is the speed of an object that exceeds the speed of sound (Mach 1). For objects traveling in dry air of a temperature of 20 °C (68 °F) at sea level, this speed is approximately . Speeds greater than five times the speed of sound (Mach 5) are often referred to as hypersonic. Flights during which only some parts of the air surrounding an object, such as the ends of rotor blades, reach supersonic speeds are called transonic. This occurs typically somewhere between Mach 0.8 and Mach 1.2.
Sounds are traveling vibrations in the form of pressure waves in an elastic medium. Objects move at supersonic speed when the objects move faster than the speed at which sound propagates through the medium. In gases, sound travels longitudinally at different speeds, mostly depending on the molecular mass and temperature of the gas, and pressure has little effect. Since air temperature and composition varies significantly with altitude, the speed of sound, and Mach numbers for a steadily moving object may change. In water at room temperature, supersonic speed means any speed greater than 1,440 m/s (4,724 ft/s). In solids, sound waves can be polarized longitudinally or transversely and have higher velocities.
Supersonic fracture is crack formation faster than the speed of sound in a brittle material.
## Early meaning
The word supersonic comes from two Latin derived words; 1) super: above and 2) sonus: sound, which together mean above sound, or faster than sound.
At the beginning of the 20th century, the term "supersonic" was used as an adjective to describe sound whose frequency is above the range of normal human hearing. The modern term for this meaning is "ultrasonic", but the older meaning sometimes still lives on, as in the word superheterodyne
## Supersonic objects
The tip of a bullwhip is generally seen as the first object designed to reach the speed of sound. This action results in its telltale "crack", which is actually just a sonic boom. The first human-made supersonic boom was likely caused by a piece of common cloth, leading to the whip's eventual development. It is the wave motion travelling through the bullwhip that makes it capable of achieving supersonic speeds.
Most modern firearm bullets are supersonic, with rifle projectiles often travelling at speeds approaching and in some cases well exceeding Mach 3.
Most spacecraft are supersonic at least during portions of their reentry, though the effects on the spacecraft are reduced by low air densities. During ascent, launch vehicles generally avoid going supersonic below 30 km (~98,400 feet) to reduce air drag.
The speed of sound decreases somewhat with altitude, due to lower temperatures found there (typically up to 25 km). At even higher altitudes the temperature starts increasing, with the corresponding increase in the speed of sound.
When an inflated balloon is burst, the torn pieces of latex contract at supersonic speed, which contributes to the sharp and loud popping noise.
### Supersonic land vehicles
To date, only one land vehicle has officially travelled at supersonic speed, the ThrustSSC. The vehicle, driven by Andy Green, holds the world land speed record, having achieved an average speed on its bi-directional run of in the Black Rock Desert on 15 October 1997.
The Bloodhound LSR project planned an attempt on the record in 2020 at Hakskeenpan in South Africa with a combination jet and hybrid rocket propelled car. The aim was to break the existing record, then make further attempts during which (the members of) the team hoped to reach speeds of up to . The effort was originally run by Richard Noble who was the leader of the ThrustSSC project, however following funding issues in 2018, the team was bought by Ian Warhurst and renamed Bloodhound LSR. Later the project was indefinitely delayed due to the COVID-19 pandemic and the vehicle was put up for sale.
### Supersonic flight
Most modern fighter aircraft are supersonic aircraft. No modern-day passenger aircraft are capable of supersonic speed, but there have been supersonic passenger aircraft, namely Concorde and the Tupolev Tu-144. Both of these passenger aircraft and some modern fighters are also capable of supercruise, a condition of sustained supersonic flight without the use of an afterburner. Due to its ability to supercruise for several hours and the relatively high frequency of flight over several decades, Concorde spent more time flying supersonically than all other aircraft combined by a considerable margin. Since Concorde's final retirement flight on November 26, 2003, there are no supersonic passenger aircraft left in service. Some large bombers, such as the Tupolev Tu-160 and Rockwell B-1 Lancer are also supersonic-capable.
The aerodynamics of supersonic aircraft is simpler than subsonic aerodynamics because the airsheets at different points along the plane often cannot affect each other. Supersonic jets and rocket vehicles require several times greater thrust to push through the extra aerodynamic drag experienced within the transonic region (around Mach 0.85–1.2). At these speeds aerospace engineers can gently guide air around the fuselage of the aircraft without producing new shock waves, but any change in cross area farther down the vehicle leads to shock waves along the body. Designers use the Supersonic area rule and the Whitcomb area rule to minimize sudden changes in size.
However, in practical applications, a supersonic aircraft must operate stably in both subsonic and supersonic profiles, hence aerodynamic design is more complex.
The main key to having low supersonic drag is to properly shape the overall aircraft to be long and thin, and close to a "perfect" shape, the von Karman ogive or Sears-Haack body. This has led to almost every supersonic cruising aircraft looking very similar to every other, with a very long and slender fuselage and large delta wings, cf. SR-71, Concorde, etc. Although not ideal for passenger aircraft, this shaping is quite adaptable for bomber use. | https://en.wikipedia.org/wiki/Supersonic_speed |
In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.
Let
$$
X
$$
follow an ordinary normal distribution,
$$
N(0,\sigma^2)
$$
. Then,
$$
Y=|X|
$$
follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.
## Properties
Using the
$$
\sigma
$$
parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by
$$
f_Y(y; \sigma) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp \left( -\frac{y^2}{2\sigma^2} \right) \quad y \geq 0,
$$
where
$$
E[Y] = \mu = \frac{\sigma\sqrt{2}}{\sqrt{\pi}}
$$
.
Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if
$$
\sigma
$$
is near zero), obtained by setting
$$
\theta=\frac{\sqrt{\pi}}{\sigma\sqrt{2}}
$$
, the probability density function is given by
$$
f_Y(y; \theta) = \frac{2\theta}{\pi}\exp \left( -\frac{y^2\theta^2}{\pi} \right) \quad y \geq 0,
$$
where
$$
E[Y] = \mu = \frac{1}{\theta}
$$
.
The cumulative distribution function (CDF) is given by
$$
F_Y(y; \sigma) = \int_0^y \frac{1}{\sigma}\sqrt{\frac{2}{\pi}} \, \exp \left( -\frac{x^2}{2\sigma^2} \right)\, dx
$$
Using the change-of-variables
$$
z = x/(\sqrt{2}\sigma)
$$
, the CDF can be written as
$$
F_Y(y; \sigma) = \frac{2}{\sqrt{\pi}} \,\int_0^{y/(\sqrt{2}\sigma)}\exp \left(-z^2\right)dz = \operatorname{erf}\left(\frac{y}{\sqrt{2}\sigma}\right),
$$
where erf is the error function, a standard function in many mathematical software packages.
The quantile function (or inverse CDF) is written:
$$
Q(F;\sigma)=\sigma\sqrt{2} \operatorname{erf}^{-1}(F)
$$
where
$$
0\le F \le 1
$$
and
$$
\operatorname{erf}^{-1}
$$
is the inverse error function
The expectation is then given by
$$
E[Y] = \sigma \sqrt{2/\pi},
$$
The variance is given by
$$
\operatorname{var}(Y) = \sigma^2\left(1 - \frac{2}{\pi}\right).
$$
Since this is proportional to the variance σ2 of X, σ can be seen as a scale parameter of the new distribution.
The differential entropy of the half-normal distribution is exactly one bit less the differential entropy of a zero-mean normal distribution with the same second moment about 0. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary. Thus,
$$
h(Y) = \frac{1}{2} \log_2 \left( \frac{\pi e \sigma^2}{2} \right) = \frac{1}{2} \log_2 \left( 2\pi e \sigma^2 \right) -1.
$$
## Applications
The half-normal distribution is commonly utilized as a prior probability distribution for variance parameters in Bayesian inference applications.
## Parameter estimation
Given numbers
$$
\{x_i\}_{i=1}^n
$$
drawn from a half-normal distribution, the unknown parameter
$$
\sigma
$$
of that distribution can be estimated by the method of maximum likelihood, giving
$$
\hat \sigma = \sqrt{\frac 1 n \sum_{i=1}^n x_i^2}
$$
The bias is equal to
$$
b \equiv \operatorname{E}\bigg[\;(\hat\sigma_\mathrm{mle} - \sigma)\;\bigg]
= - \frac{\sigma}{4n}
$$
which yields the bias-corrected maximum likelihood estimator
$$
\hat{\sigma\,}^*_\text{mle} = \hat{\sigma\,}_\text{mle} - \hat{b\,}.
$$
## Related distributions
- The distribution is a special case of the folded normal distribution with μ = 0.
- It also coincides with a zero-mean normal distribution truncated from below at zero (see truncated normal distribution)
- If Y has a half-normal distribution, then (Y/σ)2 has a chi square distribution with 1 degree of freedom, i.e. Y/σ has a chi distribution with 1 degree of freedom.
- The half-normal distribution is a special case of the generalized gamma distribution with d = 1, p = 2, a =
$$
\sqrt{2}\sigma
$$
.
- If Y has a half-normal distribution, Y -2 has a Lévy distribution
- The Rayleigh distribution is a moment-tilted and scaled generalization of the half-normal distribution.
- Modified half-normal distribution with the pdf on
$$
(0, \infty)
$$
is given as
$$
f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}
$$
, where
$$
\Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\(1,0)\end{matrix};z \right)
$$
denotes the Fox–Wright Psi function. | https://en.wikipedia.org/wiki/Half-normal_distribution |
Cosmogony is any model concerning the origin of the cosmos or the universe.
## Overview
### Scientific theories
In astronomy, cosmogony is the study of the origin of particular astrophysical objects or systems, and is most commonly used in reference to the origin of the universe, the Solar System, or the Earth–Moon system. The prevalent cosmological model of the early development of the universe is the Big Bang theory.
Sean M. Carroll, who specializes in theoretical cosmology and field theory, explains two competing explanations for the origins of the singularity, which is the center of a space in which a characteristic is limitless (one example is the singularity of a black hole, where gravity is the characteristic that becomes infinite).
It is generally accepted that the universe began at a point of singularity. When the universe started to expand, the Big Bang occurred, which evidently began the universe. The other explanation, held by proponents such as Stephen Hawking, asserts that time did not exist when it emerged along with the universe. This assertion implies that the universe does not have a beginning, as time did not exist "prior" to the universe. Hence, it is unclear whether properties such as space or time emerged with the singularity and the known universe.
Despite the research, there is currently no theoretical model that explains the earliest moments of the universe's existence (during the Planck epoch) due to a lack of a testable theory of quantum gravity. Nevertheless, researchers of string theory, its extensions (such as M-theory), and of loop quantum cosmology, like Barton Zwiebach and Washington Taylor, have proposed solutions to assist in the explanation of the universe's earliest moments. Cosmogonists have only tentative theories for the early stages of the universe and its beginning. The proposed theoretical scenarios include string theory, M-theory, the Hartle–Hawking initial state, emergent Universe, string landscape, cosmic inflation, the Big Bang, and the ekpyrotic universe. Some of these proposed scenarios, like the string theory, are compatible, whereas others are not.
### Mythology
In mythology, creation or cosmogonic myths are narratives describing the beginning of the universe or cosmos.
Some methods of the creation of the universe in mythology include:
- the will or action of a supreme being or beings,
- the process of metamorphosis,
- the copulation of female and male deities,
- from chaos,
- or via a cosmic egg.
Creation myths may be etiological, attempting to provide explanations for the origin of the universe. For instance, Eridu Genesis, the oldest known creation myth, contains an account of the creation of the world in which the universe was created out of a primeval sea (Abzu). Creation myths vary, but they may share similar deities or symbols. For instance, the ruler of the gods in Greek mythology, Zeus, is similar to the ruler of the gods in Roman mythology, Jupiter. Another example is the ruler of the gods in Tagalog mythology, Bathala, who is similar to various rulers of certain pantheons within Philippine mythology such as the Bisaya's Kaptan.Jocano, F. L. (1969). Philippine Mythology. Quezon City: Capitol Publishing House Inc.
## Compared with cosmology
In the humanities, the distinction between cosmogony and cosmology is blurred. For example, in theology, the cosmological argument for the existence of God (pre-cosmic cosmogonic bearer of personhood) is an appeal to ideas concerning the origin of the universe and is thus cosmogonical. Some religious cosmogonies have an impersonal first cause (for example Taoism).
However, in astronomy, cosmogony can be distinguished from cosmology, which studies the universe and its existence, but does not necessarily inquire into its origins. There is therefore a scientific distinction between cosmological and cosmogonical ideas. Physical cosmology is the science that attempts to explain all observations relevant to the development and characteristics of the universe on its largest scale. Some questions regarding the behaviour of the universe have been described by some physicists and cosmologists as being extra-scientific or metaphysical. Attempted solutions to such questions may include the extrapolation of scientific theories to untested regimes (such as the Planck epoch), or the inclusion of philosophical or religious ideas. | https://en.wikipedia.org/wiki/Cosmogony |
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models.
The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions
$$
\left\{ f\left( x,\cdot \right) \right\} _{x\in X}
$$
that are optimized.
## Statement
Let
$$
f(x,\alpha)
$$
and
$$
g_{j}(x,\alpha), j = 1,2, \ldots, m
$$
be real-valued continuously differentiable functions on
$$
\mathbb{R}^{n+l}
$$
, where
$$
x \in \mathbb{R}^{n}
$$
are choice variables and
$$
\alpha \in \mathbb{R}^{l}
$$
are parameters, and consider the problem of choosing
$$
x
$$
, for a given
$$
\alpha
$$
, so as to:
$$
\max_{x} f(x, \alpha)
$$
subject to
$$
g_{j}(x,\alpha) \geq 0, j = 1,2, \ldots, m
$$
and
$$
x \geq 0
$$
.
The Lagrangian expression of this problem is given by
$$
\mathcal{L} (x, \lambda, \alpha) = f(x, \alpha) + \lambda \cdot g(x, \alpha)
$$
where
$$
\lambda \in \mathbb{R}^{m}
$$
are the Lagrange multipliers. Now let
$$
x^{\ast}(\alpha)
$$
and
$$
\lambda^{\ast}(\alpha)
$$
together be the solution that maximizes the objective function f subject to the constraints (and hence are saddle points of the Lagrangian),
$$
\mathcal{L}^{\ast} (\alpha) \equiv f(x^{\ast}(\alpha), \alpha) + \lambda^{\ast}(\alpha) \cdot g(x^{\ast}(\alpha), \alpha),
$$
and define the value function
$$
V(\alpha) \equiv f(x^{\ast}(\alpha), \alpha).
$$
Then we have the following theorem.
Theorem: Assume that and are continuously differentiable. Then
$$
\frac{\partial V(\alpha)}{\partial \alpha_{k}} = \frac{\partial \mathcal{L}^{\ast} (\alpha)}{\partial \alpha_{k}} = \frac{\partial \mathcal{L} (x^{\ast} (\alpha), \lambda^{\ast} (\alpha), \alpha)}{\partial \alpha_{k}}, k = 1, 2, \ldots, l
$$
where .
## For arbitrary choice sets
Let
$$
X
$$
denote the choice set and let the relevant parameter be
$$
t\in \lbrack 0,1]
$$
. Letting
$$
f:X\times \lbrack 0,1]\rightarrow R
$$
denote the parameterized objective function, the value function
$$
V
$$
and the optimal choice correspondence (set-valued function)
$$
X^{\ast }
$$
are given by:
"Envelope theorems" describe sufficient conditions for the value function
$$
V
$$
to be differentiable in the parameter
$$
t
$$
and describe its derivative as
where
$$
f_{t}
$$
denotes the partial derivative of
$$
f
$$
with respect to
$$
t
$$
. Namely, the derivative of the value function with respect to the parameter equals the partial derivative of the objective function with respect to
$$
t
$$
holding the maximizer fixed at its optimal level.
Traditional envelope theorem derivations use the first-order condition for (), which requires that the choice set
$$
X
$$
have the convex and topological structure, and the objective function
$$
f
$$
be differentiable in the variable
$$
x
$$
. (The argument is that changes in the maximizer have only a "second-order effect" at the optimum and so can be ignored.) However, in many applications such as the analysis of incentive constraints in contract theory and game theory, nonconvex production problems, and "monotone" or "robust" comparative statics, the choice sets and objective functions generally lack the topological and convexity properties required by the traditional envelope theorems.
Paul Milgrom and Ilya Segal (2002) observe that the traditional envelope formula holds for optimization problems with arbitrary choice sets at any differentiability point of the value function, provided that the objective function is differentiable in the parameter:
Theorem 1: Let
$$
t\in \left( 0,1\right)
$$
and
$$
x\in X^{\ast }\left(t\right)
$$
. If both
$$
V^{\prime }\left( t\right)
$$
and
$$
f_{t}\left(x,t\right)
$$
exist, the envelope formula () holds.
Proof: Equation () implies that for
$$
x\in X^{\ast }\left( t\right)
$$
,
$$
\max_{s\in \left[ 0,1\right] }\left[ f\left( x,s\right) -V\left( s\right)\right] =f\left( x,t\right) -V\left( t\right) =0.
$$
Under the assumptions, the objective function of the displayed maximization problem is differentiable at
$$
s=t
$$
, and the first-order condition for this maximization is exactly equation (). Q.E.D.
While differentiability of the value function in general requires strong assumptions, in many applications weaker conditions such as absolute continuity, differentiability almost everywhere, or left- and right-differentiability, suffice. In particular, Milgrom and Segal's (2002) Theorem 2 offers a sufficient condition for
$$
V
$$
to be absolutely continuous, which means that it is differentiable almost everywhere and can be represented as an integral of its derivative:
Theorem 2: Suppose that
$$
f(x,\cdot )
$$
is absolutely continuous for all
$$
x\in X
$$
. Suppose also that there exists an integrable function
$$
b:[0,1]
$$
$$
\rightarrow
$$
$$
\mathbb{R}_{+}
$$
such that
$$
|f_{t}(x,t)|\leq b(t)
$$
for all
$$
x\in X
$$
and almost all
$$
t\in \lbrack 0,1]
$$
. Then
$$
V
$$
is absolutely continuous. Suppose, in addition, that
$$
f(x,\cdot )
$$
is differentiable for all
$$
x\in X
$$
, and that
$$
X^{\ast }(t)\neq \varnothing
$$
almost everywhere on
$$
[0,1]
$$
. Then for any selection
$$
x^{\ast }(t)\in X^{\ast }(t)
$$
,
Proof: Using ()(1), observe that for any
$$
t^{\prime},t^{\prime \prime }\in \lbrack 0,1]
$$
with
$$
t^{\prime }<t^{\prime \prime }
$$
,
$$
|V(t^{\prime \prime })-V(t^{\prime })| \leq \sup_{x\in X}|f(x,t^{\prime\prime })-f(x,t^{\prime })| =\sup_{x\in X}\left\vert \int_{t^{\prime }}^{t^{\prime \prime }}f_{t}(x,t)dt\right\vert \leq \int_{t^{\prime }}^{t^{\prime \prime
}}\sup_{x\in X}|f_{t}(x,t)|dt\leq \int_{t^{\prime }}^{t^{\prime \prime }}b(t)dt.
$$
This implies that
$$
V
$$
is absolutely continuous. Therefore,
$$
V
$$
is differentiable almost everywhere, and using () yields (). Q.E.D.
This result dispels the common misconception that nice behavior of the value function requires correspondingly nice behavior of the maximizer. Theorem 2 ensures the absolute continuity of the value function even though the maximizer may be discontinuous. In a similar vein, Milgrom and Segal's (2002) Theorem 3 implies that the value function must be differentiable at
$$
t=t_{0}
$$
and hence satisfy the envelope formula () when the family
$$
\left\{ f\left( x,\cdot \right) \right\} _{x\in X}
$$
is equi-differentiable at
$$
t_{0}\in \left( 0,1\right)
$$
and
$$
f_{t}\left(X^{\ast }\left( t\right) ,t_{0}\right)
$$
is single-valued and continuous at
$$
t=t_{0}
$$
, even if the maximizer is not differentiable at
$$
t_{0}
$$
(e.g., if
$$
X
$$
is described by a set of inequality constraints and the set of binding constraints changes at
$$
t_{0}
$$
).
## Applications
### Applications to producer theory
Theorem 1 implies Hotelling's lemma at any differentiability point of the profit function, and Theorem 2 implies the producer surplus formula. Formally, let
$$
\pi \left( p\right)
$$
denote the indirect profit function of a price-taking firm with production set
$$
X\subseteq \mathbb{R}^{L}
$$
facing prices
$$
p\in \mathbb{R}^{L}
$$
, and let
$$
x^{\ast }\left( p\right)
$$
denote the firm's supply function, i.e.,
$$
\pi (p)=\max_{x\in X}p\cdot x=p\cdot x^{\ast }\left( p\right) \text{.}
$$
Let
$$
t=p_{i}
$$
(the price of good
$$
i
$$
) and fix the other goods' prices at
$$
p_{-i}\in \mathbb{R}^{L-1}
$$
. Applying Theorem 1 to
$$
f(x,t)=tx_{i}+p_{-i}\cdot x_{-i}
$$
yields
$$
\frac{\partial \pi (p)}{\partial p_{i}}=x_{i}^{\ast }(p)
$$
(the firm's optimal supply of good
$$
i
$$
). Applying Theorem 2 (whose assumptions are verified when
$$
p_{i}
$$
is restricted to a bounded interval) yields
$$
\pi (t,p_{-i})-\pi (0,p_{-i})=\int_{0}^{p_{i}}x_{i}^{\ast }(s,p_{-i})ds,
$$
i.e. the producer surplus
$$
\pi (t,p_{-i})-\pi (0,p_{-i})
$$
can be obtained by integrating under the firm's supply curve for good
$$
i
$$
.
### Applications to mechanism design and auction theory
Consider an agent whose utility function
$$
f(x,t)
$$
over outcomes
$$
x\in \bar{X}
$$
depends on his type
$$
t\in \lbrack 0,1]
$$
. Let
$$
X\subseteq \bar{X}
$$
represent the "menu" of possible outcomes the agent could obtain in the mechanism by sending different messages. The agent's equilibrium utility
$$
V(t)
$$
in the mechanism is then given by (1), and the set
$$
X^{\ast }(t)
$$
of the mechanism's equilibrium outcomes is given by (2). Any selection
$$
x^{\ast }(t)\in X^{\ast }(t)
$$
is a choice rule implemented by the mechanism. Suppose that the agent's utility function
$$
f(x,t)
$$
is differentiable and absolutely continuous in
$$
t
$$
for all
$$
x\in Y
$$
, and that
$$
\sup_{x\in \bar{X}}|f_{t}(x,t)|
$$
is integrable on
$$
[0,1]
$$
. Then Theorem 2 implies that the agent's equilibrium utility
$$
V
$$
in any mechanism implementing a given choice rule
$$
x^{\ast }
$$
must satisfy the integral condition (4).
The integral condition (4) is a key step in the analysis of mechanism design problems with continuous type spaces. In particular, in Myerson's (1981) analysis of single-item auctions, the outcome from the viewpoint of one bidder can be described as
$$
x=\left( y,z\right)
$$
, where
$$
y
$$
is the bidder's probability of receiving the object and
$$
z
$$
is his expected payment, and the bidder's expected utility takes the form
$$
f\left( \left( y,z\right) ,t\right) =ty-z
$$
. In this case, letting
$$
\underline{t}
$$
denote the bidder's lowest possible type, the integral condition (4) for the bidder's equilibrium expected utility
$$
V
$$
takes the form
$$
V(t)-V(\underline{t})=\int_{0}^{t}y^{\ast }(s)ds.
$$
(This equation can be interpreted as the producer surplus formula for the firm whose production technology for converting numeraire
$$
z
$$
into probability
$$
y
$$
of winning the object is defined by the auction and which resells the object at a fixed price
$$
t
$$
). This condition in turn yields Myerson's (1981) celebrated revenue equivalence theorem: the expected revenue generated in an auction in which bidders have independent private values is fully determined by the bidders' probabilities
$$
y^{\ast }\left( t\right)
$$
of getting the object for all types
$$
t
$$
as well as by the expected payoffs
$$
V(\underline{t})
$$
of the bidders' lowest types. Finally, this condition is a key step in Myerson's (1981) of optimal auctions.
For other applications of the envelope theorem to mechanism design see Mirrlees (1971), Holmstrom (1979), Laffont and Maskin (1980), Riley and Samuelson (1981), Fudenberg and Tirole (1991), and Williams (1999). While these authors derived and exploited the envelope theorem by restricting attention to (piecewise) continuously differentiable choice rules or even narrower classes, it may sometimes be optimal to implement a choice rule that is not piecewise continuously differentiable. (One example is the class of trading problems with linear utility described in chapter 6.5 of Myerson (1991).) Note that the integral condition (3) still holds in this setting and implies such important results as Holmstrom's lemma (Holmstrom, 1979), Myerson's lemma (Myerson, 1981), the revenue equivalence theorem (for auctions), the Green–Laffont–Holmstrom theorem (Green and Laffont, 1979; Holmstrom, 1979), the Myerson–Satterthwaite inefficiency theorem (Myerson and Satterthwaite, 1983), the Jehiel–Moldovanu impossibility theorems (Jehiel and Moldovanu, 2001), the McAfee–McMillan weak-cartels theorem (McAfee and McMillan, 1992), and Weber's martingale theorem (Weber, 1983), etc. The details of these applications are provided in Chapter 3 of Milgrom (2004), who offers an elegant and unifying framework in auction and mechanism design analysis mainly based on the envelope theorem and other familiar techniques and concepts in demand theory.
### Applications to multidimensional parameter spaces
For a multidimensional parameter space
$$
T\subseteq \mathbb{R}^{K}
$$
, Theorem
1 can be applied to partial and directional derivatives of the value
function. If both the objective function
$$
f
$$
and the value function
$$
V
$$
are (totally) differentiable in
$$
t
$$
, Theorem 1 implies the envelope formula for their gradients:
$$
\nabla V\left( t\right) =\nabla _{t}f\left( x,t\right)
$$
for each
$$
x\in X^{\ast }\left( t\right)
$$
. While total differentiability of the value function may not be easy to ensure, Theorem 2 can be still applied along any smooth path connecting two parameter values
$$
t_{0}
$$
and
$$
t
$$
. Namely, suppose that functions
$$
f(x,\cdot )
$$
are differentiable for all
$$
x\in X
$$
with
$$
|\nabla _{t}f(x,t)|\leq B
$$
for all
$$
x\in X,
$$
$$
t\in T
$$
. A smooth path from
$$
t_{0}
$$
to
$$
t
$$
is described by a differentiable mapping
$$
\gamma :\left[ 0,1\right] \rightarrow T
$$
with a bounded derivative, such that
$$
\gamma \left( 0\right) =t_{0}
$$
and
$$
\gamma \left( 1\right) =t
$$
. Theorem 2 implies that for any such smooth path, the change of the value function can be expressed as the path integral of the partial gradient
$$
\nabla _{t}f(x^{\ast }(t),t)
$$
of the objective function along the path:
$$
V(t)-V(t_{0})=\int_{\gamma }\nabla _{t}f(x^{\ast }(s),s)\cdot ds.
$$
In particular, for
$$
t=t_{0}
$$
, this establishes that cyclic path integrals along any smooth path
$$
\gamma
$$
must be zero:
$$
\int \nabla _{t}f(x^{\ast }(s),s)\cdot ds=0.
$$
This "integrability condition" plays an important role in mechanism design with multidimensional types, constraining what kind of choice rules
$$
x^{\ast }
$$
can be sustained by mechanism-induced menus
$$
X\subseteq \bar{X}
$$
. In application to producer theory, with
$$
x\in X\subseteq \mathbb{R}^{L}
$$
being the firm's production vector and
$$
t\in \mathbb{R}^{L}
$$
being the price vector,
$$
f\left( x,t\right) =t\cdot x
$$
, and the integrability condition says that any rationalizable supply function
$$
x^{\ast }
$$
must satisfy
$$
\int x^{\ast }(s)\cdot ds=0.
$$
When
$$
x^{\ast }
$$
is continuously differentiable, this integrability condition is equivalent to the symmetry of the substitution matrix
$$
\left(\partial x_{i}^{\ast }\left( t\right) /\partial t_{j}\right) _{i,j=1}^{L}
$$
. (In consumer theory, the same argument applied to the expenditure minimization problem yields symmetry of the Slutsky matrix.)
### Applications to parameterized constraints
Suppose now that the feasible set
$$
X\left( t\right)
$$
depends on the parameter, i.e.,
$$
V(t) =\sup_{x\in X\left( t\right) }f(x,t)
$$
$$
X^{\ast }(t) =\{x\in X\left( t\right) :f(x,t)=V(t)\}\text{, }
$$
where
$$
X\left( t\right) =\left\{ x\in X:g\left( x,t\right) \geq 0\right\}
$$
for some
$$
g:X\times \left[ 0,1\right] \rightarrow \mathbb{R}^{K}.
$$
Suppose that
$$
X
$$
is a convex set,
$$
f
$$
and
$$
g
$$
are concave in
$$
x
$$
, and there exists
$$
\hat{x}\in X
$$
such that
$$
g\left( \hat{x},t\right) >0
$$
for all
$$
t\in \left[ 0,1\right]
$$
. Under these assumptions, it is well known that the above constrained optimization program can be represented as a saddle-point problem for the Lagrangian
$$
L\left( x,\lambda,t\right) =f(x,t)+\lambda\cdot g\left( x,t\right)
$$
, where
$$
\lambda \in \mathbb{R}_{+}^{K}
$$
is the vector of Lagrange multipliers chosen by the adversary to minimize the Lagrangian. This allows the application of Milgrom and Segal's (2002, Theorem 4) envelope theorem for saddle-point problems, under the additional assumptions that
$$
X
$$
is a compact set in a normed linear space,
$$
f
$$
and
$$
g
$$
are continuous in
$$
x
$$
, and
$$
f_{t}
$$
and
$$
g_{t}
$$
are continuous in
$$
\left( x,t\right)
$$
. In particular, letting
$$
\left( x^{\ast}(t),\lambda^{\ast }\left( t\right) \right)
$$
denote the Lagrangian's saddle point for parameter value
$$
t
$$
, the theorem implies that
$$
V
$$
is absolutely continuous and satisfies
$$
V(t)=V(0)+\int_{0}^{t}L_{t}(x^{\ast }(s),\lambda^{\ast }\left( s\right) ,s)ds.
$$
For the special case in which
$$
f\left( x,t\right)
$$
is independent of
$$
t
$$
,
$$
K=1
$$
, and
$$
g\left( x,t\right) =h\left( x\right) +t
$$
, the formula implies that
$$
V^{\prime }(t)=L_{t}(x^{\ast }(t),\lambda^{\ast }\left( t\right) ,t)=\lambda^{\ast}\left( t\right)
$$
for a.e.
$$
t
$$
. That is, the Lagrange multiplier
$$
\lambda^{\ast}\left( t\right)
$$
on the constraint is its "shadow price" in the optimization program.
### Other applications
Milgrom and Segal (2002) demonstrate that the generalized version of the envelope theorems can also be applied to convex programming, continuous optimization problems, saddle-point problems, and optimal stopping problems. | https://en.wikipedia.org/wiki/Envelope_theorem |
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as basic and connected by arithmetic operations (, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions.
The closed-form problem arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a closed-form expression of this object; that is, an expression of this object in terms of previous ways of specifying it.
## Example: roots of polynomials
The quadratic formula
$$
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
$$
is a closed form of the solutions to the general quadratic equation
$$
ax^2+bx+c=0.
$$
More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only th-roots and field operations
$$
(+, -, \times ,/).
$$
In fact, field theory allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.
There are expressions in radicals for all solutions of cubic equations (degree 3) and quartic equations (degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.
In higher degrees, the Abel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. A simple example is the equation
$$
x^5-x-1=0.
$$
Galois theory provides an algorithmic method for deciding whether a particular polynomial equation can be solved in radicals.
## Symbolic integration
Symbolic integration consists essentially of the search of closed forms for antiderivatives of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly logarithms, exponential function and polynomial roots. Functions that have a closed form for these basic functions are called elementary functions and include trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse hyperbolic functions.
The fundamental problem of symbolic integration is thus, given an elementary function specified by a closed-form expression, to decide whether its antiderivative is an elementary function, and, if it is, to find a closed-form expression for this antiderivative.
For rational functions; that is, for fractions of two polynomial functions; antiderivatives are not always rational fractions, but are always elementary functions that may involve logarithms and polynomial roots. This is usually proved with partial fraction decomposition. The need for logarithms and polynomial roots is illustrated by the formula
$$
\int\frac{f(x)}{g(x)}\,dx=\sum_{\alpha \in \operatorname{Roots}(g(x))} \frac{f(\alpha)}{g'(\alpha)}\ln(x-\alpha),
$$
which is valid if
$$
f
$$
and
$$
g
$$
are coprime polynomials such that
$$
g
$$
is square free and
$$
\deg f <\deg g.
$$
## Alternative definitions
Changing the basic functions to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be basic. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are basic since numerical implementations are widely available.
## Analytic expression
This is a term that is sometimes understood as a synonym for closed-form (see ) but this usage is contested (see ). It is unclear the extent to which this term is genuinely in use as opposed to the result of uncited earlier versions of this page.
## Comparison of different classes of expressions
The closed-form expressions do not include infinite series or continued fractions; neither includes integrals or limits. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.
Similarly, an equation or system of equations is said to have a closed-form solution if and only if at least one solution can be expressed as a closed-form expression; and it is said to have an analytic solution if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a "closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in and below. A closed-form or analytic solution is sometimes referred to as an explicit solution.
## Dealing with non-closed-form expressions
### Transformation into closed-form expressions
The expression:
$$
f(x) = \sum_{n=0}^\infty \frac{x}{2^n}
$$
is not in closed form because the summation entails an infinite number of elementary operations. However, by summing a geometric series this expression can be expressed in the closed form:
$$
f(x) = 2x.
$$
### Differential Galois theory
The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as differential Galois theory, by analogy with algebraic Galois theory.
The basic theorem of differential Galois theory is due to Joseph Liouville in the 1830s and 1840s and hence referred to as Liouville's theorem.
A standard example of an elementary function whose antiderivative does not have a closed-form expression is:
$$
e^{-x^2},
$$
whose one antiderivative is (up to a multiplicative constant) the error function:
$$
\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^x e^{-t^2} \, dt.
$$
### Mathematical modelling and computer simulation
Equations or systems too complex for closed-form or analytic solutions can often be analysed by mathematical modelling and computer simulation (for an example in physics, see).
## Closed-form number
Three subfields of the complex numbers have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouvillian numbers (not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouvillian numbers, denoted , form the smallest algebraically closed subfield of closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in . was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in , denoted , and referred to as EL numbers, is the smallest subfield of closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds to explicit algebraic, exponential, and logarithmic operations. "EL" stands both for "exponential–logarithmic" and as an abbreviation for "elementary".
Whether a number is a closed-form number is related to whether a number is transcendental. Formally, Liouvillian numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via transcendental number theory, in which a major result is the Gelfond–Schneider theorem, and a major open question is Schanuel's conjecture.
## Numerical computations
For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed. Some equations have no closed form solution, such as those that represent the Three-body problem or the Hodgkin–Huxley model. Therefore, the future states of these systems must be computed numerically.
## Conversion from numerical forms
There is software that attempts to find closed-form expressions for numerical values, including RIES, in Maple and SymPy, Plouffe's Inverter, and the Inverse Symbolic Calculator. | https://en.wikipedia.org/wiki/Closed-form_expression |
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games . The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.
The Folk Theorem suggests that if the players are patient enough and far-sighted (i.e. if the discount factor
$$
\delta \to 1
$$
), then repeated interaction can result in virtually any average payoff in an SPE equilibrium. "Virtually any" is here technically defined as "feasible" and "individually rational".
## Setup and definitions
We start with a basic game, also known as the stage game, which is an n-player game. In this game, each player has finitely many actions to choose from, and they make their choices simultaneously and without knowledge of the other player's choices. The collective choices of the players leads to a payoff profile, i.e. to a payoff for each of the players. The mapping from collective choices to payoff profiles is known to the players, and each player aims to maximize their payoff. If the collective choice is denoted by x, the payoff that player i receives, also known as player i's utility, will be denoted by
$$
u_i(x)
$$
.
We then consider a repetition of this stage game, finitely or infinitely many times. In each repetition, each player chooses one of their stage game options, and when making that choice, they may take into account the choices of the other players in the prior iterations. In this repeated game, a strategy for one of the players is a deterministic rule that specifies the player's choice in each iteration of the stage game, based on all other player's choices in the prior iterations. A choice of strategy for each of the players is a strategy profile, and it leads to a payout profile for the repeated game. There are a number of different ways such a strategy profile can be translated into a payout profile, outlined below.
Any Nash equilibrium payoff profile of a repeated game must satisfy two properties:
1. Individual rationality: the payoff must weakly dominate the minmax payoff profile of the constituent stage game. That is, the equilibrium payoff of each player must be at least as large as the minmax payoff of that player. This is because a player achieving less than their minmax payoff always has incentive to deviate by simply playing their minmax strategy at every history.
1. Feasibility: the payoff must be a convex combination of possible payoff profiles of the stage game. This is because the payoff in a repeated game is just a weighted average of payoffs in the basic games.
Folk theorems are partially converse claims: they say that, under certain conditions (which are different in each folk theorem), every payoff profile that is both individually rational and feasible can be realized as a Nash equilibrium payoff profile of the repeated game.
There are various folk theorems; some relate to finitely-repeated games while others relate to infinitely-repeated games.
## Infinitely-repeated games without discounting
In the undiscounted model, the players are patient. They do not differentiate between utilities in different time periods. Hence, their utility in the repeated game is represented by the sum of utilities in the basic games.
When the game is infinite, a common model for the utility in the infinitely-repeated game is the limit inferior of mean utility: If the game results in a path of outcomes
$$
x_t
$$
, where
$$
x_t
$$
denotes the collective choices of the players at iteration t (t=0,1,2,...), player i utility is defined as
$$
U_i = \liminf_{T\to \infty} \frac{1}{T} \sum_{t=0}^T u_i(x_t),
$$
where
$$
u_i
$$
is the basic-game utility function of player i.
An infinitely-repeated game without discounting is often called a "supergame".
The folk theorem in this case is very simple and contains no pre-conditions: every individually rational and feasible payoff profile in the basic game is a Nash equilibrium payoff profile in the repeated game.
The proof employs what is called a grim or grim trigger strategy. All players start by playing the prescribed action and continue to do so until someone deviates. If player i deviates, all other players switch to picking the action which minmaxes player i forever after. The one-stage gain from deviation contributes 0 to the total utility of player i. The utility of a deviating player cannot be higher than his minmax payoff. Hence all players stay on the intended path and this is indeed a Nash equilibrium.
###
### Subgame perfection
The above Nash equilibrium is not always subgame perfect. If punishment is costly for the punishers, the threat of punishment is not credible.
A subgame perfect equilibrium requires a slightly more complicated strategy. The punishment should not last forever; it should last only a finite time which is sufficient to wipe out the gains from deviation. After that, the other players should return to the equilibrium path.
The limit-of-means criterion ensures that any finite-time punishment has no effect on the final outcome. Hence, limited-time punishment is a subgame-perfect equilibrium.
- Coalition subgame-perfect equilibria: An equilibrium is called a coalition Nash equilibrium if no coalition can gain from deviating. It is called a coalition subgame-perfect equilibrium if no coalition can gain from deviating after any history. With the limit-of-means criterion, a payoff profile is attainable in coalition-Nash-equilibrium or in coalition-subgame-perfect-equilibrium, if-and-only-if it is Pareto efficient and weakly-coalition-individually-rational.
### Overtaking
Some authors claim that the limit-of-means criterion is unrealistic, because it implies that utilities in any finite time-span contribute 0 to the total utility. However, if the utilities in any finite time-span contribute a positive value, and the value is undiscounted, then it is impossible to attribute a finite numeric utility to an infinite outcome sequence. A possible solution to this problem is that, instead of defining a numeric utility for each infinite outcome sequence, we just define the preference relation between two infinite sequences. We say that agent
$$
i
$$
(strictly) prefers the sequence of outcomes
$$
y_t
$$
over the sequence
$$
x_t
$$
, if:
$$
\liminf_{T\to \infty} \sum_{t=0}^T ( u_i(y_t) - u_i(x_t)) > 0
$$
For example, consider the sequences
$$
u_i(x)=(0,0,0,0,\ldots)
$$
and
$$
u_i(y)=(-1,2,0,0,\ldots)
$$
. According to the limit-of-means criterion, they provide the same utility to player i, but according to the overtaking criterion,
$$
y
$$
is better than
$$
x
$$
for player i. See overtaking criterion for more information.
The folk theorems with the overtaking criterion are slightly weaker than with the limit-of-means criterion. Only outcomes that are strictly individually rational, can be attained in Nash equilibrium. This is because, if an agent deviates, he gains in the short run, and this gain can be wiped out only if the punishment gives the deviator strictly less utility than the agreement path. The following folk theorems are known for the overtaking criterion:
- Strict stationary equilibria: A Nash equilibrium is called strict if each player strictly prefers the infinite sequence of outcomes attained in equilibrium, over any other sequence he can deviate to. A Nash equilibrium is called stationary if the outcome is the same in each time-period. An outcome is attainable in strict-stationary-equilibrium if-and-only-if for every player the outcome is strictly better than the player's minimax outcome.
- Strict stationary subgame-perfect equilibria: An outcome is attainable in strict-stationary-subgame-perfect-equilibrium, if for every player the outcome is strictly better than the player's minimax outcome (note that this is not an "if-and-only-if" result). To achieve subgame-perfect equilibrium with the overtaking criterion, it is required to punish not only the player that deviates from the agreement path, but also every player that does not cooperate in punishing the deviant.
- The "stationary equilibrium" concept can be generalized to a "periodic equilibrium", in which a finite number of outcomes is repeated periodically, and the payoff in a period is the arithmetic mean of the payoffs in the outcomes. That mean payoff should be strictly above the minimax payoff.
- Strict stationary coalition equilibria: With the overtaking criterion, if an outcome is attainable in coalition-Nash-equilibrium, then it is Pareto efficient and weakly-coalition-individually-rational. On the other hand, if it is Pareto efficient and strongly-coalition-individually-rational it can be attained in strict-stationary-coalition-equilibrium.
## Infinitely-repeated games with discounting
Assume that the payoff of a player in an infinitely repeated game is given by the average discounted criterion with discount factor 0 < δ < 1:
$$
U_i = (1-\delta) \sum_{t \geq 0} \delta^t u_i(x_t),
$$
The discount factor indicates how patient the players are. The factor
$$
(1-\delta)
$$
is introduced so that the payoff remain bounded when
$$
\delta\rightarrow 1
$$
.
The folk theorem in this case requires that the payoff profile in the repeated game strictly dominates the minmax payoff profile (i.e., each player receives strictly more than the minmax payoff).
Let a be a strategy profile of the stage game with payoff profile u which strictly dominates the minmax payoff profile. One can define a Nash equilibrium of the game with u as resulting payoff profile as follows:
1. All players start by playing a and continue to play a if no deviation occurs.
2. If any one player, say player i, deviated, play the strategy profile m which minmaxes i forever after.
3. Ignore multilateral deviations.
If player i gets ε more than his minmax payoff each stage by following 1, then the potential loss from punishment is
$$
\frac{1}{1-\delta} \varepsilon.
$$
If δ is close to 1, this outweighs any finite one-stage gain, making the strategy a Nash equilibrium.
An alternative statement of this folk theorem allows the equilibrium payoff profile u to be any individually rational feasible payoff profile; it only requires there exist an individually rational feasible payoff profile that strictly dominates the minmax payoff profile. Then, the folk theorem guarantees that it is possible to approach u in equilibrium to any desired precision (for every ε there exists a Nash equilibrium where the payoff profile is a distance ε away from u).
Subgame perfection
Attaining a subgame perfect equilibrium in discounted games is more difficult than in undiscounted games. The cost of punishment does not vanish (as with the limit-of-means criterion).
It is not always possible to punish the non-punishers endlessly (as with the overtaking criterion) since the discount factor makes punishments far away in the future irrelevant for the present. Hence, a different approach is needed: the punishers should be rewarded.
This requires an additional assumption, that the set of feasible payoff profiles is full dimensional and the min-max profile lies in its interior. The strategy is as follows.
1. All players start by playing a and continue to play a if no deviation occurs.
2. If any one player, say player i, deviated, play the strategy profile m which minmaxes i for N periods. (Choose N and δ large enough so that no player has incentive to deviate from phase 1.)
3. If no players deviated from phase 2, all player j ≠ i gets rewarded ε above j min-max forever after, while player i continues receiving his min-max. (Full-dimensionality and the interior assumption is needed here.)
4. If player j deviated from phase 2, all players restart phase 2 with j as target.
5. Ignore multilateral deviations.
Player j ≠ i now has no incentive to deviate from the punishment phase 2. This proves the subgame perfect folk theorem.
## Finitely-repeated games without discount
Assume that the payoff of player i in a game that is repeated T times is given by a simple arithmetic mean:
$$
U_i = \frac{1}{T} \sum_{t=0}^T u_i(x_t)
$$
A folk theorem for this case has the following additional requirement:
In the basic game, for every player i, there is a Nash-equilibrium
$$
E_i
$$
that is strictly better, for i, than his minmax payoff.
This requirement is stronger than the requirement for discounted infinite games, which is in turn stronger than the requirement for undiscounted infinite games.
This requirement is needed because of the last step. In the last step, the only stable outcome is a Nash-equilibrium in the basic game. Suppose a player i gains nothing from the Nash equilibrium (since it gives him only his minmax payoff). Then, there is no way to punish that player.
On the other hand, if for every player there is a basic equilibrium which is strictly better than minmax, a repeated-game equilibrium can be constructed in two phases:
1. In the first phase, the players alternate strategies in the required frequencies to approximate the desired payoff profile.
1. In the last phase, the players play the preferred equilibrium of each of the players in turn.
In the last phase, no player deviates since the actions are already a basic-game equilibrium. If an agent deviates in the first phase, he can be punished by minmaxing him in the last phase. If the game is sufficiently long, the effect of the last phase is negligible, so the equilibrium payoff approaches the desired profile.
## Applications
Folk theorems can be applied to a diverse number of fields. For example:
- Anthropology: in a community where all behavior is well known, and where members of the community know that they will continue to have to deal with each other, then any pattern of behavior (traditions, taboos, etc.) may be sustained by social norms so long as the individuals of the community are better off remaining in the community than they would be leaving the community (the minimax condition).
- International politics: agreements between countries cannot be effectively enforced. They are kept, however, because relations between countries are long-term and countries can use "minimax strategies" against each other. This possibility often depends on the discount factor of the relevant countries. If a country is very impatient (pays little attention to future outcomes), then it may be difficult to punish it (or punish it in a credible way).
On the other hand, MIT economist Franklin Fisher has noted that the folk theorem is not a positive theory. In considering, for instance, oligopoly behavior, the folk theorem does not tell the economist what firms will do, but rather that cost and demand functions are not sufficient for a general theory of oligopoly, and the economists must include the context within which oligopolies operate in their theory.
In 2007, Borgs et al. proved that, despite the folk theorem, in the general case computing the Nash equilibria for repeated games is not easier than computing the Nash equilibria for one-shot finite games, a problem which lies in the PPAD complexity class. The practical consequence of this is that no efficient (polynomial-time) algorithm is known that computes the strategies required by folk theorems in the general case.
## Summary of folk theorems
The following table compares various folk theorems in several aspects:
- Horizon – whether the stage game is repeated finitely or infinitely many times.
- Utilities – how the utility of a player in the repeated game is determined from the player's utilities in the stage game iterations.
- Conditions on G (the stage game) – whether there are any technical conditions that should hold in the one-shot game in order for the theorem to work.
- Conditions on x (the target payoff vector of the repeated game) – whether the theorem works for any individually rational and feasible payoff vector, or only on a subset of these vectors.
- Equilibrium type – if all conditions are met, what kind of equilibrium is guaranteed by the theorem – Nash or Subgame-perfect?
- Punishment type – what kind of punishment strategy is used to deter players from deviating?
Published by Horizon Utilities Conditions on G Conditions on x Guarantee Equilibrium type Punishment type Benoit& Krishna Finite () Arithmetic mean For every player there is an equilibrium payoff strictly better than minimax. None For all there is such that, if , for every there is equilibrium with payoff -close to . Nash Aumann& Shapley Infinite Limit of means None None Payoff exactly . Nash Grim Aumann& Shapley and Rubinstein Infinite Limit of means None None Payoff exactly . Subgame-perfect Limited-time punishment. Rubinstein Infinite Overtaking None Strictly above minimax. Single outcome or a periodic sequence. Subgame-perfect Punishing non-punishers. Rubinstein Infinite Limit of means None Pareto-efficient and weakly-coalition-individually-rational None Coalition-subgame-perfect Rubinstein Infinite Overtaking None Pareto-efficient and strongly-coalition-individually-rational None Coalition-Nash Fudenberg& Maskin Infinite Sum with discount Correlated mixed strategies are allowed. Strictly above minimax. When is sufficiently near 1, there is an equilibrium with payoff exactly . Nash Grim Fudenberg& Maskin Infinite Sum with discount Only pure strategies are allowed. Strictly above minimax. For all there is such that, if , for every there is an equilibrium with payoff -close to . Nash Grim punishment. Friedman (1971, 1977) Infinite Sum with discount Correlated mixed strategies are allowed. Strictly above a Nash-equilibrium in G. When is sufficiently near 1, there is equilibrium with payoff exactly . Subgame-perfect Grim punishment using the Nash-equilibrium. Fudenberg& Maskin Infinite Sum with discount Two players Strictly above minimax. For all there is such that, if , there is equilibrium with payoff exactly . Subgame-perfect Limited-time punishment. Fudenberg& Maskin Infinite Sum with discount The IR feasible space is full-dimensional.There is a collection of IR feasible outcomes , one per player, such that for every players , and .
Strictly above minimax. For all there is such that, if , there is equilibrium with payoff exactly . Subgame-perfect Rewarding the punishers.
## Folk theorems in other settings
In allusion to the folk theorems for repeated games, some authors have used the term "folk theorem" to refer to results on the set of possible equilibria or equilibrium payoffs in other settings, especially if the results are similar in what equilibrium payoffs they allow. For instance, Tennenholtz proves a "folk theorem" for program equilibrium. Many other folk theorems have been proved in settings with commitment.
## Notes
## References
-
-
-
- A set of introductory notes to the Folk Theorem.
Category:Game theory equilibrium concepts
Category:Theorems | https://en.wikipedia.org/wiki/Folk_theorem_%28game_theory%29 |
The Capability Maturity Model (CMM) is a development model created in 1986 after a study of data collected from organizations that contracted with the U.S. Department of Defense, who funded the research. The term "maturity" relates to the degree of formality and optimization of processes, from ad hoc practices, to formally defined steps, to managed result metrics, to active optimization of the processes.
The model's aim is to improve existing software development processes, but it can also be applied to other processes.
In 2006, the Software Engineering Institute at Carnegie Mellon University developed the
### Capability Maturity Model Integration
, which has largely superseded the CMM and addresses some of its drawbacks.
## Overview
The Capability Maturity Model was originally developed as a tool for objectively assessing the ability of government contractors' processes to implement a contracted software project. The model is based on the process maturity framework first described in IEEE Software and, later, in the 1989 book Managing the Software Process by Watts Humphrey. It was later published as an article in 1993
and as a book by the same authors in 1994.
Though the model comes from the field of software development, it is also used as a model to aid in business processes generally, and has also been used extensively worldwide in government offices, commerce, and industry.
## History
### Prior need for software processes
In the 1980s, the use of computers grew more widespread, more flexible and less costly. Organizations began to adopt computerized information systems, and the demand for software development grew significantly. Many processes for software development were in their infancy, with few standard or "best practice" approaches defined.
As a result, the growth was accompanied by growing pains: project failure was common, the field of computer science was still in its early years, and the ambitions for project scale and complexity exceeded the market capability to deliver adequate products within a planned budget. Individuals such as Edward Yourdon, Larry Constantine, Gerald Weinberg, Tom DeMarco, and David Parnas began to publish articles and books with research results in an attempt to professionalize the software-development processes.
In the 1980s, several US military projects involving software subcontractors ran over-budget and were completed far later than planned, if at all. In an effort to determine why this was occurring, the United States Air Force funded a study at the Software Engineering Institute (SEI).
### Precursor
The first application of a staged maturity model to IT was not by CMU/SEI, but rather by Richard L. Nolan, who, in 1973 published the stages of growth model for IT organizations.
Watts Humphrey began developing his process maturity concepts during the later stages of his 27-year career at IBM.
### Development at Software Engineering Institute
Active development of the model by the US Department of Defense Software Engineering Institute (SEI) began in 1986 when Humphrey joined the Software Engineering Institute located at Carnegie Mellon University in Pittsburgh, Pennsylvania after retiring from IBM. At the request of the U.S. Air Force he began formalizing his Process Maturity Framework to aid the U.S. Department of Defense in evaluating the capability of software contractors as part of awarding contracts.
The result of the Air Force study was a model for the military to use as an objective evaluation of software subcontractors' process capability maturity. Humphrey based this framework on the earlier Quality Management Maturity Grid developed by Philip B. Crosby in his book "Quality is Free". Humphrey's approach differed because of his unique insight that organizations mature their processes in stages based on solving process problems in a specific order. Humphrey based his approach on the staged evolution of a system of software development practices within an organization, rather than measuring the maturity of each separate development process independently. The CMMI has thus been used by different organizations as a general and powerful tool for understanding and then improving general business process performance.
Watts Humphrey's Capability Maturity Model (CMM) was published in 1988 and as a book in 1989, in Managing the Software Process.
Organizations were originally assessed using a process maturity questionnaire and a Software Capability Evaluation method devised by Humphrey and his colleagues at the Software Engineering Institute.
The full representation of the Capability Maturity Model as a set of defined process areas and practices at each of the five maturity levels was initiated in 1991, with Version 1.1 being published in July 1993.
The CMM was published as a book in 1994 by the same authors Mark C. Paulk, Charles V. Weber, Bill Curtis, and Mary Beth Chrissis.
Capability Maturity Model Integration
The CMMI model's application in software development has sometimes been problematic. Applying multiple models that are not integrated within and across an organization could be costly in training, appraisals, and improvement activities. The Capability Maturity Model Integration (CMMI) project was formed to sort out the problem of using multiple models for software development processes, thus the CMMI model has superseded the CMM model, though the CMM model continues to be a general theoretical process capability model used in the public domain.
In 2016, the responsibility for CMMI was transferred to the Information Systems Audit and Control Association (ISACA). ISACA subsequently released CMMI v2.0 in 2021. It was upgraded again to CMMI v3.0 in 2023. CMMI now places a greater emphasis on the process architecture which is typically realized as a process diagram. Copies of CMMI are available now only by subscription.
### Adapted to other processes
The CMMI was originally intended as a tool to evaluate the ability of government contractors to perform a contracted software project. Though it comes from the area of software development, it can be, has been, and continues to be widely applied as a general model of the maturity of process (e.g., IT service management processes) in IS/IT (and other) organizations.
## Model topics
### Maturity models
A maturity model can be viewed as a set of structured levels that describe how well the behaviors, practices and processes of an organization can reliably and sustainably produce required outcomes.
A maturity model can be used as a benchmark for comparison and as an aid to understanding - for example, for comparative assessment of different organizations where there is something in common that can be used as a basis for comparison. In the case of the CMM, for example, the basis for comparison would be the organizations' software development processes.
### Structure
The model involves five aspects:
- Maturity
### Levels
: a 5-level process maturity continuum - where the uppermost (5th) level is a notional ideal state where processes would be systematically managed by a combination of process optimization and continuous process improvement.
- Key Process Areas: a Key Process Area identifies a cluster of related activities that, when performed together, achieve a set of goals considered important.
- Goals: the goals of a key process area summarize the states that must exist for that key process area to have been implemented in an effective and lasting way. The extent to which the goals have been accomplished is an indicator of how much capability the organization has established at that maturity level. The goals signify the scope, boundaries, and intent of each key process area.
- Common Features: common features include practices that implement and institutionalize a key process area. There are five types of common features: commitment to perform, ability to perform, activities performed, measurement and analysis, and verifying implementation.
- Key Practices: The key practices describe the elements of infrastructure and practice that contribute most effectively to the implementation and institutionalization of the area.
Levels
There are five levels defined along the continuum of the model and, according to the SEI: "Predictability, effectiveness, and control of an organization's software processes are believed to improve as the organization moves up these five levels. While not rigorous, the empirical evidence to date supports this belief".
1. Initial (chaotic, ad hoc, individual heroics) - the starting point for use of a new or undocumented repeat process.
1. Repeatable - the process is at least documented sufficiently such that repeating the same steps may be attempted.
1. Defined - the process is defined/confirmed as a standard business process
1. Capable - the process is quantitatively managed in accordance with agreed-upon metrics.
1. Efficient - process management includes deliberate process optimization/improvement.
Within each of these maturity levels are Key Process Areas which characterise that level, and for each such area there are five factors: goals, commitment, ability, measurement, and verification. These are not necessarily unique to CMMI, representing — as they do — the stages that organizations must go through on the way to becoming mature.
The model provides a theoretical continuum along which process maturity can be developed incrementally from one level to the next. Skipping levels is not allowed/feasible.
Level 1 - Initial It is characteristic of processes at this level that they are (typically) undocumented and in a state of dynamic change, tending to be driven in an ad hoc, uncontrolled and reactive manner by users or events. This provides a chaotic or unstable environment for the processes. (Example - a surgeon performing a new operation a small number of times - the levels of negative outcome are not known).
Level 2 - Repeatable It is characteristic of this level of maturity that some processes are repeatable, possibly with consistent results. Process discipline is unlikely to be rigorous, but where it exists it may help to ensure that existing processes are maintained during times of stress.
Level 3 - Defined It is characteristic of processes at this level that there are sets of defined and documented standard processes established and subject to some degree of improvement over time. These standard processes are in place. The processes may not have been systematically or repeatedly used - sufficient for the users to become competent or the process to be validated in a range of situations. This could be considered a developmental stage - with use in a wider range of conditions and user competence development the process can develop to next level of maturity.
Level 4 - Managed (Capable) It is characteristic of processes at this level that, using process metrics, effective achievement of the process objectives can be evidenced across a range of operational conditions. The suitability of the process in multiple environments has been tested and the process refined and adapted. Process users have experienced the process in multiple and varied conditions, and are able to demonstrate competence. The process maturity enables adaptions to particular projects without measurable losses of quality or deviations from specifications. Process Capability is established from this level. (Example - surgeon performing an operation hundreds of times with levels of negative outcome approaching zero).
Level 5 - Optimizing (Efficient)It is a characteristic of processes at this level that the focus is on continually improving process performance through both incremental and innovative technological changes/improvements. At maturity level 5, processes are concerned with addressing statistical common causes of process variation and changing the process (for example, to shift the mean of the process performance) to improve process performance. This would be done at the same time as maintaining the likelihood of achieving the established quantitative process-improvement objectives.
Between 2008 and 2019, about 12% of appraisals given were at maturity levels 4 and 5.
### Critique
The model was originally intended to evaluate the ability of government contractors to perform a software project. It has been used for and may be suited to that purpose, but critics pointed out that process maturity according to the CMM was not necessarily mandatory for successful software development.
### Software process framework
The software process framework documented is intended to guide those wishing to assess an organization's or project's consistency with the Key Process Areas. For each maturity level there are five checklist types:
{| class="wikitable"
|-
! Type
! Description
|-
| Policy
|Describes the policy contents and KPA goals recommended by the Key Process Areas.
|-
| Standard
|Describes the recommended content of select work products described in the Key Process Areas.
|-
| Process
| Describes the process information content recommended by the Key Process Areas. These are refined into checklists for:
- Roles, entry criteria, inputs, activities, outputs, exit criteria, reviews and audits, work products managed and controlled, measurements, documented procedures, training, and tools
|-
| Procedure
| Describes the recommended content of documented procedures described in the Key Process Areas.
|-
| Level overcome
| Provides an overview of an entire maturity level. These are further refined into checklists for:
- Key Process Areas purposes, goals, policies, and standards; process descriptions; procedures; training; tools; reviews and audits; work products; measurements
|} | https://en.wikipedia.org/wiki/Capability_Maturity_Model |
In mathematics,
### Hausdorff dimension
is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.
More specifically, the Hausdorff dimension is a dimensional number associated with a metric space, i.e. a set where the distances between all members are defined. The dimension is drawn from the extended real numbers,
$$
\overline{\mathbb{R}}
$$
, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
## Intuition
The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.
Every space-filling curve hits some points multiple times and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called Lebesgue covering dimension, explains why. This dimension is the greatest integer n such that in every covering of X by small open balls there is at least one point where n + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1.
But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.
The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/rd as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.
For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But Benoit Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes one sees is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
For fractals that occur in nature, the Hausdorff and box-counting dimension coincide. The packing dimension is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.
## Formal definition
The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outer measure is constructed:
Let
$$
X
$$
be a metric space. If
$$
S\subset X
$$
and
$$
d\in [0,\infty)
$$
,
$$
H^d_\delta(S)=\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S, \operatorname{diam} U_i<\delta\right \},
$$
where the infimum is taken over all countable covers
$$
U
$$
of
$$
S
$$
. The Hausdorff d-dimensional outer measure is then defined as
$$
\mathcal{H}^d(S)=\lim_{\delta\to 0}H^d_\delta(S)
$$
, and the restriction of the mapping to measurable sets justifies it as a measure, called the
$$
d
$$
-dimensional Hausdorff Measure.
Hausdorff dimension
The Hausdorff dimension
$$
\dim_{\operatorname{H}}{(X)}
$$
of
$$
X
$$
is defined by
$$
\dim_{\operatorname{H}}{(X)}:=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.
$$
This is the same as the supremum of the set of
$$
d\in [0,\infty)
$$
such that the
$$
d
$$
-dimensional Hausdorff measure of
$$
X
$$
is infinite (except that when this latter set of numbers
$$
d
$$
is empty the Hausdorff dimension is zero).
### Hausdorff content
The
$$
d
$$
-dimensional unlimited Hausdorff content of
$$
S
$$
is defined by
$$
C_H^d(S):= H_\infty^d(S) = \inf\left \{ \sum_{k=1}^\infty (\operatorname{diam} U_k)^d: \bigcup_{k=1}^\infty U_k\supseteq S \right \}
$$
In other words,
$$
C_H^d(S)
$$
has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes. (Here we use the standard convention that .) The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree.
## Examples
- Countable sets have Hausdorff dimension 0.
- The Euclidean space
$$
\R^n
$$
has Hausdorff dimension
$$
n
$$
, and the circle
$$
S^1
$$
has Hausdorff dimension 1.
- Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set, a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63. The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58. These Hausdorff dimensions are related to the "critical exponent" of the Master theorem for solving recurrence relations in the analysis of algorithms.
- Space-filling curves like the Peano curve have the same Hausdorff dimension as the space they fill.
- The trajectory of Brownian motion in dimension 2 and above is conjectured to be Hausdorff dimension 2.
- Lewis Fry Richardson performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain.
## Properties of Hausdorff dimension
### Hausdorff dimension and inductive dimension
Let X be an arbitrary separable metric space. There is a topological notion of inductive dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dimind(X).
Theorem. Suppose X is non-empty. Then
$$
\dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X).
$$
Moreover,
$$
\inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X),
$$
where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric dY of Y is topologically equivalent to dX.
These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
### Hausdorff dimension and Minkowski dimension
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
### Hausdorff dimensions and Frostman measures
If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ rs holds for some constant s > 0 and for every ball B(x, r) in X, then dimHaus(X) ≥ s. A partial converse is provided by Frostman's lemma.
### Behaviour under unions and products
If
$$
X=\bigcup_{i\in I}X_i
$$
is a finite or countable union, then
$$
\dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).
$$
This can be verified directly from the definition.
If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
$$
\dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).
$$
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of Rn, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).
## Self-similar sets
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below.
Theorem. Suppose
are each a contraction mapping on Rn with contraction constant ri < 1. Then there is a unique non-empty compact set A such that
The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance.
### The open set condition
To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition (OSC) on the sequence of contractions ψi.
There is an open set V with compact closure, such that
$$
\bigcup_{i=1}^m\psi_i (V) \subseteq V,
$$
where the sets in union on the left are pairwise disjoint.
The open set condition is a separation condition that ensures the images ψi(V) do not overlap "too much".
Theorem. Suppose the open set condition holds and each ψi is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is s where s is the unique solution of
$$
\sum_{i=1}^m r_i^s = 1.
$$
The contraction coefficient of a similitude is the magnitude of the dilation.
In general, a set E which is carried onto itself by a mapping
$$
A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A)
$$
is self-similar if and only if the intersections satisfy the following condition:
$$
H^s\left(\psi_i(E)\cap \psi_j(E)\right) =0,
$$
where s is the Hausdorff dimension of E and Hs denotes s-dimensional Hausdorff measure. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:
Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar. | https://en.wikipedia.org/wiki/Hausdorff_dimension |
The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable
$$
S = \sum_{i=1}^N X_i\,
$$
where both
$$
N\,
$$
and
$$
X_i\,
$$
are random variables and of special types. In more general cases the distribution of S is a compound distribution. The recursion for the special cases considered was introduced in a paper by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo). It is heavily used in actuarial science (see also systemic risk).
## Preliminaries
We are interested in the compound random variable
$$
S = \sum_{i=1}^N X_i\,
$$
where
$$
N\,
$$
and
$$
X_i\,
$$
fulfill the following preconditions.
### Claim size distribution
We assume the
$$
X_i\,
$$
to be i.i.d. and independent of
$$
N\,
$$
. Furthermore the
$$
X_i\,
$$
have to be distributed on a lattice
$$
h \mathbb{N}_0\,
$$
with latticewidth
$$
h>0\,
$$
.
$$
f_k = P[X_i = hk].\,
$$
In actuarial practice,
$$
X_i\,
$$
is obtained by discretisation of the claim density function (upper, lower...).
### Claim number distribution
The number of claims N is a random variable, which is said to have a "claim number distribution", and which can take values 0, 1, 2, .... etc.. For the "Panjer recursion", the probability distribution of N has to be a member of the Panjer class, otherwise known as the (a,b,0) class of distributions. This class consists of all counting random variables which fulfill the following relation:
$$
P[N=k] = p_k= \left(a + \frac{b}{k} \right) \cdot p_{k-1},~~k \ge 1.\,
$$
for some
$$
a
$$
and
$$
b
$$
which fulfill
$$
a+b \ge 0\,
$$
. The initial value
$$
p_0\,
$$
is determined such that
$$
\sum_{k=0}^\infty p_k = 1.\,
$$
The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of S. In the following
$$
W_N(x)\,
$$
denotes the probability generating function of N: for this see the table in (a,b,0) class of distributions.
In the case of claim number is known, please note the De Pril algorithm. This algorithm is suitable to compute the sum distribution of
$$
n
$$
discrete random variables.
## Recursion
The algorithm now gives a recursion to compute the
$$
g_k =P[S = hk] \,
$$
.
The starting value is
$$
g_0 = W_N(f_0)\,
$$
with the special cases
$$
g_0=p_0\cdot \exp(f_0 b) \quad \text{ if } \quad a = 0,\,
$$
and
$$
g_0=\frac{p_0}{(1-f_0a)^{1+b/a}} \quad \text{ for } \quad a \ne 0,\,
$$
and proceed with
$$
g_k=\frac{1}{1-f_0a}\sum_{j=1}^k \left( a+\frac{b\cdot j}{k} \right) \cdot f_j \cdot g_{k-j}.\,
$$
## Example
The following example shows the approximated density of
$$
\scriptstyle S \,=\, \sum_{i=1}^N X_i
$$
where
$$
\scriptstyle N\, \sim\, \text{NegBin}(3.5,0.3)\,
$$
and
$$
\scriptstyle X \,\sim \,\text{Frechet}(1.7,1)
$$
with lattice width h = 0.04. (See Fréchet distribution.)
As observed, an issue may arise at the initialization of the recursion. Guégan and Hassani (2009) have proposed a solution to deal with that issue
.
## References
## External links
- Panjer recursion and the distributions it can be used with
Category:Actuarial science
Category:Compound probability distributions
Category:Theory of probability distributions | https://en.wikipedia.org/wiki/Panjer_recursion |
A wave tank is a laboratory setup for observing the behavior of surface waves. The typical wave tank is a box filled with liquid, usually water, leaving open or air-filled space on top. At one end of the tank, an actuator generates waves; the other end usually has a wave-absorbing surface. A similar device is the ripple tank, which is flat and shallow and used for observing patterns of surface waves from above.
## Wave basin
A wave basin is a wave tank which has a width and length of comparable magnitude, often used for testing ships, offshore structures and three-dimensional models of harbors (and their breakwaters).
## Wave flume
A wave flume (or wave channel) is a special sort of wave tank: the width of the flume is much less than its length. The generated waves are therefore – more or less – two-dimensional in a vertical plane (2DV), meaning that the orbital flow velocity component in the direction perpendicular to the flume side wall is much smaller than the other two components of the three-dimensional velocity vector. This makes a wave flume a well-suited facility to study near-2DV structures, like cross-sections of a breakwater. Also (3D) constructions providing little blockage to the flow may be tested, e.g. measuring wave forces on vertical cylinders with a diameter much less than the flume width.
Wave flumes may be used to study the effects of water waves on coastal structures, offshore structures, sediment transport and other transport phenomena.
The waves are most often generated with a mechanical wavemaker, although there are also wind–wave flumes with (additional) wave generation by an air flow over the water – with the flume closed above by a roof above the free surface. The wavemaker frequently consists of a translating or rotating rigid wave board. Modern wavemakers are computer controlled, and can generate besides periodic waves also random waves, solitary waves, wave groups or even tsunami-like wave motion. The wavemaker is at one end of the wave flume, and at the other end is the construction being tested, or a wave absorber (a beach or special wave absorbing constructions).
Often, the side walls contain glass windows, or are completely made of glass, allowing for a clear visual observation of the experiment, and the easy deployment of optical instruments (e.g. by Laser Doppler velocimetry or particle image velocimetry).
## Circular wave basin
In 2014, the first circular, combined current and wave test basin, FloWaveTT, was commissioned in The University of Edinburgh. This allows for "true" 360° waves to be generated to simulate rough storm conditions as well as scientific controlled waves in the same facility. | https://en.wikipedia.org/wiki/Wave_tank |
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space
$$
(X,\Sigma)
$$
and any signed measure
$$
\mu
$$
defined on the
$$
\sigma
$$
-algebra
$$
\Sigma
$$
, there exist two
$$
\Sigma
$$
-measurable sets,
$$
P
$$
and
$$
N
$$
, of
$$
X
$$
such that:
1.
$$
P \cup N = X
$$
and
$$
P \cap N = \varnothing
$$
.
1. For every
$$
E \in \Sigma
$$
such that
$$
E \subseteq P
$$
, one has
$$
\mu(E) \geq 0
$$
, i.e.,
$$
P
$$
is a positive set for
$$
\mu
$$
.
1. For every
$$
E \in \Sigma
$$
such that
$$
E \subseteq N
$$
, one has
$$
\mu(E) \leq 0
$$
, i.e.,
$$
N
$$
is a negative set for
$$
\mu
$$
.
Moreover, this decomposition is essentially unique, meaning that for any other pair
$$
(P',N')
$$
of
$$
\Sigma
$$
-measurable subsets of
$$
X
$$
fulfilling the three conditions above, the symmetric differences
$$
P \triangle P'
$$
and
$$
N \triangle N'
$$
are
$$
\mu
$$
-null sets in the strong sense that every
$$
\Sigma
$$
-measurable subset of them has zero measure. The pair
$$
(P,N)
$$
is then called a Hahn decomposition of the signed measure
$$
\mu
$$
.
## Jordan measure decomposition
A consequence of the Hahn decomposition theorem is the , which states that every signed measure
$$
\mu
$$
defined on
$$
\Sigma
$$
has a unique decomposition into a difference
$$
\mu = \mu^{+} - \mu^{-}
$$
of two positive measures,
$$
\mu^{+}
$$
and
$$
\mu^{-}
$$
, at least one of which is finite, such that
$$
{\mu^{+}}(E) = 0
$$
for every
$$
\Sigma
$$
-measurable subset
$$
E \subseteq N
$$
and
$$
{\mu^{-}}(E) = 0
$$
for every
$$
\Sigma
$$
-measurable subset
$$
E \subseteq P
$$
, for any Hahn decomposition
$$
(P,N)
$$
of
$$
\mu
$$
. We call
$$
\mu^{+}
$$
and
$$
\mu^{-}
$$
the positive and negative part of
$$
\mu
$$
, respectively. The pair
$$
(\mu^{+},\mu^{-})
$$
is called a Jordan decomposition (or sometimes Hahn–Jordan decomposition) of
$$
\mu
$$
. The two measures can be defined as
$$
{\mu^{+}}(E) := \mu(E \cap P) \qquad \text{and} \qquad {\mu^{-}}(E) := - \mu(E \cap N)
$$
for every
$$
E \in \Sigma
$$
and any Hahn decomposition
$$
(P,N)
$$
of
$$
\mu
$$
.
Note that the Jordan decomposition is unique, while the Hahn decomposition is only essentially unique.
The Jordan decomposition has the following corollary: Given a Jordan decomposition
$$
(\mu^{+},\mu^{-})
$$
of a finite signed measure
$$
\mu
$$
, one has
$$
{\mu^{+}}(E) = \sup_{B \in \Sigma, ~ B \subseteq E} \mu(B) \quad \text{and} \quad {\mu^{-}}(E) = - \inf_{B \in \Sigma, ~ B \subseteq E} \mu(B)
$$
for any
$$
E
$$
in
$$
\Sigma
$$
. Furthermore, if
$$
\mu = \nu^{+} - \nu^{-}
$$
for a pair
$$
(\nu^{+},\nu^{-})
$$
of finite non-negative measures on
$$
X
$$
, then
$$
\nu^{+} \geq \mu^{+} \quad \text{and} \quad \nu^{-} \geq \mu^{-}.
$$
The last expression means that the Jordan decomposition is the minimal decomposition of
$$
\mu
$$
into a difference of non-negative measures. This is the minimality property of the Jordan decomposition.
Proof of the Jordan decomposition: For an elementary proof of the existence, uniqueness, and minimality of the Jordan measure decomposition see Fischer (2012).
## Proof of the Hahn decomposition theorem
Preparation: Assume that
$$
\mu
$$
does not take the value
$$
- \infty
$$
(otherwise decompose according to
$$
- \mu
$$
). As mentioned above, a negative set is a set
$$
A \in \Sigma
$$
such that
$$
\mu(B) \leq 0
$$
for every
$$
\Sigma
$$
-measurable subset
$$
B \subseteq A
$$
.
Claim: Suppose that
$$
D \in \Sigma
$$
satisfies
$$
\mu(D) \leq 0
$$
. Then there is a negative set
$$
A \subseteq D
$$
such that
$$
\mu(A) \leq \mu(D)
$$
.
Proof of the claim: Define
$$
A_{0} := D
$$
. Inductively assume for
$$
n \in \mathbb{N}_{0}
$$
that
$$
A_{n} \subseteq D
$$
has been constructed. Let
$$
t_{n} := \sup(\{ \mu(B) \mid B \in \Sigma ~ \text{and} ~ B \subseteq A_{n} \})
$$
denote the supremum of
$$
\mu(B)
$$
over all the
$$
\Sigma
$$
-measurable subsets
$$
B
$$
of
$$
A_{n}
$$
. This supremum might a priori be infinite. As the empty set
$$
\varnothing
$$
is a possible candidate for
$$
B
$$
in the definition of
$$
t_{n}
$$
, and as
$$
\mu(\varnothing) = 0
$$
, we have
$$
t_{n} \geq 0
$$
. By the definition of
$$
t_{n}
$$
, there then exists a
$$
\Sigma
$$
-measurable subset
$$
B_{n} \subseteq A_{n}
$$
satisfying
$$
\mu(B_{n}) \geq \min \! \left( 1,\frac{t_{n}}{2} \right).
$$
Set
$$
A_{n + 1} := A_{n} \setminus B_{n}
$$
to finish the induction step. Finally, define
$$
A := D \Bigg\backslash \bigcup_{n = 0}^{\infty} B_{n}.
$$
As the sets
$$
(B_{n})_{n = 0}^{\infty}
$$
are disjoint subsets of
$$
D
$$
, it follows from the sigma additivity of the signed measure
$$
\mu
$$
that
$$
\mu(D) = \mu(A) + \sum_{n = 0}^{\infty} \mu(B_{n}) \geq \mu(A) + \sum_{n = 0}^{\infty} \min \! \left( 1,\frac{t_{n}}{2} \right)\geq \mu(A).
$$
This shows that
$$
\mu(A) \leq \mu(D)
$$
. Assume
$$
A
$$
were not a negative set. This means that there would exist a
$$
\Sigma
$$
-measurable subset
$$
B \subseteq A
$$
that satisfies
$$
\mu(B) > 0
$$
. Then
$$
t_{n} \geq \mu(B)
$$
for every
$$
n \in \mathbb{N}_{0}
$$
, so the series on the right would have to diverge to
$$
+ \infty
$$
, implying that
$$
\mu(D) = + \infty
$$
, which is a contradiction, since
$$
\mu(D) \leq 0
$$
. Therefore,
$$
A
$$
must be a negative set.
Construction of the decomposition: Set
$$
N_{0} = \varnothing
$$
. Inductively, given
$$
N_{n}
$$
, define
$$
s_{n} := \inf(\{ \mu(D) \mid D \in \Sigma ~ \text{and} ~ D \subseteq X \setminus N_{n} \}).
$$
as the infimum of
$$
\mu(D)
$$
over all the
$$
\Sigma
$$
-measurable subsets
$$
D
$$
of
$$
X \setminus N_{n}
$$
. This infimum might a priori be
$$
- \infty
$$
. As
$$
\varnothing
$$
is a possible candidate for
$$
D
$$
in the definition of
$$
s_{n}
$$
, and as
$$
\mu(\varnothing) = 0
$$
, we have
$$
s_{n} \leq 0
$$
. Hence, there exists a
$$
\Sigma
$$
-measurable subset
$$
D_{n} \subseteq X \setminus N_{n}
$$
such that
$$
\mu(D_{n}) \leq \max \! \left( \frac{s_{n}}{2},- 1 \right) \leq 0.
$$
By the claim above, there is a negative set
$$
A_{n} \subseteq D_{n}
$$
such that
$$
\mu(A_{n}) \leq \mu(D_{n})
$$
. Set
$$
N_{n + 1} := N_{n} \cup A_{n}
$$
to finish the induction step. Finally, define
$$
N := \bigcup_{n = 0}^{\infty} A_{n}.
$$
As the sets
$$
(A_{n})_{n = 0}^{\infty}
$$
are disjoint, we have for every
$$
\Sigma
$$
-measurable subset
$$
B \subseteq N
$$
that
$$
\mu(B) = \sum_{n = 0}^{\infty} \mu(B \cap A_{n})
$$
by the sigma additivity of
$$
\mu
$$
. In particular, this shows that
$$
N
$$
is a negative set. Next, define
$$
P := X \setminus N
$$
. If
$$
P
$$
were not a positive set, there would exist a
$$
\Sigma
$$
-measurable subset
$$
D \subseteq P
$$
with
$$
\mu(D) < 0
$$
. Then
$$
s_{n} \leq \mu(D)
$$
for all
$$
n \in \mathbb{N}_{0}
$$
and
$$
\mu(N) = \sum_{n = 0}^{\infty} \mu(A_{n}) \leq \sum_{n = 0}^{\infty} \max \! \left( \frac{s_{n}}{2},- 1 \right) = - \infty,
$$
which is not allowed for
$$
\mu
$$
. Therefore,
$$
P
$$
is a positive set.
Proof of the uniqueness statement:
Suppose that
$$
(N',P')
$$
is another Hahn decomposition of
$$
X
$$
. Then
$$
P \cap N'
$$
is a positive set and also a negative set. Therefore, every measurable subset of it has measure zero. The same applies to
$$
N \cap P'
$$
. As
$$
P \triangle P' = N \triangle N' = (P \cap N') \cup (N \cap P'),
$$
this completes the proof. Q.E.D.
## References
-
-
## External links
- Hahn decomposition theorem at PlanetMath.
-
-
Category:Theorems in measure theory
Category:Articles containing proofs | https://en.wikipedia.org/wiki/Hahn_decomposition_theorem |
In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0, p)) to more than two outcomes.
As with the univariate negative binomial distribution, if the parameter
$$
x_0
$$
is a positive integer, the negative multinomial distribution has an urn model interpretation. Suppose we have an experiment that generates m+1≥2 possible outcomes, {X0,...,Xm}, each occurring with non-negative probabilities {p0,...,pm} respectively. If sampling proceeded until n observations were made, then {X0,...,Xm} would have been multinomially distributed. However, if the experiment is stopped once X0 reaches the predetermined value x0 (assuming x0 is a positive integer), then the distribution of the m-tuple {X1,...,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+...+Xm is not fixed, being a draw from a negative binomial distribution.
## Properties
### Marginal distributions
If m-dimensional x is partitioned as follows
$$
\mathbf{X}
=
\begin{bmatrix}
\mathbf{X}^{(1)} \\
\mathbf{X}^{(2)}
\end{bmatrix}
\text{ with sizes }\begin{bmatrix} n \times 1 \\ (m-n) \times 1 \end{bmatrix}
$$
and accordingly
$$
\boldsymbol{p}
$$
$$
\boldsymbol p
=
\begin{bmatrix}
\boldsymbol p^{(1)} \\
\boldsymbol p^{(2)}
\end{bmatrix}
\text{ with sizes }\begin{bmatrix} n \times 1 \\ (m-n) \times 1 \end{bmatrix}
$$
and let
$$
q = 1-\sum_i p_i^{(2)} = p_0+\sum_i p_i^{(1)}
$$
The marginal distribution of
$$
\boldsymbol X^{(1)}
$$
is
$$
\mathrm{NM}(x_0,p_0/q, \boldsymbol p^{(1)}/q )
$$
. That is the marginal distribution is also negative multinomial with the
$$
\boldsymbol p^{(2)}
$$
removed and the remaining p'''s properly scaled so as to add to one.
The univariate marginal is said to have a negative binomial distribution.
### Conditional distributions
The conditional distribution of given is . That is,
### Independent sums
If and If are independent, then
. Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.
### Aggregation
If
then, if the random variables with subscripts i and j'' are dropped from the vector and replaced by their sum,
$$
\mathbf{X}' = (X_1, \ldots, X_i + X_j, \ldots, X_m)\sim\operatorname{NM} (x_0, (p_1, \ldots, p_i + p_j, \ldots, p_m)).
$$
This aggregation property may be used to derive the marginal distribution of
$$
X_i
$$
mentioned above.
### Correlation matrix
The entries of the correlation matrix are
$$
\rho(X_i,X_i) = 1.
$$
$$
\rho(X_i,X_j) = \frac{\operatorname{cov}(X_i,X_j)}{\sqrt{\operatorname{var}(X_i)\operatorname{var}(X_j)}} = \sqrt{\frac{p_i p_j}{(p_0+p_i)(p_0+p_j)}}.
$$
## Parameter estimation
### Method of Moments
If we let the mean vector of the negative multinomial be
$$
\boldsymbol{\mu}=\frac{x_0}{p_0}\mathbf{p}
$$
and covariance matrix
$$
\boldsymbol{\Sigma}=\tfrac{x_0}{p_0^2}\,\mathbf{p}\mathbf{p}' + \tfrac{x_0}{p_0}\,\operatorname{diag}(\mathbf{p}),
$$
then it is easy to show through properties of determinants that
$$
|\boldsymbol{\Sigma}| = \frac{1}{p_0}\prod_{i=1}^m{\mu_i}
$$
. From this, it can be shown that
$$
x_0=\frac{\sum{\mu_i}\prod{\mu_i}}{|\boldsymbol{\Sigma}|-\prod{\mu_i}}
$$
and
$$
\mathbf{p}= \frac{|\boldsymbol{\Sigma}|-\prod{\mu_i}}{|\boldsymbol{\Sigma}|\sum{\mu_i}}\boldsymbol{\mu}.
$$
Substituting sample moments yields the method of moments estimates
$$
\hat{x}_0=\frac{(\sum_{i=1}^{m}{\bar{x_i})}\prod_{i=1}^{m}{\bar{x_i}}}{|\mathbf{S}|-\prod_{i=1}^{m}{\bar{x_i}}}
$$
and
$$
\hat{\mathbf{p}}=\left(\frac{|\boldsymbol{S}|-\prod_{i=1}^{m}{\bar{x}_i}}{|\boldsymbol{S}|\sum_{i=1}^{m}{\bar{x}_i}}\right)\boldsymbol{\bar{x}}
$$
## Related distributions
- Negative binomial distribution
- Multinomial distribution
- Inverted Dirichlet distribution, a conjugate prior for the negative multinomial
- Dirichlet negative multinomial distribution
## References
Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi-
nomial distribution. Biometrics 53: 971–82.
## Further reading
Category:Factorial and binomial topics
Category:Multivariate discrete distributions | https://en.wikipedia.org/wiki/Negative_multinomial_distribution |
A nonblocking minimal spanning switch is a device that can connect N inputs to N outputs in any combination. The most familiar use of switches of this type is in a telephone exchange. The term "non-blocking" means that if it is not defective, it can always make the connection. The term "minimal" means that it has the fewest possible components, and therefore the minimal expense.
Historically, in telephone switches, connections between callers were arranged with large, expensive banks of electromechanical relays, Strowger switches. The basic mathematical property of Strowger switches is that for each input to the switch, there is exactly one output. Much of the mathematical switching circuit theory attempts to use this property to reduce the total number of switches needed to connect a combination of inputs to a combination of outputs.
In the 1940s and 1950s, engineers in Bell Lab began an extended series of mathematical investigations into methods for reducing the size and expense of the "switched fabric" needed to implement a telephone exchange. One early, successful mathematical analysis was performed by Charles Clos (), and a switched fabric constructed of smaller switches is called a Clos network.
## Background: switching topologies
### The crossbar switch
The crossbar switch has the property of being able to connect N inputs to N outputs in any one-to-one combination, so it can connect any caller to any non-busy receiver, a property given the technical term "nonblocking". Being nonblocking it could always complete a call (to a non-busy receiver), which would maximize service availability.
However, the crossbar switch does so at the expense of using N2 (N squared) simple SPST switches. For large N (and the practical requirements of a phone switch are considered large) this growth was too expensive. Further, large crossbar switches had physical problems. Not only did the switch require too much space, but the metal bars containing the switch contacts would become so long that they would sag and become unreliable. Engineers also noticed that at any time, each bar of a crossbar switch was only making a single connection. The other contacts on the two bars were unused. This seemed to imply that most of the switching fabric of a crossbar switch was wasted.
The obvious way to emulate a crossbar switch was to find some way to build it from smaller crossbar switches. If a crossbar switch could be emulated by some arrangement of smaller crossbar switches, then these smaller crossbar switches could also, in turn be emulated by even smaller crossbar switches. The switching fabric could become very efficient, and possibly even be created from standardized parts. This is called a Clos network.
### Completely connected 3-layer switches
The next approach was to break apart the crossbar switch into three layers of smaller crossbar switches. There would be an "input layer", a "middle layer" and an "output layer." The smaller switches are less massive, more reliable, and generally easier to build, and therefore less expensive.
A telephone system only has to make a one-to-one connection. Intuitively this seems to mean that the number of inputs and the number of outputs can always be equal in each subswitch, but intuition does not prove this can be done nor does it tell us how to do so. Suppose we want to synthesize a 16 by 16 crossbar switch. The design could have 4 subswitches on the input side, each with 4 inputs, for 16 total inputs. Further, on the output side, we could also have 4 output subswitches, each with 4 outputs, for a total of 16 outputs. It is desirable that the design use as few wires as possible, because wires cost real money. The least possible number of wires that can connect two subswitches is a single wire. So, each input subswitch will have a single wire to each middle subswitch. Also, each middle subswitch will have a single wire to each output subswitch.
The question is how many middle subswitches are needed, and therefore how many total wires should connect the input layer to the middle layer. Since telephone switches are symmetric (callers and callees are interchangeable), the same logic will apply to the output layer, and the middle subswitches will be "square", having the same number of inputs as outputs.
The number of middle subswitches depends on the algorithm used to allocate connection to them. The basic algorithm for managing a three-layer switch is to search the middle subswitches for a middle subswitch that has unused wires to the needed input and output switches. Once a connectible middle subswitch is found, connecting to the correct inputs and outputs in the input and output switches is trivial.
Theoretically, in the example, only four central switches are needed, each with exactly one connection to each input switch and one connection to each output switch. This is called a "minimal spanning switch," and managing it was the holy grail of the Bell Labs' investigations.
However, a bit of work with a pencil and paper will show that it is easy to get such a minimal switch into conditions in which no single middle switch has a connection to both the needed input switch and the needed output switch. It only takes four calls to partially block the switch. If an input switch is half-full, it has connections via two middle switches. If an output switch is also half full with connections from the other two middle switches, then there is no remaining middle switch which can provide a path between that input and output.
For this reason, a "simply connected nonblocking switch" 16x16 switch with four input subswitches and four output switches was thought to require 7 middle switches; in the worst case an almost-full input subswitch would use three middle switches, an almost-full output subswitch would use three different ones, and the seventh would be guaranteed to be free to make the last connection. For this reason, sometimes this switch arrangement is called a "2n−1 switch", where n is the number of input ports of the input subswitches.
The example is intentionally small, and in such a small example, the reorganization does not save many switches. A 16×16 crossbar has 256 contacts, while a 16×16 minimal spanning switch has 4×4×4×3 = 192 contacts.
As the numbers get larger, the savings increase. For example, a 10,000 line exchange would need 100 million contacts to implement a full crossbar. But three layers of 100 100×100 subswitches would use only 300 10,000 contact subswitches, or 3 million contacts.
Those subswitches could in turn each be made of 3×10 10×10 crossbars, a total of 3000 contacts, making 900,000 for the whole exchange; that is a far smaller number than 100 million.
### Managing a minimal spanning switch
The crucial discovery was a way to reorganize connections in the middle switches to "trade wires" so that a new connection could be completed.
The first step is to find an unused link from the input subswitch to a middle-layer subswitch (which we shall call A), and an unused link from a middle-layer subswitch (which we shall call B) to the desired output subswitch. Since, prior to the arrival of the new connection, the input and output subswitches each had at least one unused connection, both of these unused links must exist.
If A and B happen to be the same middle-layer switch, then the connection can be made immediately just as in the "2n−1" switch case. However, if A and B are different middle-layer subswitches, more work is required. The algorithm finds a new arrangement of the connections through the middle subswitches A and B which includes all of the existing connections, plus the desired new connection.
Make a list of all of the desired connections that pass through A or B. That is, all of the existing connections to be maintained and the new connection. The algorithm proper only cares about the internal connections from input to output switch, although a practical implementation also has to keep track of the correct input and output switch connections.
In this list, each input subswitch can appear in at most two connections: one to subswitch A, and one to subswitch B. The options are zero, one, or two. Likewise, each output subswitch appears in at most two connections.
Each connection is linked to at most two others by a shared input or output subswitch, forming one link in a "chain" of connections.
Next, begin with the new connection. Assign it the path from its input subswitch, through middle subswitch A, to its output subswitch. If this first connection's output subswitch has a second connection, assign that second connection a path from its input subswitch through subswitch B. If that input subswitch has another connection, assign that third connection a path through subswitch A. Continue back and forth in this manner, alternating between middle subswitches A and B. Eventually one of two things must happen:
1. the chain terminates in a subswitch with only one connection, or
1. the chain loops back to the originally chosen connection.
In the first case, go back to the new connection's input subswitch and follow its chain backward, assigning connections to paths through middle subswitches B and A in the same alternating pattern.
When this is done, each input or output subswitch in the chain has at most two connections passing through it, and they are assigned to different middle switches. Thus, all the required links are available.
There may be additional connections through subswitches A and B which are not part of the chain including the new connection; those connections may be left as-is.
After the new connection pattern is designed in the software, then the electronics of the switch can actually be reprogrammed, physically moving the connections. The electronic switches are designed internally so that the new configuration can be written into the electronics without disturbing the existing connection, and then take effect with a single logic pulse. The result is that the connection moves instantaneously, with an imperceptible interruption to the conversation. In older electromechanical switches, one occasionally heard a clank of "switching noise."
This algorithm is a form of topological sort, and is the heart of the algorithm that controls a minimal spanning switch.
## Practical implementations of switches
As soon as the algorithm was discovered, Bell system engineers and managers began discussing it. After several years, Bell engineers began designing electromechanical switches that could be controlled by it. At the time, computers used tubes and were not reliable enough to control a phone system (phone system switches are safety-critical, and they are designed to have an unplanned failure about once per thirty years). Relay-based computers were too slow to implement the algorithm. However, the entire system could be designed so that when computers were reliable enough, they could be retrofitted to existing switching systems.
It's not difficult to make composite switches fault-tolerant. When a subswitch fails, the callers simply redial. So, on each new connection, the software tries the next free connection in each subswitch rather than reusing the most recently released one. The new connection is more likely to work because it uses different circuitry.
Therefore, in a busy switch, when a particular PCB lacks any connections, it is an excellent candidate for testing.
To test or remove a particular printed circuit card from service, there is a well-known algorithm. As fewer connections pass through the card's subswitch, the software routes more test signals through the subswitch to a measurement device, and then reads the measurement. This does not interrupt old calls, which remain working.
If a test fails, the software isolates the exact circuit board by reading the failure from several external switches. It then marks the free circuits in the failing circuitry as busy. As calls using the faulty circuitry are ended, those circuits are also marked busy. Some time later, when no calls pass through the faulty circuitry, the computer lights a light on the circuit board that needs replacement, and a technician can replace the circuit board. Shortly after replacement, the next test succeeds, the connections to the repaired subswitch are marked "not busy," and the switch returns to full operation.
The diagnostics on Bell's early electronic switches would actually light a green light on each good printed circuit board, and light a red light on each failed printed circuit board. The printed circuits were designed so that they could be removed and replaced without turning off the whole switch.
The eventual result was the Bell 1ESS. This was controlled by a CPU called the Central Control (CC), a lock-step, Harvard architecture dual computer using reliable diode–transistor logic. In the 1ESS CPU, two computers performed each step, checking each other. When they disagreed, they would diagnose themselves, and the correctly running computer would take up switch operation while the other would disqualify itself and request repair. The 1ESS switch was still in limited use as of 2012, and had a verified reliability of less than one unscheduled hour of failure in each thirty years of operation, validating its design.
Initially it was installed on long-distance trunks in major cities, the most heavily used parts of each telephone exchange. On the first Mother's Day that major cities operated with it, the Bell system set a record for total network capacity, both in calls completed, and total calls per second per switch. This resulted in a record for total revenue per trunk.
## Digital switches
A practical implementation of a switch can be created from an odd number of layers of smaller subswitches. Conceptually, the crossbar switches of the three-stage switch can each be further decomposed into smaller crossbar switches. Although each subswitch has limited multiplexing capability, working together they synthesize the effect of a larger N×N crossbar switch.
In a modern digital telephone switch, application of two different multiplexer approaches in alternate layers further reduces the cost of the switching fabric:
1. space-division multiplexers are something like the crossbar switches already described, or some arrangement of crossover switches or banyan switches. Any single output can select from any input. In digital switches, this is usually an arrangement of AND gates. 8000 times per second, the connection is reprogrammed to connect particular wires for the duration of a time slot. Design advantage: In space-division systems the number of space-division connections is divided by the number of time slots in the time-division multiplexing system. This dramatically reduces the size and expense of the switching fabric. It also increases the reliability, because there are far fewer physical connections to fail.
1. time-division multiplexers each have a memory which is read in a fixed order and written in a programmable order (or vice versa). This type of switch permutes time-slots in a time-division multiplexed signal that goes to the space-division multiplexers in its adjacent layers. Design advantage: Time-division switches have only one input and output wire. Since they have far fewer electrical connections to fail, they are far more reliable than space-division switches, and are therefore the preferred switches for the outer (input and output) layers of modern telephone switches.
Practical digital telephonic switches minimize the size and expense of the electronics. First, it is typical to "fold" the switch, so that both the input and output connections to a subscriber-line are handled by the same control logic. Then, a time-division switch is used in the outer layer. The outer layer is implemented in subscriber-line interface cards (SLICs) in the local presence street-side boxes. Under remote control from the central switch, the cards connect to timing-slots in a time-multiplexed line to a central switch. In the U.S. the multiplexed line is a multiple of a T-1 line. In Europe and many other countries it is a multiple of an E-1 line.
The scarce resources in a telephone switch are the connections between layers of subswitches. These connections can be either time slots or wires, depending on the type of multiplexing. The control logic has to allocate these connections, and the basic method is the algorithm already discussed. The subswitches are logically arranged so that they synthesize larger subswitches. Each subswitch, and synthesized subswitch is controlled (recursively) by logic derived from Clos's mathematics. The computer code decomposes larger multiplexers into smaller multiplexers.
If the recursion is taken to the limit, breaking down the crossbar to the minimum possible number of switching elements, the resulting device is sometimes called a crossover switch or a banyan switch depending on its topology.
Switches typically interface to other switches and fiber optic networks via fast multiplexed data lines such as SONET.
Each line of a switch may be periodically tested by the computer, by sending test data through it. If a switch's line fails, all lines of a switch are marked as in use. Multiplexer lines are allocated in a first-in-first out way, so that new connections find new switch elements. When all connections are gone from a defective switch, the defective switch can be avoided, and later replaced.
As of 2018, such switches are no longer made. They are being replaced by high-speed Internet Protocol routers.
## Example of rerouting a switch | https://en.wikipedia.org/wiki/Nonblocking_minimal_spanning_switch |
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
## The theorems
### Little Picard Theorem
: If a function is entire and non-constant, then the set of values that assumes is either the whole complex plane or the plane minus a single point.
Sketch of
## Proof
: Picard's original proof was based on properties of the modular lambda function, usually denoted by , and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If omits two values, then the composition of with the inverse of the modular function maps the plane into
the unit disc which implies that is constant by Liouville's theorem.
This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded. Many different proofs of Picard's theorem were later found and Schottky's theorem is a quantitative version of it. In the case where the values of
$$
f
$$
are missing a single point, this point is called a lacunary value of the function.
Great Picard's Theorem: If an analytic function has an essential singularity at a point , then on any punctured neighborhood of takes on all possible complex values, with at most a single exception, infinitely often.
This is a substantial strengthening of the Casorati–Weierstrass theorem, which only guarantees that the range of
$$
f
$$
is dense in the complex plane. A result of the
### Great Picard Theorem
is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception.
The "single exception" is needed in both theorems, as demonstrated here:
- ez is an entire non-constant function that is never 0,
-
$$
e^{\frac{1}{z}}
$$
has an essential singularity at 0, but still never attains 0 as a value.
Proof
Little Picard Theorem
Suppose
$$
f: \mathbb{C}\to\mathbb{C}
$$
is an entire function that omits two values
$$
z_0
$$
and
$$
z_1
$$
. Then
$$
\frac{f(z)-z_0}{z_1 - z_0}
$$
is also entire and we may assume without loss of generality that
$$
z_0 = 0
$$
and
$$
z_1=1
$$
.
Because
$$
\mathbb{C}
$$
is simply connected and the range of
$$
f
$$
omits
$$
0
$$
, f has a holomorphic logarithm. Let
$$
g
$$
be an entire function such that
$$
f(z)=e^{2\pi ig(z)}
$$
. Then the range of
$$
g
$$
omits all integers. By a similar argument using the quadratic formula, there is an entire function such that
$$
g(z)=\cos(h(z))
$$
. Then the range of
$$
h
$$
omits all complex numbers of the form
$$
2\pi n \pm i \cosh^{-1}(m)
$$
, where
$$
n
$$
is an integer and
$$
m
$$
is a nonnegative integer.
By Landau's theorem, if
$$
h'(w) \ne 0
$$
, then for all
$$
{R > 0}
$$
, the range of
$$
h
$$
contains a disk of radius
$$
|h'(w)| R/72
$$
. But from above, any sufficiently large disk contains at least one number that the range of h omits. Therefore
$$
h'(w)=0
$$
for all
$$
w
$$
. By the fundamental theorem of calculus,
$$
h
$$
is constant, so
$$
f
$$
is constant.
Great Picard Theorem
Suppose f is an analytic function on the punctured disk of radius r around the point w, and that f omits two values z0 and z1. By considering (f(p + rz) − z0)/(z1 − z0) we may assume without loss of generality that z0 = 0, z1 = 1, w = 0, and r = 1.
The function F(z) = f(e−z) is analytic in the right half-plane Re(z) > 0. Because the right half-plane is simply connected, similar to the proof of the Little Picard Theorem, there are analytic functions G and H defined on the right half-plane such that F(z) = e2πiG(z) and G(z) = cos(H(z)). For any w in the right half-plane, the open disk with radius Re(w) around w is contained in the domain of H. By Landau's theorem and the observation about the range of H in the proof of the Little Picard Theorem, there is a constant C > 0 such that |H′(w)| ≤ C / Re(w). Thus, for all real numbers x ≥ 2 and 0 ≤ y ≤ 2π,
$$
|H(x+iy)|=\left|H(2+iy)+\int_2^xH'(t+iy)\,\mathrm{d}t\right|\le|H(2+iy)|+\int_2^x\frac{C}{t}\,\mathrm{d}t\le A\log x,
$$
where A > 0 is a constant. So |G(x + iy)| ≤ xA.
Next, we observe that F(z + 2πi) = F(z) in the right half-plane, which implies that G(z + 2πi) − G(z) is always an integer. Because G is continuous and its domain is connected, the difference G(z + 2πi) − G(z) = k is a constant. In other words, the function G(z) − kz / (2πi) has period 2πi. Thus, there is an analytic function g defined in the punctured disk with radius e−2 around 0 such that G(z) − kz / (2πi) = g(e−z).
Using the bound on G above, for all real numbers x ≥ 2 and 0 ≤ y ≤ 2π,
$$
\left|G(x+iy)-\frac{k(x+iy)}{2\pi i}\right|\le x^A+\frac{|k|}{2\pi}(x+2\pi)\le C'x^{A'}
$$
holds, where A′ > A and C′ > 0 are constants. Because of the periodicity, this bound actually holds for all y. Thus, we have a bound |g(z)| ≤ C′(−log|z|)A′ for 0 < |z| < e−2. By Riemann's theorem on removable singularities, g extends to an analytic function in the open disk of radius e−2 around 0.
Hence, G(z) − kz / (2πi) is bounded on the half-plane Re(z) ≥ 3. So F(z)e−kz is bounded on the half-plane Re(z) ≥ 3, and
f(z)zk is bounded in the punctured disk of radius e−3 around 0. By Riemann's theorem on removable singularities, f(z)zk extends to an analytic function in the open disk of radius e−3 around 0. Therefore, f does not have an essential singularity at 0.
Therefore, if the function f has an essential singularity at 0, the range of f in any open disk around 0 omits at most one value. If f takes a value only finitely often, then in a sufficiently small open disk around 0, f omits that value. So f(z) takes all possible complex values, except at most one, infinitely often.
## Generalization and current research
Great Picard's theorem is true in a slightly more general form that also applies to meromorphic functions:
Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P1(C) = C ∪ {∞} denotes the Riemann sphere and f : M\{w} → P1(C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f(z) attains all but at most two points of P1(C) infinitely often.
Example: The function f(z) = 1/(1 − e1/z) is meromorphic on C* = C - {0}, the complex plane with the origin deleted. It has an essential singularity at z = 0 and attains the value ∞ infinitely often in any neighborhood of 0; however it does not attain the values 0 or 1.
With this generalization, Little Picard Theorem follows from Great Picard Theorem because an entire function is either a polynomial or it has an essential singularity at infinity. As with the little theorem, the (at most two) points that are not attained are lacunary values of the function.
The following conjecture is related to "Great Picard's Theorem":
Conjecture: Let {U1, ..., Un} be a collection of open connected subsets of C that cover the punctured unit disk D \ {0}. Suppose that on each Uj there is an injective holomorphic function fj, such that dfj = dfk on each intersection Uj ∩ Uk. Then the differentials glue together to a meromorphic 1-form on D.
It is clear that the differentials glue together to a holomorphic 1-form g dz on D \ {0}. In the special case where the residue of g at 0 is zero the conjecture follows from the "Great Picard's Theorem".
## Notes
## References
-
-
Category:Theorems in complex analysis | https://en.wikipedia.org/wiki/Picard_theorem |
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X, and input B produces response Y, then input (A + B) produces response (X + Y).
A function
$$
F(x)
$$
that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties: additivity
$$
F(x_1 + x_2) = F(x_1) + F(x_2)
$$
and homogeneity
$$
F(ax) = a F(x)
$$
for scalar .
This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency-domain linear transform methods such as Fourier and Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.
The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).
## Relation to Fourier analysis and similar methods
By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.
For example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different amplitude and phase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.
As another common example, in Green's function analysis, the stimulus is written as the superposition of infinitely many impulse functions, and the response is then a superposition of impulse responses.
Fourier analysis is particularly common for waves. For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as the superposition principle holds (which is often but not always; see nonlinear optics), the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves.
## Wave superposition
Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave and the wave itself is a function specifying the amplitude at each point.
In any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system. In many cases (for example, in the classic wave equation), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.)
### Wave diffraction vs. wave interference
With regard to wave superposition, Richard Feynman wrote:
Other authors elaborate:
Yet another source concurs:
### Wave interference
The phenomenon of interference between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in noise-canceling headphones, the summed variation has a smaller amplitude than the component variations; this is called destructive interference. In other cases, such as in a line array, the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference.
combined waveform wave 1 wave 2 Two waves in phase Two waves 180° out of phase
### Departures from linearity
In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles nonlinear optics and nonlinear acoustics.
### Quantum superposition
In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, and the equation governing its behavior is called the Schrödinger equation. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "quantum superposition") of (possibly infinitely many) other wave functions of a certain type—stationary states whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.
The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a ray in projective Hilbert space, not a vector.
According to Dirac: "if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state [italics in original]."
However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself
uses ket vector representations of states to decompose or split,
for example, a ket vector
$$
|\psi_i\rangle
$$
into superposition of component ket vectors
$$
|\phi_j\rangle
$$
as:
$$
|\psi_i\rangle = \sum_{j}{C_j}|\phi_j\rangle,
$$
where the
$$
C_j\in \textbf{C}
$$
.
The equivalence class of the
$$
|\psi_i\rangle
$$
allows a well-defined meaning to be given to the relative phases of the
$$
C_j
$$
., but an absolute (same amount
for all the
$$
C_j
$$
) phase change on the
$$
C_j
$$
does not affect the equivalence class of the
$$
|\psi_i\rangle
$$
.
There are exact correspondences between the superposition presented in the main on this page and the quantum superposition.
For example, the Bloch sphere to represent pure state of a two-level quantum mechanical system
(qubit) is also known as the Poincaré sphere representing different types of classical
pure polarization states.
Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... has no analogy in classical physics".
According to Dirac: "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory [italics in original]."
Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase,
they actually mean phase translation symmetry derived from time translation symmetry, which is also
applicable to classical states, as shown above with classical polarization states.
## Boundary-value problems
A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation
$$
F(y) = 0
$$
with some boundary specification
$$
G(y) = z.
$$
For example, in Laplace's equation with Dirichlet boundary conditions, F would be the Laplacian operator in a region R, G would be an operator that restricts y to the boundary of R, and z would be the function that y is required to equal on the boundary of R.
In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation:
$$
F(y_1) = F(y_2) = \cdots = 0 \quad \Rightarrow \quad F(y_1 + y_2 + \cdots) = 0,
$$
while the boundary values superpose:
$$
G(y_1) + G(y_2) = G(y_1 + y_2).
$$
Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems.
## Additive state decomposition
Consider a simple linear system:
$$
\dot{x} = Ax + B(u_1 + u_2), \qquad x(0) = x_0.
$$
By superposition principle, the system can be decomposed into
$$
\begin{align}
\dot{x}_1 &= Ax_1 + Bu_1, && x_1(0) = x_0,\\
\dot{x}_2 &= Ax_2 + Bu_2, && x_2(0) = 0
\end{align}
$$
with
$$
x = x_1 + x_2.
$$
Superposition principle is only available for linear systems. However, the additive state decomposition can be applied to both linear and nonlinear systems. Next, consider a nonlinear system
$$
\dot{x} = Ax + B(u_1 + u_2) + \phi\left(c^\mathsf{T} x\right), \qquad x(0) = x_0,
$$
where
$$
\phi
$$
is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into
$$
\begin{align}
\dot{x}_1 &= Ax_1 + Bu_1 + \phi(y_d), && x_1(0) = x_0, \\
\dot{x}_2 &= Ax_2 + Bu_2 + \phi\left(c^\mathsf{T} x_1 + c^\mathsf{T} x_2\right) - \phi (y_d), && x_2(0) = 0
\end{align}
$$
with
$$
x = x_1 + x_2.
$$
This decomposition can help to simplify controller design.
## Other example applications
- In electrical engineering, in a linear circuit, the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses.
- In physics, Maxwell's equations imply that the (possibly time-varying) distributions of charges and currents are related to the electric and magnetic fields by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields that arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the heat equation.
- In engineering, superposition is used to solve for beam and structure deflections of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system). Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.
- In hydrogeology, the superposition principle is applied to the drawdown of two or more water wells pumping in an ideal aquifer. This principle is used in the analytic element method to develop analytical elements capable of being combined in a single model.
- In process control, the superposition principle is used in model predictive control.
- The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization.
## History
According to Léon Brillouin, the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by Leonhard Euler and then by Joseph Lagrange. Bernoulli argued that any sonorous body could vibrate in a series of simple modes with a well-defined frequency of oscillation. As he had earlier indicated, these modes could be superposed to produce more complex vibrations. In his reaction to Bernoulli's memoirs, Euler praised his colleague for having best developed the physical part of the problem of vibrating strings, but denied the generality and superiority of the multi-modes solution.
Later it became accepted, largely through the work of Joseph Fourier. | https://en.wikipedia.org/wiki/Superposition_principle |
In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if
$$
R
$$
is a square matrix representing a rotation (rotation matrix) and
$$
\mathbf{v}
$$
is a column vector describing the position of a point in space, the product
$$
R\mathbf{v}
$$
yields another column vector describing the position of that point after that rotation. If
$$
\mathbf{v}
$$
is a row vector, the same transformation can be obtained using where
$$
R^{\mathsf T}
$$
is the transpose of
## Main diagonal
The entries
$$
a_{ii}
$$
() form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , .
The diagonal of a square matrix from the top right to the bottom left corner is called antidiagonal or counterdiagonal.
## Special kinds
Name Example with n = 3 Diagonal matrix Lower triangular matrix Upper triangular matrix
### Diagonal or triangular matrix
If all entries outside the main diagonal are zero,
$$
A
$$
is called a diagonal matrix. If all entries below (resp. above) the main diagonal are zero,
$$
A
$$
is called an upper (resp. lower) triangular matrix.
### Identity matrix
The identity matrix
$$
I_n
$$
of size
$$
n
$$
is the
$$
n \times n
$$
matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.
$$
I_1 = \begin{bmatrix} 1 \end{bmatrix}
,\
I_2 = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
,\ \ldots ,\
I_n = \begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1
\end{bmatrix}.
$$
It is a square matrix of order and also a special kind of diagonal matrix. The term identity matrix refers to the property of matrix multiplication that
$$
I_m A = A I_n = A
$$
for any
$$
m \times n
$$
matrix
### Invertible matrix and its inverse
A square matrix
$$
A
$$
is called invertible or non-singular if there exists a matrix
$$
B
$$
such that
$$
AB = BA = I_n.
$$
If
$$
B
$$
exists, it is unique and is called the inverse matrix of denoted
### Symmetric or skew-symmetric matrix
A square matrix
$$
A
$$
that is equal to its transpose, i.e., is a symmetric matrix. If instead then
$$
A
$$
is called a skew-symmetric matrix.
For a complex square matrix often the appropriate analogue of the transpose is the conjugate transpose defined as the transpose of the complex conjugate of A complex square matrix
$$
A
$$
satisfying
$$
A^*=A
$$
is called a Hermitian matrix. If instead then
$$
A
$$
is called a skew-Hermitian matrix.
By the spectral theorem, real symmetric (or complex Hermitian) matrices have an orthogonal (or unitary) eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.
### Definite matrix
Positive definite Indefinite Points such that (Ellipse). Points such that (Hyperbola).
A symmetric -matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors
$$
x \in \mathbb{R}^n
$$
the associated quadratic form given by
$$
Q(\mathbf{x}) = \mathbf{x}^\mathsf{T} A \mathbf{x}
$$
takes only positive values (respectively only negative values; both some negative and some positive values). If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive. The table at the right shows two possibilities for 2×2 matrices.
Allowing as input two different vectors instead yields the bilinear form associated to :
$$
B_A(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\mathsf{T} A \mathbf{y}.
$$
### Orthogonal matrix
An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:
$$
A^\textsf{T} = A^{-1},
$$
which entails
$$
A^\textsf{T} A = A A^\textsf{T} = I,
$$
where I is the identity matrix.
An orthogonal matrix is necessarily invertible (with inverse ), unitary (), and normal (). The determinant of any orthogonal matrix is either +1 or −1. The special orthogonal group
$$
\operatorname{SO}(n)
$$
consists of the orthogonal matrices with determinant +1.
The complex analogue of an orthogonal matrix is a unitary matrix.
### Normal matrix
A real or complex square matrix
$$
A
$$
is called normal if If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary, then it is normal. Normal matrices are of interest mainly because they include the types of matrices just listed and form the broadest class of matrices for which the spectral theorem holds.
## Operations
### Trace
The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors:
$$
\operatorname{tr}(AB) = \operatorname{tr}(BA).
$$
This is immediate from the definition of matrix multiplication:
$$
\operatorname{tr}(AB) = \sum_{i=1}^m \sum_{j=1}^n A_{ij} B_{ji} = \operatorname{tr}(BA).
$$
Also, the trace of a matrix is equal to that of its transpose, i.e.,
$$
\operatorname{tr}(A) = \operatorname{tr}(A^{\mathrm T}).
$$
### Determinant
The determinant
$$
\det(A)
$$
or
$$
|A|
$$
of a square matrix
$$
A
$$
is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in
$$
\mathbb{R}^2
$$
) or volume (in
$$
\mathbb{R}^3
$$
) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 2×2 matrices is given by
$$
\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc.
$$
The determinant of 3×3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalizes these two formulae to all dimensions.
The determinant of a product of square matrices equals the product of their determinants:
$$
\det(AB) = \det(A) \cdot \det(B)
$$
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1×1 matrix, which is its unique entry, or even the determinant of a 0×0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
### Eigenvalues and eigenvectors
A number and a non-zero vector
$$
\mathbf{v}
$$
satisfying
$$
A \mathbf{v} = \lambda \mathbf{v}
$$
are called an eigenvalue and an eigenvector of respectively. The number is an eigenvalue of an -matrix if and only if is not invertible, which is equivalent to
$$
\det(A-\lambda I) = 0.
$$
The polynomial in an indeterminate given by evaluation of the determinant is called the characteristic polynomial of . It is a monic polynomial of degree n. Therefore the polynomial equation has at most n different solutions, i.e., eigenvalues of the matrix. They may be complex even if the entries of are real. According to the Cayley–Hamilton theorem, , that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix. | https://en.wikipedia.org/wiki/Square_matrix |
Computer simulation is the running of a mathematical model on a computer, the model being designed to represent the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determined by comparing their results to the real-world outcomes they aim to predict. Computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics (computational physics), astrophysics, climatology, chemistry, biology and manufacturing, as well as human systems in economics, psychology, social science, health care and engineering. Simulation of a system is represented as the running of the system's model. It can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions.
Computer simulations are realized by running computer programs that can be either small, running almost instantly on small devices, or large-scale programs that run for hours or days on network-based groups of computers. The scale of events being simulated by computer simulations has far exceeded anything possible (or perhaps even imaginable) using traditional paper-and-pencil mathematical modeling. In 1997, a desert-battle simulation of one force invading another involved the modeling of 66,239 tanks, trucks and other vehicles on simulated terrain around Kuwait, using multiple supercomputers in the DoD High Performance Computer Modernization Program.
Other examples include a 1-billion-atom model of material deformation; a 2.64-million-atom model of the complex protein-producing organelle of all living organisms, the ribosome, in 2005;
a complete simulation of the life cycle of Mycoplasma genitalium in 2012; and the Blue Brain project at EPFL (Switzerland), begun in May 2005 to create the first computer simulation of the entire human brain, right down to the molecular level.
Because of the computational cost of simulation, computer experiments are used to perform inference such as uncertainty quantification.
## Simulation versus model
A model consists of the equations used to capture the behavior of a system. By contrast, computer simulation is the actual running of the program that perform algorithms which solve those equations, often in an approximate manner. Simulation, therefore, is the process of running a model. Thus one would not "build a simulation"; instead, one would "build a model (or a simulator)", and then either "run the model" or equivalently "run a simulation".
## History
Computer simulation developed hand-in-hand with the rapid growth of the computer, following its first large-scale deployment during the Manhattan Project in World War II to model the process of nuclear detonation. It was a simulation of 12 hard spheres using a Monte Carlo algorithm. Computer simulation is often used as an adjunct to, or substitute for, modeling systems for which simple closed form analytic solutions are not possible. There are many types of computer simulations; their common feature is the attempt to generate a sample of representative scenarios for a model in which a complete enumeration of all possible states of the model would be prohibitive or impossible.
## Data preparation
The external data requirements of simulations and models vary widely. For some, the input might be just a few numbers (for example, simulation of a waveform of AC electricity on a wire), while others might require terabytes of information (such as weather and climate models).
Input sources also vary widely:
- Sensors and other physical devices connected to the model;
- Control surfaces used to direct the progress of the simulation in some way;
- Current or historical data entered by hand;
- Values extracted as a by-product from other processes;
- Values output for the purpose by other simulations, models, or processes.
Lastly, the time at which data is available varies:
- "invariant" data is often built into the model code, either because the value is truly invariant (e.g., the value of π) or because the designers consider the value to be invariant for all cases of interest;
- data can be entered into the simulation when it starts up, for example by reading one or more files, or by reading data from a preprocessor;
- data can be provided during the simulation run, for example by a sensor network.
Because of this variety, and because diverse simulation systems have many common elements, there are a large number of specialized simulation languages. The best-known may be Simula. There are now many others.
Systems that accept data from external sources must be very careful in knowing what they are receiving. While it is easy for computers to read in values from text or binary files, what is much harder is knowing what the accuracy (compared to measurement resolution and precision) of the values are. Often they are expressed as "error bars", a minimum and maximum deviation from the value range within which the true value (is expected to) lie. Because digital computer mathematics is not perfect, rounding and truncation errors multiply this error, so it is useful to perform an "error analysis" to confirm that values output by the simulation will still be usefully accurate.
## Types
Models used for computer simulations can be classified according to several independent pairs of attributes, including:
- Stochastic or deterministic (and as a special case of deterministic, chaotic) – see external links below for examples of stochastic vs. deterministic simulations
- Steady-state or dynamic
- Continuous or discrete (and as an important special case of discrete, discrete event or DE models)
- Dynamic system simulation, e.g. electric systems, hydraulic systems or multi-body mechanical systems (described primarily by DAE:s) or dynamics simulation of field problems, e.g. CFD of FEM simulations (described by PDE:s).
- Local or distributed.
Another way of categorizing models is to look at the underlying data structures. For time-stepped simulations, there are two main classes:
- Simulations which store their data in regular grids and require only next-neighbor access are called stencil codes. Many CFD applications belong to this category.
- If the underlying graph is not a regular grid, the model may belong to the meshfree method class.
For steady-state simulations, equations define the relationships between elements of the modeled system and attempt to find a state in which the system is in equilibrium. Such models are often used in simulating physical systems, as a simpler modeling case before dynamic simulation is attempted.
- Dynamic simulations attempt to capture changes in a system in response to (usually changing) input signals.
- Stochastic models use random number generators to model chance or random events;
- A discrete event simulation (DES) manages events in time. Most computer, logic-test and fault-tree simulations are of this type. In this type of simulation, the simulator maintains a queue of events sorted by the simulated time they should occur. The simulator reads the queue and triggers new events as each event is processed. It is not important to execute the simulation in real time. It is often more important to be able to access the data produced by the simulation and to discover logic defects in the design or the sequence of events.
- A continuous dynamic simulation performs numerical solution of differential-algebraic equations or differential equations (either partial or ordinary). Periodically, the simulation program solves all the equations and uses the numbers to change the state and output of the simulation. Applications include flight simulators, construction and management simulation games, chemical process modeling, and simulations of electrical circuits. Originally, these kinds of simulations were actually implemented on analog computers, where the differential equations could be represented directly by various electrical components such as op-amps. By the late 1980s, however, most "analog" simulations were run on conventional digital computers that emulate the behavior of an analog computer.
- A special type of discrete simulation that does not rely on a model with an underlying equation, but can nonetheless be represented formally, is agent-based simulation. In agent-based simulation, the individual entities (such as molecules, cells, trees or consumers) in the model are represented directly (rather than by their density or concentration) and possess an internal state and set of behaviors or rules that determine how the agent's state is updated from one time-step to the next.
- Distributed models run on a network of interconnected computers, possibly through the Internet. Simulations dispersed across multiple host computers like this are often referred to as "distributed simulations". There are several standards for distributed simulation, including Aggregate Level Simulation Protocol (ALSP), Distributed Interactive Simulation (DIS), the High Level Architecture (simulation) (HLA) and the Test and Training Enabling Architecture (TENA).
## Visualization
Formerly, the output data from a computer simulation was sometimes presented in a table or a matrix showing how data were affected by numerous changes in the simulation parameters. The use of the matrix format was related to traditional use of the matrix concept in mathematical models. However, psychologists and others noted that humans could quickly perceive trends by looking at graphs or even moving-images or motion-pictures generated from the data, as displayed by computer-generated-imagery (CGI) animation. Although observers could not necessarily read out numbers or quote math formulas, from observing a moving weather chart they might be able to predict events (and "see that rain was headed their way") much faster than by scanning tables of rain-cloud coordinates. Such intense graphical displays, which transcended the world of numbers and formulae, sometimes also led to output that lacked a coordinate grid or omitted timestamps, as if straying too far from numeric data displays. Today, weather forecasting models tend to balance the view of moving rain/snow clouds against a map that uses numeric coordinates and numeric timestamps of events.
Similarly, CGI computer simulations of CAT scans can simulate how a tumor might shrink or change during an extended period of medical treatment, presenting the passage of time as a spinning view of the visible human head, as the tumor changes.
Other applications of CGI computer simulations are being developed to graphically display large amounts of data, in motion, as changes occur during a simulation run.
## In science
Generic examples of types of computer simulations in science, which are derived from an underlying mathematical description:
- a numerical simulation of differential equations that cannot be solved analytically, theories that involve continuous systems such as phenomena in physical cosmology, fluid dynamics (e.g., climate models, roadway noise models, roadway air dispersion models), continuum mechanics and chemical kinetics fall into this category.
- a stochastic simulation, typically used for discrete systems where events occur probabilistically and which cannot be described directly with differential equations (this is a discrete simulation in the above sense). Phenomena in this category include genetic drift, biochemical or gene regulatory networks with small numbers of molecules. (see also: Monte Carlo method).
- multiparticle simulation of the response of nanomaterials at multiple scales to an applied force for the purpose of modeling their thermoelastic and thermodynamic properties. Techniques used for such simulations are Molecular dynamics, Molecular mechanics, Monte Carlo method, and Multiscale Green's function.
Specific examples of computer simulations include:
- statistical simulations based upon an agglomeration of a large number of input profiles, such as the forecasting of equilibrium temperature of receiving waters, allowing the gamut of meteorological data to be input for a specific locale. This technique was developed for thermal pollution forecasting.
- agent based simulation has been used effectively in ecology, where it is often called "individual based modeling" and is used in situations for which individual variability in the agents cannot be neglected, such as population dynamics of salmon and trout (most purely mathematical models assume all trout behave identically).
- time stepped dynamic model. In hydrology there are several such hydrology transport models such as the SWMM and DSSAM Models developed by the U.S. Environmental Protection Agency for river water quality forecasting.
- computer simulations have also been used to formally model theories of human cognition and performance, e.g., ACT-R.
- computer simulation using molecular modeling for drug discovery.
- computer simulation to model viral infection in mammalian cells.
- computer simulation for studying the selective sensitivity of bonds by mechanochemistry during grinding of organic molecules.
- Computational fluid dynamics simulations are used to simulate the behaviour of flowing air, water and other fluids. One-, two- and three-dimensional models are used. A one-dimensional model might simulate the effects of water hammer in a pipe. A two-dimensional model might be used to simulate the drag forces on the cross-section of an aeroplane wing. A three-dimensional simulation might estimate the heating and cooling requirements of a large building.
- An understanding of statistical thermodynamic molecular theory is fundamental to the appreciation of molecular solutions. Development of the Potential Distribution Theorem (PDT) allows this complex subject to be simplified to down-to-earth presentations of molecular theory.
Notable, and sometimes controversial, computer simulations used in science include: Donella Meadows' World3 used in the Limits to Growth, James Lovelock's Daisyworld and Thomas Ray's Tierra.
In social sciences, computer simulation is an integral component of the five angles of analysis fostered by the data percolation methodology, which also includes qualitative and quantitative methods, reviews of the literature (including scholarly), and interviews with experts, and which forms an extension of data triangulation. Of course, similar to any other scientific method, replication is an important part of computational modeling
## In practical contexts
Computer simulations are used in a wide variety of practical contexts, such as:
- analysis of air pollutant dispersion using atmospheric dispersion modeling
- As a possible humane alternative to live animal testing in respect to animal rights.
- design of complex systems such as aircraft and also logistics systems.
- design of noise barriers to effect roadway noise mitigation
- modeling of application performance
- flight simulators to train pilots
- weather forecasting
- forecasting of risk
- simulation of electrical circuits
- Power system simulation
- simulation of other computers is emulation.
- forecasting of prices on financial markets (for example Adaptive Modeler)
- behavior of structures (such as buildings and industrial parts) under stress and other conditions
- design of industrial processes, such as chemical processing plants
- strategic management and organizational studies
- reservoir simulation for the petroleum engineering to model the subsurface reservoir
- process engineering simulation tools.
- robot simulators for the design of robots and robot control algorithms
- urban simulation models that simulate dynamic patterns of urban development and responses to urban land use and transportation policies.
- traffic engineering to plan or redesign parts of the street network from single junctions over cities to a national highway network to transportation system planning, design and operations. See a more detailed article on Simulation in Transportation.
- modeling car crashes to test safety mechanisms in new vehicle models.
- crop-soil systems in agriculture, via dedicated software frameworks (e.g. BioMA, OMS3, APSIM)
The reliability and the trust people put in computer simulations depends on the validity of the simulation model, therefore verification and validation are of crucial importance in the development of computer simulations. Another important aspect of computer simulations is that of reproducibility of the results, meaning that a simulation model should not provide a different answer for each execution. Although this might seem obvious, this is a special point of attention in stochastic simulations, where random numbers should actually be semi-random numbers. An exception to reproducibility are human-in-the-loop simulations such as flight simulations and computer games. Here a human is part of the simulation and thus influences the outcome in a way that is hard, if not impossible, to reproduce exactly.
Vehicle manufacturers make use of computer simulation to test safety features in new designs. By building a copy of the car in a physics simulation environment, they can save the hundreds of thousands of dollars that would otherwise be required to build and test a unique prototype. Engineers can step through the simulation milliseconds at a time to determine the exact stresses being put upon each section of the prototype.
Computer graphics can be used to display the results of a computer simulation. Animations can be used to experience a simulation in real-time, e.g., in training simulations. In some cases animations may also be useful in faster than real-time or even slower than real-time modes. For example, faster than real-time animations can be useful in visualizing the buildup of queues in the simulation of humans evacuating a building. Furthermore, simulation results are often aggregated into static images using various ways of scientific visualization.
In debugging, simulating a program execution under test (rather than executing natively) can detect far more errors than the hardware itself can detect and, at the same time, log useful debugging information such as instruction trace, memory alterations and instruction counts. This technique can also detect buffer overflow and similar "hard to detect" errors as well as produce performance information and tuning data.
## Pitfalls
Although sometimes ignored in computer simulations, it is very important to perform a sensitivity analysis to ensure that the accuracy of the results is properly understood. For example, the probabilistic risk analysis of factors determining the success of an oilfield exploration program involves combining samples from a variety of statistical distributions using the Monte Carlo method. If, for instance, one of the key parameters (e.g., the net ratio of oil-bearing strata) is known to only one significant figure, then the result of the simulation might not be more precise than one significant figure, although it might (misleadingly) be presented as having four significant figures. | https://en.wikipedia.org/wiki/Computer_simulation |
In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books that,
The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.
## Formal statement
Let K− denote the strong-operator closure of a set K in B(H), the set of bounded operators on the Hilbert space H, and let (K)1 denote the intersection of K with the unit ball of B(H).
Kaplansky density theorem. If
$$
A
$$
is a self-adjoint algebra of operators in
$$
B(H)
$$
, then each element
$$
a
$$
in the unit ball of the strong-operator closure of
$$
A
$$
is in the strong-operator closure of the unit ball of
$$
A
$$
. In other words,
$$
(A)_1^{-} = (A^{-})_1
$$
. If
$$
h
$$
is a self-adjoint operator in
$$
(A^{-})_1
$$
, then
$$
h
$$
is in the strong-operator closure of the set of self-adjoint operators in
$$
(A)_1
$$
.
The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology.
1) If h is a positive operator in (A−)1, then h is in the strong-operator closure of the set of self-adjoint operators in (A+)1, where A+ denotes the set of positive operators in A.
2) If A is a C*-algebra acting on the Hilbert space H and u is a unitary operator in A−, then u is in the strong-operator closure of the set of unitary operators in A.
In the density theorem and 1) above, the results also hold if one considers a ball of radius r > 0, instead of the unit ball.
## Proof
The standard proof uses the fact that a bounded continuous real-valued function f is strong-operator continuous. In other words, for a net {aα} of self-adjoint operators in A, the continuous functional calculus a → f(a) satisfies,
$$
\lim f(a_{\alpha}) = f (\lim a_{\alpha})
$$
in the strong operator topology. This shows that self-adjoint part of the unit ball in A− can be approximated strongly by self-adjoint elements in A. A matrix computation in M2(A) considering the self-adjoint operator with entries 0 on the diagonal and a and a* at the other positions, then removes the self-adjointness restriction and proves the theorem. | https://en.wikipedia.org/wiki/Kaplansky_density_theorem |
The quadratrix or trisectrix of Hippias (also called the quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is traced out by the crossing point of two lines, one moving by translation at a uniform speed, and the other moving by rotation around one of its points at a uniform speed. An alternative definition as a parametric curve leads to an equivalence between the quadratrix, the image of the Lambert W function, and the graph of the function
$$
y=x\cot x
$$
.
The discovery of this curve is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem, hence its name as a trisectrix. Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle, hence its name as a quadratrix.
### Dinostratus's theorem
, used in this attempt, relates an endpoint of the curve to the value of . Both angle trisection and squaring the circle can be solved using a compass, a straightedge, and a given copy of this curve, but not by compass and straightedge alone. Although a dense set of points on the curve can be constructed by compass and straightedge, allowing these problems to be approximated, the whole curve cannot be constructed in this way.
The quadratrix of Hippias is a transcendental curve. It is one of several curves used in Greek mathematics for squaring the circle.
## Definitions
### By moving lines
Consider a square
$$
ABCD
$$
, and an inscribed quarter circle arc centered at
$$
A
$$
with radius equal to the side of the square. Let
$$
E
$$
be a point that travels with a constant angular velocity along the arc from
$$
D
$$
to
$$
B
$$
, and let
$$
F
$$
be a point that travels simultaneously with a constant velocity from
$$
D
$$
to
$$
A
$$
along line segment
$$
\overline{AD}
$$
, so that
$$
E
$$
and
$$
F
$$
start at the same time at
$$
D
$$
and arrive at the same time at
$$
B
$$
and
$$
A
$$
. Then the quadratrix is defined as the locus of the intersection of line segment
$$
\overline{AE}
$$
with the parallel line to
$$
\overline{AB}
$$
through
$$
F
$$
.
### Helicoid section
If a line in three-dimensional space, perpendicular to and intersecting the rotates at a constant rate at the same time that its intersection with the moves upward at a constant rate, it will trace out a helicoid. As Pappus of Alexandria observed, the curve formed by intersecting this helicoid with a non-vertical plane that contains one of the generating lines of the helicoid, when projected onto the forms a quadratrix.
### Parametric equation
If one places square
$$
ABCD
$$
with side length
$$
a
$$
in a (Cartesian) coordinate system with the side
$$
\overline{AB}
$$
on the
$$
x
$$
-axis and with vertex
$$
A
$$
at the origin, then the quadratrix is described by a parametric equation that gives the coordinates of each point on the curve as a function of a time parameter
$$
t
$$
, as
$$
\gamma(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}=\begin{pmatrix}\frac{2a}{\pi} t\cot(t)\\\frac{2a}{\pi} t\end{pmatrix}
$$
This description can also be used to give an analytical rather than a geometric definition of the quadratrix and to extend it beyond the
$$
(0,\tfrac{\pi}{2}]
$$
interval. It does however remain undefined at the points where
$$
\cot(t)
$$
is singular, except for the case of
$$
t=0
$$
. At
$$
t=0
$$
, the singularity is removable by evaluating it using the limit
$$
\lim_{t\to 0} t \cot(t)=1
$$
, obtained as the ratio of the identity function and tangent function using l'Hôpital's rule. Removing the singularity in this way and extending the parametric definition to negative values of
$$
t
$$
yields a continuous planar curve on the range of parameter values
$$
-\pi<t<\pi
$$
.
### As the graph of a function
When reflected left to right and scaled appropriately in the complex plane, the quadratrix forms the image of the real axis for one branch of Lambert W function. The images for other branches consist of curves above and below the quadratrix, and the real axis itself. To describe the quadratrix as the graph of an unbranched function, it is advantageous to swap the
$$
y
$$
-axis and the
$$
x
$$
-axis, that is to place the side
$$
\overline{AB}
$$
on the
$$
y
$$
-axis rather than on the
$$
x
$$
-axis. Then the quadratrix forms the graph of the function
$$
f(x) = x \cdot \cot\left(\frac{\pi}{2a} \cdot x \right).
$$
## Angle trisection
The trisection of an arbitrary angle using only compass and straightedge is impossible. However, if the quadratrix is allowed as an additional tool, it is possible to divide an arbitrary angle into
$$
n
$$
equal segments and hence a trisection (
$$
n=3
$$
) becomes possible. In practical terms the quadratrix can be drawn with the help of a template or a quadratrix compass (see drawing).
By the definition of the quadratrix, the traversed angle is proportional to the traversed segment of the associated squares' side. Therefore, dividing that segment on the side into
$$
n
$$
equal parts yields a partition of the associated angle into
$$
n
$$
equal parts as well. Dividing the line segment into
$$
n
$$
equal parts with ruler and compass is possible due to the intercept theorem.
In more detail, to divide a given angle
$$
\angle BAE
$$
(at most 90°) into any desired number of equal parts, construct a square
$$
ABCD
$$
over its leg
$$
\overline{AB}
$$
. The other leg of the angle intersects the quadratrix of the square in a point
$$
G
$$
and the parallel line to the leg
$$
\overline{AB}
$$
through
$$
G
$$
intersects the side
$$
\overline{AD}
$$
of the square in
$$
F
$$
. Now the segment
$$
\overline{AF}
$$
corresponds to the angle
$$
\angle BAE
$$
and due to the definition of the quadratrix any division of the segment
$$
\overline{AF}
$$
into
$$
n
$$
equal segments yields a corresponding division of the angle
$$
\angle BAE
$$
into
$$
n
$$
equal angles. To divide the segment
$$
\overline{AF}
$$
into
$$
n
$$
equal segments, draw any ray starting at
$$
A
$$
with
$$
n
$$
equal segments (of arbitrary length) on it. Connect the endpoint
$$
O
$$
of the last segment to
$$
F
$$
and draw lines parallel to
$$
\overline{OF}
$$
through all the endpoints of the remaining
$$
n-1
$$
segments on
$$
\overline{AO}
$$
. These parallel lines divide the segment
$$
\overline{AF}
$$
into
$$
n
$$
equal segments. Now draw parallel lines to
$$
\overline{AB}
$$
through the endpoints of those segments on
$$
\overline{AF}
$$
, intersecting the trisectrix. Connecting their points of intersection to
$$
A
$$
yields a partition of angle
$$
\angle BAE
$$
into
$$
n
$$
equal angles.
Since not all points of the trisectrix can be constructed with circle and compass alone, it is really required as an additional tool beyond the compass and straightedge. However it is possible to construct a dense subset of the trisectrix by compass and straightedge. In this way, while one cannot assure an exact division of an angle into
$$
n
$$
parts without a given trisectrix, one can construct an arbitrarily close approximation to the trisectrix and therefore also to the division of the angle by compass and straightedge alone.
## Squaring the circle
Squaring the circle with compass and straightedge alone is impossible. However, if one allows the quadratrix of Hippias as an additional construction tool, the squaring of the circle becomes possible due to Dinostratus's theorem relating an endpoint of this circle to the value of . One can use this theorem to construct a square with the same area as a quarter circle. Another square with twice the side length has the same area as the full circle.
Dinostratus's theorem
According to Dinostratus's theorem the quadratrix divides one of the sides of the associated square in a ratio of
$$
\tfrac{2}{\pi}
$$
. More precisely, for the square
$$
ABCD
$$
used to define the curve, let
$$
J
$$
be the endpoint of the curve on edge
$$
AB
$$
. Then
$$
\frac{\overline{AJ}}{\overline{AB}}=\frac{2}{\pi},
$$
as can be seen from the parametric equation for the quadratrix at
$$
t=0
$$
and the limiting behavior of the function controlling its
$$
x
$$
-coordinate at that parameter value,
$$
\lim_{t\to 0}t\cot t=1
$$
.
The point
$$
J
$$
, where the quadratrix meets the side
$$
\overline{AB}
$$
of the associated square, is one of the points of the quadratrix that cannot be constructed with ruler and compass alone and not even with the help of the quadratrix compass. This is due to the fact that (as Sporus of Nicaea already observed) the two uniformly moving lines coincide and hence there exists no unique intersection point. However relying on the generalized definition of the quadratrix as a function or planar curve allows for
$$
J
$$
being a point on the quadratrix.
### Construction
For a given quarter circle with radius
$$
r
$$
one constructs the associated square
$$
ABCD
$$
with side length
$$
r
$$
. The quadratrix intersect the side
$$
\overline{AB}
$$
in
$$
J
$$
with
$$
\left|\overline{AJ}\right|=\tfrac{2}{\pi}r
$$
. Now one constructs a line segment
$$
\overline{JK}
$$
of length
$$
r
$$
being perpendicular to
$$
\overline{AB}
$$
. Then the line through
$$
A
$$
and
$$
K
$$
intersects the extension of the side
$$
\overline{BC}
$$
in
$$
L
$$
and from the intercept theorem follows
$$
\left|\overline{BL}\right|=\tfrac{\pi}{2}r
$$
. Extending
$$
\overline{AB}
$$
to the right by a new line segment
$$
\left|\overline{BO}\right|=\tfrac{r}{2}
$$
yields the rectangle
$$
BLNO
$$
with sides
$$
\overline{BL}
$$
and
$$
\overline{BO}
$$
the area of which matches the area of the quarter circle. This rectangle can be transformed into a square of the same area with the help of Euclid's geometric mean theorem. One extends the side
$$
\overline{ON}
$$
by a line segment
$$
\left|\overline{OQ}\right|=\left|\overline{BO}\right|=\tfrac{r}{2}
$$
and draws a half circle to right of
$$
\overline{NQ}
$$
, which has
$$
\overline{NQ}
$$
as its diameter. The extension of
$$
\overline{BO}
$$
meets the half circle in
$$
\overline{R}
$$
and due to Thales' theorem the line segment
$$
\overline{OR}
$$
is the altitude of the right-angled triangle
$$
QNR
$$
. Hence the geometric mean theorem can be applied, which means that
$$
\overline{OR}
$$
forms the side of a square
$$
OUSR
$$
with the same area as the rectangle
$$
BLNO
$$
and hence as the quarter circle.
## Other properties
For a quadratrix constructed from a unit square, the area under the quadratrix is
$$
\frac{2\ln 2}{\pi}\approx 0.44127.
$$
Inverting the quadratrix by a circle centered at the axis of the rotating line that defines it produces a cochleoid, and in the same way inverting the cochleoid produces a quadratrix.
## History
The quadratrix of Hippias is one of several curves used in Greek mathematics for squaring the circle, the most well-known for this purpose. Another is the Archimedean spiral, used to square the circle by Archimedes.
It is mentioned in the works of Proclus (412–485), Pappus of Alexandria (3rd and 4th centuries) and Iamblichus (c. 240 – c. 325). Proclus names Hippias as the inventor of a curve called a quadratrix and describes somewhere else how Hippias has applied the curve on the trisection problem. Pappus only mentions how a curve named a quadratrix was used by Dinostratus, Nicomedes and others to square the circle. He relays the objections of Sporus of Nicaea to this construction, but neither mentions Hippias nor attributes the invention of the quadratrix to a particular person. Iamblichus just writes in a single line, that a curve called a quadratrix was used by Nicomedes to square the circle.
From Proclus' name for the curve, it is conceivable that Hippias himself used it for squaring the circle or some other curvilinear figure. However, most historians of mathematics assume that Hippias invented the curve, but used it only for the trisection of angles. According to this theory, its use for squaring the circle only occurred decades later and was due to mathematicians like Dinostratus and Nicomedes. This interpretation of the historical sources goes back to the German mathematician and historian Moritz Cantor.
Rüdiger Thiele claims that François Viète used the trisectrix to derive Viète's formula, an infinite product of nested radicals published by Viète in 1593 that converges to
$$
2/\pi
$$
. However, other sources instead view Viète's formula as an elaboration of a method of nested polygons used by Archimedes to approximate
$$
\pi
$$
. In his 1637 book La Géométrie, René Descartes classified curves either as "geometric", admitting a precise geometric construction, or if not as "mechanical"; he gave the quadratrix as an example of a mechanical curve. In modern terminology, roughly the same distinction may be expressed by saying that it is a transcendental curve rather than an algebraic curve. Isaac Newton used trigonometric series to determine the area enclosed by the quadratrix.
## Related phenomena
When a camera with a rolling shutter takes a photograph of a quickly rotating object, such as a propeller, curves resembling the quadratrix of Hippias may appear, generated in an analogous way to the quadratrix: these curves are traced out by the points of intersection of the rotating propeller blade and the linearly moving scan line of the camera. Different curves may be generated depending on the angle of the propeller at the time when the scan line crosses its axis of rotation (rather than coinciding with the scan line at that time for the quadratrix). A similar visual phenomenon was also observed in the 19th century by Peter Mark Roget when the spoked wheel of a moving cart or train is viewed through the vertical slats of a fence or palisade; it is called Roget’s palisade illusion.
## References
## Further reading
-
- ; this unpublished preprint includes the conjecture that the quadratrix cannot be used for doubling the cube, another problem unsolvable with compass and straightedge
## External links
- Michael D. Huberty, Ko Hayashi, Chia Vang: Hippias' Quadratrix
-
Category:Euclidean plane geometry
Category:Curves
Category:Squaring the circle
Category:Area
Category:Greek mathematics | https://en.wikipedia.org/wiki/Quadratrix_of_Hippias |
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and in particle physics, for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in
### Physics
in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.
Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams and is widely used. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World, he was ranked the seventh-greatest physicist of all time.
He assisted in the development of the atomic bomb during World War II and became known to the wider public in the 1980s as a member of the Rogers Commission, the panel that investigated the Space Shuttle
### Challenger disaster
. Along with his work in theoretical physics, Feynman has been credited with having pioneered the field of quantum computing and introducing the concept of nanotechnology. He held the Richard C. Tolman professorship in theoretical physics at the California Institute of Technology.
Feynman was a keen popularizer of physics through both books and lectures, including a talk on top-down nanotechnology, "There's Plenty of Room at the Bottom" (1959) and the three-volumes of his undergraduate lectures, The Feynman Lectures on Physics (1961–1964). He delivered lectures for lay audiences, recorded in The Character of Physical Law (1965) and QED: The Strange Theory of Light and Matter (1985). Feynman also became known through his autobiographical books
### Surely You're Joking, Mr. Feynman!
(1985) and What Do You Care What Other People Think? (1988), and books written about him such as Tuva or Bust! by Ralph Leighton and the biography Genius: The Life and Science of Richard Feynman by James Gleick.
## Early life
Feynman was born on May 11, 1918, in New York City, to Lucille (; 1895–1981), a homemaker, and Melville Arthur Feynman (1890–1946), a sales manager. Feynman's father was born in Minsk, Russian Empire, and immigrated with his parents to the United States at the age of five. Feynman's mother was born in the United States. Lucille's father had emigrated from Poland, and her mother also came from a family of Polish immigrants. She trained as a primary school teacher but married Melville in 1917, before taking up a profession. Feynman was a late talker and did not speak until after his third birthday. As an adult, he spoke with a New York accent strong enough to be perceived as an affectation or exaggeration, so much so that his friends Wolfgang Pauli and Hans Bethe once commented that Feynman spoke like a "bum".
The young Feynman was heavily influenced by his father, who encouraged him to ask questions to challenge orthodox thinking, and who was always ready to teach Feynman something new. From his mother, he gained the sense of humor that he had throughout his life. As a child, he had a talent for engineering, maintained an experimental laboratory in his home, and delighted in repairing radios. This radio repairing was probably the first job Feynman had, and during this time he showed early signs of an aptitude for his later career in theoretical physics, when he would analyze the issues theoretically and arrive at the solutions. When he was in grade school, he created a home burglar alarm system while his parents were out for the day running errands.
When Richard was five, his mother gave birth to a younger brother, Henry Phillips, who died at age four weeks. Four years later, Richard's sister Joan was born and the family moved to Far Rockaway, Queens. Though separated by nine years, Joan and Richard were close, and they both shared a curiosity about the world. Though their mother thought women lacked the capacity to understand such things, Richard encouraged Joan's interest in astronomy, taking her to see the aurora borealis in Far Rockaway. As an astrophysicist, Joan would help to explain what caused the northern lights.
### Religion
Feynman's parents were both from Jewish families, and his family went to the synagogue every Friday. However, by his youth, Feynman described himself as an "avowed atheist". Many years later, in a letter to Tina Levitan, declining a request for information for her book on Jewish Nobel Prize winners, he stated, "To select, for approbation the peculiar elements that come from some supposedly Jewish heredity is to open the door to all kinds of nonsense on racial theory", adding, "at thirteen I was not only converted to other religious views, but I also stopped believing that the Jewish people are in any way 'the chosen people'".
Later in life, during a visit to the Jewish Theological Seminary, Feynman encountered the Talmud for the first time. He saw that it contained the original text in a little square on each page, and surrounding it were commentaries written over time by different people. In this way the Talmud had evolved, and everything that was discussed was carefully recorded. Despite being impressed, Feynman was disappointed with the lack of interest for nature and the outside world expressed by the rabbis, who cared about only those questions which arise from the Talmud.
## Education
Feynman attended Far Rockaway High School, which was also attended by fellow Nobel laureates Burton Richter and Baruch Samuel Blumberg. Upon starting high school, Feynman was quickly promoted to a higher math class. An IQ test administered in high school estimated his IQ at 125—high but "merely respectable", according to biographer James Gleick. His sister Joan, who scored one point higher, later jokingly claimed to an interviewer that she was smarter. Years later he declined to join Mensa International, saying that his IQ was too low.
When Feynman was 15, he taught himself trigonometry, advanced algebra, infinite series, analytic geometry, and both differential and integral calculus. Before entering college, he was experimenting with mathematical topics such as the half-derivative using his own notation. He created special symbols for logarithm, sine, cosine and tangent functions so they did not look like three variables multiplied together, and for the derivative, to remove the temptation of canceling out the
$$
d
$$
's in
$$
d/dx
$$
. A member of the Arista Honor Society, in his last year in high school he won the New York University Math Championship. His habit of direct characterization sometimes rattled more conventional thinkers; for example, one of his questions, when learning feline anatomy, was "Do you have a map of the cat?" (referring to an anatomical chart).
Feynman applied to Columbia University but was not accepted because of its quota for the number of Jews admitted. Instead, he attended the Massachusetts Institute of Technology, where he joined the Pi Lambda Phi fraternity. Although he originally majored in mathematics, he later switched to electrical engineering, as he considered mathematics to be too abstract. Noticing that he "had gone too far", he then switched to physics, which he claimed was "somewhere in between". As an undergraduate, he published two papers in the Physical Review. One of these, which was co-written with Manuel Vallarta, was titled "The Scattering of Cosmic Rays by the Stars of a Galaxy".
The other was his senior thesis, on "Forces in Molecules", based on a topic assigned by John C. Slater, who was sufficiently impressed by the paper to have it published. Its main result is known as the Hellmann–Feynman theorem.
In 1939, Feynman received a bachelor's degree and was named a Putnam Fellow. He attained a perfect score on the graduate school entrance exams to Princeton University in physics—an unprecedented feat—and an outstanding score in mathematics, but did poorly on the history and English portions. The head of the physics department there, Henry D. Smyth, had another concern, writing to Philip M. Morse to ask: "Is Feynman Jewish? We have no definite rule against Jews but have to keep their proportion in our department reasonably small because of the difficulty of placing them." Morse conceded that Feynman was indeed Jewish, but reassured Smyth that Feynman's "physiognomy and manner, however, show no trace of this characteristic".
Attendees at Feynman's first seminar, which was on the classical version of the Wheeler–Feynman absorber theory, included Albert Einstein, Wolfgang Pauli, and John von Neumann. Pauli made the prescient comment that the theory would be extremely difficult to quantize, and Einstein said that one might try to apply this method to gravity in general relativity, which Sir Fred Hoyle and Jayant Narlikar did much later as the Hoyle–Narlikar theory of gravity. Feynman received a PhD from Princeton in 1942; his thesis advisor was John Archibald Wheeler. In his doctoral thesis titled "The Principle of Least Action in Quantum Mechanics", Feynman applied the principle of stationary action to problems of quantum mechanics, inspired by a desire to quantize the Wheeler–Feynman absorber theory of electrodynamics, and laid the groundwork for the path integral formulation and Feynman diagrams. A key insight was that positrons behaved like electrons moving backwards in time. James Gleick wrote:
One of the conditions of Feynman's scholarship to Princeton was that he could not be married; nevertheless, he continued to see his high school sweetheart, Arline Greenbaum, and was determined to marry her once he had been awarded his PhD despite the knowledge that she was seriously ill with tuberculosis. This was an incurable disease at the time, and she was not expected to live more than two years. On June 29, 1942, they took the ferry to Staten Island, where they were married in the city office. The ceremony was attended by neither family nor friends and was witnessed by a pair of strangers. Feynman could kiss Arline only on the cheek. After the ceremony he took her to Deborah Hospital, where he visited her on weekends.
## Manhattan Project
In 1941, with World War II occurring in Europe but the United States not yet at war, Feynman spent the summer working on ballistics problems at the Frankford Arsenal in Pennsylvania. After the attack on Pearl Harbor brought the United States into the war, Feynman was recruited by Robert R. Wilson, who was working on means to produce enriched uranium for use in an atomic bomb, as part of what would become the Manhattan Project. At the time, Feynman had not earned a graduate degree. Wilson's team at Princeton was working on a device called an isotron, intended to electromagnetically separate uranium-235 from uranium-238. This was done in a quite different manner from that used by the calutron that was under development by a team under Wilson's former mentor, Ernest O. Lawrence, at the Radiation Laboratory of the University of California. On paper, the isotron was many times more efficient than the calutron, but Feynman and Paul Olum struggled to determine whether it was practical. Ultimately, on Lawrence's recommendation, the isotron project was abandoned.
At this juncture, in early 1943, Robert Oppenheimer was establishing the Los Alamos Laboratory, a secret laboratory on a mesa in New Mexico where atomic bombs would be designed and built. An offer was made to the Princeton team to be redeployed there. "Like a bunch of professional soldiers," Wilson later recalled, "we signed up, en masse, to go to Los Alamos." Oppenheimer recruited many young physicists, including Feynman, who he telephoned long distance from Chicago to inform that he had found a Presbyterian sanatorium in Albuquerque, New Mexico for Arline. They were among the first to depart for New Mexico, leaving on a train on March 28, 1943. The railroad supplied Arline with a wheelchair, and Feynman paid extra for a private room for her. There they spent their wedding anniversary.
At Los Alamos, Feynman was assigned to Hans Bethe's Theoretical (T) Division, and impressed Bethe enough to be made a group leader. He and Bethe developed the Bethe–Feynman formula for calculating the yield of a fission bomb, which built upon previous work by Robert Serber. As a junior physicist, he was not central to the project. He administered the computation group of human computers in the theoretical division. With Stanley Frankel and Nicholas Metropolis, he assisted in establishing a system for using IBM punched cards for computation. He invented a new method of computing logarithms that he later used on the Connection Machine. An avid drummer, Feynman figured out how to get the machine to click in musical rhythms. Other work at Los Alamos included calculating neutron equations for the Los Alamos "Water Boiler", a small nuclear reactor, to measure how close an assembly of fissile material was to criticality.
On completing this work, Feynman was sent to the Clinton Engineer
## Works
in Oak Ridge, Tennessee, where the Manhattan Project had its uranium enrichment facilities. He aided the engineers there in devising safety procedures for material storage so that criticality accidents could be avoided, especially when enriched uranium came into contact with water, which acted as a neutron moderator. He insisted on giving the rank and file a lecture on nuclear physics so that they would realize the dangers. He explained that while any amount of unenriched uranium could be safely stored, the enriched uranium had to be carefully handled. He developed a series of safety recommendations for the various grades of enrichments. He was told that if the people at Oak Ridge gave him any difficulty with his proposals, he was to inform them that Los Alamos "could not be responsible for their safety otherwise".
Returning to Los Alamos, Feynman was put in charge of the group responsible for the theoretical work and calculations on the proposed uranium hydride bomb, which ultimately proved to be infeasible. He was sought out by physicist Niels Bohr for one-on-one discussions. He later discovered the reason: most of the other physicists were too much in awe of Bohr to argue with him. Feynman had no such inhibitions, vigorously pointing out anything he considered to be flawed in Bohr's thinking. He said he felt as much respect for Bohr as anyone else, but once anyone got him talking about physics, he would become so focused he forgot about social niceties. Perhaps because of this, Bohr never warmed to Feynman.
At Los Alamos, which was isolated for security, Feynman amused himself by investigating the combination locks on the cabinets and desks of physicists. He often found that they left the lock combinations on the factory settings, wrote the combinations down, or used easily guessable combinations like dates. He found one cabinet's combination by trying numbers he thought a physicist might use (it proved to be 27–18–28 after the base of natural logarithms, e = 2.71828 ...), and found that the three filing cabinets where a colleague kept research notes all had the same combination. He left notes in the cabinets as a prank, spooking his colleague, Frederic de Hoffmann, into thinking a spy had gained access to them.
Feynman's $380 () monthly salary was about half the amount needed for his modest living expenses and Arline's medical bills, and they were forced to dip into her $3,300 () in savings. On weekends he borrowed a car from his friend Klaus Fuchs to drive to Albuquerque to see Arline. Asked who at Los Alamos was most likely to be a spy, Fuchs mentioned Feynman's safe-cracking and frequent trips to Albuquerque; Fuchs himself later confessed to spying for the Soviet Union. The FBI would compile a bulky file on Feynman, particularly in view of Feynman's Q clearance.
Informed that Arline was dying, Feynman drove to Albuquerque and sat with her for hours until she died on June 16, 1945. He then immersed himself in work on the project and was present at the Trinity nuclear test. Feynman claimed to be the only person to see the explosion without the very dark glasses or welder's lenses provided, reasoning that it was safe to look through a truck windshield, as it would screen out the harmful ultraviolet radiation. The immense brightness of the explosion made him duck to the truck's floor, where he saw a temporary "purple splotch" afterimage.
## Cornell (1945–1949)
Feynman nominally held an appointment at the University of Wisconsin–Madison as an assistant professor of physics, but was on unpaid leave during his involvement in the Manhattan Project. In 1945, he received a letter from Dean Mark Ingraham of the College of Letters and Science requesting his return to the university to teach in the coming academic year. His appointment was not extended when he did not commit to returning. In a talk given there several years later, Feynman quipped, "It's great to be back at the only university that ever had the good sense to fire me."
As early as October 30, 1943, Bethe had written to the chairman of the physics department of his university, Cornell, to recommend that Feynman be hired. On February 28, 1944, this was endorsed by Robert Bacher, also from Cornell, and one of the most senior scientists at Los Alamos. This led to an offer being made in August 1944, which Feynman accepted. Oppenheimer had also hoped to recruit Feynman to the University of California, but the head of the physics department, Raymond T. Birge, was reluctant. He made Feynman an offer in May 1945, but Feynman turned it down. Cornell matched its salary offer of $3,900 () per annum. Feynman became one of the first of the Los Alamos Laboratory's group leaders to depart, leaving for Ithaca, New York, in October 1945.
Because Feynman was no longer working at the Los Alamos Laboratory, he was no longer exempt from the draft. At his induction physical, Army psychiatrists diagnosed Feynman as suffering from a mental illness and the Army gave him a 4-F exemption on mental grounds. His father died suddenly on October 8, 1946, and Feynman suffered from depression. On October 17, 1946, he wrote a letter to Arline, expressing his deep love and heartbreak. The letter was sealed and only opened after his death. "Please excuse my not mailing this," the letter concluded, "but I don't know your new address." Unable to focus on research problems, Feynman began tackling physics problems, not for utility, but for self-satisfaction. One of these involved analyzing the physics of a twirling, nutating disk as it is moving through the air, inspired by an incident in the cafeteria at Cornell when someone tossed a dinner plate in the air. He read the work of Sir William Rowan Hamilton on quaternions, and tried unsuccessfully to use them to formulate a relativistic theory of electrons. His work during this period, which used equations of rotation to express various spinning speeds, ultimately proved important to his Nobel Prize–winning work, yet because he felt burned out and had turned his attention to less immediately practical problems, he was surprised by the offers of professorships from other renowned universities, including the Institute for Advanced Study, the University of California, Los Angeles, and the University of California, Berkeley.
Feynman was not the only frustrated theoretical physicist in the early post-war years. Quantum electrodynamics suffered from infinite integrals in perturbation theory. These were clear mathematical flaws in the theory, which Feynman and Wheeler had tried, unsuccessfully, to work around. "Theoreticians", noted Murray Gell-Mann, "were in disgrace". In June 1947, leading American physicists met at the Shelter Island Conference. For Feynman, it was his "first big conference with big men ... I had never gone to one like this one in peacetime." The problems plaguing quantum electrodynamics were discussed, but the theoreticians were completely overshadowed by the achievements of the experimentalists, who reported the discovery of the Lamb shift, the measurement of the magnetic moment of the electron, and Robert Marshak's two-meson hypothesis.
Bethe took the lead from the work of Hans Kramers, and derived a renormalized non-relativistic quantum equation for the Lamb shift. The next step was to create a relativistic version. Feynman thought that he could do this, but when he went back to Bethe with his solution, it did not converge. Feynman carefully worked through the problem again, applying the path integral formulation that he had used in his thesis. Like Bethe, he made the integral finite by applying a cut-off term. The result corresponded to Bethe's version. Feynman presented his work to his peers at the Pocono Conference in 1948. It did not go well. Julian Schwinger gave a long presentation of his work in quantum electrodynamics, and Feynman then offered his version, entitled "Alternative Formulation of Quantum Electrodynamics". The unfamiliar Feynman diagrams, used for the first time, puzzled the audience. Feynman failed to get his point across, and Paul Dirac, Edward Teller and Niels Bohr all raised objections.
To Freeman Dyson, one thing at least was clear: Shin'ichirō Tomonaga, Schwinger and Feynman understood what they were talking about even if no one else did, but had not published anything. He was convinced that Feynman's formulation was easier to understand, and ultimately managed to convince Oppenheimer that this was the case. Dyson published a paper in 1949, which added new rules to Feynman's that told how to implement renormalization. Feynman was prompted to publish his ideas in the Physical Review in a series of papers over three years. His 1948 papers on "A Relativistic Cut-Off for Classical Electrodynamics" attempted to explain what he had been unable to get across at Pocono. His 1949 paper on "The Theory of Positrons" addressed the Schrödinger equation and Dirac equation, and introduced what is now called the Feynman propagator. Finally, in papers on the "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction" in 1950 and "An Operator Calculus Having Applications in Quantum Electrodynamics" in 1951, he developed the mathematical basis of his ideas, derived familiar formulae and advanced new ones.
While papers by others initially cited Schwinger, papers citing Feynman and employing Feynman diagrams appeared in 1950, and soon became prevalent. Students learned and used the powerful new tool that Feynman had created. Computer programs were later written to evaluate Feynman diagrams, enabling physicists to use quantum field theory to make high-precision predictions. Marc Kac adapted Feynman's technique of summing over possible histories of a particle to the study of parabolic partial differential equations, yielding what is now known as the Feynman–Kac formula, the use of which extends beyond physics to many applications of stochastic processes. To Schwinger, however, the Feynman diagram was "pedagogy, not physics".
Looking back on this period, Feynman would reflect fondly on his time at the Telluride House, where he resided for a large period of his Cornell career. In an interview, he described the House as "a group of boys that have been specially selected because of their scholarship, because of their cleverness or whatever it is, to be given free board and lodging and so on, because of their brains". He enjoyed the house's convenience and said that "it's there that I did the fundamental work" for which he won the Nobel Prize.
However, Feynman was also reported to have been quite restless during his time at Cornell. By 1949, as the period was coming to a close, he had never settled into a particular house or apartment, moving instead between guest houses or student residences. While he did spend some time living with various married friends, these situations were reported to frequently end because the "arrangements became sexually volatile". The renowned 31 year old was known to frequently pursue his married female friends, undergraduate girls and women, and to hire sex workers, which would sour many of his friendships. Additionally, Feynman was not fond of Ithaca's cold winter weather or feeling as though he lived in the shadow of Hans Bethe while at Cornell.
## Brazil (1949–1952)
Feynman spent several weeks in Rio de Janeiro in July 1949. That year, the Soviet Union detonated its first atomic bomb, generating concerns about espionage. Fuchs was arrested as a Soviet spy in 1950 and the FBI questioned Bethe about Feynman's loyalty. Physicist David Bohm was arrested on December 4, 1950, and emigrated to Brazil in October 1951. Because of the fears of a nuclear war, a girlfriend told Feynman that he should also consider moving to South America. He had a sabbatical coming for 1951–1952, and elected to spend it in Brazil, where he gave courses at the Centro Brasileiro de Pesquisas Físicas.
In Brazil, Feynman was impressed with samba music, and learned to play the , a metal percussion instrument based on a frying pan. He was an enthusiastic amateur player of bongo and conga drums and often played them in the pit orchestra in musicals. He spent time in Rio with his friend Bohm, but Bohm could not convince Feynman to investigate Bohm's ideas on physics.
## Caltech and later years (1952–1978)
### Personal and political life
Feynman did not return to Cornell. Bacher, who had been instrumental in bringing Feynman to Cornell, had lured him to the California Institute of Technology (Caltech). Part of the deal was that he could spend his first year on sabbatical in Brazil. He had become smitten by Mary Louise Bell from Neodesha, Kansas. They had met in a cafeteria in Cornell, where she had studied the history of Mexican art and textiles. She later followed him to Caltech, where he gave a lecture. While he was in Brazil, she taught classes on the history of furniture and interiors at Michigan State University. He proposed to her by mail from Rio de Janeiro, and they married in Boise, Idaho, on June 28, 1952, shortly after he returned. They frequently quarreled and she was frightened by what she described as "a violent temper". Their politics were different; although he registered and voted as a Republican, she was more conservative, and her opinion on the 1954 Oppenheimer security hearing ("Where there's smoke there's fire") offended him. They separated on May 20, 1956. An interlocutory decree of divorce was entered on June 19, 1956, on the grounds of "extreme cruelty". The divorce became final on May 5, 1958.
In the wake of the 1957 Sputnik crisis, the U.S. government's interest in science rose for a time. Feynman was considered for a seat on the President's Science Advisory Committee, but was not appointed. At this time, the FBI interviewed a woman close to Feynman, possibly his ex-wife Bell, who sent a written statement to J. Edgar Hoover on August 8, 1958:
The U.S. government nevertheless sent Feynman to Geneva for the September 1958 Atoms for Peace Conference. On the beach at Lake Geneva, he met Gweneth Howarth, who was from Ripponden, West Yorkshire, and working in Switzerland as an au pair. Feynman's love life had been turbulent since his divorce; his previous girlfriend had walked off with his Albert Einstein Award medal and, on the advice of an earlier girlfriend, had feigned pregnancy and extorted him into paying for an abortion, then used the money to buy furniture. When Feynman found that Howarth was being paid only $25 a month, he offered her $20 (equivalent to $202 in 2022) a week to be his live-in maid. Feynman knew that this sort of behavior was illegal under the Mann Act, so he had a friend, Matthew Sands, act as her sponsor. Howarth pointed out that she already had two boyfriends, but decided to take Feynman up on his offer, and arrived in Altadena, California, in June 1959. She made a point of dating other men, but Feynman proposed in early 1960. They were married on September 24, 1960, at the Huntington Hotel in Pasadena. They had a son, Carl, in 1962, and adopted a daughter, Michelle, in 1968. Besides their home in Altadena, they had a beach house in Baja California, purchased with the money from Feynman's Nobel Prize.
### Allegations of sexism
There were protests over his alleged sexism at Caltech in 1968, and again in 1972. Protesters "objected to his use of sexist stories about 'lady drivers' and clueless women in his lectures." Feynman recalled protesters entering a hall and picketing a lecture he was about to make in San Francisco, calling him a "sexist pig". He later reflected on the incident claiming that it prompted him to address the protesters, saying that "women do indeed suffer prejudice and discrimination in physics, and your presence here today serves to remind us of these difficulties and the need to remedy them".
In his 1985 memoir, Surely You're Joking, Mr. Feynman!, he recalled holding meetings in strip clubs, drawing naked portraits of his female students while lecturing at Caltech, and pretending to be an undergraduate to deceive younger women into sleeping with him.
### Feynman diagram van
In 1975, in Long Beach, CA, Feynman bought a Dodge Tradesman Maxivan with a bronze-khaki exterior and yellow-green interior, with custom Feynman diagram exterior murals. After Feynman's death, Gweneth sold the van for $1 to one of Feynman's friends, film producer Ralph Leighton, who later put it into storage, where it began to rust. In 2012, video game designer Seamus Blackley, a father of the Xbox, bought the van. Qantum was the license plate ID.
Physics
At Caltech, Feynman investigated the physics of the superfluidity of supercooled liquid helium, where helium seems to display a complete lack of viscosity when flowing. Feynman provided a quantum-mechanical explanation for the Soviet physicist Lev Landau's theory of superfluidity. Applying the Schrödinger equation to the question showed that the superfluid was displaying quantum mechanical behavior observable on a macroscopic scale. This helped with the problem of superconductivity, but the solution eluded Feynman. It was solved with the BCS theory of superconductivity, proposed by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer in 1957.
Feynman, inspired by a desire to quantize the Wheeler–Feynman absorber theory of electrodynamics, laid the groundwork for the path integral formulation and Feynman diagrams.
With Murray Gell-Mann, Feynman developed a model of weak decay, which showed that the current coupling in the process is a combination of vector and axial currents (an example of weak decay is the decay of a neutron into an electron, a proton, and an antineutrino). Although E. C. George Sudarshan and Robert Marshak developed the theory nearly simultaneously, Feynman's collaboration with Gell-Mann was seen as seminal because the weak interaction was neatly described by the vector and axial currents. It thus combined the 1933 beta decay theory of Enrico Fermi with an explanation of parity violation.
Feynman attempted an explanation, called the parton model, of the strong interactions governing nucleon scattering. The parton model emerged as a complement to the quark model developed by Gell-Mann. The relationship between the two models was murky; Gell-Mann referred to Feynman's partons derisively as "put-ons". In the mid-1960s, physicists believed that quarks were just a bookkeeping device for symmetry numbers, not real particles; the statistics of the omega-minus particle, if it were interpreted as three identical strange quarks bound together, seemed impossible if quarks were real.
The SLAC National Accelerator Laboratory deep inelastic scattering experiments of the late 1960s showed that nucleons (protons and neutrons) contained point-like particles that scattered electrons. It was natural to identify these with quarks, but Feynman's parton model attempted to interpret the experimental data in a way that did not introduce additional hypotheses. For example, the data showed that some 45% of the energy momentum was carried by electrically neutral particles in the nucleon. These electrically neutral particles are now seen to be the gluons that carry the forces between the quarks, and their three-valued color quantum number solves the omega-minus problem. Feynman did not dispute the quark model; for example, when the fifth quark was discovered in 1977, Feynman immediately pointed out to his students that the discovery implied the existence of a sixth quark, which was discovered in the decade after his death.
After the success of quantum electrodynamics, Feynman turned to quantum gravity. By analogy with the photon, which has spin 1, he investigated the consequences of a free massless spin 2 field and derived the Einstein field equation of general relativity, but little more. The computational device that Feynman discovered then for gravity, "ghosts", which are "particles" in the interior of his diagrams that have the "wrong" connection between spin and statistics, have proved invaluable in explaining the quantum particle behavior of the Yang–Mills theories, for example, quantum chromodynamics and the electro-weak theory. He did work on all four of the fundamental interactions of nature: electromagnetic, the weak force, the strong force and gravity. John and Mary Gribbin state in their book on Feynman that "Nobody else has made such influential contributions to the investigation of all four of the interactions".
Partly as a way to bring publicity to progress in physics, Feynman offered $1,000 prizes for two of his challenges in nanotechnology; one was claimed by William McLellan and the other by Tom Newman.
Feynman was also interested in the relationship between physics and computation. He was also one of the first scientists to conceive the possibility of quantum computers. In the 1980s he began to spend his summers working at Thinking Machines Corporation, helping to build some of the first parallel supercomputers and considering the construction of quantum computers.
Between 1984 and 1986, he developed a variational method for the approximate calculation of path integrals, which has led to a powerful method of converting divergent perturbation expansions into convergent strong-coupling expansions (variational perturbation theory) and, as a consequence, to the most accurate determination of critical exponents measured in satellite experiments. At Caltech, he once chalked "What I cannot create I do not understand" on his blackboard.
### Machine technology
Feynman had studied the ideas of John von Neumann while researching quantum field theory. His most famous lecture on the subject was delivered in 1959 at the California Institute of Technology, published under the title "There's Plenty of Room at the Bottom" a year later. In this lecture he theorized on future opportunities for designing miniaturized machines, which could build smaller reproductions of themselves. This lecture is frequently cited in technical literature on microtechnology, and nanotechnology.
Feynman also suggested that it should be possible, in principle, to make nanoscale machines that "arrange the atoms the way we want" and do chemical synthesis by mechanical manipulation.
He also presented the possibility of "swallowing the doctor", an idea that he credited in the essay to his friend and graduate student Albert Hibbs. This concept involved building a tiny, swallowable surgical robot.
### Pedagogy
In the early 1960s, Feynman acceded to a request to "spruce up" the teaching of undergraduates at the California Institute of Technology, also called Caltech. After three years devoted to the task, he produced a series of lectures that later became The Feynman Lectures on Physics. Accounts vary about how successful the original lectures were. Feynman's own preface, written just after an exam on which the students did poorly, was somewhat pessimistic. His colleagues David L. Goodstein and Gerry Neugebauer said later that the intended audience of first-year students found the material intimidating while older students and faculty found it inspirational, so the lecture hall remained full even as the first-year students dropped away. In contrast, physicist Matthew Sands recalled the student attendance as being typical for a large lecture course.
Converting the lectures into books occupied Matthew Sands and Robert B. Leighton as part-time co-authors for several years. Feynman suggested that the book cover should have a picture of a drum with mathematical diagrams about vibrations drawn upon it, in order to illustrate the application of mathematics to understanding the world. Instead, the publishers gave the books plain red covers, though they included a picture of Feynman playing drums in the foreword. Even though the books were not adopted by universities as textbooks, they continue to sell well because they provide a deep understanding of physics.
Many of Feynman's lectures and miscellaneous talks were turned into other books, including The Character of Physical Law, QED: The Strange Theory of Light and Matter, Statistical Mechanics, Lectures on Gravitation, and the Feynman Lectures on Computation.
Feynman wrote about his experiences teaching physics undergraduates in Brazil. The students' studying habits and the Portuguese language textbooks were so devoid of any context or applications for their information that, in Feynman's opinion, the students were not learning physics at all. At the end of the year, Feynman was invited to give a lecture on his teaching experiences, and he agreed to do so, provided he could speak frankly, which he did.
Feynman opposed rote learning, or unthinking memorization, as well as other teaching methods that emphasized form over function. In his mind, clear thinking and clear presentation were fundamental prerequisites for his attention. It could be perilous even to approach him unprepared, and he did not forget fools and pretenders.
In 1964, he served on the California State Curriculum Commission, which was responsible for approving textbooks to be used by schools in California. He was not impressed with what he found. Many of the mathematics texts covered subjects of use only to pure mathematicians as part of the "New Math". Elementary students were taught about sets, but:
In April 1966, Feynman delivered an address to the National Science Teachers Association, in which he suggested how students could be made to think like scientists, be open-minded, curious, and especially, to doubt. In the course of the lecture, he gave a definition of science, which he said came about by several stages. The evolution of intelligent life on planet Earth—creatures such as cats that play and learn from experience. The evolution of humans, who came to use language to pass knowledge from one individual to the next, so that the knowledge was not lost when an individual died. Unfortunately, incorrect knowledge could be passed down as well as correct knowledge, so another step was needed. Galileo and others started doubting the truth of what was passed down and to investigate ab initio, from experience, what the true situation was—this was science.
In 1974, Feynman delivered the Caltech commencement address on the topic of cargo cult science, which has the semblance of science, but is only pseudoscience due to a lack of "a kind of scientific integrity, a principle of scientific thought that corresponds to a kind of utter honesty" on the part of the scientist. He instructed the graduating class that "The first principle is that you must not fool yourself—and you are the easiest person to fool. So you have to be very careful about that. After you've not fooled yourself, it's easy not to fool other scientists. You just have to be honest in a conventional way after that."
Feynman served as doctoral advisor to 30 students.
### Case before the Equal Employment Opportunity Commission
In 1977, Feynman supported his English literature colleague Jenijoy La Belle, who had been hired as Caltech's first female professor in 1969, and filed suit with the Equal Employment Opportunity Commission after she was refused tenure in 1974. The EEOC ruled against Caltech in 1977, adding that La Belle had been paid less than male colleagues. La Belle finally received tenure in 1979. Many of Feynman's colleagues were surprised that he took her side, but he had gotten to know La Belle and liked and admired her.
Surely You're Joking, Mr. Feynman!
In the 1960s, Feynman began thinking of writing an autobiography, and he began granting interviews to historians. In the 1980s, working with Ralph Leighton (Robert Leighton's son), he recorded chapters on audio tape that Ralph transcribed. The book was published in 1985 as Surely You're Joking, Mr. Feynman! and became a best-seller.
Gell-Mann was upset by Feynman's account in the book of the weak interaction work, and threatened to sue, resulting in a correction being inserted in later editions. This incident was just the latest provocation in decades of bad feeling between the two scientists. Gell-Mann often expressed frustration at the attention Feynman received; he remarked: was a great scientist, but he spent a great deal of his effort generating anecdotes about himself."
Feynman has been criticized for a chapter in the book entitled "You Just Ask Them?", where he describes how he learned to seduce women at a bar he went to in the summer of 1946. A mentor taught him to ask a woman if she would sleep with him before buying her anything. He describes seeing women at the bar as "bitches" in his thoughts, and tells a story of how he told a woman named Ann that she was "worse than a whore" after Ann persuaded him to buy her sandwiches by telling him he could eat them at her place, but then, after he bought them, saying they actually could not eat together because another man was coming over. Later on that same evening, Ann returned to the bar to take Feynman to her place. Feynman states at the end of the chapter that this behaviour was not typical of him: "So it worked even with an ordinary girl! But no matter how effective the lesson was, I never really used it after that. I didn't enjoy doing it that way. But it was interesting to know that things worked much differently from how I was brought up."
Challenger disaster
Feynman played an important role on the Presidential Rogers Commission, which investigated the 1986 Space Shuttle Challenger disaster. He had been reluctant to participate, but was persuaded by advice from his wife. Feynman clashed several times with commission chairman William P. Rogers. During a break in one hearing, Rogers told commission member Neil Armstrong, "Feynman is becoming a pain in the ass."
During a televised hearing, Feynman demonstrated that the material used in the shuttle's O-rings became less resilient in cold weather by compressing a sample of the material in a clamp and immersing it in ice-cold water. The commission ultimately determined that the disaster was caused by the primary O-ring not properly sealing in unusually cold weather at Cape Canaveral.
Feynman devoted the latter half of his 1988 book What Do You Care What Other People Think? to his experience on the Rogers Commission, straying from his usual convention of brief, light-hearted anecdotes to deliver an extended and sober narrative. Feynman's account reveals a disconnect between NASA's engineers and executives that was far more striking than he expected. His interviews of NASA's high-ranking managers revealed startling misunderstandings of elementary concepts. For instance, NASA managers claimed that there was a 1 in 100,000 probability of a catastrophic failure aboard the Shuttle, but Feynman discovered that NASA's own engineers estimated the probability of a catastrophe at closer to 1 in 200. He concluded that NASA management's estimate of the reliability of the Space Shuttle was unrealistic, and he was particularly angered that NASA used it to recruit Christa McAuliffe into the Teacher-in-Space program. He warned in his appendix to the commission's report (which was included only after he threatened not to sign the report), "For a successful technology, reality must take precedence over public relations, for nature cannot be fooled."
### Recognition and awards
The first public recognition of Feynman's work came in 1954, when Lewis Strauss, the chairman of the Atomic Energy Commission (AEC) notified him that he had won the Albert Einstein Award, which was worth $15,000 and came with a gold medal. Because of Strauss's actions in stripping Oppenheimer of his security clearance, Feynman was reluctant to accept the award, but Isidor Isaac Rabi cautioned him: "You should never turn a man's generosity as a sword against him. Any virtue that a man has, even if he has many vices, should not be used as a tool against him." It was followed by the AEC's Ernest Orlando Lawrence Award in 1962. Schwinger, Tomonaga and Feynman shared the 1965 Nobel Prize in Physics "for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles". He was elected a Foreign Member of the Royal Society in 1965, received the Oersted Medal in 1972, and the National Medal of Science in 1979. He was elected a Member of the National Academy of Sciences, but ultimately resigned and is no longer listed by them. Schwinger called him "an honest man, the outstanding intuitionist of our age, and a prime example of what may lie in store for anyone who dares follow the beat of a different drum."
## Death
In 1978, Feynman sought medical treatment for abdominal pains and was diagnosed with liposarcoma, a rare form of cancer. Surgeons removed a "very large" tumor that had crushed one kidney and his spleen. In 1986 doctors discovered another cancer, Waldenström macroglobulinemia. Further operations were performed in October 1986 and October 1987. He was again hospitalized at the UCLA Medical Center on February 3, 1988. A ruptured duodenal ulcer caused kidney failure, and he declined to undergo the dialysis that might have prolonged his life for a few months. Feynman's wife Gweneth, sister Joan, and cousin Frances Lewine watched over him during the final days of his life until he died on February 15, 1988.
When Feynman was nearing death, he asked his friend and colleague Danny Hillis why Hillis appeared so sad. Hillis replied that he thought Feynman was going to die soon. Hillis quotes Feynman as replying:
Near the end of his life, Feynman attempted to visit the Tuvan Autonomous Soviet Socialist Republic (ASSR) in the Soviet Union, a dream thwarted by Cold War bureaucratic issues. The letter from the Soviet government authorizing the trip was not received until the day after he died. His daughter Michelle later made the journey. Ralph Leighton chronicled the attempt in Tuva or Bust!, published in 1991.
His burial was at Mountain View Cemetery and Mausoleum in Altadena, California. His last words were: "I'd hate to die twice. It's so boring."
## Popular legacy
Aspects of Feynman's life have been portrayed in various media. Feynman was portrayed by Matthew Broderick in the 1996 biopic Infinity. Actor Alan Alda commissioned playwright Peter Parnell to write a two-character play about a fictional day in the life of Feynman set two years before Feynman's death. The play, QED, premiered at the Mark Taper Forum in Los Angeles in 2001 and was later presented at the Vivian Beaumont Theater on Broadway, with both productions starring Alda as Richard Feynman. Real Time Opera premiered its opera Feynman at the Norfolk (Connecticut) Chamber Music Festival in June 2005. In 2011, Feynman was the subject of a biographical graphic novel entitled simply Feynman, written by Jim Ottaviani and illustrated by Leland Myrick. In 2013, Feynman's role on the Rogers Commission was dramatised by the BBC in The Challenger (US title: The Challenger Disaster), with William Hurt playing Feynman. In 2016, Oscar Isaac performed a public reading of Feynman's 1946 love letter to the late Arline. In the 2023 American film Oppenheimer, directed by Christopher Nolan and based on American Prometheus, Feynman is portrayed by actor Jack Quaid.
Feynman is commemorated in various ways. On May 4, 2005, the United States Postal Service issued the "American Scientists" commemorative set of four 37-cent self-adhesive stamps in several configurations. The scientists depicted were Richard Feynman, John von Neumann, Barbara McClintock, and Josiah Willard Gibbs. Feynman's stamp, sepia-toned, features a photograph of Feynman in his thirties and eight small Feynman diagrams. The stamps were designed by Victor Stabin under the artistic direction of Carl T. Herrman. The main building for the Computing Division at Fermilab is named the "Feynman Computing Center" in his honor. Two photographs of Feynman were used in Apple Computer's "Think Different" advertising campaign, which launched in 1997. Sheldon Cooper, a fictional theoretical physicist from the television series The Big Bang Theory, was depicted as a Feynman fan, even emulating him by playing the bongo drums. On January 27, 2016, co-founder of Microsoft Bill Gates wrote an article describing Feynman's talents as a teacher ("The Best Teacher I Never Had"), which inspired Gates to create Project Tuva to place the videos of Feynman's Messenger Lectures, The Character of Physical Law, on a website for public viewing. In 2015 Gates made a video in response to Caltech's request for thoughts on Feynman for the 50th anniversary of Feynman's 1965 Nobel Prize, on why he thought Feynman was special. At CERN (the European Organization for Nuclear Research, home of the Large Hadron Collider), a street on the Meyrin site is named "Route Feynman".
Works
### Selected scientific works
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- Lecture presented at the fifteenth annual meeting of the National Science Teachers Association, 1966 in New York City.
-
-
-
-
-
- Proceedings of the International Workshop at Wangerooge Island, Germany; Sept 1–4, 1987.
-
### Textbooks and lecture notes
The Feynman Lectures on Physics is perhaps his most accessible work for anyone with an interest in physics, compiled from lectures to Caltech undergraduates in 1961–1964. As news of the lectures' lucidity grew, professional physicists and graduate students began to drop in to listen. Co-authors Robert B. Leighton and Matthew Sands, colleagues of Feynman, edited and illustrated them into book form. The work has endured and is useful to this day. They were edited and supplemented in 2005 with Feynman's Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics by Michael Gottlieb and Ralph Leighton (Robert Leighton's son), with support from Kip Thorne and other physicists.
- Includes Feynman's Tips on Physics (with Michael Gottlieb and Ralph Leighton), which includes four previously unreleased lectures on problem solving, exercises by Robert Leighton and Rochus Vogt, and a historical essay by Matthew Sands. Three volumes; originally published as separate volumes in 1964 and 1966.
-
-
-
-
-
-
-
-
-
-
- .
### Popular works
-
-
- No Ordinary Genius: The Illustrated Richard Feynman, ed. Christopher Sykes, W. W. Norton & Company, 1996, .
- Six Easy Pieces: Essentials of Physics Explained by Its Most Brilliant Teacher, Perseus
### Books
, 1994, . Listed by the board of directors of the Modern Library as one of the 100 best nonfiction books.
- Six Not So Easy Pieces: Einstein's Relativity, Symmetry and Space-Time, Addison Wesley, 1997, .
-
-
- Classic Feynman: All the Adventures of a Curious Character, edited by Ralph Leighton, W. W. Norton & Company, 2005, . Chronologically reordered omnibus volume of Surely You're Joking, Mr. Feynman! and What Do You Care What Other People Think?, with a bundled CD containing one of Feynman's signature lectures.
### Audio and video recordings
- Safecracker Suite (a collection of drum pieces interspersed with Feynman telling anecdotes)
- Los Alamos From Below (audio, talk given by Feynman at Santa Barbara on February 6, 1975)
- The Feynman Lectures on Physics: The Complete Audio Collection, selections from which were also released as Six Easy Pieces and Six Not So Easy Pieces
- The Messenger Lectures (link), given at Cornell in 1964, in which he explains basic topics in physics; they were also adapted into the book The Character of Physical Law
- The Douglas Robb Memorial Lectures, four public lectures of which the four chapters of the book QED: The Strange Theory of Light and Matter are transcripts. (1979)
- The Pleasure of Finding Things Out, BBC Horizon episode (1981) (not to be confused with the later published book of the same title)
- Richard Feynman: Fun to Imagine Collection, BBC Archive of six short films of Feynman talking in a style that is accessible to all about the physics behind common to all experiences. (1983)
- Elementary Particles and the Laws of Physics, from the 1986 Dirac Memorial Lectures (video, 1986)
- Tiny Machines: The Feynman Talk on Nanotechnology (video, 1984)
- Computers From the Inside Out (video)
- Quantum Mechanical View of Reality: Workshop at Esalen (video, 1983)
- Idiosyncratic Thinking Workshop (video, 1985)
- Bits and Pieces—From Richard's Life and Times (video, 1988)
- Strangeness Minus Three (video, BBC Horizon 1964)
- No Ordinary Genius (video, Cristopher Sykes Documentary)
- Four NOVA episodes are made about or with him. (TV program, 1975, 1983, 1989, 1993)
- The Motion of Planets Around the Sun (audio, sometimes titled "Feynman's Lost Lecture")
- Nature of Matter (audio)
## References
## Sources
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
## Further reading
### Articles
- Physics Today, American Institute of Physics magazine, February 1989 Issue. (Vol. 42, No. 2.) Special Feynman memorial issue containing non-technical articles on Feynman's life and work in physics.
-
Books
- Brown, Laurie M. and Rigden, John S. (editors) (1993) Most of the Good Stuff: Memories of Richard Feynman Simon & Schuster, New York, . Commentary by Joan Feynman, John Wheeler, Hans Bethe, Julian Schwinger, Murray Gell-Mann, Daniel Hillis, David Goodstein, Freeman Dyson, and Laurie Brown
- Dyson, Freeman (1979) Disturbing the Universe. Harper and Row. . Dyson's autobiography. The chapters "A Scientific Apprenticeship" and "A Ride to Albuquerque" describe his impressions of Feynman in the period 1947–1948 when Dyson was a graduate student at Cornell
-
-
- for high school readers
-
- Published in the United Kingdom as Some Time With Feynman
-
### Films and plays
- Infinity (1996), a movie both directed by and starring Matthew Broderick as Feynman, depicting his love affair with his first wife and ending with the Trinity test.
- Parnell, Peter (2002), QED, Applause Books, (play)
- Whittell, Crispin (2006), Clever Dick, Oberon Books, (play)
- "The Quest for Tannu Tuva", with Richard Feynman and Ralph Leighton. 1987, BBC Horizon and PBS Nova (entitled "Last Journey of a Genius").
- No Ordinary Genius, a two-part documentary about Feynman's life and work, with contributions from colleagues, friends and family. 1993, BBC Horizon and PBS Nova (a one-hour version, under the title The Best Mind Since Einstein) (2 × 50-minute films)
- The Challenger (2013), a BBC Two factual drama starring William Hurt, tells the story of American Nobel prize-winning physicist Richard Feynman's determination to reveal the truth behind the 1986 Space Shuttle Challenger disaster.
- The Fantastic Mr Feynman. One hour documentary. 2013, BBC TV
- How We Built The Bomb, a docudrama about The Manhattan Project at Los Alamos. Feynman is played by actor/playwright Michael Raver. 2015
## External links
-
-
- Online edition of The Feynman Lectures on Physics by California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer
- Oral history interview transcript with Richard Feynman on 4 March 1966 – Session I from Oral History Interviews, Niels Bohr Library & Archives, American Institute of Physics
- Oral history interview transcript with Richard Feynman on 5 March 1966 – Session II from Oral History Interviews, Niels Bohr Library & Archives, American Institute of Physics
- Oral history interview transcript with Richard Feynman on 27 June 1966 – Session III from Oral History Interviews, Niels Bohr Library & Archives, American Institute of Physics
- Oral history interview transcript with Richard Feynman on 28 June 1966 – Session IV from Oral History Interviews, Niels Bohr Library & Archives, American Institute of Physics
- Oral history interview transcript with Richard Feynman on 4 February 1973 – Session V from Oral History Interviews, Niels Bohr Library & Archives, American Institute of Physics
- Richard Feynman – Scientist. Teacher. Raconteur. Musician A site dedicated to Richard Feynman
Category:1918 births
Category:1988 deaths
Category:20th-century American physicists
Category:American atheists
Category:20th-century atheists
Category:American people of Russian-Jewish descent
Category:American people of Polish-Jewish descent
Category:American Nobel laureates
Category:American skeptics
Category:California Institute of Technology faculty
Category:Deaths from cancer in California
Category:Cellular automatists
Category:Cornell University faculty
Category:Deaths from liposarcoma
Category:American experimental physicists
Category:Far Rockaway High School alumni
Category:Foreign members of the Royal Society
Category:Manhattan Project people
Category:American nanotechnologists
Category:National Medal of Science laureates
Category:Niels Bohr International Gold Medal recipients
Category:Nobel laureates in Physics
Category:Nuclear weapons scientists and engineers
Category:American particle physicists
Category:People from Far Rockaway, Queens
Category:Princeton University alumni
Category:Putnam Fellows
Category:Quantum computing
Category:Scientists from California
Category:Scientists from New York (state)
Category:Sloan Research Fellows
Category:Space Shuttle Challenger disaster
Category:American textbook writers
Category:American quantum physicists
Category:American relativity theorists
Category:Quantum gravity physicists
Category:United States Army civilians
Category:Massachusetts Institute of Technology School of Science alumni
Category:Fellows of the American Physical Society
Category:American science writers
Category:Jewish American atheists
Category:Burials at Mountain View Cemetery (Altadena, California) | https://en.wikipedia.org/wiki/Richard_Feynman |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.