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Taniyama group
In mathematics, the Taniyama group is a group that is an extension of the absolute Galois group of the rationals by the Serre group. It was introduced by Langlands (1977) using an observation by Deligne, and named after Yutaka Taniyama. It was intended to be the group scheme whose representations correspond to the (hypothetical) CM motives over the field Q of rational numbers.
References
• Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen (1982), "Langlands's Construction of the Taniyama Group", Hodge cycles, motives, and Shimura varieties. (PDF), Lecture Notes in Mathematics, vol. 900, Berlin-New York: Springer-Verlag, doi:10.1007/978-3-540-38955-2_14, ISBN 3-540-11174-3, MR 0654325
• Langlands, R. P. (1977), "Automorphic representations, Shimura varieties, and motives. Ein Märchen", Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., vol. 33, pp. 205–246, ISBN 0-8218-1437-0, MR 0546619
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Tanja Bergkvist
Tanja-Helena Dessislava Bergkvist (born January 3, 1974, in Lund) is a Swedish mathematician and blogger. She earned a Ph.D. in mathematics in 2007 at Stockholm University,[1] and has served as a professor at the KTH Royal Institute of Technology, Uppsala University, and the Sigtunaskolan Humanistiska Läroverket. She has also worked as a researcher at the Swedish Defence Research Agency. She has gained notoriety for her conservative approach towards gender studies.[2][3][4][5][6][7]
References
1. Tanja Bergkvist at the Mathematics Genealogy Project
2. Lena, Martinsson; Gabriele, Griffin; Nygren, Katarina Giritli (16 March 2016). Challenging the Myth of Gender Equality in Sweden. Policy Press. p. 122. ISBN 978-1-4473-2596-3.
3. Radio, Sveriges. "Genusperspektivet och forskningens frihet - Filosofiska rummet". sverigesradio.se (in Swedish). Retrieved 2 January 2022.
4. "Genusdebatt i riksdagen". www.varldenidag.se (in Swedish). March 12, 2010.
5. ""Genusvetare förvirrar barnen" | SvD Debatt". Svenska Dagbladet (in Swedish). 19 December 2008.
6. "Tanja Bergkvist i debatt mot "genuspedagog" - TV4 2008(OBS: Läs texten)". TV4. December 27, 2008.
7. "Föreningen Heimdal – Sveriges största och äldsta borgerliga studentförening" (in Swedish). Retrieved 2 January 2022.
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| Wikipedia |
German tank problem
In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number of items which are sequentially numbered from 1 to N. A random sample of these items is taken and their sequence numbers observed; the problem is to estimate N from these observed numbers.
The problem can be approached using either frequentist inference or Bayesian inference, leading to different results. Estimating the population maximum based on a single sample yields divergent results, whereas estimation based on multiple samples is a practical estimation question whose answer is simple (especially in the frequentist setting) but not obvious (especially in the Bayesian setting).
The problem is named after its historical application by Allied forces in World War II to the estimation of the monthly rate of German tank production from very limited data. This exploited the manufacturing practice of assigning and attaching ascending sequences of serial numbers to tank components (chassis, gearbox, engine, wheels), with some of the tanks eventually being captured in battle by Allied forces.
Suppositions
The adversary is presumed to have manufactured a series of tanks marked with consecutive whole numbers, beginning with serial number 1. Additionally, regardless of a tank's date of manufacture, history of service, or the serial number it bears, the distribution over serial numbers becoming revealed to analysis is uniform, up to the point in time when the analysis is conducted.
Example
Assuming tanks are assigned sequential serial numbers starting with 1, suppose that four tanks are captured and that they have the serial numbers: 19, 40, 42 and 60.
The frequentist approach predicts the total number of tanks produced will be:
$N\approx 74$
The Bayesian approach predicts that the median number of tanks produced will be very similar to the frequentist prediction:
$N_{med}\approx 74.5$
whereas the Bayesian mean predicts that the number of tanks produced would be:
$N_{av}\approx 89$
Let N equal the total number of tanks predicted to have been produced, m equal the highest serial number observed and k equal the number of tanks captured.
The frequentist prediction is calculated as:
$N\approx m+{\frac {m}{k}}-1=74$
The Bayesian median is calculated as:
$N_{med}\approx m+{\frac {m\ln(2)}{k-1}}=74.5$
The Bayesian mean is calculated as:
$N_{av}\approx (m-1){\frac {k-1}{k-2}}=89$
Both Bayesian computations are based on the following probability mass function:
$\Pr(N=n)={\begin{cases}0&{\text{if }}n<m\\{\frac {k-1}{k}}{\frac {\binom {m-1}{k-1}}{\binom {n}{k}}}&{\text{if }}n\geq m,\end{cases}}$
This distribution has a positive skewness, related to the fact that there are at least 60 tanks. Because of this skewness, the mean may not be the most meaningful estimate. The median in this example is 74.5, in close agreement with the frequentist formula. Using Stirling's approximation, the Bayesian probability function may be approximated as
$\Pr(N=n)\approx {\begin{cases}0&{\text{if }}n<m\\(k-1)m^{k-1}n^{-k}&{\text{if }}n\geq m,\end{cases}}$
which results in the following approximation for the median:
$N_{med}\approx m+{\frac {m\ln(2)}{k-1}}$
Finally, the average estimate by Bayesians, and its deviation, are computed as:
${\begin{aligned}N&\approx \mu \pm \sigma =89\pm 50,\\[5pt]\mu &=(m-1){\frac {k-1}{k-2}},\\[5pt]\sigma &={\sqrt {\frac {(k-1)(m-1)(m-k+1)}{(k-3)(k-2)^{2}}}}.\end{aligned}}$
Historical example of the problem
During the course of the Second World War, the Western Allies made sustained efforts to determine the extent of German production and approached this in two major ways: conventional intelligence gathering and statistical estimation. In many cases, statistical analysis substantially improved on conventional intelligence. In some cases, conventional intelligence was used in conjunction with statistical methods, as was the case in estimation of Panther tank production just prior to D-Day.
The allied command structure had thought the Panzer V (Panther) tanks seen in Italy, with their high velocity, long-barreled 75 mm/L70 guns, were unusual heavy tanks and would only be seen in northern France in small numbers, much the same way as the Tiger I was seen in Tunisia. The US Army was confident that the Sherman tank would continue to perform well, as it had versus the Panzer III and Panzer IV tanks in North Africa and Sicily.[lower-alpha 1] Shortly before D-Day, rumors indicated that large numbers of Panzer V tanks were being used.
To determine whether this was true, the Allies attempted to estimate the number of tanks being produced. To do this, they used the serial numbers on captured or destroyed tanks. The principal numbers used were gearbox numbers, as these fell in two unbroken sequences. Chassis and engine numbers were also used, though their use was more complicated. Various other components were used to cross-check the analysis. Similar analyses were done on wheels, which were observed to be sequentially numbered (i.e., 1, 2, 3, ..., N).[2][lower-alpha 2][3][4]
The analysis of tank wheels yielded an estimate for the number of wheel molds that were in use. A discussion with British road wheel makers then estimated the number of wheels that could be produced from this many molds, which yielded the number of tanks that were being produced each month. Analysis of wheels from two tanks (32 road wheels each, 64 road wheels total) yielded an estimate of 270 tanks produced in February 1944, substantially more than had previously been suspected.[5]
German records after the war showed production for the month of February 1944 was 276.[6][lower-alpha 3] The statistical approach proved to be far more accurate than conventional intelligence methods, and the phrase "German tank problem" became accepted as a descriptor for this type of statistical analysis.
Estimating production was not the only use of this serial-number analysis. It was also used to understand German production more generally, including number of factories, relative importance of factories, length of supply chain (based on lag between production and use), changes in production, and use of resources such as rubber.
Specific data
According to conventional Allied intelligence estimates, the Germans were producing around 1,400 tanks a month between June 1940 and September 1942. Applying the formula below to the serial numbers of captured tanks, the number was calculated to be 246 a month. After the war, captured German production figures from the ministry of Albert Speer showed the actual number to be 245.[3]
Estimates for some specific months are given as:[7]
MonthStatistical estimateIntelligence estimateGerman records
June 19401691,000122
June 19412441,550271
August 19423271,550342
Similar analyses
Similar serial-number analysis was used for other military equipment during World War II, most successfully for the V-2 rocket.[8]
Factory markings on Soviet military equipment were analyzed during the Korean War, and by German intelligence during World War II.[9]
In the 1980s, some Americans were given access to the production line of Israel's Merkava tanks. The production numbers were classified, but the tanks had serial numbers, allowing estimation of production.[10]
The formula has been used in non-military contexts, for example to estimate the number of Commodore 64 computers built, where the result (12.5 million) matches the low-end estimates.[11]
Countermeasures
To confound serial-number analysis, serial numbers can be excluded, or usable auxiliary information reduced. Alternatively, serial numbers that resist cryptanalysis can be used, most effectively by randomly choosing numbers without replacement from a list that is much larger than the number of objects produced, or by producing random numbers and checking them against the list of already assigned numbers; collisions are likely to occur unless the number of digits possible is more than twice the number of digits in the number of objects produced (where the serial number can be in any base); see birthday problem.[lower-alpha 4] For this, a cryptographically secure pseudorandom number generator may be used. All these methods require a lookup table (or breaking the cypher) to back out from serial number to production order, which complicates use of serial numbers: a range of serial numbers cannot be recalled, for instance, but each must be looked up individually, or a list generated.
Alternatively, sequential serial numbers can be encrypted with a simple substitution cipher, which allows easy decoding, but is also easily broken by frequency analysis: even if starting from an arbitrary point, the plaintext has a pattern (namely, numbers are in sequence). One example is given in Ken Follett's novel Code to Zero, where the encryption of the Jupiter-C rocket serial numbers is given by:
HUNTSVILEX
1234567890
The code word here is Huntsville (with repeated letters omitted) to get a 10-letter key.[12] The rocket number 13 was therefore "HN", and the rocket number 24 was "UT".
Frequentist analysis
Minimum-variance unbiased estimator
For point estimation (estimating a single value for the total, ${\widehat {N}}$), the minimum-variance unbiased estimator (MVUE, or UMVU estimator) is given by:[lower-alpha 5]
${\widehat {N}}=m(1+k^{-1})-1,$
where m is the largest serial number observed (sample maximum) and k is the number of tanks observed (sample size).[10][13] Note that once a serial number has been observed, it is no longer in the pool and will not be observed again.
This has a variance[10]
$\operatorname {var} \left({\widehat {N}}\right)={\frac {1}{k}}{\frac {(N-k)(N+1)}{(k+2)}}\approx {\frac {N^{2}}{k^{2}}}{\text{ for small samples }}k\ll N,$
so the standard deviation is approximately N/k, the expected size of the gap between sorted observations in the sample.
The formula may be understood intuitively as the sample maximum plus the average gap between observations in the sample, the sample maximum being chosen as the initial estimator, due to being the maximum likelihood estimator,[lower-alpha 6] with the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum,[lower-alpha 7] and written as
${\widehat {N}}=m+{\frac {m-k}{k}}=m+mk^{-1}-1=m(1+k^{-1})-1.$
This can be visualized by imagining that the observations in the sample are evenly spaced throughout the range, with additional observations just outside the range at 0 and N + 1. If starting with an initial gap between 0 and the lowest observation in the sample (the sample minimum), the average gap between consecutive observations in the sample is $(m-k)/k$; the $-k$ being because the observations themselves are not counted in computing the gap between observations.[lower-alpha 8]. A derivation of the expected value and the variance of the sample maximum are shown in the page of the discrete uniform distribution.
This philosophy is formalized and generalized in the method of maximum spacing estimation; a similar heuristic is used for plotting position in a Q–Q plot, plotting sample points at k / (n + 1), which is evenly on the uniform distribution, with a gap at the end.
Confidence intervals
Instead of, or in addition to, point estimation, interval estimation can be carried out, such as confidence intervals. These are easily computed, based on the observation that the probability that k observations in the sample will fall in an interval covering p of the range (0 ≤ p ≤ 1) is pk (assuming in this section that draws are with replacement, to simplify computations; if draws are without replacement, this overstates the likelihood, and intervals will be overly conservative).
Thus the sampling distribution of the quantile of the sample maximum is the graph x1/k from 0 to 1: the p-th to q-th quantile of the sample maximum m are the interval [p1/kN, q1/kN]. Inverting this yields the corresponding confidence interval for the population maximum of [m/q1/k, m/p1/k].
For example, taking the symmetric 95% interval p = 2.5% and q = 97.5% for k = 5 yields 0.0251/5 ≈ 0.48, 0.9751/5 ≈ 0.995, so the confidence interval is approximately [1.005m, 2.08m]. The lower bound is very close to m, thus more informative is the asymmetric confidence interval from p = 5% to 100%; for k = 5 this yields 0.051/5 ≈ 0.55 and the interval [m, 1.82m].
More generally, the (downward biased) 95% confidence interval is [m, m/0.051/k] = [m, m·201/k]. For a range of k values, with the UMVU point estimator (plus 1 for legibility) for reference, this yields:
kPoint estimateConfidence interval
12m[m, 20m]
21.5m[m, 4.5m]
51.2m[m, 1.82m]
101.1m[m, 1.35m]
201.05m[m, 1.16m]
Immediate observations are:
• For small sample sizes, the confidence interval is very wide, reflecting great uncertainty in the estimate.
• The range shrinks rapidly, reflecting the exponentially decaying probability that all observations in the sample will be significantly below the maximum.
• The confidence interval exhibits positive skew, as N can never be below the sample maximum, but can potentially be arbitrarily high above it.
Note that m/k cannot be used naively (or rather (m + m/k − 1)/k) as an estimate of the standard error SE, as the standard error of an estimator is based on the population maximum (a parameter), and using an estimate to estimate the error in that very estimate is circular reasoning.
Bayesian analysis
The Bayesian approach to the German tank problem is to consider the credibility $(N=n\mid M=m,K=k)$ that the number of enemy tanks $N$ is equal to the number $n$, when the number of observed tanks, $K$ is equal to the number $k$, and the maximum observed serial number $M$ is equal to the number $m$. The answer to this problem depends on the choice of prior for $N$. One can proceed using a proper prior, e.g., the Poisson or Negative Binomial distribution, where closed formula for the posterior mean and posterior variance can be obtained.[14] An alternative is to proceed using direct calculations as shown below.
For brevity, in what follows, $(N=n\mid M=m,K=k)$ is written $(n\mid m,k)$
Conditional probability
The rule for conditional probability gives
$(n\mid m,k)(m\mid k)=(m\mid n,k)(n\mid k)=(m,n\mid k)$
Probability of M knowing N and K
The expression
$(m\mid n,k)=(M=m\mid N=n,K=k)$
is the conditional probability that the maximum serial number observed, $M$, is equal to $m$, when the number of enemy tanks, $N$, is known to be equal to $n$, and the number of enemy tanks observed, $K$, is known to be equal to $k$.
It is
$(m\mid n,k)={\binom {m-1}{k-1}}{\binom {n}{k}}^{-1}[k\leq m][m\leq n]$
where ${\binom {n}{k}}$ is a binomial coefficient and $[k\leq n]$ is an Iverson bracket.
The expression can be derived as follows: $(m\mid n,k)$ answers the question: "What is the probability of a specific serial number $m$ being the highest number observed in a sample of $k$ tanks, given there are $n$ tanks in total?"
One can think of the sample of size $k$ to be the result of $k$ individual draws. Assume $m$ is observed on draw number $d$. The probability of this occurring is:
$\underbrace {{\frac {m-1}{n}}\cdot {\frac {m-2}{n-1}}\cdot {\frac {m-3}{n-2}}\cdots {\frac {m-d+1}{n-d+2}}} _{\text{d-1 - times}}\cdot \underbrace {\frac {1}{n-d+1}} _{\text{draw no. d}}\cdot \underbrace {{\frac {m-d}{n-d}}\cdot {\frac {m-d-1}{n-d-1}}\cdots {\frac {m-d-(k-d-1)}{n-d-(k-d-1)}}} _{k-d-times}={\frac {(n-k)!}{n!}}\cdot {\frac {(m-1)!}{(m-k)!}}.$
As can be seen from the right-hand side, this expression is independent of $d$ and therefore the same for each $d\leq k$. As $m$ can be drawn on $k$ different draws, the probability of any specific $m$ being the largest one observed is $k$ times the above probability:
$(m\mid n,k)=k\cdot {\frac {(n-k)!}{n!}}\cdot {\frac {(m-1)!}{(m-k)!}}={\binom {m-1}{k-1}}{\binom {n}{k}}^{-1}.$
Probability of M knowing only K
The expression $(m\mid k)=(M=m\mid K=k)$ is the probability that the maximum serial number is equal to $m$ once $k$ tanks have been observed but before the serial numbers have actually been observed.
The expression $(m\mid k)$ can be re-written in terms of the other quantities by marginalizing over all possible $n$.
${\begin{aligned}(m\mid k)&=\sum _{n=0}^{\infty }(m,n\mid k)\\&=\sum _{n=0}^{\infty }(m\mid n,k)(n\mid k)\end{aligned}}$
Credibility of N knowing only K
The expression
$(n\mid k)=(N=n\mid K=k)$
is the credibility that the total number of tanks, $N$, is equal to $n$ when the number $K$ tanks observed is known to be $k$, but before the serial numbers have been observed. Assume that it is some discrete uniform distribution
$(n\mid k)=(\Omega -k)^{-1}[k\leq n][n<\Omega ]$
The upper limit $\Omega $ must be finite, because the function
$f(n)=\lim _{\Omega \rightarrow \infty }(\Omega -k)^{-1}[k\leq n][n<\Omega ]=0$
is not a mass distribution function.
Credibility of N knowing M and K
$(n\mid m,k)=(m\mid n,k)\left(\sum _{n=m}^{\Omega -1}(m\mid n,k)\right)^{-1}[m\leq n][n<\Omega ]$
If k ≥ 2, then $\sum _{n=m}^{\infty }(m\mid n,k)<\infty $, and the unwelcome variable $\Omega $ disappears from the expression.
$(n\mid m,k)=(m\mid n,k)\left(\sum _{n=m}^{\infty }(m\mid n,k)\right)^{-1}[m\leq n]$
For k ≥ 1 the mode of the distribution of the number of enemy tanks is m.
For k ≥ 2, the credibility that the number of enemy tanks is equal to $n$, is
$(N=n\mid m,k)=(k-1){\binom {m-1}{k-1}}k^{-1}{\binom {n}{k}}^{-1}[m\leq n]$
The credibility that the number of enemy tanks, N, is greater than n, is
$(N>n\mid m,k)={\begin{cases}1&{\text{if }}n<m\\{\frac {\binom {m-1}{k-1}}{\binom {n}{k-1}}}&{\text{if }}n\geq m\end{cases}}$
Mean value and standard deviation
For k ≥ 3, N has the finite mean value:
$(m-1)(k-1)(k-2)^{-1}$
For k ≥ 4, N has the finite standard deviation:
$(k-1)^{1/2}(k-2)^{-1}(k-3)^{-1/2}(m-1)^{1/2}(m+1-k)^{1/2}$
These formulas are derived below.
Summation formula
The following binomial coefficient identity is used below for simplifying series relating to the German Tank Problem.
$\sum _{n=m}^{\infty }{\frac {1}{\binom {n}{k}}}={\frac {k}{k-1}}{\frac {1}{\binom {m-1}{k-1}}}$
This sum formula is somewhat analogous to the integral formula
$\int _{n=m}^{\infty }{\frac {dn}{n^{k}}}={\frac {1}{k-1}}{\frac {1}{m^{k-1}}}$
These formulas apply for k > 1.
One tank
Observing one tank randomly out of a population of n tanks gives the serial number m with probability 1/n for m ≤ n, and zero probability for m > n. Using Iverson bracket notation this is written
$(M=m\mid N=n,K=1)=(m\mid n)={\frac {[m\leq n]}{n}}$
This is the conditional probability mass distribution function of $m$.
When considered a function of n for fixed m this is a likelihood function.
${\mathcal {L}}(n)={\frac {[n\geq m]}{n}}$
The maximum likelihood estimate for the total number of tanks is N0 = m, clearly a biased estimate since the true number can be more than this, potentially many more, but cannot be fewer.
The marginal likelihood (i.e. marginalized over all models) is infinite, being a tail of the harmonic series.
$\sum _{n}{\mathcal {L}}(n)=\sum _{n=m}^{\infty }{\frac {1}{n}}=\infty $
but
${\begin{aligned}\sum _{n}{\mathcal {L}}(n)[n<\Omega ]&=\sum _{n=m}^{\Omega -1}{\frac {1}{n}}\\[5pt]&=H_{\Omega -1}-H_{m-1}\end{aligned}}$
where $H_{n}$ is the harmonic number.
The credibility mass distribution function depends on the prior limit $\Omega $:
${\begin{aligned}&(N=n\mid M=m,K=1)\\[5pt]={}&(n\mid m)={\frac {[m\leq n]}{n}}{\frac {[n<\Omega ]}{H_{\Omega -1}-H_{m-1}}}\end{aligned}}$
The mean value of $N$ is
${\begin{aligned}\sum _{n}n\cdot (n\mid m)&=\sum _{n=m}^{\Omega -1}{\frac {1}{H_{\Omega -1}-H_{m-1}}}\\[5pt]&={\frac {\Omega -m}{H_{\Omega -1}-H_{m-1}}}\\[5pt]&\approx {\frac {\Omega -m}{\log \left({\frac {\Omega -1}{m-1}}\right)}}\end{aligned}}$
Two tanks
If two tanks rather than one are observed, then the probability that the larger of the observed two serial numbers is equal to m, is
$(M=m\mid N=n,K=2)=(m\mid n)=[m\leq n]{\frac {m-1}{\binom {n}{2}}}$
When considered a function of n for fixed m this is a likelihood function
${\mathcal {L}}(n)=[n\geq m]{\frac {m-1}{\binom {n}{2}}}$
The total likelihood is
${\begin{aligned}\sum _{n}{\mathcal {L}}(n)&={\frac {m-1}{1}}\sum _{n=m}^{\infty }{\frac {1}{\binom {n}{2}}}\\[4pt]&={\frac {m-1}{1}}\cdot {\frac {2}{2-1}}\cdot {\frac {1}{\binom {m-1}{2-1}}}\\[4pt]&=2\end{aligned}}$
and the credibility mass distribution function is
${\begin{aligned}&(N=n\mid M=m,K=2)\\[4pt]={}&(n\mid m)\\[4pt]={}&{\frac {{\mathcal {L}}(n)}{\sum _{n}{\mathcal {L}}(n)}}\\[4pt]={}&[n\geq m]{\frac {m-1}{n(n-1)}}\end{aligned}}$
The median ${\tilde {N}}$ satisfies
$\sum _{n}[n\geq {\tilde {N}}](n\mid m)={\frac {1}{2}}$
so
${\frac {m-1}{{\tilde {N}}-1}}={\frac {1}{2}}$
and so the median is
${\tilde {N}}=2m-1$
but the mean value of $N$ is infinite
$\mu =\sum _{n}n\cdot (n\mid m)={\frac {m-1}{1}}\sum _{n=m}^{\infty }{\frac {1}{n-1}}=\infty $
Credibility mass distribution function
The conditional probability that the largest of k observations taken from the serial numbers {1,...,n}, is equal to m, is
${\begin{aligned}&(M=m\mid N=n,K=k\geq 2)\\={}&(m\mid n,k)\\={}&[m\leq n]{\frac {\binom {m-1}{k-1}}{\binom {n}{k}}}\end{aligned}}$
The likelihood function of n is the same expression
${\mathcal {L}}(n)=[n\geq m]{\frac {\binom {m-1}{k-1}}{\binom {n}{k}}}$
The total likelihood is finite for k ≥ 2:
${\begin{aligned}\sum _{n}{\mathcal {L}}(n)&={\frac {\binom {m-1}{k-1}}{1}}\sum _{n=m}^{\infty }{1 \over {\binom {n}{k}}}\\&={\frac {\binom {m-1}{k-1}}{1}}\cdot {\frac {k}{k-1}}\cdot {\frac {1}{\binom {m-1}{k-1}}}\\&={\frac {k}{k-1}}\end{aligned}}$
The credibility mass distribution function is
${\begin{aligned}&(N=n\mid M=m,K=k\geq 2)=(n\mid m,k)\\={}&{\frac {{\mathcal {L}}(n)}{\sum _{n}{\mathcal {L}}(n)}}\\={}&[n\geq m]{\frac {k-1}{k}}{\frac {\binom {m-1}{k-1}}{\binom {n}{k}}}\\={}&[n\geq m]{\frac {m-1}{n}}{\frac {\binom {m-2}{k-2}}{\binom {n-1}{k-1}}}\\={}&[n\geq m]{\frac {m-1}{n}}{\frac {m-2}{n-1}}{\frac {k-1}{k-2}}{\frac {\binom {m-3}{k-3}}{\binom {n-2}{k-2}}}\end{aligned}}$
The complementary cumulative distribution function is the credibility that N > x
${\begin{aligned}&(N>x\mid M=m,K=k)\\[4pt]={}&{\begin{cases}1&{\text{if }}x<m\\\sum _{n=x+1}^{\infty }(n\mid m,k)&{\text{if }}x\geq m\end{cases}}\\={}&[x<m]+[x\geq m]\sum _{n=x+1}^{\infty }{\frac {k-1}{k}}{\frac {\binom {m-1}{k-1}}{\binom {N}{k}}}\\[4pt]={}&[x<m]+[x\geq m]{\frac {k-1}{k}}{\frac {\binom {m-1}{k-1}}{1}}\sum _{n=x+1}^{\infty }{\frac {1}{\binom {n}{k}}}\\[4pt]={}&[x<m]+[x\geq m]{\frac {k-1}{k}}{\frac {\binom {m-1}{k-1}}{1}}\cdot {\frac {k}{k-1}}{\frac {1}{\binom {x}{k-1}}}\\[4pt]={}&[x<m]+[x\geq m]{\frac {\binom {m-1}{k-1}}{\binom {x}{k-1}}}\end{aligned}}$
The cumulative distribution function is the credibility that N ≤ x
${\begin{aligned}&(N\leq x\mid M=m,K=k)\\[4pt]={}&1-(N>x\mid M=m,K=k)\\[4pt]={}&[x\geq m]\left(1-{\frac {\binom {m-1}{k-1}}{\binom {x}{k-1}}}\right)\end{aligned}}$
Order of magnitude
The order of magnitude of the number of enemy tanks is
${\begin{aligned}\mu &=\sum _{n}n\cdot (N=n\mid M=m,K=k)\\[4pt]&=\sum _{n}n[n\geq m]{\frac {m-1}{n}}{\frac {\binom {m-2}{k-2}}{\binom {n-1}{k-1}}}\\[4pt]&={\frac {m-1}{1}}{\frac {\binom {m-2}{k-2}}{1}}\sum _{n=m}^{\infty }{\frac {1}{\binom {n-1}{k-1}}}\\[4pt]&={\frac {m-1}{1}}{\frac {\binom {m-2}{k-2}}{1}}\cdot {\frac {k-1}{k-2}}{\frac {1}{\binom {m-2}{k-2}}}\\[4pt]&={\frac {m-1}{1}}{\frac {k-1}{k-2}}\end{aligned}}$
Statistical uncertainty
The statistical uncertainty is the standard deviation $\sigma $, satisfying the equation
$\sigma ^{2}+\mu ^{2}=\sum _{n}n^{2}\cdot (N=n\mid M=m,K=k)$
So
${\begin{aligned}\sigma ^{2}+\mu ^{2}-\mu &=\sum _{n}n(n-1)\cdot (N=n\mid M=m,K=k)\\[4pt]&=\sum _{n=m}^{\infty }n(n-1){\frac {m-1}{n}}{\frac {m-2}{n-1}}{\frac {k-1}{k-2}}{\frac {\binom {m-3}{k-3}}{\binom {n-2}{k-2}}}\\[4pt]&={\frac {m-1}{1}}{\frac {m-2}{1}}{\frac {k-1}{k-2}}\cdot {\frac {\binom {m-3}{k-3}}{1}}\sum _{n=m}^{\infty }{\frac {1}{\binom {n-2}{k-2}}}\\[4pt]&={\frac {m-1}{1}}{\frac {m-2}{1}}{\frac {k-1}{k-2}}{\frac {\binom {m-3}{k-3}}{1}}{\frac {k-2}{k-3}}{\frac {1}{\binom {m-3}{k-3}}}\\[4pt]&={\frac {m-1}{1}}{\frac {m-2}{1}}{\frac {k-1}{k-3}}\end{aligned}}$
and
${\begin{aligned}\sigma &={\sqrt {{\frac {m-1}{1}}{\frac {m-2}{1}}{\frac {k-1}{k-3}}+\mu -\mu ^{2}}}\\[4pt]&={\sqrt {\frac {(k-1)(m-1)(m-k+1)}{(k-3)(k-2)^{2}}}}\end{aligned}}$
The variance-to-mean ratio is simply
${\frac {\sigma ^{2}}{\mu }}={\frac {m-k+1}{(k-3)(k-2)}}$
See also
• Mark and recapture, other method of estimating population size
• Maximum spacing estimation, which generalizes the intuition of "assume uniformly distributed"
• Copernican principle and Lindy effect, analogous predictions of lifetime assuming just one observation in the sample (current age).
• The Doomsday argument, application to estimate expected survival time of the human race.
• Generalized extreme value distribution, possible limit distributions of sample maximum (opposite question).
• Maximum likelihood
• Bias of an estimator
• Likelihood function
Further reading
• Goodman, L. A. (1954). "Some Practical Techniques in Serial Number Analysis". Journal of the American Statistical Association. American Statistical Association. 49 (265): 97–112. doi:10.2307/2281038. JSTOR 2281038.
Notes
1. An Armored Ground Forces policy statement of November 1943 concluded: "The recommendation of a limited proportion of tanks carrying a 90 mm gun is not concurred in for the following reasons: The M4 tank has been hailed widely as the best tank of the battlefield today. ... There appears to be no fear on the part of our forces of the German Mark VI (Tiger) tank. There can be no basis for the T26 tank other than the conception of a tank-vs.-tank duel – which is believed to be unsound and unnecessary."[1]
2. The lower bound was unknown, but to simplify the discussion, this detail is generally omitted, taking the lower bound as known to be 1.
3. Ruggles & Brodie is largely a practical analysis and summary, not a mathematical one – the estimation problem is only mentioned in footnote 3 on page 82, where they estimate the maximum as "sample maximum + average gap".
4. As discussed in birthday attack, one can expect a collision after 1.25√H numbers, if choosing from H possible outputs. This square root corresponds to half the digits. For example, in any base, the square root of a number with 100 digits is approximately a number with 50 digits.
5. In a continuous distribution, there is no −1 term.
6. Given a particular set of observations, this set is most likely to occur if the population maximum is the sample maximum, not a higher value (it cannot be lower).
7. The sample maximum is never more than the population maximum, but can be less, hence it is a biased estimator: it will tend to underestimate the population maximum.
8. For example, the gap between 2 and 7 is (7 − 2) − 1 = 4, consisting of 3, 4, 5, and 6.
References
1. AGF policy statement. Chief of staff AGF. November 1943. MHI
2. Ruggles & Brodie 1947, pp. 73–74.
3. "Gavyn Davies does the maths – How a statistical formula won the war". The Guardian. 20 July 2006. Retrieved 6 July 2014.
4. Matthews, Robert (23 May 1998), "Data sleuths go to war, sidebar in feature "Hidden truths"", New Scientist, archived from the original on 18 April 2001
5. Bob Carruthers (1 March 2012). Panther V in Combat. Coda Books. pp. 94–. ISBN 978-1-908538-15-4.
6. Ruggles & Brodie 1947, pp. 82–83.
7. Ruggles & Brodie 1947, p. 89.
8. Ruggles & Brodie 1947, pp. 90–91.
9. Volz 2008.
10. Johnson 1994.
11. "How many Commodore 64 computers were really sold?". pagetable.com. 1 February 2011. Archived from the original on 6 March 2016. Retrieved 6 July 2014.
12. "Rockets and Missiles". www.spaceline.org.
13. Joyce, Smart. "German Tank Problem". Logan High School. Archived from the original on 24 April 2012. Retrieved 8 July 2014.
14. Höhle, M.; Held, L. (2006). "Bayesian Estimation of the Size of a Population" (PDF). Technical Report SFB 386, No. 399, Department of Statistics, University of Munich. Retrieved 17 April 2016.
Works cited
• Johnson, R. W. (Summer 1994). "Estimating the Size of a Population" (PDF). Teaching Statistics. 16 (2): 50–52. doi:10.1111/j.1467-9639.1994.tb00688.x. Archived from the original (PDF) on 23 February 2014.
• Ruggles, R.; Brodie, H. (1947). "An Empirical Approach to Economic Intelligence in World War II". Journal of the American Statistical Association. 42 (237): 72. doi:10.1080/01621459.1947.10501915. JSTOR 2280189.
• Volz, A. G. (July 2008). "A Soviet Estimate of German Tank Production". The Journal of Slavic Military Studies. 21 (3): 588–590. doi:10.1080/13518040802313902. S2CID 144483708.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
| Wikipedia |
Dieudonné determinant
In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by Dieudonné (1943).
If K is a division ring, then the Dieudonné determinant is a homomorphism of groups from the group GLn(K) of invertible n by n matrices over K onto the abelianization K×/[K×, K×] of the multiplicative group K× of K.
For example, the Dieudonné determinant for a 2-by-2 matrix is the residue class, in K×/[K×, K×], of
$\det \left({\begin{array}{*{20}c}a&b\\c&d\end{array}}\right)=\left\lbrace {\begin{array}{*{20}c}-cb&{\text{if }}a=0\\ad-aca^{-1}b&{\text{if }}a\neq 0\end{array}}\right..$
Properties
Let R be a local ring. There is a determinant map from the matrix ring GL(R) to the abelianised unit group R×ab with the following properties:[1]
• The determinant is invariant under elementary row operations
• The determinant of the identity is 1
• If a row is left multiplied by a in R× then the determinant is left multiplied by a
• The determinant is multiplicative: det(AB) = det(A)det(B)
• If two rows are exchanged, the determinant is multiplied by −1
• If R is commutative, then the determinant is invariant under transposition
Tannaka–Artin problem
Assume that K is finite over its centre F. The reduced norm gives a homomorphism Nn from GLn(K) to F×. We also have a homomorphism from GLn(K) to F× obtained by composing the Dieudonné determinant from GLn(K) to K×/[K×, K×] with the reduced norm N1 from GL1(K) = K× to F× via the abelianization.
The Tannaka–Artin problem is whether these two maps have the same kernel SLn(K). This is true when F is locally compact[2] but false in general.[3]
See also
• Moore determinant over a division algebra
References
1. Rosenberg (1994) p.64
2. Nakayama, Tadasi; Matsushima, Yozô (1943). "Über die multiplikative Gruppe einer p-adischen Divisionsalgebra". Proc. Imp. Acad. Tokyo (in German). 19: 622–628. doi:10.3792/pia/1195573246. Zbl 0060.07901.
3. Platonov, V.P. (1976). "The Tannaka-Artin problem and reduced K-theory". Izv. Akad. Nauk SSSR, Ser. Mat. (in Russian). 40: 227–261. Zbl 0338.16005.
• Dieudonné, Jean (1943), "Les déterminants sur un corps non commutatif", Bulletin de la Société Mathématique de France, 71: 27–45, doi:10.24033/bsmf.1345, ISSN 0037-9484, MR 0012273, Zbl 0028.33904
• Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94248-3, MR 1282290, Zbl 0801.19001. Errata
• Serre, Jean-Pierre (2003), Trees, Springer, p. 74, ISBN 3-540-44237-5, Zbl 1013.20001
• Suprunenko, D.A. (2001) [1994], "Determinant", Encyclopedia of Mathematics, EMS Press
| Wikipedia |
Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named after Tadao Tannaka and Mark Grigorievich Krein. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(G) with some additional structure, formed by the finite-dimensional representations of G.
Duality theorems of Tannaka and Krein describe the converse passage from the category Π(G) back to the group G, allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise from a group in this fashion. Alexander Grothendieck later showed that by a similar process, Tannaka duality can be extended to the case of algebraic groups via Tannakian formalism. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physicists. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual Hopf algebroids.
The idea of Tannaka–Krein duality: category of representations of a group
In Pontryagin duality theory for locally compact commutative groups, the dual object to a group G is its character group ${\hat {G}},$ which consists of its one-dimensional unitary representations. If we allow the group G to be noncommutative, the most direct analogue of the character group is the set of equivalence classes of irreducible unitary representations of G. The analogue of the product of characters is the tensor product of representations. However, irreducible representations of G in general fail to form a group, or even a monoid, because a tensor product of irreducible representations is not necessarily irreducible. It turns out that one needs to consider the set $\Pi (G)$ of all finite-dimensional representations, and treat it as a monoidal category, where the product is the usual tensor product of representations, and the dual object is given by the operation of the contragredient representation.
A representation of the category $\Pi (G)$ is a monoidal natural transformation from the identity functor $\operatorname {id} _{\Pi (G)}$ to itself. In other words, it is a non-zero function $\varphi $ that associates with any $T\in \operatorname {Ob} \Pi (G)$ an endomorphism of the space of T and satisfies the conditions of compatibility with tensor products, $\varphi (T\otimes U)=\varphi (T)\otimes \varphi (U)$, and with arbitrary intertwining operators $f\colon T\to U$, namely, $f\circ \varphi (T)=\varphi (U)\circ f$. The collection $\Gamma (\Pi (G))$ of all representations of the category $\Pi (G)$ can be endowed with multiplication $\varphi \psi (T)=\varphi (T)\psi (T)$ and topology, in which convergence is defined pointwise, i.e., a sequence $\{\varphi _{a}\}$ converges to some $\varphi $ if and only if $\{\varphi _{a}(T)\}$ converges to $\varphi (T)$ for all $T\in \operatorname {Ob} \Pi (G)$. It can be shown that the set $\Gamma (\Pi (G))$ thus becomes a compact (topological) group.
Theorems of Tannaka and Krein
Tannaka's theorem provides a way to reconstruct the compact group G from its category of representations Π(G).
Let G be a compact group and let F: Π(G) → VectC be the forgetful functor from finite-dimensional complex representations of G to complex finite-dimensional vector spaces. One puts a topology on the natural transformations τ: F → F by setting it to be the coarsest topology possible such that each of the projections End(F) → End(V) given by $\tau \mapsto \tau _{V}$ (taking a natural transformation $\tau $ to its component $\tau _{V}$ at $V\in \Pi (G)$) is a continuous function. We say that a natural transformation is tensor-preserving if it is the identity map on the trivial representation of G, and if it preserves tensor products in the sense that $\tau _{V\otimes W}=\tau _{V}\otimes \tau _{W}$. We also say that τ is self-conjugate if ${\overline {\tau }}=\tau $ where the bar denotes complex conjugation. Then the set ${\mathcal {T}}(G)$ of all tensor-preserving, self-conjugate natural transformations of F is a closed subset of End(F), which is in fact a (compact) group whenever G is a (compact) group. Every element x of G gives rise to a tensor-preserving self-conjugate natural transformation via multiplication by x on each representation, and hence one has a map $G\to {\mathcal {T}}(G)$. Tannaka's theorem then says that this map is an isomorphism.
Krein's theorem answers the following question: which categories can arise as a dual object to a compact group?
Let Π be a category of finite-dimensional vector spaces, endowed with operations of tensor product and involution. The following conditions are necessary and sufficient in order for Π to be a dual object to a compact group G.
1. There exists an object $I$ with the property that $I\otimes A\approx A$ for all objects A of Π (which will necessarily be unique up to isomorphism).
2. Every object A of Π can be decomposed into a sum of minimal objects.
3. If A and B are two minimal objects then the space of homomorphisms HomΠ(A, B) is either one-dimensional (when they are isomorphic) or is equal to zero.
If all these conditions are satisfied then the category Π = Π(G), where G is the group of the representations of Π.
Generalization
Interest in Tannaka–Krein duality theory was reawakened in the 1980s with the discovery of quantum groups in the work of Drinfeld and Jimbo. One of the main approaches to the study of a quantum group proceeds through its finite-dimensional representations, which form a category akin to the symmetric monoidal categories Π(G), but of more general type, braided monoidal category. It turned out that a good duality theory of Tannaka–Krein type also exists in this case and plays an important role in the theory of quantum groups by providing a natural setting in which both the quantum groups and their representations can be studied. Shortly afterwards different examples of braided monoidal categories were found in rational conformal field theory. Tannaka–Krein philosophy suggests that braided monoidal categories arising from conformal field theory can also be obtained from quantum groups, and in a series of papers, Kazhdan and Lusztig proved that it was indeed so. On the other hand, braided monoidal categories arising from certain quantum groups were applied by Reshetikhin and Turaev to construction of new invariants of knots.
Doplicher–Roberts theorem
The Doplicher–Roberts theorem (due to Sergio Doplicher and John E. Roberts) characterises Rep(G) in terms of category theory, as a type of subcategory of the category of Hilbert spaces.[1] Such subcategories of compact group unitary representations on Hilbert spaces are:
1. a strict symmetric monoidal C*-category with conjugates
2. a subcategory having subobjects and direct sums, such that the C*-algebra of endomorphisms of the monoidal unit contains only scalars.
See also
• Gelfand–Naimark theorem
Notes
1. Doplicher, S.; Roberts, J. (1989). "A new duality theory for compact groups". Inventiones Mathematicae. 98 (1): 157–218. Bibcode:1989InMat..98..157D. doi:10.1007/BF01388849.
External links
• Durkdević, Mićok (December 1996). "Quantum principal bundles and Tannaka-Krein duality theory". Reports on Mathematical Physics. 38 (3): 313–324. arXiv:q-alg/9507018. Bibcode:1996RpMP...38..313K. CiteSeerX 10.1.1.269.3027. doi:10.1016/S0034-4877(97)84884-7.
• Van Daele, Alfons (2000). "Quantum groups with invariant integrals". Proceedings of the National Academy of Sciences. 97 (2): 541–6. Bibcode:2000PNAS...97..541V. doi:10.1073/pnas.97.2.541. JSTOR 121658. PMC 33963. PMID 10639115.
• Joyal, A.; Street, R. (1991), "An introduction to Tannaka duality and quantum groups" (PDF), Category Theory, Lecture Notes in Mathematics, vol. 1488, Springer, doi:10.1007/BFb0084235, ISBN 978-3-540-46435-8
| Wikipedia |
Tannakian formalism
In mathematics, a Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory.
The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups G and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups G which are profinite groups.
The gist of the theory is that the fiber functor Φ of the Galois theory is replaced by a tensor functor T from C to K-Vect. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group (a priori only a monoid) of natural transformations of T into itself, that respect the tensor structure. This is by nature not an algebraic group, but an inverse limit of algebraic groups (pro-algebraic group).
Formal definition
A neutral Tannakian category is a rigid abelian tensor category, such that there exists a K-tensor functor to the category of finite dimensional K-vector spaces that is exact and faithful.[1]
Applications
The construction is used in cases where a Hodge structure or l-adic representation is to be considered in the light of group representation theory. For example, the Mumford–Tate group and motivic Galois group are potentially to be recovered from one cohomology group or Galois module, by means of a mediating Tannakian category it generates.
Those areas of application are closely connected to the theory of motives. Another place in which Tannakian categories have been used is in connection with the Grothendieck–Katz p-curvature conjecture; in other words, in bounding monodromy groups.
The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group ${}^{L}G$ of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G. This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with ${}^{L}G$.
Extensions
Wedhorn (2004) has established partial Tannaka duality results in the situation where the category is R-linear, where R is no longer a field (as in classical Tannakian duality), but certain valuation rings.
References
1. Deligne & Milne (1982)
• Deligne, Pierre (2007) [1990], "Catégories tannakiennes", The Grothendieck Festschrift, vol. II, Birkhauser, pp. 111–195, ISBN 9780817645755
• Deligne, Pierre; Milne, James (1982), "Tannakian categories", in Deligne, Pierre; Milne, James; Ogus, Arthur; Shih, Kuang-yen (eds.), Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer, pp. 101–228, ISBN 978-3-540-38955-2
• Saavedra Rivano, Neantro (1972), Catégories Tannakiennes, Lecture Notes in Mathematics, vol. 265, Springer, ISBN 978-3-540-37477-0, MR 0338002
• Wedhorn, Torsten (2004), "On Tannakian duality over valuation rings", Journal of Algebra, 282 (2): 575–609, doi:10.1016/j.jalgebra.2004.07.024, MR 2101076
Further reading
• M. Larsen and R. Pink. Determining representations from invariant dimensions. Invent. math., 102:377–389, 1990.
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
Inequalities in information theory
Inequalities are very important in the study of information theory. There are a number of different contexts in which these inequalities appear.
Entropic inequalities
Main article: Entropic vector
Consider a tuple $X_{1},X_{2},\dots ,X_{n}$ of $n$ finitely (or at most countably) supported random variables on the same probability space. There are 2n subsets, for which (joint) entropies can be computed. For example, when n = 2, we may consider the entropies $H(X_{1}),$ $H(X_{2}),$ and $H(X_{1},X_{2})$. They satisfy the following inequalities (which together characterize the range of the marginal and joint entropies of two random variables):
• $H(X_{1})\geq 0$
• $H(X_{2})\geq 0$
• $H(X_{1})\leq H(X_{1},X_{2})$
• $H(X_{2})\leq H(X_{1},X_{2})$
• $H(X_{1},X_{2})\leq H(X_{1})+H(X_{2}).$
In fact, these can all be expressed as special cases of a single inequality involving the conditional mutual information, namely
$I(A;B|C)\geq 0,$
where $A$, $B$, and $C$ each denote the joint distribution of some arbitrary (possibly empty) subset of our collection of random variables. Inequalities that can be derived as linear combinations of this are known as Shannon-type inequalities.
For larger $n$ there are further restrictions on possible values of entropy. To make this precise, a vector $h$ in $\mathbb {R} ^{2^{n}}$ indexed by subsets of $\{1,\dots ,n\}$ is said to be entropic if there is a joint, discrete distribution of n random variables $X_{1},\dots ,X_{n}$ such that $h_{I}=H(X_{i}\colon i\in I)$ is their joint entropy, for each subset $I$. The set of entropic vectors is denoted $\Gamma _{n}^{*}$, following the notation of Yeung.[1] It is not closed nor convex for $n\geq 3$, but its topological closure ${\overline {\Gamma _{n}^{*}}}$ is known to be convex and hence it can be characterized by the (infinitely many) linear inequalities satisfied by all entropic vectors, called entropic inequalities.
The set of all vectors that satisfy Shannon-type inequalities (but not necessarily other entropic inequalities) contains ${\overline {\Gamma _{n}^{*}}}$. This containment is strict for $n\geq 4$ and further inequalities are known as non-Shannon type inequalities. Zhang and Yeung reported the first non-Shannon-type inequality,[2] often referred to as the Zhang-Yeung inequality. Matus[3] proved that no finite set of inequalities can characterize (by linear combinations) all entropic inequalities. In other words, the region ${\overline {\Gamma _{n}^{*}}}$ is not a polytope.
Lower bounds for the Kullback–Leibler divergence
A great many important inequalities in information theory are actually lower bounds for the Kullback–Leibler divergence. Even the Shannon-type inequalities can be considered part of this category, since the interaction information can be expressed as the Kullback–Leibler divergence of the joint distribution with respect to the product of the marginals, and thus these inequalities can be seen as a special case of Gibbs' inequality.
On the other hand, it seems to be much more difficult to derive useful upper bounds for the Kullback–Leibler divergence. This is because the Kullback–Leibler divergence DKL(P||Q) depends very sensitively on events that are very rare in the reference distribution Q. DKL(P||Q) increases without bound as an event of finite non-zero probability in the distribution P becomes exceedingly rare in the reference distribution Q, and in fact DKL(P||Q) is not even defined if an event of non-zero probability in P has zero probability in Q. (Hence the requirement that P be absolutely continuous with respect to Q.)
Gibbs' inequality
Main article: Gibbs' inequality
This fundamental inequality states that the Kullback–Leibler divergence is non-negative.
Kullback's inequality
Main article: Kullback's inequality
Another inequality concerning the Kullback–Leibler divergence is known as Kullback's inequality.[4] If P and Q are probability distributions on the real line with P absolutely continuous with respect to Q, and whose first moments exist, then
$D_{KL}(P\parallel Q)\geq \Psi _{Q}^{*}(\mu '_{1}(P)),$
where $\Psi _{Q}^{*}$ is the large deviations rate function, i.e. the convex conjugate of the cumulant-generating function, of Q, and $\mu '_{1}(P)$ is the first moment of P.
The Cramér–Rao bound is a corollary of this result.
Pinsker's inequality
Main article: Pinsker's inequality
Pinsker's inequality relates Kullback–Leibler divergence and total variation distance. It states that if P, Q are two probability distributions, then
${\sqrt {{\frac {1}{2}}D_{KL}^{(e)}(P\parallel Q)}}\geq \sup\{|P(A)-Q(A)|:A{\text{ is an event to which probabilities are assigned.}}\}.$
where
$D_{KL}^{(e)}(P\parallel Q)$
is the Kullback–Leibler divergence in nats and
$\sup _{A}|P(A)-Q(A)|$
is the total variation distance.
Other inequalities
Hirschman uncertainty
In 1957,[5] Hirschman showed that for a (reasonably well-behaved) function $f:\mathbb {R} \rightarrow \mathbb {C} $ such that $\int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1,$ and its Fourier transform $g(y)=\int _{-\infty }^{\infty }f(x)e^{-2\pi ixy}\,dx,$ the sum of the differential entropies of $|f|^{2}$ and $|g|^{2}$ is non-negative, i.e.
$-\int _{-\infty }^{\infty }|f(x)|^{2}\log |f(x)|^{2}\,dx-\int _{-\infty }^{\infty }|g(y)|^{2}\log |g(y)|^{2}\,dy\geq 0.$
Hirschman conjectured, and it was later proved,[6] that a sharper bound of $\log(e/2),$ which is attained in the case of a Gaussian distribution, could replace the right-hand side of this inequality. This is especially significant since it implies, and is stronger than, Weyl's formulation of Heisenberg's uncertainty principle.
Tao's inequality
Given discrete random variables $X$, $Y$, and $Y'$, such that $X$ takes values only in the interval [−1, 1] and $Y'$ is determined by $Y$ (such that $H(Y'|Y)=0$), we have[7][8]
$\operatorname {E} {\big (}{\big |}\operatorname {E} (X|Y')-\operatorname {E} (X\mid Y){\big |}{\big )}\leq {\sqrt {I(X;Y\mid Y')\,2\log 2}},$
relating the conditional expectation to the conditional mutual information. This is a simple consequence of Pinsker's inequality. (Note: the correction factor log 2 inside the radical arises because we are measuring the conditional mutual information in bits rather than nats.)
Machine based proof checker of information-theoretic inequalities
Several machine based proof checker algorithms are now available. Proof checker algorithms typically verify the inequalities as either true or false. More advanced proof checker algorithms can produce proof or counterexamples.[9]ITIP is a Matlab based proof checker for all Shannon type Inequalities. Xitip is an open source, faster version of the same algorithm implemented in C with a graphical front end. Xitip also has a built in language parsing feature which support a broader range of random variable descriptions as input. AITIP and oXitip are cloud based implementations for validating the Shannon type inequalities. oXitip uses GLPK optimizer and has a C++ backend based on Xitip with a web based user interface. AITIP uses Gurobi solver for optimization and a mix of python and C++ in the backend implementation. It can also provide the canonical break down of the inequalities in terms of basic Information measures.[9]
See also
• Cramér–Rao bound
• Entropy power inequality
• Entropic vector
• Fano's inequality
• Jensen's inequality
• Kraft inequality
• Pinsker's inequality
References
1. Yeung, R.W. (1997). "A framework for linear information inequalities". IEEE Transactions on Information Theory. 43 (6): 1924–1934. doi:10.1109/18.641556.)
2. Zhang, Z.; Yeung, R. W. (1998). "On characterization of entropy function via information inequalities". IEEE Transactions on Information Theory. 44 (4): 1440–1452. doi:10.1109/18.681320.
3. Matus, F. (2007). Infinitely many information inequalities. 2007 IEEE International Symposium on Information Theory.
4. Fuchs, Aimé; Letta, Giorgio (1970). "L'Inégalité de KULLBACK. Application à la théorie de l'estimation". Séminaire de Probabilités IV Université de Strasbourg. pp. 108–131. doi:10.1007/bfb0059338. ISBN 978-3-540-04913-5. MR 0267669. {{cite book}}: |journal= ignored (help)CS1 maint: location missing publisher (link)
5. Hirschman, I. I. (1957). "A Note on Entropy". American Journal of Mathematics. 79 (1): 152–156. doi:10.2307/2372390. JSTOR 2372390.
6. Beckner, W. (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (6): 159–182. doi:10.2307/1970980. JSTOR 1970980.
7. Tao, T. (2006). "Szemerédi's regularity lemma revisited". Contrib. Discrete Math. 1: 8–28. arXiv:math/0504472. Bibcode:2005math......4472T.
8. Ahlswede, Rudolf (2007). "The final form of Tao's inequality relating conditional expectation and conditional mutual information". Advances in Mathematics of Communications. 1 (2): 239–242. doi:10.3934/amc.2007.1.239.
9. Ho, S.W.; Ling, L.; Tan, C.W.; Yeung, R.W. (2020). "Proving and Disproving Information Inequalities: Theory and Scalable Algorithms". IEEE Transactions on Information Theory. 66 (9): 5525–5536. doi:10.1109/TIT.2020.2982642. S2CID 216530139.
External links
• Thomas M. Cover, Joy A. Thomas. Elements of Information Theory, Chapter 16, "Inequalities in Information Theory" John Wiley & Sons, Inc. 1991 Print ISBN 0-471-06259-6 Online ISBN 0-471-20061-1 pdf
• Amir Dembo, Thomas M. Cover, Joy A. Thomas. Information Theoretic Inequalities. IEEE Transactions on Information Theory, Vol. 37, No. 6, November 1991. pdf
• ITIP: http://user-www.ie.cuhk.edu.hk/~ITIP/
• XITIP: http://xitip.epfl.ch
• N. R. Pai, Suhas Diggavi, T. Gläßle, E. Perron, R.Pulikkoonattu, R. W. Yeung, Y. Yan, oXitip: An Online Information Theoretic Inequalities Prover http://www.oxitip.com
• Siu Wai Ho, Lin Ling, Chee Wei Tan and Raymond W. Yeung, AITIP (Information Theoretic Inequality Prover): https://aitip.org
• Nivedita Rethnakar, Suhas Diggavi, Raymond. W. Yeung, InformationInequalities.jl: Exploring Information-Theoretic Inequalities, Julia Package, 2021
| Wikipedia |
Linear speedup theorem
In computational complexity theory, the linear speedup theorem for Turing machines states that given any real c > 0 and any k-tape Turing machine solving a problem in time f(n), there is another k-tape machine that solves the same problem in time at most f(n)/c + 2n + 3, where k > 1.[1][2] If the original machine is non-deterministic, then the new machine is also non-deterministic. The constants 2 and 3 in 2n + 3 can be lowered, for example, to n + 2.[1]
Proof
The construction is based on packing several tape symbols of the original machine M into one tape symbol of the new machine N. It has a similar effect as using longer words and commands in processors: it speeds up the computations but increases the machine size. How many old symbols are packed into a new symbol depends on the desired speed-up.
Suppose the new machine packs three old symbols into a new symbol. Then the alphabet of the new machine is $\Sigma \cup \Sigma ^{3}$: it consists of the original symbols and the packed symbols. The new machine has the same number k > 1 of tapes. A state of N consists of the following components:
• the state of M;
• for each tape, three packed symbols that describe the packed symbol under the head, the packed symbol on the left, and the packed symbol on the right; and
• for each tape, the original head position within the packed symbol under the head of N.
The new machine N starts with encoding the given input into a new alphabet (that is why its alphabet must include $\Sigma $). For example, if the input to 2-tape M is on the left, then after the encoding the tape configuration of N is on the right:
[ #_abbabba_...] [ #(_,_,_)(_,_,_)(_,_,_)...]
[ #_________...] [ #(_,a,b)(b,a,b)(b,a,_)...]
The new machine packs three old symbols (e.g., the blank symbol _, the symbol a, and the symbol b) into a new symbol (here (_,a,b)) and copies it the second tape, while erasing the first tape. At the end of the initialization, the new machine directs its head to the beginning. Overall, this takes 2n + 3 steps.
After the initialization, the state of N is $(q_{0};~~~?,(\_,\_,\_),?;~~~?,(\_,a,b),?;~~~[1,1])$, where the symbol $?$ ?} means that it will be filled in by the machine later; the symbol $[1,1]$ means that the head of the original machine points to the first symbols inside $(\_,\_,\_)$ and $(\_,a,b)$. Now the machine starts simulating m = 3 transitions of M using six of its own transitions (in this concrete case, there will be no speed up, but in general m can be much larger than six). Let the configurations of M and N be:
[ #__bbabba_...] [ #(_,a,b)(b,a,b)(b,_,_)...]
[ #_abbabb__...] [ #(_,_,b)(b,a,b)(b,a,_)...]
where the bold symbols indicate the head position. The state of N is $(q;~~~?,(\_,\_,b),?;~~~?,(b,\_,\_),?;~~~[3,1])$. Now the following happens:
• N moves right, left, left, right. After the four moves, the machine N has all its $?$ ?} filled, and its state becomes $(q;~~~\#,(\_,\_,b),(b,a,b);~~~(b,a,b),(b,\_,\_),(\_,\_,\_);~~~[3,1])$
• Now N updates its symbols and state according to m = 3 transitions of the original machine. This may require two moves (update the current symbol and update one of its adjacent symbols). Suppose the original machine moves as follows (with the corresponding configuration of N on the right):
[ #_____bba_...] [ #(_,a,b)(b,_,_)(_,_,_)...]
[ #_abb_____...] [ #(_,_,_)(_,_,b)(b,a,_)...]
Thus, the state of N becomes $(q';~~~?,(\_,\_,b),?;~~~?,(b,\_,\_),?;~~~[3,1])$.
Complexity
Initialization requires 2n + 3 steps. In the simulation, 6 steps of N simulate m steps of M. Choosing m > 6c produces the running time bounded by $f(n)/c+2n+3.$
References
1. Christos Papadimitriou (1994). "2.4. Linear speedup". Computational Complexity. Addison-Wesley.
2. Thomas A. Sudkamp (1994). "14.2 Linear Speedup". Languages and Machines: An Introduction to the Theory of Computer Science. Addison-Wesley.
| Wikipedia |
Tape correction (surveying)
In surveying, tape correction(s) refer(s) to correcting measurements for the effect of slope angle, expansion or contraction due to temperature, and the tape's sag, which varies with the applied tension. Not correcting for these effects gives rise to systematic errors, i.e. effects which act in a predictable manner and therefore can be corrected by mathematical methods.
Correction due to slope
$C_{v}=2L*sin^{2}+{\frac {A}{2}}$
Where
L= Inclined length measured
A= Inclined angle
When distances are measured along the slope, the equivalent horizontal distance may be determined by applying a slope correction.
The vertical slope angle of the length measured must be measured. (Refer to the figure on the other side) Thus,
• For gentle slopes, $m<20\%$
$C_{h}={\frac {h^{2}}{2s}}$
• For steep slopes, $20\%\leq m\leq 30\%$
$C_{h}={\frac {h^{2}}{2s}}+{\frac {h^{4}}{8s^{3}}}$
• For very steep slopes, $m>30\%$
$C_{h}=s(1-\cos \theta )$,
Or, more simply, $d=s*\cos \theta $
Where:
$C_{h}$ is the correction of measured slope distance due to slope;
$\theta $ is the angle between the measured slope line and horizontal line;
s is the measured slope distance.
d is the horizontal distance.
The correction $C_{h}$ is subtracted from $s$ to obtain the equivalent horizontal distance on the slope line:
$d=s-C_{h}$
Correction due to temperature
When measuring or laying out distances, the standard temperature of the tape and the temperature of the tape at time of measurement are usually different. A difference in temperature will cause the tape to lengthen or shorten, so the measurement taken will not be exactly correct. A correction can be applied to the measured length to obtain the correct length.
The correction of the tape length due to change in temperature is given by:
$C_{t}=C\cdot L(T_{m}-T_{s})$
Where:
$C_{t}$ is the correction to be applied to the tape due to temperature;
C is the coefficient of thermal expansion of the metal that forms the tape;
L is the length of the tape or length of the line measured.
$T_{m}$ is the observed temperature of the tape at the time of measurement;
$T_{s}$ is the standard temperature, when the tape is at the correct length, often 20 °C;
The correction $C_{t}$ is added to $L$ to obtain the corrected distance:
$d=L+C_{t}$
For common tape measurements, the tape used is a steel tape with coefficient of thermal expansion C equal to 0.000,011,6 units per unit length per degree Celsius change. This means that the tape changes length by 1.16 mm per 10 m tape per 10 °C change from the standard temperature of the tape. For a 30 meter long tape with standard temperature of 20 °C used at 40 °C, the change in length is 7 mm over the length of the tape.
Correction due to sag
A tape not supported along its length will sag and form a catenary between end supports. According to the section of tension correction some tapes are calibrated for sag at standard tension. These tapes will require complex sag and tension corrections if used at non-standard tensions. The correction due to sag must be calculated separately for each unsupported stretch separately and is given by:
$C_{s}={\frac {{\omega }^{2}L^{3}}{24P^{2}}}$
Where:
$C_{s}$ is the correction applied to the tape due to sag; meters;
$\omega $ is the weight of the tape per unit length; newtons per meter;
L is the length between the two ends of the catenary; meters;
P is the tension or pull applied to the tape; newtons.
A tape held in catenary will record a value larger than the correct measurement. Thus, the correction $C_{s}$ is subtracted from $L$ to obtain the corrected distance:
$d=L-C_{s}$
Note that the weight of the tape per unit length is equal to the weight of the tape divided by the length of the tape:
$\omega ={\frac {W}{L}}$
so: $W=\omega L$
Therefore, we can rewrite the formula for correction due to sag as:
$C_{s}={\frac {W^{2}L}{24P^{2}}}$
Derivation (sag)
The general formula for a catenary formed by a tape supported only at its ends is
$y={\frac {P}{\omega g}}\cosh \left({\frac {x\omega g}{P}}\right)$.
Here, $g$ is the gravitational acceleration. The arc length between two support points at $x=-k/2$ and $x=+k/2$ is found by usual methods via integration:
$L=\int _{-k/2}^{+k/2}{\sqrt {1+\left(dy/dx\right)^{2}}}\,dx$
For convenience set $a={\frac {P}{\omega g}}$. The integrand is simplified as follows using hyperbolic function identities:
${\sqrt {1+\left(dy/dx\right)^{2}}}={\sqrt {1+\left({\frac {d}{dx}}\left(a\cosh \left({\frac {x}{a}}\right)\right)\right)^{2}}}={\sqrt {1+\sinh ^{2}\left({\frac {x}{a}}\right)}}=\cosh \left({\frac {x}{a}}\right)$
The tape length $L$ is then found by integrating:
$L=\int _{-k/2}^{+k/2}\cosh \left({\frac {x}{a}}\right)dx=\left[a\sinh \left({\frac {x}{a}}\right)\right]_{x=-k/2}^{x=+k/2}=\left(2a\right)\sinh \left({\frac {k}{2a}}\right)$
Now the correction for tape sag is the difference between the actual span between the supports, $k$, and the arc length of the tape's catenary, $L$. Call this correction $\delta =k-L$. The absolute value of this $\delta $ correction is $C_{s}$ above, the amount you would subtract from the tape measurement to get the true span distance.
A Taylor series expansion of $\delta $ in terms of the quantity $L$ is desired to give a good first approximation to the correction. In fact, the first nonvanishing term in the Taylor series is cubic in $L$, and the next nonvanishing term is to the fifth power of L; thus, a series expansion for $\delta $ is reasonable. To this end, we need to find an expression for $\delta $ that contains $L$ but not $k$. We already have an expression for $L$ in terms of $k$, but now need to find the inverse function (for $k$ in terms of $L$):
${\frac {L}{2a}}=\sinh \left({\frac {k}{2a}}\right)$
$\sinh ^{-1}\left({\frac {L}{2a}}\right)={\frac {k}{2a}}$
$k=\left(2a\right)\sinh ^{-1}\left({\frac {L}{2a}}\right)$
$\delta =k-L=\left(2a\right)\sinh ^{-1}\left({\frac {L}{2a}}\right)-L$
Evaluating $\delta $ at $L=0$ yields zero, so there is no zero-order term in the Taylor series. The first derivative of this function with respect to L is
${\frac {d\delta }{dL}}={\frac {1}{\sqrt {{\frac {L^{2}}{4a^{2}}}+1}}}-1$.
Evaluated at L=0, it vanishes and so does not contribute a Taylor series term. The second derivative of $\delta $ is
${\frac {d^{2}\delta }{{dL}^{2}}}=-{\frac {L}{4a^{2}\left({\frac {L^{2}}{4a^{2}}}+1\right)^{3/2}}}$.
Again, when evaluated at L=0 it vanishes. When evaluated at L=0, the third derivative survives, however.
${\frac {d^{3}\delta }{{dL}^{3}}}=-{\frac {\left(8a^{3}-4{\text{aL}}^{2}\right)}{\left(4a^{2}+L^{2}\right)^{5/2}}}$
Thus, the first surviving term in the Taylor series is:
$\delta \cong \left[{\frac {d^{3}\delta }{{dL}^{3}}}\right]_{L=0}{\frac {L^{3}}{3!}}=-{\frac {1}{4a^{2}}}{\frac {L^{3}}{6}}={\frac {-L^{3}}{24a^{2}}}={\frac {-L^{3}\omega ^{2}g^{2}}{24P^{2}}}$
Notice that the variable $P$ here is the tension on the cable, whereas above, $P$ is the mass whose gravitational force (mass times gravitational acceleration) equals the tension on the cable. The only conversion necessary then is to take $P/g$ here and equate it to $P$ above. Also, this formula is the tape sag correction to be added to the measured distance, so the negative sign in front can be removed and the tape sag correction can be made instead by subtracting the absolute value as is done in the preceding section.
Correction due to tension
Some tapes are already calibrated to account for the sag at a standard tension.[1][2] In this case, errors arise when the tape is pulled at a Tension which differs from the standard tension used at standardization. The tape will pulled less than its standard length when a tension less than the standard tension is applied, making the tape too long.
A tape stretches in an elastic manner until it reaches its elastic limit, when it will deform permanently and ruin the tape.
The correction due to tension is given by:
$C_{p}={\frac {(P_{m}-P_{s})L}{AE}}$
Where:
$C_{p}$ is the elongation in tape length due to pull; or the correction to be applied due to applying a tension which differs from standard tension; meters;
$P_{m}$ is the tension applied to the tape during measurement; newtons;
$P_{s}$ is the standard tension, when the tape is the correct length, often 50 newtons; newtons;
L is the measured or erroneous length of the line; meters
A is the cross-sectional area of the tape; square centimeters;
E is the modulus of elasticity of the tape material; newtons per square centimeter;
The correction $C_{p}$ is added to $L$ to obtain the corrected distance:
$d=L+C_{p}$
The value for A is given by:
$A={\frac {W}{(L)(U_{w})}}$
Where:
W is the total weight of the tape; kilograms;
$U_{w}$ is the unit weight of the tape; kilograms per cubic centimeter.
For steel tapes, the value for $U_{w}$ is $7.866\times 10^{-3}kg/cm^{3}$.
Correction due to incorrect tape length
Manufacturers of measuring tapes do not usually guarantee the exact length of tapes, and standardization is a process where a standard temperature and tension are determined at which the tape is the exact length. The nominal length of tapes can be affected by physical imperfections, stretching or wear. Constant use of tapes cause wear, tapes can become kinked and may be improperly repaired when breaks occur.
The correction due to tape length is given by:
$C_{L}=Corr\times M_{L}$
Where:
CL is the corrected length of the line to be measured or laid out;
ML is the measured length or length to be laid out;
NL is the nominal length of the tape as specified by its mark;
KL is a known length;
Corr is the ratio of measured to actual length ${\frac {M_{L}}{K_{L}}}$, determined by measuring a known length.
In the U.S., some tapes come with United States Bureau of Standards certifications establishing the correction needed per 100' of tape.
Note that incorrect tape length introduces a systematic error that must be calibrated periodically.
See also
• Local attraction
References
Mostly in pdf:
• La Putt, Juny (2008). Elementary Surveying (3rd ed.). Mandaluyong: National Bookstore. ISBN 978-971-08-5581-0. Originally published by Baguio Research and Publishing Center, Baguio, Philippines in 1981.
• "Taping corrections" (PDF). Ferris State University.
• Burch, Robert. "Tape correction" (PDF). Ferris State University.
• Snelgrove, Ken (2007-09-24). "Taping Distance Errors Quantified" (PDF). University of British Columbia.
• "Information to Examinees Sitting for the Fundamentals of Surveying Examination" (PDF).
1. Unknown (2014-10-22). "GMP 8 - Good Measurement Practice for Reporting Tape Calibrations" (PDF). NIST.gov National Institute of Standards and Technology. Retrieved 2020-07-27.{{cite web}}: CS1 maint: url-status (link)
2. Unknown (2014-10-22). "SOP No. 12 Recommended Standard Operating Procedure for Calibration of Metal Tapes Tape-to-Tape Method" (PDF). NIST.gov National Institute of Standards & Technology.{{cite web}}: CS1 maint: url-status (link)
| Wikipedia |
Tape diagram
A tape diagram is a rectangular visual model resembling a piece of tape, that is used to assist with the calculation of ratios and addition, subtraction, and commonly multiplication. It is also known as a divided bar model, fraction strip, length model or strip diagram. In mathematics education, it is used to solve word problems for children in elementary school.[1]
Example
If, for example, a boy has won fifteen games, and the ratio of his wins to losses is 3:2, a tape diagram can be used to determine his number of losses, such as by doing
15 ?
Total games lost = ?
3 2
Total games lost = ?
Since the ratio between his wins and losses is 3:2, and he has won fifteen games, it can be concluded that the boy has lost ten of these games.
References
1. Lisa Watts-Lawton; Colleen Sheeron (2015). Eureka Math Curriculum Guide: A Story of Units, Grade 2 yes. Wiley. p. 128. ISBN 978-1-118-81261-7.
| Wikipedia |
Hole
A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of engineering. Depending on the material and the placement, a hole may be an indentation in a surface (such as a hole in the ground), or may pass completely through that surface (such as a hole created by a hole puncher in a piece of paper).
Types
Holes can occur for a number of reasons, including natural processes and intentional actions by humans or animals. Holes in the ground that are made intentionally, such as holes made while searching for food, for replanting trees, or postholes made for securing an object, are usually made through the process of digging. Unintentional holes in an object are often a sign of damage. Potholes and sinkholes can damage human settlements.[1]
Holes can occur in a wide variety of materials, and at a wide range of scales. The smallest holes observable by humans include pinholes and perforations, but the smallest phenomenon described as a hole is an electron hole, which is a position in an atom or atomic lattice where an electron is missing. The largest phenomenon described as a hole is a supermassive black hole, an astronomical object which can be billions of times more massive than Earth's sun.
The deepest hole on Earth is the man-made Kola Superdeep Borehole, with a true vertical drill-depth of more than 7.5 miles (12 kilometers), which is only a fraction of the nearly 4,000 mile (6,400 kilometer) distance to the center of the Earth.[2]
In mathematics
In mathematics, holes are examined in a number of ways. One of these is in homology, which is a general way of associating certain algebraic objects to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology, and homology was originally a rigorous mathematical method for defining and categorizing holes in a mathematical object called a manifold. The initial motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.[3] For instance, a circle is not a disk because the circle has a hole through it while the disk is solid,[4] and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. Because a hole is immaterial, it is not immediately obvious how to define one or distinguish it from others.
Another is the notion of homotopy group: these are invariants of a topological space that, when non-trivial (one also says in this case that the space is not k-connected), detect the presence of "holes" in the sense that the space contains a sphere that cannot be contracted to a point. The term of hole is often used informally when discussing these objects.[5]
For surfaces a notion closer to the intuitive meaning exists: the genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.[6] In layman's terms, it is exactly the number of "holes" the surface has, when represented as a submanifold in 3-space.
In physics
In physics, antimatter is pervasively described as a hole, a location that, when brought together with ordinary matter to fill the hole, results in both the hole and the matter cancelling each-other out. This is analogous to patching a pothole with asphalt, or filling a bubble below the surface of water with an equal amount of water to cancel it out. The most direct example is the electron hole; a fairly general theoretical description is provided by the Dirac sea, which treats positrons (or anti-particles in general) as holes. Holes provide one of the two primary forms of conduction in a semi-conductor, that is, the material from which transistors are made; without holes, current could not flow, and transistors turn on and off by enabling or disabling the creation of holes.
In biology
Animal bodies tend to contain specialized holes which serve various biological functions, such as the intake of oxygen or food, the excretion of waste, and the intake or expulsion of other fluids for reproductive purposes. In some simple animals, a single hole serves all of these purposes.[7] The formation of holes is a significant event in the development of an animal:
All animals start out in development with one hole, the blastopore. If there are two holes, the second hole forms later. The blastopore can arise at the top or the bottom of the embryo.[7]
Gramicidin A, a polypeptide with a helical shape, has been described as a portable hole. When it forms a dimer, it can embed itself in cellular bilayer membranes and form a hole through which water molecules can pass.[8]
Blind and through
In engineering, machining, and tooling, a hole may be a blind hole or a through hole (also called a thru-hole or clearance hole). A blind hole is a hole that is reamed, drilled, or milled to a specified depth without breaking through to the other side of the workpiece. A through hole is a hole that is made to go completely through the material of an object. In other words, a through hole is a hole that goes all the way through something. Taps used for through holes are generally tapered since it will tap faster and the chips will be released when the tap exits the hole.
The etymology of the blind hole is that it is not possible to see through it. It may also refer to any feature that is taken to a specific depth, more specifically referring to internally threaded hole (tapped holes). Not considering the drill point, the depth of the blind hole, conventionally, may be slightly deeper than that of the threaded depth.
There are three accepted methods of threading blind holes:
1. Conventional tapping, especially with bottom taps
2. Single-point threading, where the workpiece is rotated, and a pointed cutting tool is fed into the workpiece at the same rate as the pitch of the internal thread. Single-pointing inside a blind hole, like boring inside one, is inherently more challenging than doing so in a through hole. This was especially true in the era when manual machining was the only method of control. Today, CNC makes these tasks less stressful, but nevertheless still more challenging than with through holes.
3. Helical interpolation, where the workpiece remains stationary and Computer Numerical Control (CNC) moves a milling cutter in the correct helical path for a given thread, milling the thread.
At least two U.S. tool manufacturers have manufactured tools for thread milling in blind holes: Ingersoll Cutting Tools of Rockford, Illinois, and Tooling Systems of Houston, Texas, who introduced the Thread Mill in 1977, a device that milled large internal threads in the blind holes of oil well blowout preventers. Today many CNC milling machines can run such a thread milling cycle (see a video of such a cut in the "External links" section).
One use of through holes in electronics is with through-hole technology, a mounting scheme involving the use of leads on the components that are inserted into holes drilled in printed circuit boards (PCB) and soldered to pads on the opposite side either by manual assembly (hand placement) or by the use of automated insertion mount machines.[9][10]
Pinholes
A pinhole is a small hole, usually made by pressing a thin, pointed object such as a pin through an easily penetrated material such as a fabric or a very thin layer of metal. Similar holes made by other means are also often called pinholes. Pinholes may be intentionally made for various reasons. For example, in optics pinholes are used as apertures to select certain rays of light. This is used in pinhole cameras to form an image without the use of a lens.[11] Pinholes on produce packaging have been used to control the atmosphere and relative humidity within the packaging.[12]
In many fields, pinholes are a harmful side effect of manufacturing processes. For example, in the assembly of microcircuits, pinholes in the dielectric insulator layer coating the circuit can cause the circuit to fail. Therefore, "[t]o avoid pinholes that might protrude through the entire thickness of the dielectric layer, it is a common practice to screen several layers of dielectric with drying and firing after each screening", thereby preventing the pinholes from becoming continuous.[13]
Philosophy and psychology
It has been noted that holes occupy an unusual ontological position in philosophy, as people tend to refer to them as tangible and countable objects, when in fact they are the absence of something in another object.[14][15]
In the study of visual perception, a hole is a special case of figure-ground, because the ground region is entirely surrounded by the figure. For a region to be perceived as a visual hole three factors are important: depth factors indicating that the enclosed region lies behind; grouping between the enclosed region and the surround; and figural factors (for example symmetry, convexity, or familiarity) that lead to the perception of a figure rather than a hole.[16][17]
There is a debate on whether holes are special and whether they are perceived as having their own shape. They may be special in some cases, but not in the ownership of the contours.[18]
Some people have an aversion to the sight of irregular patterns or clusters of small holes, a condition called trypophobia.[19][20] Researchers hypothesize that this is the result of a biological revulsion that associates trypophobic shapes with danger or disease, and may therefore have an evolutionary basis.[21][19]
In culture and as a metaphor
An example of the use of holes in popular culture can be found in the Beatles lyric from the song, "A Day in the Life", from their 1967 album Sgt. Pepper's Lonely Hearts Club Band:
I read the news today, oh boy:
Four thousand holes in Blackburn Lancashire.
And though the holes were rather small,
They had to count them all,
Now they know how many holes it takes to fill the Albert Hall.[22]
The reference to 4,000 holes was written by John Lennon, and inspired by a Far & Near news brief from the same 17 January edition of the Daily Mail, which had also provided inspiration for previous verses of the song. Under the headline "The holes in our roads", the brief stated: "There are 4,000 holes in the road in Blackburn, Lancashire, or one twenty-sixth of a hole per person, according to a council survey. If Blackburn is typical, there are two million holes in Britain's roads and 300,000 in London".[23] Holes have also been described as ontological parasites because they can only exist as aspects of another object.[14] The psychological concept of a hole as a physical object is taken to its logical extreme in the fictional concept of a portable hole, exemplified in role-playing games and characterized as a "hole" that a person can carry with them, keep things in, and enter themselves as needed.[24]
In art holes are sometimes referred to as negative space, as in the case of the Japanese concept of Ma.
Holes can also be referenced metaphorically as existing in non-tangible things. For example, a person who provides an account of an event that lacks important details can be said to have "holes in their story", and a fictional work with unexplained narrative elements can be said to have plot holes.[25] A person who has suffered loss is often referred to as having a "hole in their heart". The concept of a "God-shaped hole" occurs in religious discourse:
[H]umans are commonly said to have “a God-shaped hole” in our souls. If you are a religious person, you can explain the hole by saying that God put it there in order to make it easier for us to receive Him. If you are a naturalist or an atheist, you believe the God-shaped hole is in our minds, not our souls. You then look for reasons that the concept of God might have evolved in our species.[26]
Unicode
Hole Emoji
Unicode symbol for HOLE, U+1F573 ( 🕳 )
The Unicode symbol for HOLE, U+1F573, was approved in 2014 as part of the Miscellaneous Symbols and Pictographs chart in Unicode 7.0,[27] and was part of Emoji 1.0, published in 2015.[28] As pictorial representations for emoji are platform-dependent, Emojipedia shows images of the hole symbol as depicted on various platforms.[29]
Gallery
• Hole in the ground dug by a fox as its burrow.
• Hole in a Eucalyptus tree used as a nest by Lorikeets.
• Trees visible through a large hole in a tree trunk in Laos.
• Hole in a cloud over Karawanks, Slovenia.
• Sock with a hole in it.
• Simulated view of a black hole in front of the Large Magellanic Cloud.
• First image of a black hole by the Event Horizon Telescope.
• Close-up view of an electronic circuit board showing component lead holes (gold-plated) with through-hole plating.
• Sound holes precisely carved into the surface of a guitar can facilitate a desired sound.
• First World War cartoon Well, if you knows of a better 'ole, go to it by Bruce Bairnsfather, 1915.
• A hole in the Berlin wall, 2019.
• A handheld hole punch, used to make holes in paper and similar materials.
See also
• Annulus (mathematics)
• Depression (geology)
• Law of holes
• Sinus
• Tunnel
• Watering hole
• Trypophobia
References
1. "The hole story". The Economist. 2016-06-11. Retrieved 2017-02-11.
2. "What's At The Bottom Of The Deepest Hole On Earth?". 2016-03-11. Retrieved 2016-08-17.
3. Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University (2008), p. 254.
4. Arnold, Bradford Henry (2013). Intuitive Concepts in Elementary Topology. Dover Books on Mathematics. Courier / Dover Publications. p. 58. ISBN 978-0-48627576-5.
5. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
6. Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
7. Dunn, Casey (2009-11-06). "A tale of two holes – Creature Cast – Learn Science at Scitable". www.nature.com.
8. Mouritsen, Ole G. (2005). Life - As a Matter of Fat: The Emerging Science of Lipidomics. Springer. p. 186. ISBN 978-3-54023248-3. OCLC 1156049123.
9. Electronic Packaging: Solder Mounting Technologies in K. H. Buschow et al (eds.), Encyclopedia of Materials: Science and Technology, Elsevier, 2001 ISBN 0-08-043152-6, pp. 2708–2709
10. Horowitz, Paul; Hill, Winfield (1989). The art of electronics (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-0-52137095-0.
11. What is a Pinhole Camera?, pinhole.cz.
12. Enrique Ortega-Rivas, Processing Effects on Safety and Quality of Foods (2010), p. 280.
13. James J. Licari, Leonard R. Enlow, Hybrid Microcircuit Technology Handbook, 2nd Edition (2008), p. 162.
14. Casati, Roberto; Varzi, Achille (2018-12-18). Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University – via Stanford Encyclopedia of Philosophy.
15. Lewis, DK, Lewis, SR (1970). "Holes". Australasian Journal of Philosophy. 48: 206–212. doi:10.1093/0195032047.003.0001. ISBN 0195032047.
16. Rock, Irv (1973). Orientation and Form. New York: Academic Press. ISBN 978-0125912501.
17. Abrahams, Marc (2005). "A hole full of surprises".
18. Bertamini M (2006). "Who owns the contour of a visual hole?". Perception. 35 (7): 883–894. doi:10.1068/p5496. PMID 16970198. S2CID 8073234.
19. Martínez-Aguayo, Juan Carlos; Lanfranco, Renzo C.; Arancibia, Marcelo; Sepúlveda, Elisa; Madrid, Eva (2018). "Trypophobia: What Do We Know So Far? A Case Report and Comprehensive Review of the Literature". Frontiers in Psychiatry. 9: 15. doi:10.3389/fpsyt.2018.00015. ISSN 1664-0640. PMC 5811467. PMID 29479321. This article incorporates text by Juan Carlos Martínez-Aguay, Renzo C. Lanfranco, Marcelo Arancibia, Elisa Sepúlveda and Eva Madrid available under the CC BY 4.0 license.
20. Le, An T. D.; Cole, Geoff G.; Wilkins, Arnold J. (2015-01-30). "Assessment of trypophobia and an analysis of its visual precipitation". Quarterly Journal of Experimental Psychology. 68 (11): 2304–2322. doi:10.1080/17470218.2015.1013970. PMID 25635930. S2CID 42086559.
21. Milosevic, Irena; McCabe, Randi E. (2015). Phobias: The Psychology of Irrational Fear. ABC-CLIO. pp. 401–402. ISBN 978-1-61069576-3. Retrieved 2017-10-25.
22. "Photographic impressions of Beatle songs by Art Kane", LIFE, Vol. 65, No. 12 (1968-09-20), p. 67.
23. "Far & Near: The holes in our roads". The Daily Mail. No. 21994. 1967-01-17. p. 7.
24. David Zeb Cook, Jean Rabe, Warren Spector, Dungeon Master Guide for the AD&D Game (1995), p. 235.
25. "plot hole | Definition of plot hole in English by Oxford Dictionaries". Oxford Dictionaries. Archived from the original on 2017-09-07. Retrieved 2017-09-07.
26. Blakey Vermeule, Why Do We Care about Literary Characters? (2010), p. 10.
27. "Unicode: Miscellaneous Symbols and Pictographs". Retrieved 2018-08-20.
28. "Emoji Version 1.0". Retrieved 2018-08-20.
29. "Emojipedia: Hole". Retrieved 2018-08-20.
| Wikipedia |
Tara E. Brendle
Tara Elise Brendle FRSE is an American mathematician who works in geometric group theory, which involves the intersection of algebra and low-dimensional topology. In particular, she studies mapping class group of surfaces, including braid groups, and their relationship to automorphism groups of free groups and arithmetic groups. She is a professor of mathematics and head of mathematics at the University of Glasgow.[1]
Education and career
Brendle received her B.S. in mathematics, magna cum laude, from Haverford College in 1995.[2] At Haverford, she won All Middle-Atlantic Conference honors in 1992 for her volleyball playing,[3] and won honorable mention in the 1995 Alice T. Schafer Prize for Excellence in Mathematics by an Undergraduate Woman of the Association for Women in Mathematics for her undergraduate research in knot theory.[4] She received her M.A. in mathematics from Columbia University in 1996 and went on to complete her Ph.D. at Columbia under the supervision of Joan Birman in 2002.[5]. After receiving her Ph.D. from Columbia, Brendle was a National Science Foundation VIGRE Assistant Professor at Cornell University and an assistant professor at Louisiana State University. She moved to her present position at the University of Glasgow in 2008.[6]
Recognition
Brendle became a member of the Young Academy of Scotland in 2014.[7] She was elected a Fellow of the American Mathematical Society in the 2020 class, "for contributions to topology and geometry, for expository lectures, and for service to the profession aimed at the full participation of women in mathematics."[8] She became a Fellow of the Royal Society of Edinburgh in 2021,[9] and in the same year won the Senior Whitehead Prize "for her fundamental work in geometric group theory, concentrating on the study of groups arising in low-dimensional topology, and for her exemplary record of work in support of mathematics and mathematicians".[10]
References
1. School of Mathematics & Statistics:Professor Tara Brendle, University of Glasgow, retrieved 3 November 2019
2. Curriculum vitae (PDF), Cornell University Mathematics, 2004, retrieved 2019-11-03
3. Volleyball All-Time Honors, Haverford Athletics, retrieved 2019-11-03
4. Alice T. Schafer Prize for Excellence in Mathematics by an Undergraduate Woman 1995, Association for Women in Mathematics, retrieved 2019-11-03
5. Tara E. Brendle at the Mathematics Genealogy Project
6. Tara E. Brendle, University of Glasgow, retrieved 3 November 2019
7. Webb, Sam (30 June 2014), Scotland's Young Academy welcomes 43 new members, Young Academy of Scotland, retrieved 2019-11-03
8. 2020 Class of the Fellows of the AMS, American Mathematical Society, retrieved 3 November 2019
9. Professor Tara Brendle FRSE, Royal Society of Edinburgh, 5 May 2021, retrieved 2021-05-19
10. Senior Whitehead Prize: citation for Tara Brendle (PDF), London Mathematical Society, retrieved 2022-02-03
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| Wikipedia |
Tara S. Holm
Tara Suzanne Holm is a mathematician at Cornell University specializing in algebraic geometry and symplectic geometry.[1]
Tara S. Holm
CitizenshipUnited States
Alma materMassachusetts Institute of Technology
Known forAbstract algebra
Awards
• Simons Fellow
• Fellow of the American Mathematical Society
Scientific career
FieldsMathematics
InstitutionsCornell University
ThesisEquivariant Cohomology, Homogeneous Spaces and Graphs (2002)
Doctoral advisorVictor William Guillemin
Life and career
Holm graduated summa cum laude from Dartmouth College.[2] Holm received her Ph.D. from the Massachusetts Institute of Technology in 2002 under the supervision of Victor Guillemin.[3] She went on to a three-year postdoc at the University of California, Berkeley, before eventually joining the faculty at Cornell.
Awards and honors
In 2012, Holm became a fellow of the American Mathematical Society.[4]
In 2013, Holm was awarded a Simons Fellowship.[5]
In 2019, Holm was awarded the Sze/Hernandez Teaching prize at Cornell.[6]
In 2019, Holm was the AWM/MAA Falconer Lecturer at MAA MathFest.[7]
From 2011-2013, Holm was an AMS Council member at large.[8]
Selected publications
• Harada, Megumi; Henriques, André; Holm, Tara S. (2005). "Computation of generalized equivariant cohomologies of Kac-Moody flag varieties". Advances in Mathematics. 197 (1): 198–221. doi:10.1016/j.aim.2004.10.003.
• Guillemin, V.; Holm, T.; Zara, C. (2006). "A GKM description of the equivariant cohomology ring of a homogeneous space". Journal of Algebraic Combinatorics. 23 (1): 21–41. doi:10.1007/s10801-006-6027-4.
• Hausmann, Jean-Claude; Holm, Tara S.; Puppe, Volker (2005). "Conjugation spaces". Algebraic & Geometric Topology. 5: 923–964. doi:10.2140/agt.2005.5.923.
• Biss, Daniel; Guillemin, Victor W.; Holm, Tara S. (2004). "The mod 2 cohomology of fixed point sets of anti-symplectic involutions". Advances in Mathematics. 185 (2): 370–399. doi:10.1016/j.aim.2003.07.007.
References
1. "Home page of Tara S. Holm". Cornell University. Retrieved Feb 2, 2015.
2. "Tara Holm, Timothy Riley". New York Times. Oct 31, 2008. Retrieved Feb 2, 2015.
3. Tara Suzanne Holm at the Mathematics Genealogy Project
4. List of Fellows of the American Mathematical Society
5. "2013 Simons Fellows Awardees: Mathematics". Simons Foundation. Archived from the original on January 4, 2015. Retrieved Feb 2, 2015.
6. "Sze/Hernandez Teaching Prize | Cornell University College of Arts and Sciences Cornell Arts & Sciences". as.cornell.edu. Retrieved 2019-10-06.
7. "AWM at MAA MathFest 2019". awm-math.org. Retrieved 2020-01-17.
8. "AMS Committees". American Mathematical Society. Retrieved 2023-03-29.
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Tak (function)
In computer science, the Tak function is a recursive function, named after Ikuo Takeuchi (ja:竹内郁雄). It is defined as follows:
$\tau (x,y,z)={\begin{cases}\tau (\tau (x-1,y,z),\tau (y-1,z,x),\tau (z-1,x,y))&{\text{if }}y<x\\z&{\text{otherwise}}\end{cases}}$
def tak(x, y, z):
if y < x:
return tak(
tak(x-1, y, z),
tak(y-1, z, x),
tak(z-1, x, y)
)
else:
return z
This function is often used as a benchmark for languages with optimization for recursion.[1][2][3][4]
tak() vs. tarai()
The original definition by Takeuchi was as follows:
def tarai(x, y, z):
if y < x:
return tarai(
tarai(x-1, y, z),
tarai(y-1, z, x),
tarai(z-1, x, y)
)
else:
return y # not z!
tarai is short for たらい回し tarai mawashi, "to pass around" in Japanese.
John McCarthy named this function tak() after Takeuchi.[5]
However, in certain later references, the y somehow got turned into the z. This is a small, but significant difference because the original version benefits significantly by lazy evaluation. Though written in exactly the same manner as others, the Haskell code below runs much faster.
tarai :: Int -> Int -> Int -> Int
tarai x y z
| x <= y = y
| otherwise = tarai (tarai (x-1) y z)
(tarai (y-1) z x)
(tarai (z-1) x y)
One can easily accelerate this function via memoization yet lazy evaluation still wins.
The best known way to optimize tarai is to use mutually recursive helper function as follows.
def laziest_tarai(x, y, zx, zy, zz):
if not y < x:
return y
else:
return laziest_tarai(
tarai(x-1, y, z),
tarai(y-1, z, x),
tarai(zx, zy, zz)-1, x, y)
def tarai(x, y, z):
if not y < x:
return y
else:
return laziest_tarai(
tarai(x-1, y, z),
tarai(y-1, z, x),
z-1, x, y)
Here is an efficient implementation of tarai() in C:
int tarai(int x, int y, int z)
{
while (x > y) {
int oldx = x, oldy = y;
x = tarai(x - 1, y, z);
y = tarai(y - 1, z, oldx);
if (x <= y) break;
z = tarai(z - 1, oldx, oldy);
}
return y;
}
Note the additional check for (x <= y) before z (the third argument) is evaluated, avoiding unnecessary recursive evaluation.
References
1. Peter Coffee (1996). "Tak test stands the test of time". PC Week. 13 (39).
2. "Recursive Methods" by Elliotte Rusty Harold
3. Johnson-Davies, David (June 1986). "Six of the Best Against the Clock". Acorn User. pp. 179, 181–182. Retrieved 28 October 2020.
4. Johnson-Davies, David (November 1986). "Testing the Tak". Acorn User. pp. 197, 199. Retrieved 28 October 2020.
5. John McCarthy (December 1979). "An Interesting LISP Function". ACM Lisp Bulletin (3): 6–8. doi:10.1145/1411829.1411833. S2CID 31639459.
External links
• Weisstein, Eric W. "TAK Function". MathWorld.
• TAK Function
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| Wikipedia |
Function approximation
In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way.[1] The need for function approximations arises in many branches of applied mathematics, and computer science in particular , such as predicting the growth of microbes in microbiology.[2] Function approximations are used where theoretical models are unavailable or hard to compute.[2]
Not to be confused with Function fitting.
One can distinguish two major classes of function approximation problems:
First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).[3]
Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead.[4]
To some extent, the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.
References
1. Lakemeyer, Gerhard; Sklar, Elizabeth; Sorrenti, Domenico G.; Takahashi, Tomoichi (2007-09-04). RoboCup 2006: Robot Soccer World Cup X. Springer. ISBN 978-3-540-74024-7.
2. Basheer, I.A.; Hajmeer, M. (2000). "Artificial neural networks: fundamentals, computing, design, and application" (PDF). Journal of Microbiological Methods. 43 (1): 3–31. doi:10.1016/S0167-7012(00)00201-3. PMID 11084225. S2CID 18267806.
3. Mhaskar, Hrushikesh Narhar; Pai, Devidas V. (2000). Fundamentals of Approximation Theory. CRC Press. ISBN 978-0-8493-0939-7.
4. Charte, David; Charte, Francisco; García, Salvador; Herrera, Francisco (2019-04-01). "A snapshot on nonstandard supervised learning problems: taxonomy, relationships, problem transformations and algorithm adaptations". Progress in Artificial Intelligence. 8 (1): 1–14. arXiv:1811.12044. doi:10.1007/s13748-018-00167-7. ISSN 2192-6360. S2CID 53715158.
See also
• Approximation theory
• Fitness approximation
• Kriging
• Least squares (function approximation)
• Radial basis function network
| Wikipedia |
Leakage (machine learning)
In statistics and machine learning, leakage (also known as data leakage or target leakage) is the use of information in the model training process which would not be expected to be available at prediction time, causing the predictive scores (metrics) to overestimate the model's utility when run in a production environment.[1]
Part of a series on
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Leakage is often subtle and indirect, making it hard to detect and eliminate. Leakage can cause a statistician or modeler to select a suboptimal model, which could be outperformed by a leakage-free model.[1]
Leakage modes
Leakage can occur in many steps in the machine learning process. The leakage causes can be sub-classified into two possible sources of leakage for a model: features and training examples.[1]
Feature leakage
Feature or column-wise leakage is caused by the inclusion of columns which are one of the following: a duplicate label, a proxy for the label, or the label itself. These features, known as anachronisms, will not be available when the model is used for predictions, and result in leakage if included when the model is trained.[2]
For example, including a "MonthlySalary" column when predicting "YearlySalary"; or "MinutesLate" when predicting "IsLate"; or more subtly "NumOfLatePayments" when predicting "ShouldGiveLoan".
Training example leakage
Row-wise leakage is caused by improper sharing of information between rows of data. Types of row-wise leakage include:
• Premature featurization; leaking from premature featurization before CV/Train/Test split (must fit MinMax/ngrams/etc on only the train split, then transform the test set)
• Duplicate rows between train/validation/test (e.g. oversampling a dataset to pad its size before splitting; e.g. different rotations/augmentations of a single image; bootstrap sampling before splitting; or duplicating rows to up sample the minority class)
• Non-i.i.d. data
• Time leakage (e.g. splitting a time-series dataset randomly instead of newer data in test set using a TrainTest split or rolling-origin cross validation)
• Group leakage -- not including a grouping split column (e.g. Andrew Ng's group had 100k x-rays of 30k patients, meaning ~3 images per patient. The paper used random splitting instead of ensuring that all images of a patient was in the same split. Hence the model partially memorized the patients instead of learning to recognize pneumonia in chest x-rays.[3][4])
For time-dependent datasets, the structure of the system being studied evolves over time (i.e. it is "non-stationary"). This can introduce systematic differences between the training and validation sets. For example, if a model for predicting stock values is trained on data for a certain five-year period, it is unrealistic to treat the subsequent five-year period as a draw from the same population. As another example, suppose a model is developed to predict an individual's risk for being diagnosed with a particular disease within the next year.
Detection
See also
• AutoML
• Cross-validation
• Overfitting
• Resampling (statistics)
• Supervised learning
• Training, validation, and test sets
References
1. Shachar Kaufman; Saharon Rosset; Claudia Perlich (January 2011). "Leakage in Data Mining: Formulation, Detection, and Avoidance". Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. 6: 556–563. doi:10.1145/2020408.2020496. S2CID 9168804. Retrieved 13 January 2020.
2. Soumen Chakrabarti (2008). "9". Data Mining: Know it All. Morgan Kaufmann Publishers. p. 383. ISBN 978-0-12-374629-0. Anachronistic variables are a pernicious mining problem. However, they aren't any problem at all at deployment time—unless someone expects the model to work! Anachronistic variables are out of place in time. Specifically, at data modeling time, they carry information back from the future to the past.
3. Guts, Yuriy (30 October 2018). Yuriy Guts. TARGET LEAKAGE IN MACHINE LEARNING (Talk). AI Ukraine Conference. Ukraine – via YouTube.
• Yuriy Guts. "Target Leakage in ML" (PDF). AI Ukraine Online Conference.
4. Nick, Roberts (16 November 2017). "Replying to @AndrewYNg @pranavrajpurkar and 2 others". Brooklyn, NY, USA: Twitter. Archived from the original on 10 June 2018. Retrieved 13 January 2020. Replying to @AndrewYNg @pranavrajpurkar and 2 others ... Were you concerned that the network could memorize patient anatomy since patients cross train and validation? "ChestX-ray14 dataset contains 112,120 frontal-view X-ray images of 30,805 unique patients. We randomly split the entire dataset into 80% training, and 20% validation."
| Wikipedia |
Codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.
A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph.[1] The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution.
A codomain is not part of a function f if f is defined as just a graph.[2][3] For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: X → Y.[4]
Examples
For a function
$f\colon \mathbb {R} \rightarrow \mathbb {R} $
defined by
$f\colon \,x\mapsto x^{2},$ or equivalently $f(x)\ =\ x^{2},$
the codomain of f is $\textstyle \mathbb {R} $, but f does not map to any negative number. Thus the image of f is the set $\textstyle \mathbb {R} _{0}^{+}$; i.e., the interval [0, ∞).
An alternative function g is defined thus:
$g\colon \mathbb {R} \rightarrow \mathbb {R} _{0}^{+}$
$g\colon \,x\mapsto x^{2}.$
While f and g map a given x to the same number, they are not, in this view, the same function because they have different codomains. A third function h can be defined to demonstrate why:
$h\colon \,x\mapsto {\sqrt {x}}.$
The domain of h cannot be $\textstyle \mathbb {R} $ but can be defined to be $\textstyle \mathbb {R} _{0}^{+}$:
$h\colon \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} .$
The compositions are denoted
$h\circ f,$
$h\circ g.$
On inspection, h ∘ f is not useful. It is true, unless defined otherwise, that the image of f is not known; it is only known that it is a subset of $\textstyle \mathbb {R} $. For this reason, it is possible that h, when composed with f, might receive an argument for which no output is defined – negative numbers are not elements of the domain of h, which is the square root function.
Function composition therefore is a useful notion only when the codomain of the function on the right side of a composition (not its image, which is a consequence of the function and could be unknown at the level of the composition) is a subset of the domain of the function on the left side.
The codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. In the example, g is a surjection while f is not. The codomain does not affect whether a function is an injection.
A second example of the difference between codomain and image is demonstrated by the linear transformations between two vector spaces – in particular, all the linear transformations from $\textstyle \mathbb {R} ^{2}$ to itself, which can be represented by the 2×2 matrices with real coefficients. Each matrix represents a map with the domain $\textstyle \mathbb {R} ^{2}$ and codomain $\textstyle \mathbb {R} ^{2}$. However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with rank 2) but many do not, instead mapping into some smaller subspace (the matrices with rank 1 or 0). Take for example the matrix T given by
$T={\begin{pmatrix}1&0\\1&0\end{pmatrix}}$
which represents a linear transformation that maps the point (x, y) to (x, x). The point (2, 3) is not in the image of T, but is still in the codomain since linear transformations from $\textstyle \mathbb {R} ^{2}$ to $\textstyle \mathbb {R} ^{2}$ are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that T does not have full rank since its image is smaller than the whole codomain.
See also
• Bijection – One-to-one correspondence
• Morphism#Codomain
Notes
1. Bourbaki 1970, p. 76
2. Bourbaki 1970, p. 77
3. Forster 2003, pp. 10–11
4. Eccles 1997, p. 91 (quote 1, quote 2); Mac Lane 1998, p. 8; Mac Lane, in Scott & Jech 1967, p. 232; Sharma 2004, p. 91; Stewart & Tall 1977, p. 89
References
• Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN 9783540340348.
• Eccles, Peter J. (1997), An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions, Cambridge University Press, ISBN 978-0-521-59718-0
• Forster, Thomas (2003), Logic, Induction and Sets, Cambridge University Press, ISBN 978-0-521-53361-4
• Mac Lane, Saunders (1998), Categories for the working mathematician (2nd ed.), Springer, ISBN 978-0-387-98403-2
• Scott, Dana S.; Jech, Thomas J. (1967), Axiomatic set theory, Symposium in Pure Mathematics, American Mathematical Society, ISBN 978-0-8218-0245-8
• Sharma, A.K. (2004), Introduction To Set Theory, Discovery Publishing House, ISBN 978-81-7141-877-0
• Stewart, Ian; Tall, David Orme (1977), The foundations of mathematics, Oxford University Press, ISBN 978-0-19-853165-4
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| Wikipedia |
Tarjan's off-line lowest common ancestors algorithm
In computer science, Tarjan's off-line lowest common ancestors algorithm is an algorithm for computing lowest common ancestors for pairs of nodes in a tree, based on the union-find data structure. The lowest common ancestor of two nodes d and e in a rooted tree T is the node g that is an ancestor of both d and e and that has the greatest depth in T. It is named after Robert Tarjan, who discovered the technique in 1979. Tarjan's algorithm is an offline algorithm; that is, unlike other lowest common ancestor algorithms, it requires that all pairs of nodes for which the lowest common ancestor is desired must be specified in advance. The simplest version of the algorithm uses the union-find data structure, which unlike other lowest common ancestor data structures can take more than constant time per operation when the number of pairs of nodes is similar in magnitude to the number of nodes. A later refinement by Gabow & Tarjan (1983) speeds the algorithm up to linear time.
Not to be confused with Tarjan's strongly connected components algorithm.
Pseudocode
The pseudocode below determines the lowest common ancestor of each pair in P, given the root r of a tree in which the children of node n are in the set n.children. For this offline algorithm, the set P must be specified in advance. It uses the MakeSet, Find, and Union functions of a disjoint-set forest. MakeSet(u) removes u to a singleton set, Find(u) returns the standard representative of the set containing u, and Union(u,v) merges the set containing u with the set containing v. TarjanOLCA(r) is first called on the root r.
function TarjanOLCA(u) is
MakeSet(u)
u.ancestor := u
for each v in u.children do
TarjanOLCA(v)
Union(u, v)
Find(u).ancestor := u
u.color := black
for each v such that {u, v} in P do
if v.color == black then
print "Tarjan's Lowest Common Ancestor of " + u +
" and " + v + " is " + Find(v).ancestor + "."
Each node is initially white, and is colored black after it and all its children have been visited.
For each node pair {u,v} to be investigated:
• When v is already black (viz. when v comes before u in a post-order traversal of the tree): After u is colored black, the lowest common ancestor of this pair is available as Find(v).ancestor, but only while the LCA of u and v is not colored black.
• Otherwise: Once v is colored black, the LCA will be available as Find(u).ancestor, while the LCA is not colored black.
For reference, here are optimized versions of MakeSet, Find, and Union for a disjoint-set forest:
function MakeSet(x) is
x.parent := x
x.rank := 1
function Union(x, y) is
xRoot := Find(x)
yRoot := Find(y)
if xRoot.rank > yRoot.rank then
yRoot.parent := xRoot
else if xRoot.rank < yRoot.rank then
xRoot.parent := yRoot
else if xRoot.rank == yRoot.rank then
yRoot.parent := xRoot
xRoot.rank := xRoot.rank + 1
function Find(x) is
if x.parent != x then
x.parent := Find(x.parent)
return x.parent
References
• Gabow, H. N.; Tarjan, R. E. (1983), "A linear-time algorithm for a special case of disjoint set union", Proceedings of the 15th ACM Symposium on Theory of Computing (STOC), pp. 246–251, doi:10.1145/800061.808753.
• Tarjan, R. E. (1979), "Applications of path compression on balanced trees", Journal of the ACM, 26 (4): 690–715, doi:10.1145/322154.322161.
| Wikipedia |
Tarjan's strongly connected components algorithm
Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph. It runs in linear time, matching the time bound for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm. The algorithm is named for its inventor, Robert Tarjan.[1]
Tarjan's strongly connected components algorithm
Tarjan's algorithm animation
Data structureGraph
Worst-case performance$O(|V|+|E|)$
Overview
The algorithm takes a directed graph as input, and produces a partition of the graph's vertices into the graph's strongly connected components. Each vertex of the graph appears in exactly one of the strongly connected components. Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, a vertex whose in-degree or out-degree is 0, or any vertex of an acyclic graph.
The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent depth-first searches are conducted on any nodes that have not yet been found). As usual with depth-first search, the search visits every node of the graph exactly once, declining to revisit any node that has already been visited. Thus, the collection of search trees is a spanning forest of the graph. The strongly connected components will be recovered as certain subtrees of this forest. The roots of these subtrees are called the "roots" of the strongly connected components. Any node of a strongly connected component might serve as a root, if it happens to be the first node of a component that is discovered by search.
Stack invariant
Nodes are placed on a stack in the order in which they are visited. When the depth-first search recursively visits a node v and its descendants, those nodes are not all necessarily popped from the stack when this recursive call returns. The crucial invariant property is that a node remains on the stack after it has been visited if and only if there exists a path in the input graph from it to some node earlier on the stack. In other words, it means that in the DFS a node would be only removed from the stack after all its connected paths have been traversed. When the DFS will backtrack it would remove the nodes on a single path and return to the root in order to start a new path.
At the end of the call that visits v and its descendants, we know whether v itself has a path to any node earlier on the stack. If so, the call returns, leaving v on the stack to preserve the invariant. If not, then v must be the root of its strongly connected component, which consists of v together with any nodes later on the stack than v (such nodes all have paths back to v but not to any earlier node, because if they had paths to earlier nodes then v would also have paths to earlier nodes which is false). The connected component rooted at v is then popped from the stack and returned, again preserving the invariant.
Bookkeeping
Each node v is assigned a unique integer v.index, which numbers the nodes consecutively in the order in which they are discovered. It also maintains a value v.lowlink that represents the smallest index of any node on the stack known to be reachable from v through v's DFS subtree, including v itself. Therefore v must be left on the stack if v.lowlink < v.index, whereas v must be removed as the root of a strongly connected component if v.lowlink == v.index. The value v.lowlink is computed during the depth-first search from v, as this finds the nodes that are reachable from v.
Note that the lowlink is different from the lowpoint, which is the smallest index reachable from v through any part of the graph.[1]: 156 [2]
The algorithm in pseudocode
algorithm tarjan is
input: graph G = (V, E)
output: set of strongly connected components (sets of vertices)
index := 0
S := empty stack
for each v in V do
if v.index is undefined then
strongconnect(v)
function strongconnect(v)
// Set the depth index for v to the smallest unused index
v.index := index
v.lowlink := index
index := index + 1
S.push(v)
v.onStack := true
// Consider successors of v
for each (v, w) in E do
if w.index is undefined then
// Successor w has not yet been visited; recurse on it
strongconnect(w)
v.lowlink := min(v.lowlink, w.lowlink)
else if w.onStack then
// Successor w is in stack S and hence in the current SCC
// If w is not on stack, then (v, w) is an edge pointing to an SCC already found and must be ignored
// Note: The next line may look odd - but is correct.
// It says w.index not w.lowlink; that is deliberate and from the original paper
v.lowlink := min(v.lowlink, w.index)
// If v is a root node, pop the stack and generate an SCC
if v.lowlink = v.index then
start a new strongly connected component
repeat
w := S.pop()
w.onStack := false
add w to current strongly connected component
while w ≠ v
output the current strongly connected component
The index variable is the depth-first search node number counter. S is the node stack, which starts out empty and stores the history of nodes explored but not yet committed to a strongly connected component. Note that this is not the normal depth-first search stack, as nodes are not popped as the search returns up the tree; they are only popped when an entire strongly connected component has been found.
The outermost loop searches each node that has not yet been visited, ensuring that nodes which are not reachable from the first node are still eventually traversed. The function strongconnect performs a single depth-first search of the graph, finding all successors from the node v, and reporting all strongly connected components of that subgraph.
When each node finishes recursing, if its lowlink is still set to its index, then it is the root node of a strongly connected component, formed by all of the nodes above it on the stack. The algorithm pops the stack up to and including the current node, and presents all of these nodes as a strongly connected component.
Note that v.lowlink := min(v.lowlink, w.index) is the correct way to update v.lowlink if w is on stack. Because w is on the stack already, (v, w) is a back-edge in the DFS tree and therefore w is not in the subtree of v. Because v.lowlink takes into account nodes reachable only through the nodes in the subtree of v we must stop at w and use w.index instead of w.lowlink.
Complexity
Time Complexity: The Tarjan procedure is called once for each node; the forall statement considers each edge at most once. The algorithm's running time is therefore linear in the number of edges and nodes in G, i.e. $O(|V|+|E|)$.
In order to achieve this complexity, the test for whether w is on the stack should be done in constant time. This may be done, for example, by storing a flag on each node that indicates whether it is on the stack, and performing this test by examining the flag.
Space Complexity: The Tarjan procedure requires two words of supplementary data per vertex for the index and lowlink fields, along with one bit for onStack and another for determining when index is undefined. In addition, one word is required on each stack frame to hold v and another for the current position in the edge list. Finally, the worst-case size of the stack S must be $|V|$ (i.e. when the graph is one giant component). This gives a final analysis of $O(|V|\cdot (2+5w))$ where $w$ is the machine word size. The variation of Nuutila and Soisalon-Soininen reduced this to $O(|V|\cdot (1+4w))$ and, subsequently, that of Pearce requires only $O(|V|\cdot (1+3w))$.[3][4]
Additional remarks
While there is nothing special about the order of the nodes within each strongly connected component, one useful property of the algorithm is that no strongly connected component will be identified before any of its successors. Therefore, the order in which the strongly connected components are identified constitutes a reverse topological sort of the DAG formed by the strongly connected components.[5]
Donald Knuth described Tarjan's SCC algorithm as one of his favorite implementations in the book The Stanford GraphBase.[6]
He also wrote:[7]
The data structures that he devised for this problem fit together in an amazingly beautiful way, so that the quantities you need to look at while exploring a directed graph are always magically at your fingertips. And his algorithm also does topological sorting as a byproduct.
References
1. Tarjan, R. E. (1972), "Depth-first search and linear graph algorithms", SIAM Journal on Computing, 1 (2): 146–160, CiteSeerX 10.1.1.327.8418, doi:10.1137/0201010
2. "Lecture #19: Depth First Search and Strong Components" (PDF). 15-451/651: Algorithms Fall 2018. Carnegie Mellon University. pp. 7–8. Retrieved 9 August 2021.
3. Nuutila, Esko (1994). "On Finding the Strongly Connected Components in a Directed Graph". Information Processing Letters. 49 (1): 9–14. doi:10.1016/0020-0190(94)90047-7.
4. Pearce, David. "A Space Efficient Algorithm for Detecting Strongly Connected Components". Information Processing Letters. 116 (1): 47–52. doi:10.1016/j.ipl.2015.08.010.
5. Harrison, Paul. "Robust topological sorting and Tarjan's algorithm in Python". Retrieved 9 February 2011.
6. Knuth, The Stanford GraphBase, pages 512–519.
7. Knuth, Donald (2014-05-20). Twenty Questions for Donald Knuth.
External links
• Rosetta Code, showing implementations in different languages
• PHP implementation of Tarjan's strongly connected components algorithm
• JavaScript implementation of Tarjan's strongly connected components algorithm
| Wikipedia |
Prouhet–Tarry–Escott problem
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations
$\sum _{a\in A}a^{i}=\sum _{b\in B}b^{i}$
for each integer i from 1 to a given k. It has been shown that n must be strictly greater than k. Solutions with $k=n-1$ are called ideal solutions. Ideal solutions are known for $3\leq n\leq 10$ and for $n=12$. No ideal solution is known for $n=11$ or for $n\geq 13$.[1]
This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Edward B. Escott, who studied it in the early 1910s. The problem originates from letters of Christian Goldbach and Leonhard Euler (1750/1751).
Examples
Ideal solutions
An ideal solution for n = 6 is given by the two sets { 0, 5, 6, 16, 17, 22 } and { 1, 2, 10, 12, 20, 21 }, because:
01 + 51 + 61 + 161 + 171 + 221 = 11 + 21 + 101 + 121 + 201 + 211
02 + 52 + 62 + 162 + 172 + 222 = 12 + 22 + 102 + 122 + 202 + 212
03 + 53 + 63 + 163 + 173 + 223 = 13 + 23 + 103 + 123 + 203 + 213
04 + 54 + 64 + 164 + 174 + 224 = 14 + 24 + 104 + 124 + 204 + 214
05 + 55 + 65 + 165 + 175 + 225 = 15 + 25 + 105 + 125 + 205 + 215.
For n = 12, an ideal solution is given by A = {±22, ±61, ±86, ±127, ±140, ±151} and B = {±35, ±47, ±94, ±121, ±146, ±148}.[2]
Other solutions
Prouhet used the Thue–Morse sequence to construct a solution with $n=2^{k}$ for any $k$. Namely, partition the numbers from 0 to $2^{k+1}-1$ into a) the numbers each with an even number of ones in its binary expansion and b) the numbers each with an odd number of ones in its binary expansion; then the two sets of the partition give a solution to the problem.[3] For instance, for $n=8$ and $k=3$, Prouhet's solution is:
01 + 31 + 51 + 61 + 91 + 101 + 121 + 151 = 11 + 21 + 41 + 71 + 81 + 111 + 131 + 141
02 + 32 + 52 + 62 + 92 + 102 + 122 + 152 = 12 + 22 + 42 + 72 + 82 + 112 + 132 + 142
03 + 33 + 53 + 63 + 93 + 103 + 123 + 153 = 13 + 23 + 43 + 73 + 83 + 113 + 133 + 143.
Generalizations
A higher dimensional version of the Prouhet–Tarry–Escott problem has been introduced and studied by Andreas Alpers and Robert Tijdeman in 2007: Given parameters $n,k\in \mathbb {N} $, find two different multi-sets $\{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}$, $\{(x_{1}',y_{1}'),\dots ,(x_{n}',y_{n}')\}$ of points from $\mathbb {Z} ^{2}$ such that
$\sum _{i=1}^{n}x_{i}^{j}y_{i}^{d-j}=\sum _{i=1}^{n}{x'}_{i}^{j}{y'}_{i}^{d-j}$
for all $d,j\in \{0,\dots ,k\}$ with $j\leq d.$ This problem is related to discrete tomography and also leads to special Prouhet-Tarry-Escott solutions over the Gaussian integers (though solutions to the Alpers-Tijdeman problem do not exhaust the Gaussian integer solutions to Prouhet-Tarry-Escott).
A solution for $n=6$ and $k=5$ is given, for instance, by:
$\{(x_{1},y_{1}),\dots ,(x_{6},y_{6})\}=\{(2,1),(1,3),(3,6),(6,7),(7,5),(5,2)\}$ and
$\{(x'_{1},y'_{1}),\dots ,(x'_{6},y'_{6})\}=\{(1,2),(2,5),(5,7),(7,6),(6,3),(3,1)\}$.
No solutions for $n=k+1$ with $k\geq 6$ are known.
See also
• Euler's sum of powers conjecture
• Beal's conjecture
• Jacobi–Madden equation
• Lander, Parkin, and Selfridge conjecture
• Taxicab number
• Pythagorean quadruple
• Sums of powers, a list of related conjectures and theorems
• Discrete tomography
Notes
1. Borwein 2002, p. 85.
2. Solution found by Nuutti Kuosa, Jean-Charles Meyrignac and Chen Shuwen, in 1999.
3. Wright, E. M. (1959), "Prouhet's 1851 solution of the Tarry–Escott problem of 1910", The American Mathematical Monthly, 66: 199–201, doi:10.2307/2309513, MR 0104622.
References
• Borwein, Peter B. (2002), "The Prouhet–Tarry–Escott problem", Computational Excursions in Analysis and Number Theory, CMS Books in Mathematics, Springer-Verlag, pp. 85–96, ISBN 0-387-95444-9, retrieved 2009-06-16 Chap.11.
• Alpers, Andreas; Rob Tijdeman (2007), "The Two-Dimensional Prouhet-Tarry-Escott Problem" (PDF), Journal of Number Theory, 123 (2): 403–412, doi:10.1016/j.jnt.2006.07.001, retrieved 2015-04-01.
External links
• Weisstein, Eric W., "Prouhet-Tarry-Escott problem", MathWorld
| Wikipedia |
Tarry point
In geometry, the Tarry point T for a triangle △ABC is a point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle △DEF. The Tarry point lies on the other endpoint of the diameter of the circumcircle drawn through the Steiner point.[1] The point is named for Gaston Tarry.[2]
See also
• Concurrent lines
Notes
1. Weisstein, Eric W. "Tarry Point". MathWorld.
2. "Gaston Tarry". August 2006.
| Wikipedia |
Tarski's axioms
Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory (Tarski 1959) (i.e., that part of Euclidean geometry that is formulable as an elementary theory). Other modern axiomizations of Euclidean geometry are Hilbert's axioms and Birkhoff's axioms.
This article is about axioms for Euclidean geometry. For Tarski's axioms for the real numbers, see Tarski's axiomatization of the reals. For Tarski's axioms for set theory, see Tarski–Grothendieck set theory.
Overview
Early in his career Tarski taught geometry and researched set theory. His coworker Steven Givant (1999) explained Tarski's take-off point:
From Enriques, Tarski learned of the work of Mario Pieri, an Italian geometer who was strongly influenced by Peano. Tarski preferred Pieri's system [of his Point and Sphere memoir], where the logical structure and the complexity of the axioms were more transparent.
Givant then says that "with typical thoroughness" Tarski devised his system:
What was different about Tarski's approach to geometry? First of all, the axiom system was much simpler than any of the axiom systems that existed up to that time. In fact the length of all of Tarski's axioms together is not much more than just one of Pieri's 24 axioms. It was the first system of Euclidean geometry that was simple enough for all axioms to be expressed in terms of the primitive notions only, without the help of defined notions. Of even greater importance, for the first time a clear distinction was made between full geometry and its elementary — that is, its first order — part.
Like other modern axiomatizations of Euclidean geometry, Tarski's employs a formal system consisting of symbol strings, called sentences, whose construction respects formal syntactical rules, and rules of proof that determine the allowed manipulations of the sentences. Unlike some other modern axiomatizations, such as Birkhoff's and Hilbert's, Tarski's axiomatization has no primitive objects other than points, so a variable or constant cannot refer to a line or an angle. Because points are the only primitive objects, and because Tarski's system is a first-order theory, it is not even possible to define lines as sets of points. The only primitive relations (predicates) are "betweenness" and "congruence" among points.
Tarski's axiomatization is shorter than its rivals, in a sense Tarski and Givant (1999) make explicit. It is more concise than Pieri's because Pieri had only two primitive notions while Tarski introduced three: point, betweenness, and congruence. Such economy of primitive and defined notions means that Tarski's system is not very convenient for doing Euclidean geometry. Rather, Tarski designed his system to facilitate its analysis via the tools of mathematical logic, i.e., to facilitate deriving its metamathematical properties. Tarski's system has the unusual property that all sentences can be written in universal-existential form, a special case of the prenex normal form. This form has all universal quantifiers preceding any existential quantifiers, so that all sentences can be recast in the form $\forall u\forall v\ldots \exists a\exists b\dots .$ This fact allowed Tarski to prove that Euclidean geometry is decidable: there exists an algorithm which can determine the truth or falsity of any sentence. Tarski's axiomatization is also complete. This does not contradict Gödel's first incompleteness theorem, because Tarski's theory lacks the expressive power needed to interpret Robinson arithmetic (Franzén 2005, pp. 25–26).
The axioms
Alfred Tarski worked on the axiomatization and metamathematics of Euclidean geometry intermittently from 1926 until his death in 1983, with Tarski (1959) heralding his mature interest in the subject. The work of Tarski and his students on Euclidean geometry culminated in the monograph Schwabhäuser, Szmielew, and Tarski (1983), which set out the 10 axioms and one axiom schema shown below, the associated metamathematics, and a fair bit of the subject. Gupta (1965) made important contributions, and Tarski and Givant (1999) discuss the history.
Fundamental relations
These axioms are a more elegant version of a set Tarski devised in the 1920s as part of his investigation of the metamathematical properties of Euclidean plane geometry. This objective required reformulating that geometry as a first-order theory. Tarski did so by positing a universe of points, with lower case letters denoting variables ranging over that universe. Equality is provided by the underlying logic (see First-order logic#Equality and its axioms).[1] Tarski then posited two primitive relations:
• Betweenness, a triadic relation. The atomic sentence Bxyz or (y)B(x,z) denotes that y is "between" x and z, in other words, that y is a point on the line segment xz. (This relation is interpreted inclusively, so that Bxyz is trivially true whenever x=y or y=z).
• Congruence (or "equidistance"), a tetradic relation. The atomic sentence Cwxyz or (w,x)C(y,z) or commonly wx ≡ yz can be interpreted as wx is congruent to yz, in other words, that the length of the line segment wx is equal to the length of the line segment yz.
Betweenness captures the affine aspect (such as the parallelism of lines) of Euclidean geometry; congruence, its metric aspect (such as angles and distances). The background logic includes identity, a binary relation. The axioms invoke identity (or its negation) on five occasions.
The axioms below are grouped by the types of relation they invoke, then sorted, first by the number of existential quantifiers, then by the number of atomic sentences. The axioms should be read as universal closures; hence any free variables should be taken as tacitly universally quantified.
Congruence axioms
Reflexivity of Congruence
$xy\equiv yx\,.$
Identity of Congruence
$xy\equiv zz\rightarrow x=y.$
Transitivity of Congruence
$(xy\equiv zu\land xy\equiv vw)\rightarrow zu\equiv vw.$
Commentary
While the congruence relation $xy\equiv zw$ is, formally, a 4-way relation among points, it may also be thought of, informally, as a binary relation between two line segments $xy$ and $zw$. The "Reflexivity" and "Transitivity" axioms above, combined, prove both:
• that this binary relation is in fact an equivalence relation
• it is reflexive: $xy\equiv xy$.
• it is symmetric $xy\equiv zw\rightarrow zw\equiv xy$.
• it is transitive $(xy\equiv zu\land zu\equiv vw)\rightarrow xy\equiv vw$.
• and that the order in which the points of a line segment are specified is irrelevant.
• $xy\equiv zw\rightarrow xy\equiv wz$.
• $xy\equiv zw\rightarrow yx\equiv zw$.
• $xy\equiv zw\rightarrow yx\equiv wz$.
The "transitivity" axiom asserts that congruence is Euclidean, in that it respects the first of Euclid's "common notions".
The "Identity of Congruence" axiom states, intuitively, that if xy is congruent with a segment that begins and ends at the same point, x and y are the same point. This is closely related to the notion of reflexivity for binary relations.
Betweenness axioms
Identity of Betweenness
$Bxyx\rightarrow x=y.$
The only point on the line segment $xx$ is $x$ itself.
Axiom of Pasch
$(Bxuz\land Byvz)\rightarrow \exists a\,(Buay\land Bvax).$
Axiom schema of Continuity
Let φ(x) and ψ(y) be first-order formulae containing no free instances of either a or b. Let there also be no free instances of x in ψ(y) or of y in φ(x). Then all instances of the following schema are axioms:
$\exists a\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Baxy]\rightarrow \exists b\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Bxby].$
Let r be a ray with endpoint a. Let the first order formulae φ and ψ define subsets X and Y of r, such that every point in Y is to the right of every point of X (with respect to a). Then there exists a point b in r lying between X and Y. This is essentially the Dedekind cut construction, carried out in a way that avoids quantification over sets.
Lower Dimension
$\exists a\,\exists b\,\exists c\,[\neg Babc\land \neg Bbca\land \neg Bcab].$
There exist three noncollinear points. Without this axiom, the theory could be modeled by the one-dimensional real line, a single point, or even the empty set.
Congruence and betweenness
Upper Dimension
$(xu\equiv xv)\land (yu\equiv yv)\land (zu\equiv zv)\land (u\neq v)\rightarrow (Bxyz\lor Byzx\lor Bzxy).$
Three points equidistant from two distinct points form a line. Without this axiom, the theory could be modeled by three-dimensional or higher-dimensional space.
Axiom of Euclid
Each of the three variants of this axiom, all equivalent over the remaining Tarski's axioms to Euclid's parallel postulate, has an advantage over the others:
• A dispenses with existential quantifiers;
• B has the fewest variables and atomic sentences;
• C requires but one primitive notion, betweenness. This variant is the usual one given in the literature.
A: $((Bxyw\land xy\equiv yw)\land (Bxuv\land xu\equiv uv)\land (Byuz\land yu\equiv uz))\rightarrow yz\equiv vw.$
Let a line segment join the midpoint of two sides of a given triangle. That line segment will be half as long as the third side. This is equivalent to the interior angles of any triangle summing to two right angles.
B: $Bxyz\lor Byzx\lor Bzxy\lor \exists a\,(xa\equiv ya\land xa\equiv za).$
Given any triangle, there exists a circle that includes all of its vertices.
C: $(Bxuv\land Byuz\land x\neq u)\rightarrow \exists a\,\exists b\,(Bxya\land Bxzb\land Bavb).$
Given any angle and any point v in its interior, there exists a line segment including v, with an endpoint on each side of the angle.
Five Segment
${(x\neq y\land Bxyz\land Bx'y'z'\land xy\equiv x'y'\land yz\equiv y'z'\land xu\equiv x'u'\land yu\equiv y'u')}\rightarrow zu\equiv z'u'.$
Begin with two triangles, xuz and x'u'z'. Draw the line segments yu and y'u', connecting a vertex of each triangle to a point on the side opposite to the vertex. The result is two divided triangles, each made up of five segments. If four segments of one triangle are each congruent to a segment in the other triangle, then the fifth segments in both triangles must be congruent.
This is equivalent to the side-angle-side rule for determining that two triangles are congruent; if the angles uxz and u'x'z' are congruent (there exist congruent triangles xuz and x'u'z'), and the two pairs of incident sides are congruent (xu ≡ x'u' and xz ≡ x'z'), then the remaining pair of sides is also congruent (uz ≡ u'z').
Segment Construction
$\exists z\,[Bxyz\land yz\equiv ab].$
For any point y, it is possible to draw in any direction (determined by x) a line congruent to any segment ab.
Discussion
Starting from two primitive relations whose fields are a dense universe of points, Tarski built a geometry of line segments. According to Tarski and Givant (1999: 192-93), none of the above axioms are fundamentally new. The first four axioms establish some elementary properties of the two primitive relations. For instance, Reflexivity and Transitivity of Congruence establish that congruence is an equivalence relation over line segments. The Identity of Congruence and of Betweenness govern the trivial case when those relations are applied to nondistinct points. The theorem xy≡zz ↔ x=y ↔ Bxyx extends these Identity axioms.
A number of other properties of Betweenness are derivable as theorems including:
• Reflexivity: Bxxy ;
• Symmetry: Bxyz → Bzyx ;
• Transitivity: (Bxyw ∧ Byzw) → Bxyz ;
• Connectivity: (Bxyw ∧ Bxzw) → (Bxyz ∨ Bxzy).
The last two properties totally order the points making up a line segment.
Upper and Lower Dimension together require that any model of these axioms have a specific finite dimensionality. Suitable changes in these axioms yield axiom sets for Euclidean geometry for dimensions 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8(1), 8(n), 9(0), 9(1), 9(n) ). Note that solid geometry requires no new axioms, unlike the case with Hilbert's axioms. Moreover, Lower Dimension for n dimensions is simply the negation of Upper Dimension for n - 1 dimensions.
When the number of dimensions is greater than 1, Betweenness can be defined in terms of congruence (Tarski and Givant, 1999). First define the relation "≤" (where $ab\leq cd$ is interpreted "the length of line segment $ab$ is less than or equal to the length of line segment $cd$"):
$xy\leq zu\leftrightarrow \forall v(zv\equiv uv\rightarrow \exists w(xw\equiv yw\land yw\equiv uv)).$
In the case of two dimensions, the intuition is as follows: For any line segment xy, consider the possible range of lengths of xv, where v is any point on the perpendicular bisector of xy. It is apparent that while there is no upper bound to the length of xv, there is a lower bound, which occurs when v is the midpoint of xy. So if xy is shorter than or equal to zu, then the range of possible lengths of xv will be a superset of the range of possible lengths of zw, where w is any point on the perpendicular bisector of zu.
Betweenness can then be defined by using the intuition that the shortest distance between any two points is a straight line:
$Bxyz\leftrightarrow \forall u((ux\leq xy\land uz\leq zy)\rightarrow u=y).$
The Axiom Schema of Continuity assures that the ordering of points on a line is complete (with respect to first-order definable properties). The Axioms of Pasch and Euclid are well known. Remarkably, Euclidean geometry requires just the following further axioms:
• Segment Construction. This axiom makes measurement and the Cartesian coordinate system possible—simply assign the real number of 1 to some arbitrary non-empty line segment;
Let wff stand for a well-formed formula (or syntactically correct formula) of elementary geometry. Tarski and Givant (1999: 175) proved that elementary geometry is:
• Consistent: There is no wff such that it and its negation are both theorems;
• Complete: Every sentence or its negation is a theorem provable from the axioms;
• Decidable: There exists an algorithm that assigns a truth value to every sentence. This follows from Tarski's:
• Decision procedure for the real closed field, which he found by quantifier elimination (the Tarski–Seidenberg theorem);
• Axioms admitting of a (multi-dimensional) faithful interpretation as a real closed field.
Gupta (1965) proved the above axioms independent, Pasch and Reflexivity of Congruence excepted.
Negating the Axiom of Euclid yields hyperbolic geometry, while eliminating it outright yields absolute geometry. Full (as opposed to elementary) Euclidean geometry requires giving up a first order axiomatization: replace φ(x) and ψ(y) in the axiom schema of Continuity with x ∈ A and y ∈ B, where A and B are universally quantified variables ranging over sets of points.
Comparison with Hilbert
Hilbert's axioms for plane geometry number 16, and include Transitivity of Congruence and a variant of the Axiom of Pasch. The only notion from intuitive geometry invoked in the remarks to Tarski's axioms is triangle. (Versions B and C of the Axiom of Euclid refer to "circle" and "angle," respectively.) Hilbert's axioms also require "ray," "angle," and the notion of a triangle "including" an angle. In addition to betweenness and congruence, Hilbert's axioms require a primitive binary relation "on," linking a point and a line. The Axiom schema of Continuity plays a role similar to Hilbert's two axioms of Continuity. This schema is indispensable; Euclidean geometry in Tarski's (or equivalent) language cannot be finitely axiomatized as a first-order theory. Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic.
The first four groups of axioms of Hilbert's axioms for plane geometry are bi-interpretable with Tarski's axioms minus continuity.
See also
• Euclidean geometry
• Euclidean space
Notes
1. Tarski and Givant, 1999, page 177
References
• Franzén, Torkel (2005), Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, A K Peters, ISBN 1-56881-238-8
• Givant, Steven (1 December 1999). "Unifying threads in Alfred Tarski's Work". The Mathematical Intelligencer. 21 (1): 47–58. doi:10.1007/BF03024832. ISSN 1866-7414.
• Gupta, H. N. (1965). Contributions to the Axiomatic Foundations of Geometry (Ph.D. thesis). University of California-Berkeley.
• Tarski, Alfred (1959), "What is elementary geometry?", in Leon Henkin, Patrick Suppes and Alfred Tarski (ed.), The axiomatic method. With special reference to geometry and physics. Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, pp. 16–29, MR 0106185.
• Available as a 2007 reprint, Brouwer Press, ISBN 1-4437-2812-8
• Tarski, Alfred; Givant, Steven (1999), "Tarski's system of geometry", The Bulletin of Symbolic Logic, 5 (2): 175–214, CiteSeerX 10.1.1.27.9012, doi:10.2307/421089, ISSN 1079-8986, JSTOR 421089, MR 1791303, S2CID 18551419
• Schwabhäuser, W.; Szmielew, W.; Tarski, Alfred (1983). Metamathematische Methoden in der Geometrie. Springer-Verlag.
• Szczerba, L. W. (1986). "Tarski and Geometry". Journal of Symbolic Logic. 51 (4): 907–12. doi:10.2307/2273904. JSTOR 2273904. S2CID 35275962.
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| Wikipedia |
Tarski's exponential function problem
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable.
The problem
The ordered real field R is a structure over the language of ordered rings Lor = (+,·,−,<,0,1), with the usual interpretation given to each symbol. It was proved by Tarski that the theory of the real field, Th(R), is decidable. That is, given any Lor-sentence φ there is an effective procedure for determining whether
$\mathbb {R} \models \varphi .$
He then asked whether this was still the case if one added a unary function exp to the language that was interpreted as the exponential function on R, to get the structure Rexp.
Conditional and equivalent results
The problem can be reduced to finding an effective procedure for determining whether any given exponential polynomial in n variables and with coefficients in Z has a solution in Rn. Macintyre & Wilkie (1995) showed that Schanuel's conjecture implies such a procedure exists, and hence gave a conditional solution to Tarski's problem. Schanuel's conjecture deals with all complex numbers so would be expected to be a stronger result than the decidability of Rexp, and indeed, Macintyre and Wilkie proved that only a real version of Schanuel's conjecture is required to imply the decidability of this theory.
Even the real version of Schanuel's conjecture is not a necessary condition for the decidability of the theory. In their paper, Macintyre and Wilkie showed that an equivalent result to the decidability of Th(Rexp) is what they dubbed the Weak Schanuel's Conjecture. This conjecture states that there is an effective procedure that, given n ≥ 1 and exponential polynomials in n variables with integer coefficients f1,..., fn, g, produces an integer η ≥ 1 that depends on n, f1,..., fn, g, and such that if α ∈ Rn is a non-singular solution of the system
$f_{1}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=\ldots =f_{n}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=0$
then either g(α) = 0 or |g(α)| > η−1.
Workaround
Recently there are attempts at handling the theory of the real numbers with functions such as the exponential function or sine by relaxing decidability to the weaker notion of quasi-decidability. A theory is quasi-decidable if there is a procedure that decides satisfiability but that may run forever for inputs that are not robust in a certain, well-defined sense. Such a procedure exists for systems of n equations in n variables (Franek, Ratschan & Zgliczynski 2011).
References
• Kuhlmann, S. (2001) [1994], "Model theory of the real exponential function", Encyclopedia of Mathematics, EMS Press
• Macintyre, A.J.; Wilkie, A.J. (1995), "On the decidability of the real exponential field", in Odifreddi, P.G. (ed.), Kreisel 70th Birthday Volume, CLSI
• Franek, Peter; Ratschan, Stefan; Zgliczynski, Piotr (2011), "Satisfiability of Systems of Equations of Real Analytic Functions Is Quasi-decidable", Mathematical Foundations of Computer Science 2011, LNCS, vol. 6907, Springer, pp. 315–326, doi:10.1007/978-3-642-22993-0_30, ISBN 978-3-642-22992-3
| Wikipedia |
Tarski's high school algebra problem
In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist.
Statement of the problem
Tarski considered the following eleven axioms about addition ('+'), multiplication ('·'), and exponentiation to be standard axioms taught in high school:
1. x + y = y + x
2. (x + y) + z = x + (y + z)
3. x · 1 = x
4. x · y = y · x
5. (x · y) · z = x · (y · z)
6. x · (y + z) = x · y + x ·z
7. 1x = 1
8. x1 = x
9. xy + z = xy · xz
10. (x · y)z = xz · yz
11. (xy)z = xy · z
These eleven axioms, sometimes called the high school identities,[1] are related to the axioms of a bicartesian closed category or an exponential ring.[2] Tarski's problem then becomes: are there identities involving only addition, multiplication, and exponentiation, that are true for all positive integers, but that cannot be proved using only the axioms 1–11?
Example of a provable identity
Since the axioms seem to list all the basic facts about the operations in question, it is not immediately obvious that there should be anything provably true one can state using only the three operations, but cannot prove with the axioms. However, proving seemingly innocuous statements can require long proofs using only the above eleven axioms. Consider the following proof that $(x+1)^{2}=x^{2}+2\cdot x+1:$
${\begin{aligned}(x+1)^{2}&=(x+1)^{1+1}\\&=(x+1)^{1}\cdot (x+1)^{1}&&{\text{by 9.}}\\&=(x+1)\cdot (x+1)&&{\text{by two applications of 8.}}\\&=(x+1)\cdot x+(x+1)\cdot 1&&{\text{by 6.}}\\&=x\cdot (x+1)+(x+1)&&{\text{by 4. and 3.}}\\&=(x\cdot x+x\cdot 1)+(x\cdot 1+1)&&{\text{by 6. and 3.}}\\&=x\cdot x+(x\cdot 1+x\cdot 1)+1&&{\text{by two applications of 2.}}\\&=x^{1}\cdot x^{1}+x\cdot (1+1)+1&&{\text{by 6. and two applications of 8.}}\\&=x^{1+1}+x\cdot 2+1&&{\text{by 9.}}\\&=x^{2}+2\cdot x+1&&{\text{by 4.}}\end{aligned}}$
Strictly we should not write sums of more than two terms without brackets, and therefore a completely formal proof would prove the identity $(x+1)^{2}=\left(x^{2}+2\cdot x\right)+1$ (or $(x+1)^{2}=x^{2}+(2\cdot x+1)$) and would have an extra set of brackets in each line from $x\cdot x+(x\cdot 1+x\cdot 1)+1$ onwards.
The length of proofs is not an issue; proofs of similar identities to that above for things like $(x+y)^{100}$ would take a lot of lines, but would really involve little more than the above proof.
History of the problem
The list of eleven axioms can be found explicitly written down in the works of Richard Dedekind,[3] although they were obviously known and used by mathematicians long before then. Dedekind was the first, though, who seemed to be asking if these axioms were somehow sufficient to tell us everything we could want to know about the integers. The question was put on a firm footing as a problem in logic and model theory sometime in the 1960s by Alfred Tarski,[1][4] and by the 1980s it had become known as Tarski's high school algebra problem.
Solution
In 1980 Alex Wilkie proved that not every identity in question can be proved using the axioms above.[5] He did this by explicitly finding such an identity. By introducing new function symbols corresponding to polynomials that map positive numbers to positive numbers he proved this identity, and showed that these functions together with the eleven axioms above were both sufficient and necessary to prove it. The identity in question is
${\begin{aligned}&\left((1+x)^{y}+(1+x+x^{2})^{y}\right)^{x}\cdot \left((1+x^{3})^{x}+(1+x^{2}+x^{4})^{x}\right)^{y}\\={}&\left((1+x)^{x}+(1+x+x^{2})^{x}\right)^{y}\cdot \left((1+x^{3})^{y}+(1+x^{2}+x^{4})^{y}\right)^{x}.\end{aligned}}$
This identity is usually denoted $W(x,y)$ and is true for all positive integers $x$ and $y,$ as can be seen by factoring $(1-x+x^{2})^{xy}$ out of the second factor on each side; yet it cannot be proved true using the eleven high school axioms.
Intuitively, the identity cannot be proved because the high school axioms can't be used to discuss the polynomial $1-x+x^{2}.$ Reasoning about that polynomial and the subterm $-x$ requires a concept of negation or subtraction, and these are not present in the high school axioms. Lacking this, it is then impossible to use the axioms to manipulate the polynomial and prove true properties about it. Wilkie's results from his paper show, in more formal language, that the "only gap" in the high school axioms is the inability to manipulate polynomials with negative coefficients.
R. Gurevič showed in 1988 that there is no finite axiomatization for the valid equations for the positive natural numbers with 1, addition, multiplication, and exponentiation.[6][7]
Generalisations
Wilkie proved that there are statements about the positive integers that cannot be proved using the eleven axioms above and showed what extra information is needed before such statements can be proved. Using Nevanlinna theory it has also been proved that if one restricts the kinds of exponential one takes then the above eleven axioms are sufficient to prove every true statement.[8]
Another problem stemming from Wilkie's result, which remains open, is that which asks what the smallest algebra is for which $W(x,y)$ is not true but the eleven axioms above are. In 1985 an algebra with 59 elements was found that satisfied the axioms but for which $W(x,y)$ was false.[4] Smaller such algebras have since been found, and it is now known that the smallest such one must have either 11 or 12 elements.[9]
See also
• Elementary function – Mathematical function
• Elementary function arithmetic – System of arithmetic in proof theory
• Liouville's theorem (differential algebra) – Says when antiderivatives of elementary functions can expressed as elementary functions
• Nonelementary integral – Integrals not expressible in closed-form from elementary functions
• Richardson's theorem – Undecidability of equality of real numbers
Notes
1. Stanley Burris, Simon Lee, Tarski's high school identities, American Mathematical Monthly, 100, (1993), no.3, pp. 231–236.
2. Strictly speaking an exponential ring has an exponential function E that takes each element x to something that acts like ax for a fixed number a. But a slight generalisation gives the axioms for the binary operation of exponentiation. The lack of axioms about additive inverses means the axioms would have described an exponential commutative semiring, except there are no axioms about additive identities in Tarski's axioms either. However, some authors use the term rig to mean a semiring with additive identities, and reserve the term semiring for the general case not necessarily having additive identities. To those authors, the axioms do describe an exponential commutative semiring.
3. Richard Dedekind, Was sind und was sollen die Zahlen?, 8te unveränderte Aufl. Friedr. Vieweg & Sohn, Braunschweig (1960). English translation: What are numbers and what should they be? Revised, edited, and translated from the German by H. A. Pogorzelski, W. Ryan, and W. Snyder, RIM Monographs in Mathematics, Research Institute for Mathematics, (1995).
4. R. Gurevič, Equational theory of positive numbers with exponentiation, Proc. Amer. Math. Soc. 94 no.1, (1985), pp.135–141.
5. A.J. Wilkie, On exponentiation – a solution to Tarski's high school algebra problem, Connections between model theory and algebraic and analytic geometry, Quad. Mat., 6, Dept. Math., Seconda Univ. Napoli, Caserta, (2000), pp. 107–129.
6. R. Gurevič, Equational theory of positive numbers with exponentiation is not finitely axiomatizable, Annals of Pure and Applied Logic, 49:1–30, 1990.
7. Fiore, Cosmo, and Balat. Remarks on Isomorphisms in Typed Lambda Calculi with Empty and Sum Types
8. C. Ward Henson, Lee A. Rubel, Some applications of Nevanlinna theory to mathematical logic: Identities of exponential functions, Transactions of the American Mathematical Society, vol.282 1, (1984), pp. 1–32.
9. Jian Zhang, Computer search for counterexamples to Wilkie's identity, Automated Deduction – CADE-20, Springer (2005), pp. 441–451, doi:10.1007/11532231_32.
References
• Stanley N. Burris, Karen A. Yeats, The saga of the high school identities, Algebra Universalis 52 no.2–3, (2004), pp. 325–342, MR2161657.
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| Wikipedia |
Tarski's undefinability theorem
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic".[1]
The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.
History
In 1931, Kurt Gödel published the incompleteness theorems, which he proved in part by showing how to represent the syntax of formal logic within first-order arithmetic. Each expression of the formal language of arithmetic is assigned a distinct number. This procedure is known variously as Gödel numbering, coding and, more generally, as arithmetization. In particular, various sets of expressions are coded as sets of numbers. For various syntactic properties (such as being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences.
The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language (e.g. a predicate is definable in Zermelo-Fraenkel set theory for whether formulae in the language of Peano arithmetic are true in the standard model of arithmetic[2]) must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.
The undefinability theorem is conventionally attributed to Alfred Tarski. Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1933 publication of Tarski's work (Murawski 1998). While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to John von Neumann. Tarski had obtained almost all results of his 1933 monograph "The Concept of Truth in the Languages of the Deductive Sciences" between 1929 and 1931, and spoke about them to Polish audiences. However, as he emphasized in the paper, the undefinability theorem was the only result he did not obtain earlier. According to the footnote to the undefinability theorem (Twierdzenie I) of the 1933 monograph, the theorem and the sketch of the proof were added to the monograph only after the manuscript had been sent to the printer in 1931. Tarski reports there that, when he presented the content of his monograph to the Warsaw Academy of Science on March 21, 1931, he expressed at this place only some conjectures, based partly on his own investigations and partly on Gödel's short report on the incompleteness theorems "Einige metamathematische Resultate über Entscheidungsdefinitheit und Widerspruchsfreiheit" [Some metamathematical results on the definiteness of decision and consistency], Austrian Academy of Sciences, Vienna, 1930.
Statement
We will first state a simplified version of Tarski's theorem, then state and prove in the next section the theorem Tarski proved in 1933.
Let $L$ be the language of first-order arithmetic. This is the theory of the natural numbers, including their addition and multiplication, axiomatized by the first-order Peano axioms. This is a "first-order" theory: the quantifiers extend over natural numbers, but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci sequence.
Let $N$ be the standard structure for $L,$ i.e. $N$ consists of the ordinary set of natural numbers and their addition and multiplication. Each sentence in $L$ can be interpreted in $N$ and then becomes either true or false. Thus $(L,N)$ is the "interpreted first-order language of arithmetic".
Each formula $\varphi $ in $L$ has a Gödel number $g(\varphi ).$ This is a natural number that "encodes" $\varphi .$ In that way, the language $L$ can talk about formulas in $L,$ not just about numbers. Let $T$ denote the set of $L$-sentences true in $N,$ and $T^{*}$ the set of Gödel numbers of the sentences in $T.$ The following theorem answers the question: Can $T^{*}$ be defined by a formula of first-order arithmetic?
Tarski's undefinability theorem: There is no $L$-formula $True(n)$ that defines $T^{*}.$ That is, there is no $L$-formula $True(n)$ such that for every $L$-sentence $A,$ $True(g(A))\iff A$ holds in $N.$
Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic. This implies a major limitation on the scope of "self-representation". It is possible to define a formula $True(n)$ whose extension is $T^{*},$ but only by drawing on a metalanguage whose expressive power goes beyond that of $L.$ For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic. However, this formula would only be able to define a truth predicate for formulas in the original language $L.$ To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on.
To prove the theorem, we proceed by contradiction and assume that an $L$-formula $True(n)$ exists which is true for the natural number $n$ in $N$ if and only if $n$ is the Gödel number of a sentence in $L$ that is true in $N.$ We could then use $True(n)$ to define a new $L$-formula $S(m)$ which is true for the natural number $m$ if and only if $m$ is the Gödel number of a formula $\varphi (x)$ (with a free variable $x$) such that $\varphi (m)$ is false when interpreted in $N$ (i.e. the formula $\varphi (x),$ when applied to its own Gödel number, yields a false statement). If we now consider the Gödel number $g$ of the formula $S(m),$ and ask whether the sentence $S(g)$ is true in $N,$ we obtain a contradiction. (This is known as a diagonal argument.)
The theorem is a corollary of Post's theorem about the arithmetical hierarchy, proved some years after Tarski (1933). A semantic proof of Tarski's theorem from Post's theorem is obtained by reductio ad absurdum as follows. Assuming $T^{*}$ is arithmetically definable, there is a natural number $n$ such that $T^{*}$ is definable by a formula at level $\Sigma _{n}^{0}$ of the arithmetical hierarchy. However, $T^{*}$ is $\Sigma _{k}^{0}$-hard for all $k.$ Thus the arithmetical hierarchy collapses at level $n,$ contradicting Post's theorem.
General form
Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting theorem applies to any formal language with negation, and with sufficient capability for self-reference that the diagonal lemma holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more general formal systems, such as ZFC.
Tarski's undefinability theorem (general form): Let $(L,N)$ be any interpreted formal language which includes negation and has a Gödel numbering $g(\varphi )$ satisfying the diagonal lemma, i.e. for every $L$-formula $B(x)$ (with one free variable $x$) there is a sentence $A$ such that $A\iff B(g(A))$ holds in $N.$ Then there is no $L$-formula $True(n)$ with the following property: for every $L$-sentence $A,$ $True(g(A))\iff A$ is true in $N.$
The proof of Tarski's undefinability theorem in this form is again by reductio ad absurdum. Suppose that an $L$-formula $True(n)$ as above existed, i.e., if $A$ is a sentence of arithmetic, then $True(g(A))$ holds in $N$ if and only if $A$ holds in $N.$ Hence for all $A,$ the formula $True(g(A))\iff A$ holds in $N.$ But the diagonal lemma yields a counterexample to this equivalence, by giving a "liar" formula $S$ such that $S\iff \lnot True(g(S))$ holds in $N.$ This is a contradiction. QED.
Discussion
The formal machinery of the proof given above is wholly elementary except for the diagonalization which the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for example, it does not invoke recursive functions in any way. The proof does assume that every $L$-formula has a Gödel number, but the specifics of a coding method are not required. Hence Tarski's theorem is much easier to motivate and prove than the more celebrated theorems of Gödel about the metamathematical properties of first-order arithmetic.
Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., Lucas 1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough self-reference for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.
An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates and function symbols defining all the semantic concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula $A$ to its truth value $||A||,$ and the "semantic denotation function" mapping a term $t$ to the object it denotes. Tarski's theorem then generalizes as follows: No sufficiently powerful language is strongly-semantically-self-representational.
The undefinability theorem does not prevent truth in one theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order Peano arithmetic that are true in $N$ is definable by a formula in second order arithmetic. Similarly, the set of true formulas of the standard model of second order arithmetic (or $n$-th order arithmetic for any $n$) can be defined by a formula in first-order ZFC.
See also
• Chaitin's incompleteness theorem – Measure of algorithmic complexityPages displaying short descriptions of redirect targets
• Gödel's completeness theorem – Fundamental theorem in mathematical logic
• Gödel's incompleteness theorems – Limitative results in mathematical logic
References
1. Cezary Cieśliński, "How Tarski Defined the Undefinable," European Review 23.1 (2015): 139–149.
2. Joel David Hamkins; Yang, Ruizhi (2013). "Satisfaction is not absolute". arXiv:1312.0670 [math.LO].
Primary sources
• Tarski, A. (1933). Pojęcie Prawdy w Językach Nauk Dedukcyjnych (in Polish). Nakładem Towarzystwa Naukowego Warszawskiego.
• Tarski, A. (1936). "Der Wahrheitsbegriff in den formalisierten Sprachen" (PDF). Studia Philosophica (in German). 1: 261–405. Archived from the original (PDF) on 9 January 2014. Retrieved 26 June 2013.
• Tarski, A. (1983). "The Concept of Truth in Formalized Languages" (PDF). In Corcoran, J. (ed.). Logic, Semantics, Metamathematics. Translated by J. H. Woodger. Hackett. English translation of Tarski's 1936 article.
Further reading
• Bell, J. L.; Machover, M. (1977). A Course in Mathematical Logic. North-Holland.
• Boolos, G.; Burgess, J.; Jeffrey, R. (2002). Computability and Logic (4th ed.). Cambridge University Press.
• Lucas, J. R. (1961). "Mind, Machines, and Gödel". Philosophy. 36 (137): 112–27. doi:10.1017/S0031819100057983. S2CID 55408480. Archived from the original on 2007-08-19.
• Murawski, R. (1998). "Undefinability of truth. The problem of the priority: Tarski vs. Gödel". History and Philosophy of Logic. 19 (3): 153–160. doi:10.1080/01445349808837306. Archived from the original on 2011-06-08.
• Smullyan, R. (1991). Godel's Incompleteness Theorems. Oxford Univ. Press.
• Smullyan, R. (2001). "Gödel's Incompleteness Theorems". In Goble, L. (ed.). The Blackwell Guide to Philosophical Logic. Blackwell. pp. 72–89.
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| Wikipedia |
Tarski's theorem about choice
In mathematics, Tarski's theorem, proved by Alfred Tarski (1924), states that in ZF the theorem "For every infinite set $A$, there is a bijective map between the sets $A$ and $A\times A$" implies the axiom of choice. The opposite direction was already known, thus the theorem and axiom of choice are equivalent.
Tarski told Jan Mycielski (2006) that when he tried to publish the theorem in Comptes Rendus de l'Académie des Sciences de Paris, Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest.
Proof
The goal is to prove that the axiom of choice is implied by the statement "for every infinite set $A:$ $|A|=|A\times A|$". It is known that the well-ordering theorem is equivalent to the axiom of choice; thus it is enough to show that the statement implies that for every set $B$ there exists a well-order.
Since the collection of all ordinals such that there exists a surjective function from $B$ to the ordinal is a set, there exists an infinite ordinal, $\beta ,$ such that there is no surjective function from $B$ to $\beta .$ We assume without loss of generality that the sets $B$ and $\beta $ are disjoint. By the initial assumption, $|B\cup \beta |=|(B\cup \beta )\times (B\cup \beta )|,$ thus there exists a bijection $f:B\cup \beta \to (B\cup \beta )\times (B\cup \beta ).$
For every $x\in B,$ it is impossible that $\beta \times \{x\}\subseteq f[B],$ because otherwise we could define a surjective function from $B$ to $\beta .$ Therefore, there exists at least one ordinal $\gamma \in \beta $ such that $f(\gamma )\in \beta \times \{x\},$ so the set $S_{x}=\{\gamma :f(\gamma )\in \beta \times \{x\}\}$ is not empty.
We can define a new function: $g(x)=\min S_{x}.$ This function is well defined since $S_{x}$ is a non-empty set of ordinals, and so has a minimum. For every $x,y\in B,x\neq y$ the sets $S_{x}$ and $S_{y}$ are disjoint. Therefore, we can define a well order on $B,$ for every $x,y\in B$ we define $x\leq y\iff g(x)\leq g(y),$ since the image of $g,$ that is, $g[B]$ is a set of ordinals and therefore well ordered.
References
• Rubin, Herman; Rubin, Jean E. (1985), Equivalents of the Axiom of Choice II, North Holland/Elsevier, ISBN 0-444-87708-8
• Mycielski, Jan (2006), "A system of axioms of set theory for the rationalists" (PDF), Notices of the American Mathematical Society, 53 (2): 209
• Tarski, A. (1924), "Sur quelques theorems qui equivalent a l'axiome du choix", Fundamenta Mathematicae, 5: 147–154, doi:10.4064/fm-5-1-147-154
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| Wikipedia |
Tarski–Grothendieck set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a Grothendieck universe it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than ZFC. For example, adding this axiom supports category theory.
The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.
Axioms
Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds “Tarski's axiom”. We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's basic objects and processes are fully formal; they are described informally below. First, let us assume that:
• Given any set $A$, the singleton $\{A\}$ exists.
• Given any two sets, their unordered and ordered pairs exist.
• Given any set of sets, its union exists.
TG includes the following axioms, which are conventional because they are also part of ZFC:
• Set axiom: Quantified variables range over sets alone; everything is a set (the same ontology as ZFC).
• Axiom of extensionality: Two sets are identical if they have the same members.
• Axiom of regularity: No set is a member of itself, and circular chains of membership are impossible.
• Axiom schema of replacement: Let the domain of the class function $F$ be the set $A$. Then the range of $F$ (the values of $F(x)$ for all members $x$ of $A$) is also a set.
It is Tarski's axiom that distinguishes TG from other axiomatic set theories. Tarski's axiom also implies the axioms of infinity, choice,[1][2] and power set.[3][4] It also implies the existence of inaccessible cardinals, thanks to which the ontology of TG is much richer than that of conventional set theories such as ZFC.
• Tarski's axiom (adapted from Tarski 1939[5]). For every set $x$, there exists a set $y$ whose members include:
- $x$ itself;
- every element of every member of $y$;
- every subset of every member of $y$;
- the power set of every member of $y$;
- every subset of $y$ of cardinality less than that of $y$.
More formally:
$\forall x\exists y[x\in y\land \forall z\in y(z\subseteq y\land {\mathcal {P}}(z)\subseteq y\land {\mathcal {P}}(z)\in y)\land \forall z\in {\mathcal {P}}(y)(\neg (z\approx y)\to z\in y)]$
where “${\mathcal {P}}(x)$” denotes the power class of x and “$\approx $” denotes equinumerosity. What Tarski's axiom states (in the vernacular) is that for each set $x$ there is a Grothendieck universe it belongs to.
That $y$ looks much like a “universal set” for $x$ – it not only has as members the powerset of $x$, and all subsets of $x$, it also has the powerset of that powerset and so on – its members are closed under the operations of taking powerset or taking a subset. It's like a “universal set” except that of course it is not a member of itself and is not a set of all sets. That's the guaranteed Grothendieck universe it belongs to. And then any such $y$ is itself a member of an even larger “almost universal set” and so on. It's one of the strong cardinality axioms guaranteeing vastly more sets than one normally assumes to exist.
Implementation in the Mizar system
The Mizar language, underlying the implementation of TG and providing its logical syntax, is typed and the types are assumed to be non-empty. Hence, the theory is implicitly taken to be non-empty. The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors.
The system includes equality, the membership predicate and the following standard definitions:
• Singleton: A set with one member;
• Unordered pair: A set with two distinct members. $\{a,b\}=\{b,a\}$;
• Ordered pair: The set $\{\{a,b\},\{a\}\}=(a,b)\neq (b,a)$;
• Subset: A set all of whose members are members of another given set;
• The union of a set of sets $Y$: The set of all members of any member of $Y$.
Implementation in Metamath
The Metamath system supports arbitrary higher-order logics, but it is typically used with the "set.mm" definitions of axioms. The ax-groth axiom adds Tarski's axiom, which in Metamath is defined as follows:
⊢ ∃y(x ∈ y ∧ ∀z ∈ y (∀w(w ⊆ z → w ∈ y) ∧ ∃w ∈ y ∀v(v ⊆ z → v ∈ w)) ∧ ∀z(z ⊆ y → (z ≈ y ∨ z ∈ y)))
See also
• Axiom of limitation of size
Notes
1. Tarski (1938)
2. "WELLORD2: Zermelo Theorem and Axiom of Choice. The correspondence of well ordering relations and ordinal numbers".
3. Robert Solovay, Re: AC and strongly inaccessible cardinals.
4. Metamath grothpw.
5. Tarski (1939)
References
• Andreas Blass, I.M. Dimitriou, and Benedikt Löwe (2007) "Inaccessible Cardinals without the Axiom of Choice," Fundamenta Mathematicae 194: 179-89.
• Bourbaki, Nicolas (1972). "Univers". In Michael Artin; Alexandre Grothendieck; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 1 (Lecture Notes in Mathematics 269) (in French). Berlin; New York: Springer-Verlag. pp. 185–217. Archived from the original on 2003-08-26.
• Patrick Suppes (1960) Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972.
• Tarski, Alfred (1938). "Über unerreichbare Kardinalzahlen" (PDF). Fundamenta Mathematicae. 30: 68–89. doi:10.4064/fm-30-1-68-89.
• Tarski, Alfred (1939). "On the well-ordered subsets of any set" (PDF). Fundamenta Mathematicae. 32: 176–183. doi:10.4064/fm-32-1-176-783.
External links
• Trybulec, Andrzej, 1989, "Tarski–Grothendieck Set Theory", Journal of Formalized Mathematics.
• Metamath: "Proof Explorer Home Page." Scroll down to "Grothendieck's Axiom."
• PlanetMath: "Tarski's Axiom"
Set theory
Overview
• Set (mathematics)
Axioms
• Adjunction
• Choice
• countable
• dependent
• global
• Constructibility (V=L)
• Determinacy
• Extensionality
• Infinity
• Limitation of size
• Pairing
• Power set
• Regularity
• Union
• Martin's axiom
• Axiom schema
• replacement
• specification
Operations
• Cartesian product
• Complement (i.e. set difference)
• De Morgan's laws
• Disjoint union
• Identities
• Intersection
• Power set
• Symmetric difference
• Union
• Concepts
• Methods
• Almost
• Cardinality
• Cardinal number (large)
• Class
• Constructible universe
• Continuum hypothesis
• Diagonal argument
• Element
• ordered pair
• tuple
• Family
• Forcing
• One-to-one correspondence
• Ordinal number
• Set-builder notation
• Transfinite induction
• Venn diagram
Set types
• Amorphous
• Countable
• Empty
• Finite (hereditarily)
• Filter
• base
• subbase
• Ultrafilter
• Fuzzy
• Infinite (Dedekind-infinite)
• Recursive
• Singleton
• Subset · Superset
• Transitive
• Uncountable
• Universal
Theories
• Alternative
• Axiomatic
• Naive
• Cantor's theorem
• Zermelo
• General
• Principia Mathematica
• New Foundations
• Zermelo–Fraenkel
• von Neumann–Bernays–Gödel
• Morse–Kelley
• Kripke–Platek
• Tarski–Grothendieck
• Paradoxes
• Problems
• Russell's paradox
• Suslin's problem
• Burali-Forti paradox
Set theorists
• Paul Bernays
• Georg Cantor
• Paul Cohen
• Richard Dedekind
• Abraham Fraenkel
• Kurt Gödel
• Thomas Jech
• John von Neumann
• Willard Quine
• Bertrand Russell
• Thoralf Skolem
• Ernst Zermelo
Mathematical logic
General
• Axiom
• list
• Cardinality
• First-order logic
• Formal proof
• Formal semantics
• Foundations of mathematics
• Information theory
• Lemma
• Logical consequence
• Model
• Theorem
• Theory
• Type theory
Theorems (list)
& Paradoxes
• Gödel's completeness and incompleteness theorems
• Tarski's undefinability
• Banach–Tarski paradox
• Cantor's theorem, paradox and diagonal argument
• Compactness
• Halting problem
• Lindström's
• Löwenheim–Skolem
• Russell's paradox
Logics
Traditional
• Classical logic
• Logical truth
• Tautology
• Proposition
• Inference
• Logical equivalence
• Consistency
• Equiconsistency
• Argument
• Soundness
• Validity
• Syllogism
• Square of opposition
• Venn diagram
Propositional
• Boolean algebra
• Boolean functions
• Logical connectives
• Propositional calculus
• Propositional formula
• Truth tables
• Many-valued logic
• 3
• Finite
• ∞
Predicate
• First-order
• list
• Second-order
• Monadic
• Higher-order
• Free
• Quantifiers
• Predicate
• Monadic predicate calculus
Set theory
• Set
• Hereditary
• Class
• (Ur-)Element
• Ordinal number
• Extensionality
• Forcing
• Relation
• Equivalence
• Partition
• Set operations:
• Intersection
• Union
• Complement
• Cartesian product
• Power set
• Identities
Types of Sets
• Countable
• Uncountable
• Empty
• Inhabited
• Singleton
• Finite
• Infinite
• Transitive
• Ultrafilter
• Recursive
• Fuzzy
• Universal
• Universe
• Constructible
• Grothendieck
• Von Neumann
Maps & Cardinality
• Function/Map
• Domain
• Codomain
• Image
• In/Sur/Bi-jection
• Schröder–Bernstein theorem
• Isomorphism
• Gödel numbering
• Enumeration
• Large cardinal
• Inaccessible
• Aleph number
• Operation
• Binary
Set theories
• Zermelo–Fraenkel
• Axiom of choice
• Continuum hypothesis
• General
• Kripke–Platek
• Morse–Kelley
• Naive
• New Foundations
• Tarski–Grothendieck
• Von Neumann–Bernays–Gödel
• Ackermann
• Constructive
Formal systems (list),
Language & Syntax
• Alphabet
• Arity
• Automata
• Axiom schema
• Expression
• Ground
• Extension
• by definition
• Conservative
• Relation
• Formation rule
• Grammar
• Formula
• Atomic
• Closed
• Ground
• Open
• Free/bound variable
• Language
• Metalanguage
• Logical connective
• ¬
• ∨
• ∧
• →
• ↔
• =
• Predicate
• Functional
• Variable
• Propositional variable
• Proof
• Quantifier
• ∃
• !
• ∀
• rank
• Sentence
• Atomic
• Spectrum
• Signature
• String
• Substitution
• Symbol
• Function
• Logical/Constant
• Non-logical
• Variable
• Term
• Theory
• list
Example axiomatic
systems
(list)
• of arithmetic:
• Peano
• second-order
• elementary function
• primitive recursive
• Robinson
• Skolem
• of the real numbers
• Tarski's axiomatization
• of Boolean algebras
• canonical
• minimal axioms
• of geometry:
• Euclidean:
• Elements
• Hilbert's
• Tarski's
• non-Euclidean
• Principia Mathematica
Proof theory
• Formal proof
• Natural deduction
• Logical consequence
• Rule of inference
• Sequent calculus
• Theorem
• Systems
• Axiomatic
• Deductive
• Hilbert
• list
• Complete theory
• Independence (from ZFC)
• Proof of impossibility
• Ordinal analysis
• Reverse mathematics
• Self-verifying theories
Model theory
• Interpretation
• Function
• of models
• Model
• Equivalence
• Finite
• Saturated
• Spectrum
• Submodel
• Non-standard model
• of arithmetic
• Diagram
• Elementary
• Categorical theory
• Model complete theory
• Satisfiability
• Semantics of logic
• Strength
• Theories of truth
• Semantic
• Tarski's
• Kripke's
• T-schema
• Transfer principle
• Truth predicate
• Truth value
• Type
• Ultraproduct
• Validity
Computability theory
• Church encoding
• Church–Turing thesis
• Computably enumerable
• Computable function
• Computable set
• Decision problem
• Decidable
• Undecidable
• P
• NP
• P versus NP problem
• Kolmogorov complexity
• Lambda calculus
• Primitive recursive function
• Recursion
• Recursive set
• Turing machine
• Type theory
Related
• Abstract logic
• Category theory
• Concrete/Abstract Category
• Category of sets
• History of logic
• History of mathematical logic
• timeline
• Logicism
• Mathematical object
• Philosophy of mathematics
• Supertask
Mathematics portal
| Wikipedia |
Tarski Lectures
The Alfred Tarski Lectures are an annual distinction in mathematical logic and series of lectures held at the University of California, Berkeley. Established in tribute to Alfred Tarski on the fifth anniversary of his death, the award has been given every year since 1989.[1][2] Following a 2-year hiatus after the 2020 lecture was not given due to the COVID-19 pandemic, the lectures resumed in 2023.[3]
Tarski Lecturers
The list of past Tarski lecturers is maintained by UC Berkeley.[3][4]
• 1989 Dana S. Scott
• 1990 Willard Van Orman Quine
• 1991 Bjarni Jónsson, Howard Jerome Keisler
• 1992 Donald A. Martin
• 1993 Alex Wilkie
• 1994 Michael O. Rabin
• 1995 Hilary Putnam
• 1996 Ehud Hrushovski
• 1997 Menachem Magidor
• 1998 Angus Macintyre
• 1999 Patrick Suppes
• 2000 Alexander Razborov
• 2001 Ronald Jensen
• 2002 Boris Zilber
• 2003 Ralph McKenzie
• 2004 Alexander S. Kechris
• 2005 Zlil Sela
• 2006 Solomon Feferman
• 2007 Harvey Friedman
• 2008 Yiannis N. Moschovakis
• 2009 Anand Pillay
• 2010 Greg Hjorth
• 2011 Johan van Benthem
• 2012 Per Martin-Löf
• 2013 Jonathan Pila
• 2014 Stevo Todorčević
• 2015 Julia F. Knight
• 2016 William W. Tait
• 2017 Lou van den Dries
• 2018 Hugh Woodin
• 2019 Thomas Hales
• 2020 Zoé Chatzidakis, not delivered due to the COVID-19 pandemic[3]
• 2023 Richard Shore
See also
• Center for New Media Lectures
• Howison Lectures
• Gödel Lecture
• List of mathematics awards
• List of philosophy awards
• List of logicians
External links
• Site of the Alfred Tarski Lectures at UC Berkeley Mathematics
• Site of the Alfred Tarski Lectures at UC Berkeley Logic
• List of past Alfred Tarski Lectures
References
1. Feferman, Anita Burdman; Feferman, Solomon (2004-10-04). Alfred Tarski: Life and Logic. Cambridge University Press. p. 266. ISBN 978-0-521-80240-6.
2. Tarski Committee (December 1988). "Announcement". Synthese. 77 (3): Back matter. JSTOR 20116602 – via JSTOR. On the occasion of the fifth anniversary of the death of Alfred Tarski we announce the inauguration of an annual series of Alfred Tarski Lectures to be delivered at the University of California, Berkeley.
3. "The Tarski Lectures | Department of Mathematics at University of California Berkeley". math.berkeley.edu. Retrieved 2021-11-02.
4. "Group in Logic and the Methodology of Science - Past Tarski Lectures". logic.berkeley.edu. Retrieved 2021-11-02.
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| Wikipedia |
Tarski's axiomatization of the reals
In 1936, Alfred Tarski set out an axiomatization of the real numbers and their arithmetic, consisting of only the 8 axioms shown below and a mere four primitive notions:[1] the set of reals denoted R, a binary total order over R, denoted by the infix operator <, a binary operation of addition over R, denoted by the infix operator +, and the constant 1.
The literature occasionally mentions this axiomatization but never goes into detail, notwithstanding its economy and elegant metamathematical properties. This axiomatization appears little known, possibly because of its second-order nature. Tarski's axiomatization can be seen as a version of the more usual definition of real numbers as the unique Dedekind-complete ordered field; it is however made much more concise by using unorthodox variants of standard algebraic axioms and other subtle tricks (see e.g. axioms 4 and 5, which combine the usual four axioms of abelian groups).
The term "Tarski's axiomatization of real numbers" also refers to the theory of real closed fields, which Tarski showed completely axiomatizes the first-order theory of the structure 〈R, +, ·, <〉.
The axioms
Axioms of order (primitives: R, <):
Axiom 1
If x < y, then not y < x. That is, "<" is an asymmetric relation. This implies that "<" is not a reflexive relationship, i.e. for all x, x < x is false.
Axiom 2
If x < z, there exists a y such that x < y and y < z. In other words, "<" is dense in R.
Axiom 3
"<" is Dedekind-complete. More formally, for all X, Y ⊆ R, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y.
To clarify the above statement somewhat, let X ⊆ R and Y ⊆ R. We now define two common English verbs in a particular way that suits our purpose:
X precedes Y if and only if for every x ∈ X and every y ∈ Y, x < y.
The real number z separates X and Y if and only if for every x ∈ X with x ≠ z and every y ∈ Y with y ≠ z, x < z and z < y.
Axiom 3 can then be stated as:
"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."
The three axioms imply that R is a linear continuum.
Axioms of addition (primitives: R, <, +):
Axiom 4
x + (y + z) = (x + z) + y.
Axiom 5
For all x, y, there exists a z such that x + z = y.
Axiom 6
If x + y < z + w, then x < z or y < w.
Axioms for one (primitives: R, <, +, 1):
Axiom 7
1 ∈ R.
Axiom 8
1 < 1 + 1.
This axiomatization does not give rise to a first-order theory, because the formal statement of axiom 3 includes two universal quantifiers over all possible subsets of R. Tarski proved that these 8 axioms and 4 primitive notions are independent.
How these axioms imply a field
Theorem — $\mathbb {R} $ is a Archimedean ordered abelian group.
Proof
Tarski's axioms imply that $\mathbb {R} $ is a totally ordered[2] abelian group under addition with distinguished element $1$. Since $\mathbb {R} $ is Dedekind-complete and a totally ordered abelian group, $\mathbb {R} $ is Archimedean, because the Dedekind-completion of any totally ordered abelian group with infinite elements or infinitesimals is not an abelian group, and the Dedekind-completion of any Archimedean ordered abelian group is still Archimedean.
Theorem — $\mathbb {R} $ has a complete metric
Proof
Tarski's axioms imply that $\mathbb {R} $ is a totally ordered abelian group under addition with distinguished element $1$. As a result, there exist maximum and minimum binary functions $\max :\mathbb {R} \times \mathbb {R} \to \mathbb {R} $ :\mathbb {R} \times \mathbb {R} \to \mathbb {R} } and $\min :\mathbb {R} \times \mathbb {R} \to \mathbb {R} $ :\mathbb {R} \times \mathbb {R} \to \mathbb {R} } , with the absolute value function defined as $\vert x\vert =\max(x,-x)$.
Since $\mathbb {R} $ is Dedekind-complete, Archimedean, and a totally ordered abelian group, $\mathbb {R} $ is a metric space with respect to the absolute value $\vert x\vert $ and thus a Hausdorff space, and every Cauchy net in $\mathbb {R} $ converges to a unique element of $\mathbb {R} $, and thus the absolute value $\vert x\vert $ is a complete metric on $\mathbb {R} $.
Theorem — $\mathbb {Q} $ embeds in $\mathbb {R} $.
Proof
Since $\mathbb {R} $ is an abelian group, it is a $\mathbb {Z} $-module, and since $\mathbb {R} $ is totally ordered, it is a torsion-free module and thus a torsion-free abelian group, which means that the integers $\mathbb {Z} $ embed in $\mathbb {R} $, with injective group homomorphism $f:\mathbb {Z} \to \mathbb {R} $ where $f(0)=0$ and $f(1)=1$. As a result, for every integer $a\in \mathbb {Z} $ and $b\in \mathbb {Z} $ the affine functions $x\mapsto ax+b$ are well defined in $\mathbb {R} $.
Since $\mathbb {R} $ is Dedekind-complete, Archimedean, and a totally ordered abelian group, any closed interval $[a,b]$ on $\mathbb {R} $ is compact and connected. Since $\mathbb {R} $ is also a complete metric space, the intermediate value theorem is satisfied for every function from a closed interval $[a,b]$ to $\mathbb {R} $. Because $x\mapsto ax+b$ are monotonic for $a>0$, and for $a<0$ the function is just the negation of a monotonic function, $x\mapsto ax+b$ have a root for $\vert a\vert >0$. Thus $\mathbb {R} $ is a divisible group and a $\mathbb {Q} $-vector space, with an injective group homomorphism $f:\mathbb {Q} \to \mathbb {R} $ where $f(0)=0$ and $f(1)=1$.
Theorem — $\mathbb {R} $ is a commutative ring.
Proof
Since every Cauchy net in $\mathbb {R} $ converges to a unique element of $\mathbb {R} $, for every directed set $A$ and Cauchy net $(a_{i})_{i\in A}$ in the rational numbers, there exists a Cauchy net of linear functions $(f_{i})_{i\in A}$ defined as $f_{i}(x)=a_{i}x$. The limit of the Cauchy net $\lim _{i\in A}(f_{i})_{i}$ exists and is a unique function $g(x)=\lim _{i\in A}(a_{i})_{i}x$. Since every real number is the limit of a Cauchy net of rational numbers, there is an $\mathbb {R} $-action $\mu :\mathbb {R} \to (\mathbb {R} \to \mathbb {R} )$ :\mathbb {R} \to (\mathbb {R} \to \mathbb {R} )} which takes a real number $r$ to the linear function $x\mapsto rx$, with $\alpha (1)=x\mapsto x$ being the idenitity function. The uncurrying of $\alpha $ leads to a bilinear function $(-)(-):\mathbb {R} \times \mathbb {R} \to \mathbb {R} $ called multiplication of the real numbers, defined on the entire domain of the binary function. Since linear functions in the function space with function composition and the identity function is a commutative monoid, $\mathbb {R} $ multiplication and the multiplicative identity element $1$ is also commutative monoid, which means that $\mathbb {R} $ is a commutative ring.
Theorem — $\mathbb {R} $ is a field
Proof
Since $\mathbb {R} $ is a commutative ring, power series are well defined, and because all Cauchy nets converge in $\mathbb {R} $, all Cauchy sequences and all Cauchy power series converge in $\mathbb {R} $. In particular, every geometric series is a Cauchy power series and the limit of the geometric series $\sum _{n=0}^{\infty }x^{n}$ and $\sum _{n=0}^{\infty }(-1)^{n}x^{n}$ converges in the open interval $(-1,1)$. Thus let us define functions $f:(-1,1)\to \mathbb {R} $ and $g:(-1,1)\to \mathbb {R} $ as
$f(x):=\sum _{n=0}^{\infty }x^{n}$
$g(x):=\sum _{n=0}^{\infty }(-1)^{n}x^{n}$
Let us define the function
$h(x,a):=(-a)\sum _{n=0}^{\infty }a^{n}(x+f(a+1))^{n}$
for $a<0$ and
$k(x,a):=a\sum _{n=0}^{\infty }(-a)^{n}(x+g(a-1))^{n}$
for $a>0$. These are functions which converge on the open interval $(1/a,0)$ for $h(x,a)$ and $(0,1/a)$ for $k(x,a)$, and satisfy the identity $h(x,a)x=1$ for all $a<0$ and $x\in (1/a,0)$, and $k(x,a)x=1$ for all $a>0$ and $x\in (0,1/a)$, by definition of the geometric series.
The reciprocal function is piecewise defined as ${\frac {1}{x}}={\begin{cases}\lim _{a\to 0^{-}}h(x,a)&{\text{if }}x<0\\\lim _{a\to 0^{+}}k(x,a)&{\text{if }}x>0\\\end{cases}}$
As limits preserve multiplication, ${\frac {1}{x}}x=1$. Thus, $\mathbb {R} $ is a field.
Otto Hölder showed that every Archimedean group is isomorphic (as an ordered group) to a subgroup of the Dedekind-complete Archimedean group with distinguished element $1>0$, $\mathbb {R} $.[3][4][5][6] Because $\mathbb {Q} $ is an Archimedean ordered field, let us define $\mathbb {R} ^{'}$ as the Dedekind completion of $\mathbb {Q} $. The Dedekind completion of any Archimedean ordered field is terminal in the concrete category of Dedekind complete Archimedean ordered fields,[7] Because $\mathbb {R} ^{'}$ is a Dedekind-complete Archimedean ordered field, every Archimedean group embeds into $\mathbb {R} ^{'}$ as well. As a result, the two sets $\mathbb {R} ^{'}$ and $\mathbb {R} $ are isomorphic to each other, which means that $\mathbb {R} $ is a field.
Tarski stated, without proof, that these axioms gave a total ordering. The missing component was supplied in 2008 by Stefanie Ucsnay.[2]
See also
• Construction of the real numbers – Axiomatic definitions of the real numbersPages displaying wikidata descriptions as a fallback
• Decidability of first-order theories of the real numbers
References
1. Tarski, Alfred (24 March 1994). Introduction to Logic and to the Methodology of Deductive Sciences (4 ed.). Oxford University Press. ISBN 978-0-19-504472-0.
2. Ucsnay, Stefanie (Jan 2008). "A Note on Tarski's Note". The American Mathematical Monthly. 115 (1): 66–68. doi:10.1080/00029890.2008.11920497. JSTOR 27642393. S2CID 42194493.
3. Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: American Mathematical Society, p. 61, ISBN 978-0-8218-1963-0, MR 1794715
4. Fuchs, László (2011) [1963]. Partially ordered algebraic systems. Mineola, New York: Dover Publications. pp. 45–46. ISBN 978-0-486-48387-0.
5. Kopytov, V. M.; Medvedev, N. Ya. (1996), Right-Ordered Groups, Siberian School of Algebra and Logic, Springer, pp. 33–34, ISBN 9780306110603.
6. For a proof for abelian groups, see Ribenboim, Paulo (1999), The Theory of Classical Valuations, Monographs in Mathematics, Springer, p. 60, ISBN 9780387985251.
7. Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study.
Real numbers
• 0.999...
• Absolute difference
• Cantor set
• Cantor–Dedekind axiom
• Completeness
• Construction
• Decidability of first-order theories
• Extended real number line
• Gregory number
• Irrational number
• Normal number
• Rational number
• Rational zeta series
• Real coordinate space
• Real line
• Tarski axiomatization
• Vitali set
| Wikipedia |
Tarski's circle-squaring problem
Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklós Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich estimated the number of pieces in his decomposition at roughly 1050. A constructive solution was given by Łukasz Grabowski, András Máthé and Oleg Pikhurko in 2016[1] which worked everywhere except for a set of measure zero. More recently, Andrew Marks and Spencer Unger (2017) gave a completely constructive solution using about $10^{200}$ Borel pieces.[2] In 2021 Máthé, Noel and Pikhurko improved the properties of the pieces.[3][4]
In particular, Lester Dubins, Morris W. Hirsch & Jack Karush proved it is impossible to dissect a circle and make a square using pieces that could be cut with an idealized pair of scissors (that is, having Jordan curve boundary).[5] The pieces used in Laczkovich's proof are non-measurable subsets.
Laczkovich actually proved the reassembly can be done using translations only; rotations are not required. Along the way, he also proved that any simple polygon in the plane can be decomposed into finitely many pieces and reassembled using translations only to form a square of equal area. The Bolyai–Gerwien theorem is a related but much simpler result: it states that one can accomplish such a decomposition of a simple polygon with finitely many polygonal pieces if both translations and rotations are allowed for the reassembly.
It follows from a result of Wilson (2005) that it is possible to choose the pieces in such a way that they can be moved continuously while remaining disjoint to yield the square. Moreover, this stronger statement can be proved as well to be accomplished by means of translations only.
These results should be compared with the much more paradoxical decompositions in three dimensions provided by the Banach–Tarski paradox; those decompositions can even change the volume of a set. However, in the plane, a decomposition into finitely many pieces must preserve the sum of the Banach measures of the pieces, and therefore cannot change the total area of a set (Wagon 1993).
See also
• Squaring the circle, a different problem: the task (which has been proven to be impossible) of constructing, for a given circle, a square of equal area with straightedge and compass alone.
References
1. Grabowski, Łukasz; Máthé, András; Pikhurko, Oleg (27 April 2022). "Measurable equidecompositions for group actions with an expansion property". Journal of the European Mathematical Society. 24 (12): 4277–4326. arXiv:1601.02958. doi:10.4171/JEMS/1189.
2. Marks, Andrew; Unger, Spencer (25 Aug 2017). "A constructive solution to Tarski's circle squaring problem (presentation)" (PDF). Retrieved 12 Jul 2021.{{cite web}}: CS1 maint: url-status (link)
3. Máthé, András; Noel, Jonathan A.; Pikhurko, Oleg (2022-02-03). "Circle Squaring with Pieces of Small Boundary and Low Borel Complexity". arXiv:2202.01412 [math.MG].
4. Nadis, Steve (2022-02-08). "An Ancient Geometry Problem Falls to New Mathematical Techniques". Quanta Magazine. Retrieved 2022-02-18.
5. Dubins, Lester; Hirsch, Morris W.; Karush, Jack (December 1963). "Scissor congruence". Israel Journal of Mathematics. 1 (4): 239–247. doi:10.1007/BF02759727. ISSN 1565-8511.
• Hertel, Eike; Richter, Christian (2003), "Squaring the circle by dissection" (PDF), Beiträge zur Algebra und Geometrie, 44 (1): 47–55, MR 1990983.
• Laczkovich, Miklos (1990), "Equidecomposability and discrepancy: a solution to Tarski's circle squaring problem", Journal für die Reine und Angewandte Mathematik, 1990 (404): 77–117, doi:10.1515/crll.1990.404.77, MR 1037431, S2CID 117762563.
• Laczkovich, Miklos (1994), "Paradoxical decompositions: a survey of recent results", Proc. First European Congress of Mathematics, Vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Basel: Birkhäuser, pp. 159–184, MR 1341843.
• Marks, Andrew; Unger, Spencer (2017), "Borel circle squaring", Annals of Mathematics, 186 (2): 581–605, arXiv:1612.05833, doi:10.4007/annals.2017.186.2.4, S2CID 738154.
• Tarski, Alfred (1925), "Probléme 38", Fundamenta Mathematicae, 7: 381.
• Wilson, Trevor M. (2005), "A continuous movement version of the Banach–Tarski paradox: A solution to De Groot's problem" (PDF), Journal of Symbolic Logic, 70 (3): 946–952, doi:10.2178/jsl/1122038921, MR 2155273, S2CID 15825008.
• Wagon, Stan (1993), The Banach–Tarski Paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, p. 169, ISBN 9780521457040.
| Wikipedia |
Tarski monster group
In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.
This article is about the kind of infinite group known as a Tarski monster group. For the largest of the sporadic finite simple groups, see Monster group.
Definition
Let $p$ be a fixed prime number. An infinite group $G$ is called a Tarski monster group for $p$ if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has $p$ elements.
Properties
• $G$ is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
• $G$ is simple. If $N\trianglelefteq G$ and $U\leq G$ is any subgroup distinct from $N$ the subgroup $NU$ would have $p^{2}$ elements.
• The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime $p>10^{75}$.
• Tarski monster groups are an example of non-amenable groups not containing a free subgroup.
References
• A. Yu. Olshanskii, An infinite group with subgroups of prime orders, Math. USSR Izv. 16 (1981), 279–289; translation of Izvestia Akad. Nauk SSSR Ser. Matem. 44 (1980), 309–321.
• A. Yu. Olshanskii, Groups of bounded period with subgroups of prime order, Algebra and Logic 21 (1983), 369–418; translation of Algebra i Logika 21 (1982), 553–618.
• Ol'shanskiĭ, A. Yu. (1991), Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-1394-6
| Wikipedia |
Tarski's plank problem
In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by Thøger Bang (1950, 1951).[1]
Statement
Given a convex body C in Rn and a hyperplane H, the width of C parallel to H, w(C,H), is the distance between the two supporting hyperplanes of C that are parallel to H. The smallest such distance (i.e. the infimum over all possible hyperplanes) is called the minimal width of C, w(C).
The (closed) set of points P between two distinct, parallel hyperplanes in Rn is called a plank, and the distance between the two hyperplanes is called the width of the plank, w(P). Tarski conjectured that if a convex body C of minimal width w(C) was covered by a collection of planks, then the sum of the widths of those planks must be at least w(C). That is, if P1,…,Pm are planks such that
$C\subseteq P_{1}\cup \ldots \cup P_{m}\subset \mathbb {R} ^{n},$
then
$\sum _{i=1}^{m}w(P_{i})\geq w(C).$
Bang proved this is indeed the case.
Nomenclature
The name of the problem, specifically for the sets of points between parallel hyperplanes, comes from the visualisation of the problem in R2. Here, hyperplanes are just straight lines and so planks become the space between two parallel lines. Thus the planks can be thought of as (infinitely long) planks of wood, and the question becomes how many planks does one need to completely cover a convex tabletop of minimal width w? Bang's theorem shows that, for example, a circular table of diameter d feet can't be covered by fewer than d planks of wood of width one foot each.
References
1. King, Jonathan L. (1994). "Three problems in search of a measure". Amer. Math. Monthly. 101 (7): 609–628. doi:10.2307/2974690. JSTOR 2974690.
• Bang, Thøger (1950), "On covering by parallel-strips.", Mat. Tidsskr. B.: 49–53, MR 0038085
• Bang, Thøger (1951), "A solution of the "plank problem"", Proc. Amer. Math. Soc., 2 (6): 990–993, doi:10.2307/2031721, JSTOR 2031721, MR 0046672
| Wikipedia |
List of things named after Alfred Tarski
In the history of mathematics, Alfred Tarski (1901–1983) is one of the most important logicians. His name is now associated with a number of theorems and concepts in that field.
Theorems
• Łoś–Tarski preservation theorem
• Knaster–Tarski theorem (sometimes referred to as Tarski's fixed point theorem)
• Tarski's undefinability theorem
• Tarski–Seidenberg theorem
• Some fixed point theorems, usually variants of the Kleene fixed-point theorem, are referred to the Tarski–Kantorovitch fixed–point principle or the Tarski–Kantorovitch theorem although the use of this terminology is limited.
• Tarski's theorem
Other mathematics-related work
• Bernays-Tarski axiom system
• Banach–Tarski paradox
• Lindenbaum–Tarski algebra
• Łukasiewicz-Tarski logic
• Jónsson–Tarski duality
• Jónsson–Tarski algebra
• Gödel–McKinsey–Tarski translation
• The semantic theory of truth is sometimes referred to as Tarski's definition of truth or Tarski's truth definitions.
• Tarski's axiomatization of the reals
• Tarski's axioms for plane geometry
• Tarski's circle-squaring problem
• Tarski's exponential function problem
• Tarski–Grothendieck set theory
• Tarski's high school algebra problem
• Tarski–Kuratowski algorithm
• Tarski monster group
• Tarski's plank problem
• Tarski's problems for free groups
• Tarski–Vaught test
• Tarski's World
Other
• 13672 Tarski, a main-belt asteroid
| Wikipedia |
Knaster–Tarski theorem
In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following:
Let (L, ≤) be a complete lattice and let f : L → L be an monotonic function (w.r.t. ≤ ). Then the set of fixed points of f in L forms a complete lattice under ≤ .
It was Tarski who stated the result in its most general form,[1] and so the theorem is often known as Tarski's fixed-point theorem. Some time earlier, Knaster and Tarski established the result for the special case where L is the lattice of subsets of a set, the power set lattice.[2]
The theorem has important applications in formal semantics of programming languages and abstract interpretation.
A kind of converse of this theorem was proved by Anne C. Davis: If every order preserving function f : L → L on a lattice L has a fixed point, then L is a complete lattice.[3]
Consequences: least and greatest fixed points
Since complete lattices cannot be empty (they must contain a supremum and infimum of the empty set), the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least fixed point (or greatest fixed point). In many practical cases, this is the most important implication of the theorem.
The least fixpoint of f is the least element x such that f(x) = x, or, equivalently, such that f(x) ≤ x; the dual holds for the greatest fixpoint, the greatest element x such that f(x) = x.
If f(lim xn) = lim f(xn) for all ascending sequences xn, then the least fixpoint of f is lim f n(0) where 0 is the least element of L, thus giving a more "constructive" version of the theorem. (See: Kleene fixed-point theorem.) More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of f α(0), taking α over the ordinals, where f α is defined by transfinite induction: f α+1 = f (f α) and f γ for a limit ordinal γ is the least upper bound of the f β for all β ordinals less than γ.[4] The dual theorem holds for the greatest fixpoint.
For example, in theoretical computer science, least fixed points of monotonic functions are used to define program semantics. Often a more specialized version of the theorem is used, where L is assumed to be the lattice of all subsets of a certain set ordered by subset inclusion. This reflects the fact that in many applications only such lattices are considered. One then usually is looking for the smallest set that has the property of being a fixed point of the function f. Abstract interpretation makes ample use of the Knaster–Tarski theorem and the formulas giving the least and greatest fixpoints.
The Knaster–Tarski theorem can be used to give a simple proof of the Cantor–Bernstein–Schroeder theorem.[5][6]
Weaker versions of the theorem
Weaker versions of the Knaster–Tarski theorem can be formulated for ordered sets, but involve more complicated assumptions. For example:
Let L be a partially ordered set with a least element (bottom) and let f : L → L be an monotonic function. Further, suppose there exists u in L such that f(u) ≤ u and that any chain in the subset $\{x\in L\mid x\leq f(x),x\leq u\}$ has a supremum. Then f admits a least fixed point.
This can be applied to obtain various theorems on invariant sets, e.g. the Ok's theorem:
For the monotone map F : P(X ) → P(X ) on the family of (closed) nonempty subsets of X, the following are equivalent: (o) F admits A in P(X ) s.t. $A\subseteq F(A)$, (i) F admits invariant set A in P(X ) i.e. $A=F(A)$, (ii) F admits maximal invariant set A, (iii) F admits the greatest invariant set A.
In particular, using the Knaster-Tarski principle one can develop the theory of global attractors for noncontractive discontinuous (multivalued) iterated function systems. For weakly contractive iterated function systems the Kantorovitch fixpoint theorem (known also as Tarski-Kantorovitch fixpoint principle) suffices.
Other applications of fixed-point principles for ordered sets come from the theory of differential, integral and operator equations.
Proof
Let us restate the theorem.
For a complete lattice $\langle L,\leq \rangle $ and a monotone function $f\colon L\rightarrow L$ on L, the set of all fixpoints of f is also a complete lattice $\langle P,\leq \rangle $, with:
• $\bigvee P=\bigvee \{x\in L\mid x\leq f(x)\}$ as the greatest fixpoint of f
• $\bigwedge P=\bigwedge \{x\in L\mid x\geq f(x)\}$ as the least fixpoint of f.
Proof. We begin by showing that P has both a least element and a greatest element. Let D = {x | x ≤ f(x)} and x ∈ D (we know that at least 0L belongs to D). Then because f is monotone we have f(x) ≤ f(f(x)), that is f(x) ∈ D.
Now let $u=\bigvee D$ (u exists because D ⊆ L and L is a complete lattice). Then for all x ∈ D it is true that x ≤ u and f(x) ≤ f(u), so x ≤ f(x) ≤ f(u). Therefore, f(u) is an upper bound of D, but u is the least upper bound, so u ≤ f(u), i.e. u ∈ D. Then f(u) ∈ D (because f(u) ≤ f(f(u))) and so f(u) ≤ u from which follows f(u) = u. Because every fixpoint is in D we have that u is the greatest fixpoint of f.
The function f is monotone on the dual (complete) lattice $\langle L^{op},\geq \rangle $. As we have just proved, its greatest fixpoint exists. It is the least fixpoint of L, so P has least and greatest elements, that is more generally, every monotone function on a complete lattice has a least fixpoint and a greatest fixpoint.
For a, b in L we write [a, b] for the closed interval with bounds a and b: {x ∈ L | a ≤ x ≤ b}. If a ≤ b, then $\langle $[a, b], ≤$\rangle $ is a complete lattice.
It remains to be proven that P is a complete lattice. Let $1_{L}=\bigvee L$, W ⊆ P and $w=\bigvee W$. We show that f([w, 1L]) ⊆ [w, 1L]. Indeed, for every x ∈ W we have x = f(x) and since w is the least upper bound of W, x ≤ f(w). In particular w ≤ f(w). Then from y ∈ [w, 1L] follows that w ≤ f(w) ≤ f(y), giving f(y) ∈ [w, 1L] or simply f([w, 1L]) ⊆ [w, 1L]. This allows us to look at f as a function on the complete lattice [w, 1L]. Then it has a least fixpoint there, giving us the least upper bound of W. We've shown that an arbitrary subset of P has a supremum, that is, P is a complete lattice.
See also
• Modal μ-calculus
Notes
1. Alfred Tarski (1955). "A lattice-theoretical fixpoint theorem and its applications". Pacific Journal of Mathematics. 5:2: 285–309.
2. B. Knaster (1928). "Un théorème sur les fonctions d'ensembles". Ann. Soc. Polon. Math. 6: 133–134. With A. Tarski.
3. Anne C. Davis (1955). "A characterization of complete lattices". Pacific Journal of Mathematics. 5 (2): 311–319. doi:10.2140/pjm.1955.5.311.
4. Cousot, Patrick; Cousot, Radhia (1979). "Constructive versions of tarski's fixed point theorems". Pacific Journal of Mathematics. 82: 43–57. doi:10.2140/pjm.1979.82.43.
5. Example 3 in R. Uhl, "Tarski's Fixed Point Theorem", from MathWorld--a Wolfram Web Resource, created by Eric W. Weisstein.
6. Davey, Brian A.; Priestley, Hilary A. (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. pp. 63, 4. ISBN 9780521784511.
References
• Andrzej Granas and James Dugundji (2003). Fixed Point Theory. Springer-Verlag, New York. ISBN 978-0-387-00173-9.
• Forster, T. (2003-07-21). Logic, Induction and Sets. ISBN 978-0-521-53361-4.
Further reading
• S. Hayashi (1985). "Self-similar sets as Tarski's fixed points". Publications of the Research Institute for Mathematical Sciences. 21 (5): 1059–1066. doi:10.2977/prims/1195178796.
• J. Jachymski; L. Gajek; K. Pokarowski (2000). "The Tarski-Kantorovitch principle and the theory of iterated function systems". Bull. Austral. Math. Soc. 61 (2): 247–261. doi:10.1017/S0004972700022243.
• E.A. Ok (2004). "Fixed set theory for closed correspondences with applications to self-similarity and games". Nonlinear Anal. 56 (3): 309–330. CiteSeerX 10.1.1.561.4581. doi:10.1016/j.na.2003.08.001.
• B.S.W. Schröder (1999). "Algorithms for the fixed point property". Theoret. Comput. Sci. 217 (2): 301–358. doi:10.1016/S0304-3975(98)00273-4.
• S. Heikkilä (1990). "On fixed points through a generalized iteration method with applications to differential and integral equations involving discontinuities". Nonlinear Anal. 14 (5): 413–426. doi:10.1016/0362-546X(90)90082-R.
• R. Uhl (2003). "Smallest and greatest fixed points of quasimonotone increasing mappings". Mathematische Nachrichten. 248–249: 204–210. doi:10.1002/mana.200310016.
External links
• J. B. Nation, Notes on lattice theory.
• An application to an elementary combinatorics problem: Given a book with 100 pages and 100 lemmas, prove that there is some lemma written on the same page as its index
| Wikipedia |
Tarski–Kuratowski algorithm
In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithm that produces an upper bound for the complexity of a given formula in the arithmetical hierarchy and analytical hierarchy.
The algorithm is named after Alfred Tarski and Kazimierz Kuratowski.
Algorithm
The Tarski–Kuratowski algorithm for the arithmetical hierarchy consists of the following steps:
1. Convert the formula to prenex normal form. (This is the non-deterministic part of the algorithm, as there may be more than one valid prenex normal form for the given formula.)
2. If the formula is quantifier-free, it is in $\Sigma _{0}^{0}$ and $\Pi _{0}^{0}$.
3. Otherwise, count the number of alternations of quantifiers; call this k.
4. If the first quantifier is ∃, the formula is in $\Sigma _{k+1}^{0}$.
5. If the first quantifier is ∀, the formula is in $\Pi _{k+1}^{0}$.
References
• Rogers, Hartley The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1; ISBN 0-07-053522-1
| Wikipedia |
Tarski–Seidenberg theorem
In mathematics, the Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities. The theorem—also known as the Tarski–Seidenberg projection property—is named after Alfred Tarski and Abraham Seidenberg.[1] It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations and inequalities by logical connectives ∨ (or), ∧ (and), ¬ (not) and quantifiers ∀ (for all), ∃ (exists) is equivalent to a similar formula without quantifiers. An important consequence is the decidability of the theory of real-closed fields.
Although the original proof of the theorem was constructive, the resulting algorithm has a computational complexity that is too high for using the method on a computer. George E. Collins introduced the algorithm of cylindrical algebraic decomposition, which allows quantifier elimination over the reals in double exponential time. This complexity is optimal, as there are examples where the output has a double exponential number of connected components. This algorithm is therefore fundamental, and it is widely used in computational algebraic geometry.
Statement
A semialgebraic set in Rn is a finite union of sets defined by a finite number of polynomial equations and inequalities, that is by a finite number of statements of the form
$p(x_{1},\ldots ,x_{n})=0\,$
and
$q(x_{1},\ldots ,x_{n})>0\,$
for polynomials p and q. We define a projection map π : Rn +1 → Rn by sending a point (x1, ..., xn, xn +1) to (x1, ..., xn). Then the Tarski–Seidenberg theorem states that if X is a semialgebraic set in Rn +1 for some n ≥ 1, then π(X) is a semialgebraic set in Rn.
Failure with algebraic sets
If we only define sets using polynomial equations and not inequalities then we define algebraic sets rather than semialgebraic sets. For these sets the theorem fails, i.e. projections of algebraic sets need not be algebraic. As a simple example consider the hyperbola in R2 defined by the equation
$xy-1=0.\,$
This is a perfectly good algebraic set, but projecting it down by sending (x, y) in R2 to x in R produces the set of points satisfying x ≠ 0. This is a semialgebraic set, but it is not an algebraic set as the algebraic sets in R are R itself, the empty set and the finite sets.
This example shows also that, over the complex numbers, the projection of an algebraic set may be non-algebraic. Thus the existence of real algebraic sets with non-algebraic projections does not rely on the fact that the field of real numbers is not algebraically closed.
Another example is the parabola in R2, which is defined by the equation
$y^{2}-x=0.$
Its projection onto the x-axis is the half-line [0, ∞), a semialgebraic set that cannot be obtained from algebraic sets by (finite) intersections, unions, and set complements.
Relation to structures
This result confirmed that semialgebraic sets in Rn form what is now known as an o-minimal structure on R. These are collections of subsets Sn of Rn for each n ≥ 1 such that we can take finite unions and complements of the subsets in Sn and the result will still be in Sn, moreover the elements of S1 are simply finite unions of intervals and points. The final condition for such a collection to be an o-minimal structure is that the projection map on the first n coordinates from Rn +1 to Rn must send subsets in Sn +1 to subsets in Sn. The Tarski–Seidenberg theorem tells us that this holds if Sn is the set of semialgebraic sets in Rn.
See also
• Decidability of first-order theories of the real numbers
References
1. Mishra, Bhubaneswar (1993). Algorithmic Algebra. New York: Springer. pp. 345–347. ISBN 0-387-94090-1.
• van den Dries, L. (1998). Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series. Vol. 248. Cambridge University Press. Zbl 0953.03045.
• Khovanskii, Askold G. (1991). Fewnomials. Translations of Mathematical Monographs. Vol. 88. Translated from the Russian by Smilka Zdravkovska. Providence, RI: American Mathematical Society. ISBN 0-8218-4547-0. Zbl 0728.12002.
• Neyman, Abraham (2003). "Real Algebraic Tools in Stochastic Games". Stochastic Games and Applications. Dordrecht: Kluwer. pp. 57–75. ISBN 1-4020-1492-9.
External links
• Tarski–Seidenberg theorem at PlanetMath.org
| Wikipedia |
Tasha Inniss
Tasha Rose Inniss is an American mathematician and the director of education and industry outreach for the Institute for Operations Research and the Management Sciences (INFORMS).
Tasha Inniss
Jane Garvey (left) presents Tasha Inniss with an FAA Student-of-the-Year award in 2002
Alma materXavier University of Louisiana
University of Maryland, College Park
Scientific career
InstitutionsTrinity Washington University
Spelman College
ThesisStochastic Models for the Estimation of Airport Arrival Capacity Distributions (2000)
Early life and education
Inniss was born in New Orleans and grew up without a father.[1] She became interested in mathematics in fourth grade, and decided she would study it as a freshman in high school.[2] She studied mathematics at Xavier University of Louisiana, graduating summa cum laude.[3] In 1992 she was listed in the Who's Who Among Colleges and Universities for her academic achievements.[1] She earned a master's degree in applied mathematics from Georgia Institute of Technology.[4]
She moved to the University of Maryland for her PhD, funded by the David and Lucile Packard Foundation.[1][3][5][6] In 2000, Inniss became the first African American woman to obtain a Ph.D. from the University of Maryland, together with Sherry Scott and Kimberly Weems.[7] Her dissertation was Stochastic Models for the Estimation of Airport Arrival Capacity Distributions. She was part of the National Center of Excellence for Aviation Operators and advised by Michael Owen Ball.[4][8] Her brother, Enos Inniss, also completed his PhD in 2000.[1]
Research and career
In 2001 she was appointed the Clare Boothe Luce Professor of Mathematics at Trinity Washington University.[1] Her doctoral thesis described programming methods to calibrate models to estimate airport capacity.[9] She remains a consultant for the Federal Aviation Administration.[10]
She joined the department of mathematics at Spelman College in 2005 as an assistant professor.[11][12][13]
Throughout her career she has worked to recruit, support and mentor underrepresented minority students.[14][15][16][17] She led a National Science Foundation project that looked to increase the quality and quantity of underrepresented minorities matriculating and completing doctoral degrees.[18] She has contributed to the EDGE Foundation (Enhancing Diversity in Graduate Education) program.[19]
In 2017 she joined the Institute for Operations Research and the Management Sciences as Director of Education.[4]
Inniss' work earned her recognition by Mathematically Gifted & Black as a Black History Month 2017 Honoree.[20]
In 2022, Inniss was added to the American Mathematical Society (AMS) Committee on Professional Ethics.[21]
References
1. "Underrepresented Minorities in Science: A Statistical Anomaly: The Story of an African-American Woman Battling the Odds to Become a Mathematician". Science | AAAS. 2001-03-02. Retrieved 2018-05-05.
2. Case, Bettye Anne; Leggett, Anne M., eds. (2005). Complexities : women in mathematics. Princeton, N.J.: Princeton University Press. ISBN 978-1400880164. OCLC 949753960.
3. "Student Essay Contest: 2001 Results". awm-math.org. Retrieved 2019-02-24.
4. INFORMS. "INFORMS adds new Director of Education and Industry Outreach to Leadership Team". INFORMS. Retrieved 2018-05-05.
5. "Delta SEE: About Us". www.deltasee.org. Retrieved 2018-05-05.
6. "Packard Foundation Graduate Scholars Program: HBCU Directory". ehrweb.aaas.org. Retrieved 2018-05-05.
7. "The First Three African American Women to Receive Doctorates in the Mathematics Department". www.math.umd.edu. Retrieved 2018-05-05.
8. Tasha Inniss at the Mathematics Genealogy Project
9. Tasha, Inniss (2001). "Stochastic Models for the Estimation of Airport Arrival Capacity Distributions". {{cite journal}}: Cite journal requires |journal= (help)
10. "Tasha R. Inniss". Duchess International Magazine. Retrieved 2018-05-05.
11. Inniss, Tasha R.; Lee, John R.; Light, Marc; Grassi, Michael A.; Thomas, George; Williams, Andrew B. (2006-11-10). Towards applying text mining and natural language processing for biomedical ontology acquisition. ACM. pp. 7–14. doi:10.1145/1183535.1183539. ISBN 978-1595935267. S2CID 2365000.
12. "Interviewees". livingthinkers. Retrieved 2018-05-05.
13. Muicahy, Colm (2017). "A Century of Mathematical Excellence at Spelman College". A Century of Mathematical Excellence at Spelman College. Atlanta University Center, Robert W. Woodruff Library. doi:10.22595/scpubs.00013.
14. L., Joiner, Lottie (2003-01-05). "Success to the Third Degree". Black Issues in Higher Education. 18 (11). ISSN 0742-0277.{{cite journal}}: CS1 maint: multiple names: authors list (link)
15. ""HBCUs' Relevance in Diversifying the STEM Workforce" by Carter-Johnson, Frances; Inniss, Tasha; Lee, Mark E. - Diverse Issues in Higher Education, Vol. 35, Issue 2, February 22, 2018". Archived from the original on May 6, 2018.
16. "Against the Odds: Three African-American Women to Discuss the Road to Math Ph.D.s". Office of News & Media Relations | UMass Amherst. Retrieved 2018-05-05.
17. Review, Peer (2014-04-29). "Who Is Minding the Gap?". Association of American Colleges & Universities. Retrieved 2018-05-05.
18. "NSF Award Search: Award#1249262 - Bridge to the Doctorate at UMCP, 2012-2014". www.nsf.gov. Retrieved 2018-05-05.
19. "EDGE 2009". THE EDGE PROGRAM. Retrieved 2018-05-05.
20. "Tasha Inniss". Mathematically Gifted & Black.
21. "AMS Committees". American Mathematical Society. Retrieved 2023-03-29.
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Tate's algorithm
In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over $\mathbb {Q} $, or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent fp of p in the conductor of E, the type of reduction at p, the local index
$c_{p}=[E(\mathbb {Q} _{p}):E^{0}(\mathbb {Q} _{p})],$
where $E^{0}(\mathbb {Q} _{p})$ is the group of $\mathbb {Q} _{p}$-points whose reduction mod p is a non-singular point. Also, the algorithm determines whether or not the given integral model is minimal at p, and, if not, returns an integral model with integral coefficients for which the valuation at p of the discriminant is minimal.
Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surfaces: in turn this determines the exponent fp of the conductor E.
Tate's algorithm can be greatly simplified if the characteristic of the residue class field is not 2 or 3; in this case the type and c and f can be read off from the valuations of j and Δ (defined below).
Tate's algorithm was introduced by John Tate (1975) as an improvement of the description of the Néron model of an elliptic curve by Néron (1964).
Notation
Assume that all the coefficients of the equation of the curve lie in a complete discrete valuation ring R with perfect residue field K and maximal ideal generated by a prime π. The elliptic curve is given by the equation
$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.$
Define:
$v(\Delta )=$ the p-adic valuation of $\pi $ in $\Delta $, that is, exponent of $\pi $ in prime factorization of $\Delta $, or infinity if $\Delta =0$
$a_{i,m}=a_{i}/\pi ^{m}$
$b_{2}=a_{1}^{2}+4a_{2}$
$b_{4}=a_{1}a_{3}+2a_{4}^{}$
$b_{6}=a_{3}^{2}+4a_{6}$
$b_{8}=a_{1}^{2}a_{6}-a_{1}a_{3}a_{4}+4a_{2}a_{6}+a_{2}a_{3}^{2}-a_{4}^{2}$
$c_{4}=b_{2}^{2}-24b_{4}$
$c_{6}=-b_{2}^{3}+36b_{2}b_{4}-216b_{6}$
$\Delta =-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}$
$j=c_{4}^{3}/\Delta .$
The algorithm
• Step 1: If π does not divide Δ then the type is I0, c=1 and f=0.
• Step 2: If π divides Δ but not c4 then the type is Iv with v = v(Δ), c=v, and f=1.
• Step 3. Otherwise, change coordinates so that π divides a3,a4,a6. If π2 does not divide a6 then the type is II, c=1, and f=v(Δ);
• Step 4. Otherwise, if π3 does not divide b8 then the type is III, c=2, and f=v(Δ)−1;
• Step 5. Otherwise, let Q1 be the polynomial
$Q_{1}(Y)=Y^{2}+a_{3,1}Y-a_{6,2}.$.
If π3 does not divide b6 then the type is IV, c=3 if $Q_{1}(Y)$ has two roots in K and 1 if it has two roots outside of K, and f=v(Δ)−2.
• Step 6. Otherwise, change coordinates so that π divides a1 and a2, π2 divides a3 and a4, and π3 divides a6. Let P be the polynomial
$P(T)=T^{3}+a_{2,1}T^{2}+a_{4,2}T+a_{6,3}.$
If $P(T)$ has 3 distinct roots modulo π then the type is I0*, f=v(Δ)−4, and c is 1+(number of roots of P in K).
• Step 7. If P has one single and one double root, then the type is Iν* for some ν>0, f=v(Δ)−4−ν, c=2 or 4: there is a "sub-algorithm" for dealing with this case.
• Step 8. If P has a triple root, change variables so the triple root is 0, so that π2 divides a2 and π3 divides a4, and π4 divides a6. Let Q2 be the polynomial
$Q_{2}(Y)=Y^{2}+a_{3,2}Y-a_{6,4}.$.
If $Q_{2}(Y)$ has two distinct roots modulo π then the type is IV*, f=v(Δ)−6, and c is 3 if the roots are in K, 1 otherwise.
• Step 9. If $Q_{2}(Y)$ has a double root, change variables so the double root is 0. Then π3 divides a3 and π5 divides a6. If π4 does not divide a4 then the type is III* and f=v(Δ)−7 and c = 2.
• Step 10. Otherwise if π6 does not divide a6 then the type is II* and f=v(Δ)−8 and c = 1.
• Step 11. Otherwise the equation is not minimal. Divide each an by πn and go back to step 1.
Implementations
The algorithm is implemented for algebraic number fields in the PARI/GP computer algebra system, available through the function elllocalred.
References
• Cremona, John (1997), Algorithms for modular elliptic curves (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-59820-6, Zbl 0872.14041, retrieved 2007-12-20
• Laska, Michael (1982), "An Algorithm for Finding a Minimal Weierstrass Equation for an Elliptic Curve", Mathematics of Computation, 38 (157): 257–260, doi:10.2307/2007483, JSTOR 2007483, Zbl 0493.14016
• Néron, André (1964), "Modèles minimaux des variétés abèliennes sur les corps locaux et globaux", Publications Mathématiques de l'IHÉS (in French), 21: 5–128, doi:10.1007/BF02684271, MR 0179172, Zbl 0132.41403
• Silverman, Joseph H. (1994), Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, ISBN 0-387-94328-5, Zbl 0911.14015
• Tate, John (1975), "Algorithm for determining the type of a singular fiber in an elliptic pencil", in Birch, B.J.; Kuyk, W. (eds.), Modular Functions of One Variable IV, Lecture Notes in Mathematics, vol. 476, Berlin / Heidelberg: Springer, pp. 33–52, doi:10.1007/BFb0097582, ISBN 978-3-540-07392-5, ISSN 1617-9692, MR 0393039, Zbl 1214.14020
| Wikipedia |
Tate's isogeny theorem
In mathematics, Tate's isogeny theorem, proved by Tate (1966), states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations).
References
• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 9788185931869, MR 0282985, OCLC 138290
• Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2 (2): 134–144, Bibcode:1966InMat...2..134T, doi:10.1007/BF01404549, ISSN 0020-9910, MR 0206004, S2CID 245902
| Wikipedia |
Tate duality
In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by John Tate (1962) and Georges Poitou (1967).
Local Tate duality
Main article: local Tate duality
For a p-adic local field $k$, local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology:
$\displaystyle H^{r}(k,M)\times H^{2-r}(k,M')\rightarrow H^{2}(k,\mathbb {G} _{m})=\mathbb {Q} /\mathbb {Z} $
where $M$ is a finite group scheme, $M'$ its dual $\operatorname {Hom} (M,G_{m})$, and $\mathbb {G} _{m}$ is the multiplicative group. For a local field of characteristic $p>0$, the statement is similar, except that the pairing takes values in $H^{2}(k,\mu )=\bigcup _{p\nmid n}{\tfrac {1}{n}}\mathbb {Z} /\mathbb {Z} $.[1] The statement also holds when $k$ is an Archimedean field, though the definition of the cohomology groups looks somewhat different in this case.
Global Tate duality
Given a finite group scheme $M$ over a global field $k$, global Tate duality relates the cohomology of $M$ with that of $M'=\operatorname {Hom} (M,G_{m})$ using the local pairings constructed above. This is done via the localization maps
$\alpha _{r,M}:H^{r}(k,M)\rightarrow {\prod _{v}}'H^{r}(k_{v},M),$
where $v$ varies over all places of $k$, and where $\prod '$ denotes a restricted product with respect to the unramified cohomology groups. Summing the local pairings gives a canonical perfect pairing
${\prod _{v}}'H^{r}(k_{v},M)\times {\prod _{v}}'H^{2-r}(k_{v},M')\rightarrow \mathbb {Q} /\mathbb {Z} .$
One part of Poitou-Tate duality states that, under this pairing, the image of $H^{r}(k,M)$ has annihilator equal to the image of $H^{2-r}(k,M')$ for $r=0,1,2$.
The map $\alpha _{r,M}$ has a finite kernel for all $r$, and Tate also constructs a canonical perfect pairing
${\text{ker}}(\alpha _{1,M})\times \ker(\alpha _{2,M'})\rightarrow \mathbb {Q} /\mathbb {Z} .$
These dualities are often presented in the form of a nine-term exact sequence
$0\rightarrow H^{0}(k,M)\rightarrow {\prod _{v}}'H^{0}(k_{v},M)\rightarrow H^{2}(k,M')^{*}$
$\rightarrow H^{1}(k,M)\rightarrow {\prod _{v}}'H^{1}(k_{v},M)\rightarrow H^{1}(k,M')^{*}$
$\rightarrow H^{2}(k,M)\rightarrow {\prod _{v}}'H^{2}(k_{v},M)\rightarrow H^{0}(k,M')^{*}\rightarrow 0.$
Here, the asterisk denotes the Pontryagin dual of a given locally compact abelian group.
All of these statements were presented by Tate in a more general form depending on a set of places $S$ of $k$, with the above statements being the form of his theorems for the case where $S$ contains all places of $k$. For the more general result, see e.g. Neukirch, Schmidt & Wingberg (2000, Theorem 8.4.4).
Poitou–Tate duality
Among other statements, Poitou–Tate duality establishes a perfect pairing between certain Shafarevich groups. Given a global field $k$, a set S of primes, and the maximal extension $k_{S}$ which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of $\operatorname {Gal} (k_{S}/k)$ which vanish in the Galois cohomology of the local fields pertaining to the primes in S.[2]
An extension to the case where the ring of S-integers ${\mathcal {O}}_{S}$ is replaced by a regular scheme of finite type over $\operatorname {Spec} {\mathcal {O}}_{S}$ was shown by Geisser & Schmidt (2017) harvtxt error: no target: CITEREFGeisserSchmidt2017 (help).
See also
• Artin–Verdier duality
• Tate pairing
References
1. Neukirch, Schmidt & Wingberg (2000, Theorem 7.2.6)
2. See Neukirch, Schmidt & Wingberg (2000, Theorem 8.6.8) for a precise statement.
• Geisser, Thomas H.; Schmidt, Alexander (2018), "Poitou-Tate duality for arithmetic schemes", Compositio Mathematica, 154 (9): 2020–2044, arXiv:1709.06913, Bibcode:2017arXiv170906913G, doi:10.1112/S0010437X18007340, S2CID 119735104
• Haberland, Klaus (1978), Galois cohomology of algebraic number fields, VEB Deutscher Verlag der Wissenschaften, ISBN 9780685872048, MR 0519872
• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of number fields, Springer, ISBN 3-540-66671-0, MR 1737196
• Poitou, Georges (1967), "Propriétés globales des modules finis", Cohomologie galoisienne des modules finis, Séminaire de l'Institut de Mathématiques de Lille, sous la direction de G. Poitou. Travaux et Recherches Mathématiques, vol. 13, Paris: Dunod, pp. 255–277, MR 0219591
• Tate, John (1963), "Duality theorems in Galois cohomology over number fields", Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 288–295, MR 0175892, archived from the original on 2011-07-17
| Wikipedia |
Tate–Shafarevich group
In arithmetic geometry, the Tate–Shafarevich group Ш(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group $\mathrm {WC} (A/K)=H^{1}(G_{K},A)$, where $G_{K}=\mathrm {Gal} (K/\mathbb {Q} )$ is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from K by completing with respect to all its Archimedean and non Archimedean valuations v). Thus, in terms of Galois cohomology, Ш(A/K) can be defined as
$\bigcap _{v}\mathrm {ker} \left(H^{1}\left(G_{K},A\right)\rightarrow H^{1}\left(G_{K_{v}},A_{v}\right)\right).$
This group was introduced by Serge Lang and John Tate[1] and Igor Shafarevich.[2] Cassels introduced the notation Ш(A/K), where Ш is the Cyrillic letter "Sha", for Shafarevich, replacing the older notation TS or TŠ.
Elements of the Tate–Shafarevich group
Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of A that have Kv-rational points for every place v of K, but no K-rational point. Thus, the group measures the extent to which the Hasse principle fails to hold for rational equations with coefficients in the field K. Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve x4 − 17 = 2y2 has solutions over the reals and over all p-adic fields, but has no rational points.[3] Ernst S. Selmer gave many more examples, such as 3x3 + 4y3 + 5z3 = 0.[4]
The special case of the Tate–Shafarevich group for the finite group scheme consisting of points of some given finite order n of an abelian variety is closely related to the Selmer group.
Tate-Shafarevich conjecture
The Tate–Shafarevich conjecture states that the Tate–Shafarevich group is finite. Karl Rubin proved this for some elliptic curves of rank at most 1 with complex multiplication.[5] Victor A. Kolyvagin extended this to modular elliptic curves over the rationals of analytic rank at most 1 (The modularity theorem later showed that the modularity assumption always holds).[6]
Cassels–Tate pairing
The Cassels–Tate pairing is a bilinear pairing Ш(A) × Ш(Â) → Q/Z, where A is an abelian variety and  is its dual. Cassels introduced this for elliptic curves, when A can be identified with  and the pairing is an alternating form.[7] The kernel of this form is the subgroup of divisible elements, which is trivial if the Tate–Shafarevich conjecture is true. Tate extended the pairing to general abelian varieties, as a variation of Tate duality.[8] A choice of polarization on A gives a map from A to Â, which induces a bilinear pairing on Ш(A) with values in Q/Z, but unlike the case of elliptic curves this need not be alternating or even skew symmetric.
For an elliptic curve, Cassels showed that the pairing is alternating, and a consequence is that if the order of Ш is finite then it is a square. For more general abelian varieties it was sometimes incorrectly believed for many years that the order of Ш is a square whenever it is finite; this mistake originated in a paper by Swinnerton-Dyer,[9] who misquoted one of the results of Tate.[8] Poonen and Stoll gave some examples where the order is twice a square, such as the Jacobian of a certain genus 2 curve over the rationals whose Tate–Shafarevich group has order 2,[10] and Stein gave some examples where the power of an odd prime dividing the order is odd.[11] If the abelian variety has a principal polarization then the form on Ш is skew symmetric which implies that the order of Ш is a square or twice a square (if it is finite), and if in addition the principal polarization comes from a rational divisor (as is the case for elliptic curves) then the form is alternating and the order of Ш is a square (if it is finite).
See also
• Birch and Swinnerton-Dyer conjecture
Citations
1. Lang & Tate 1958.
2. Shafarevich 1959.
3. Lind 1940.
4. Selmer 1951.
5. Rubin 1987.
6. Kolyvagin 1988.
7. Cassels 1962.
8. Tate 1963.
9. Swinnerton-Dyer 1967.
10. Poonen & Stoll 1999.
11. Stein 2004.
References
• Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", Proceedings of the London Mathematical Society, Third Series, 12: 259–296, doi:10.1112/plms/s3-12.1.259, ISSN 0024-6115, MR 0163913
• Cassels, John William Scott (1962b), "Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung", Journal für die reine und angewandte Mathematik, 211 (211): 95–112, doi:10.1515/crll.1962.211.95, ISSN 0075-4102, MR 0163915
• Cassels, John William Scott (1991), Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, doi:10.1017/CBO9781139172530, ISBN 978-0-521-41517-0, MR 1144763
• Hindry, Marc; Silverman, Joseph H. (2000), Diophantine geometry: an introduction, Graduate Texts in Mathematics, vol. 201, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98981-5
• Greenberg, Ralph (1994), "Iwasawa Theory and p-adic Deformation of Motives", in Serre, Jean-Pierre; Jannsen, Uwe; Kleiman, Steven L. (eds.), Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0
• Kolyvagin, V. A. (1988), "Finiteness of E(Q) and SH(E,Q) for a subclass of Weil curves", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 52 (3): 522–540, 670–671, ISSN 0373-2436, 954295
• Lang, Serge; Tate, John (1958), "Principal homogeneous spaces over abelian varieties", American Journal of Mathematics, 80 (3): 659–684, doi:10.2307/2372778, ISSN 0002-9327, JSTOR 2372778, MR 0106226
• Lind, Carl-Erik (1940). Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins (Thesis). Vol. 1940. University of Uppsala. 97 pp. MR 0022563.
• Poonen, Bjorn; Stoll, Michael (1999), "The Cassels-Tate pairing on polarized abelian varieties", Annals of Mathematics, Second Series, 150 (3): 1109–1149, arXiv:math/9911267, doi:10.2307/121064, ISSN 0003-486X, JSTOR 121064, MR 1740984
• Rubin, Karl (1987), "Tate–Shafarevich groups and L-functions of elliptic curves with complex multiplication", Inventiones Mathematicae, 89 (3): 527–559, Bibcode:1987InMat..89..527R, doi:10.1007/BF01388984, ISSN 0020-9910, MR 0903383
• Selmer, Ernst S. (1951), "The Diophantine equation ax³+by³+cz³=0", Acta Mathematica, 85: 203–362, doi:10.1007/BF02395746, ISSN 0001-5962, MR 0041871
• Shafarevich, I. R. (1959), "The group of principal homogeneous algebraic manifolds", Doklady Akademii Nauk SSSR (in Russian), 124: 42–43, ISSN 0002-3264, MR 0106227 English translation in his collected mathematical papers
• Stein, William A. (2004), "Shafarevich–Tate groups of nonsquare order" (PDF), Modular curves and abelian varieties, Progr. Math., vol. 224, Basel, Boston, Berlin: Birkhäuser, pp. 277–289, MR 2058655
• Swinnerton-Dyer, P. (1967), "The conjectures of Birch and Swinnerton-Dyer, and of Tate", in Springer, Tonny A. (ed.), Proceedings of a Conference on Local Fields (Driebergen, 1966), Berlin, New York: Springer-Verlag, pp. 132–157, MR 0230727
• Tate, John (1958), WC-groups over p-adic fields, Séminaire Bourbaki; 10e année: 1957/1958, vol. 13, Paris: Secrétariat Mathématique, MR 0105420
• Tate, John (1963), "Duality theorems in Galois cohomology over number fields", Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 288–295, MR 0175892, archived from the original on 2011-07-17
• Weil, André (1955), "On algebraic groups and homogeneous spaces", American Journal of Mathematics, 77 (3): 493–512, doi:10.2307/2372637, ISSN 0002-9327, JSTOR 2372637, MR 0074084
| Wikipedia |
Tate cohomology group
In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by John Tate (1952, p. 297), and are used in class field theory.
Definition
If G is a finite group and A a G-module, then there is a natural map N from $H_{0}(G,A)$ to $H^{0}(G,A)$ taking a representative a to $\sum _{g\in G}ga$ (the sum over all G-conjugates of a). The Tate cohomology groups ${\hat {H}}^{n}(G,A)$ are defined by
• ${\hat {H}}^{n}(G,A)=H^{n}(G,A)$ for $n\geq 1$,
• ${\hat {H}}^{0}(G,A)=\operatorname {coker} N=$ quotient of $H^{0}(G,A)$ by norms of elements of A,
• ${\hat {H}}^{-1}(G,A)=\ker N=$ quotient of norm 0 elements of A by principal elements of A,
• ${\hat {H}}^{n}(G,A)=H_{-n-1}(G,A)$ for $n\leq -2$.
Properties
• If
$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$
is a short exact sequence of G-modules, then we get the usual long exact sequence of Tate cohomology groups:
$\cdots \longrightarrow {\hat {H}}^{n}(G,A)\longrightarrow {\hat {H}}^{n}(G,B)\longrightarrow {\hat {H}}^{n}(G,C)\longrightarrow {\hat {H}}^{n+1}(G,A)\longrightarrow {\hat {H}}^{n+1}(G,B)\cdots $
• If A is an induced G module then all Tate cohomology groups of A vanish.
• The zeroth Tate cohomology group of A is
(Fixed points of G on A)/(Obvious fixed points of G acting on A)
where by the "obvious" fixed point we mean those of the form $\sum ga$. In other words, the zeroth cohomology group in some sense describes the non-obvious fixed points of G acting on A.
The Tate cohomology groups are characterized by the three properties above.
Tate's theorem
Tate's theorem (Tate 1952) gives conditions for multiplication by a cohomology class to be an isomorphism between cohomology groups. There are several slightly different versions of it; a version that is particularly convenient for class field theory is as follows:
Suppose that A is a module over a finite group G and a is an element of $H^{2}(G,A)$, such that for every subgroup E of G
• $H^{1}(E,A)$ is trivial, and
• $H^{2}(E,A)$ is generated by $\operatorname {Res} (a)$, which has order E.
Then cup product with a is an isomorphism:
• ${\hat {H}}^{n}(G,\mathbb {Z} )\longrightarrow {\hat {H}}^{n+2}(G,A)$
for all n; in other words the graded Tate cohomology of A is isomorphic to the Tate cohomology with integral coefficients, with the degree shifted by 2.
Tate-Farrell cohomology
F. Thomas Farrell extended Tate cohomology groups to the case of all groups G of finite virtual cohomological dimension. In Farrell's theory, the groups ${\hat {H}}^{n}(G,A)$ are isomorphic to the usual cohomology groups whenever n is greater than the virtual cohomological dimension of the group G. Finite groups have virtual cohomological dimension 0, and in this case Farrell's cohomology groups are the same as those of Tate.
See also
• Herbrand quotient
• Class formation
References
• M. F. Atiyah and C. T. C. Wall, "Cohomology of Groups", in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2, Chapter IV. See section 6.
• Brown, Kenneth S. (1982). Cohomology of Groups. Graduate Texts in Mathematics. Vol. 87. New York-Berlin: Springer-Verlag. ISBN 0-387-90688-6. MR 0672956.
• Farrell, F. Thomas (1977). "An extension of tate cohomology to a class of infinite groups". Journal of Pure and Applied Algebra. 10 (2): 153–161. doi:10.1016/0022-4049(77)90018-4. MR 0470103.
• Tate, John (1952), "The higher dimensional cohomology groups of class field theory", Annals of Mathematics, 2, 56: 294–297, doi:10.2307/1969801, JSTOR 1969801, MR 0049950
| Wikipedia |
Tate conjecture
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.
Tate conjecture
John Tate in 1993
FieldAlgebraic geometry and number theory
Conjectured byJohn Tate
Conjectured in1963
Known casesdivisors on abelian varieties
ConsequencesStandard conjectures on algebraic cycles
Statement of the conjecture
Let V be a smooth projective variety over a field k which is finitely generated over its prime field. Let ks be a separable closure of k, and let G be the absolute Galois group Gal(ks/k) of k. Fix a prime number ℓ which is invertible in k. Consider the ℓ-adic cohomology groups (coefficients in the ℓ-adic integers Zℓ, scalars then extended to the ℓ-adic numbers Qℓ) of the base extension of V to ks; these groups are representations of G. For any i ≥ 0, a codimension-i subvariety of V (understood to be defined over k) determines an element of the cohomology group
$H^{2i}(V_{k_{s}},\mathbf {Q} _{\ell }(i))=W$
which is fixed by G. Here Qℓ(i ) denotes the ith Tate twist, which means that this representation of the Galois group G is tensored with the ith power of the cyclotomic character.
The Tate conjecture states that the subspace WG of W fixed by the Galois group G is spanned, as a Qℓ-vector space, by the classes of codimension-i subvarieties of V. An algebraic cycle means a finite linear combination of subvarieties; so an equivalent statement is that every element of WG is the class of an algebraic cycle on V with Qℓ coefficients.
Known cases
The Tate conjecture for divisors (algebraic cycles of codimension 1) is a major open problem. For example, let f : X → C be a morphism from a smooth projective surface onto a smooth projective curve over a finite field. Suppose that the generic fiber F of f, which is a curve over the function field k(C), is smooth over k(C). Then the Tate conjecture for divisors on X is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian variety of F.[1] By contrast, the Hodge conjecture for divisors on any smooth complex projective variety is known (the Lefschetz (1,1)-theorem).
Probably the most important known case is that the Tate conjecture is true for divisors on abelian varieties. This is a theorem of Tate for abelian varieties over finite fields, and of Faltings for abelian varieties over number fields, part of Faltings's solution of the Mordell conjecture. Zarhin extended these results to any finitely generated base field. The Tate conjecture for divisors on abelian varieties implies the Tate conjecture for divisors on any product of curves C1 × ... × Cn.[2]
The (known) Tate conjecture for divisors on abelian varieties is equivalent to a powerful statement about homomorphisms between abelian varieties. Namely, for any abelian varieties A and B over a finitely generated field k, the natural map
${\text{Hom}}(A,B)\otimes _{\mathbf {Z} }\mathbf {Q} _{\ell }\to {\text{Hom}}_{G}\left(H_{1}\left(A_{k_{s}},\mathbf {Q} _{\ell }\right),H_{1}\left(B_{k_{s}},\mathbf {Q} _{\ell }\right)\right)$
is an isomorphism.[3] In particular, an abelian variety A is determined up to isogeny by the Galois representation on its Tate module H1(Aks, Zℓ).
The Tate conjecture also holds for K3 surfaces over finitely generated fields of characteristic not 2.[4] (On a surface, the nontrivial part of the conjecture is about divisors.) In characteristic zero, the Tate conjecture for K3 surfaces was proved by André and Tankeev. For K3 surfaces over finite fields of characteristic not 2, the Tate conjecture was proved by Nygaard, Ogus, Charles, Madapusi Pera, and Maulik.
Totaro (2017) surveys known cases of the Tate conjecture.
Related conjectures
Let X be a smooth projective variety over a finitely generated field k. The semisimplicity conjecture predicts that the representation of the Galois group G = Gal(ks/k) on the ℓ-adic cohomology of X is semisimple (that is, a direct sum of irreducible representations). For k of characteristic 0, Moonen (2017) showed that the Tate conjecture (as stated above) implies the semisimplicity of
$H^{i}\left(X\times _{k}{\overline {k}},\mathbf {Q} _{\ell }(n)\right).$
For k finite of order q, Tate showed that the Tate conjecture plus the semisimplicity conjecture would imply the strong Tate conjecture, namely that the order of the pole of the zeta function Z(X, t) at t = q−j is equal to the rank of the group of algebraic cycles of codimension j modulo numerical equivalence.[5]
Like the Hodge conjecture, the Tate conjecture would imply most of Grothendieck's standard conjectures on algebraic cycles. Namely, it would imply the Lefschetz standard conjecture (that the inverse of the Lefschetz isomorphism is defined by an algebraic correspondence); that the Künneth components of the diagonal are algebraic; and that numerical equivalence and homological equivalence of algebraic cycles are the same.
Notes
1. D. Ulmer. Arithmetic Geometry over Global Function Fields (2014), 283-337. Proposition 5.1.2 and Theorem 6.3.1.
2. J. Tate. Motives (1994), Part 1, 71-83. Theorem 5.2.
3. J. Tate. Arithmetical Algebraic Geometry (1965), 93-110. Equation (8).
4. K. Madapusi Pera. Inventiones Mathematicae. Theorem 1.
5. J. Tate. Motives (1994), Part 1, 71-83. Theorem 2.9.
References
• André, Yves (1996), "On the Shafarevich and Tate conjectures for hyper-Kähler varieties", Mathematische Annalen, 305: 205–248, doi:10.1007/BF01444219, MR 1391213, S2CID 122949797
• Faltings, Gerd (1983), "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern", Inventiones Mathematicae, 73 (3): 349–366, Bibcode:1983InMat..73..349F, doi:10.1007/BF01388432, MR 0718935, S2CID 121049418
• Madapusi Pera, K. (2013), "The Tate conjecture for K3 surfaces in odd characteristic", Inventiones Mathematicae, 201 (2): 625–668, arXiv:1301.6326, Bibcode:2013arXiv1301.6326M, doi:10.1007/s00222-014-0557-5, S2CID 253746655
• Moonen, Ben (2017), A remark on the Tate conjecture, arXiv:1709.04489v1
• Tate, John (1965), "Algebraic cycles and poles of zeta functions", in Schilling, O. F. G. (ed.), Arithmetical Algebraic Geometry, New York: Harper and Row, pp. 93–110, MR 0225778
• Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2 (2): 134–144, Bibcode:1966InMat...2..134T, doi:10.1007/bf01404549, MR 0206004, S2CID 245902
• Tate, John (1994), "Conjectures on algebraic cycles in ℓ-adic cohomology", Motives, Proceedings of Symposia in Pure Mathematics, vol. 55, American Mathematical Society, pp. 71–83, ISBN 0-8218-1636-5, MR 1265523
• Ulmer, Douglas (2014), "Curves and Jacobians over function fields", Arithmetic Geometry over Global Function Fields, Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, pp. 283–337, doi:10.1007/978-3-0348-0853-8, ISBN 978-3-0348-0852-1
• Totaro, Burt (2017), "Recent progress on the Tate conjecture", Bulletin of the American Mathematical Society, New Series, 54 (4): 575–590, doi:10.1090/bull/1588
External links
• James Milne, The Tate conjecture over finite fields (AIM talk).
| Wikipedia |
Tate module
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G(Ks) (the Ks-valued points of G). In this case, the Tate module of A is equipped with an action of the absolute Galois group of K, and it is referred to as the Tate module of G.
Not to be confused with Hodge–Tate module.
Definition
Given an abelian group A and a prime number p, the p-adic Tate module of A is
$T_{p}(A)={\underset {\longleftarrow }{\lim }}A[p^{n}]$
where A[pn] is the pn torsion of A (i.e. the kernel of the multiplication-by-pn map), and the inverse limit is over positive integers n with transition morphisms given by the multiplication-by-p map A[pn+1] → A[pn]. Thus, the Tate module encodes all the p-power torsion of A. It is equipped with the structure of a Zp-module via
$z(a_{n})_{n}=((z{\text{ mod }}p^{n})a_{n})_{n}.$
Examples
The Tate module
When the abelian group A is the group of roots of unity in a separable closure Ks of K, the p-adic Tate module of A is sometimes referred to as the Tate module (where the choice of p and K are tacitly understood). It is a free rank one module over Zp with a linear action of the absolute Galois group GK of K. Thus, it is a Galois representation also referred to as the p-adic cyclotomic character of K. It can also be considered as the Tate module of the multiplicative group scheme Gm,K over K.
The Tate module of an abelian variety
Given an abelian variety G over a field K, the Ks-valued points of G are an abelian group. The p-adic Tate module Tp(G) of G is a Galois representation (of the absolute Galois group, GK, of K).
Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then Tp(G) is a free module over Zp of rank 2d, where d is the dimension of G.[1] In the other case, it is still free, but the rank may take any value from 0 to d (see for example Hasse–Witt matrix).
In the case where p is not equal to the characteristic of K, the p-adic Tate module of G is the dual of the étale cohomology $H_{\text{et}}^{1}(G\times _{K}K^{s},\mathbf {Z} _{p})$.
A special case of the Tate conjecture can be phrased in terms of Tate modules.[2] Suppose K is finitely generated over its prime field (e.g. a finite field, an algebraic number field, a global function field), of characteristic different from p, and A and B are two abelian varieties over K. The Tate conjecture then predicts that
$\mathrm {Hom} _{K}(A,B)\otimes \mathbf {Z} _{p}\cong \mathrm {Hom} _{G_{K}}(T_{p}(A),T_{p}(B))$
where HomK(A, B) is the group of morphisms of abelian varieties from A to B, and the right-hand side is the group of GK-linear maps from Tp(A) to Tp(B). The case where K is a finite field was proved by Tate himself in the 1960s.[3] Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".[4]
In the case of a Jacobian over a curve C over a finite field k of characteristic prime to p, the Tate module can be identified with the Galois group of the composite extension
$k(C)\subset {\hat {k}}(C)\subset A^{(p)}\ $
where ${\hat {k}}$ is an extension of k containing all p-power roots of unity and A(p) is the maximal unramified abelian p-extension of ${\hat {k}}(C)$.[5]
Tate module of a number field
The description of the Tate module for the function field of a curve over a finite field suggests a definition for a Tate module of an algebraic number field, the other class of global field, introduced by Kenkichi Iwasawa. For a number field K we let Km denote the extension by pm-power roots of unity, ${\hat {K}}$ the union of the Km and A(p) the maximal unramified abelian p-extension of ${\hat {K}}$. Let
$T_{p}(K)=\mathrm {Gal} (A^{(p)}/{\hat {K}})\ .$
Then Tp(K) is a pro-p-group and so a Zp-module. Using class field theory one can describe Tp(K) as isomorphic to the inverse limit of the class groups Cm of the Km under norm.[5]
Iwasawa exhibited Tp(K) as a module over the completion Zp[[T]] and this implies a formula for the exponent of p in the order of the class groups Cm of the form
$\lambda m+\mu p^{m}+\kappa \ .$
The Ferrero–Washington theorem states that μ is zero.[6]
See also
• Tate conjecture
• Tate twist
• Iwasawa theory
Notes
1. Murty 2000, Proposition 13.4
2. Murty 2000, §13.8
3. Tate 1966
4. Faltings 1983
5. Manin & Panchishkin 2007, p. 245
6. Manin & Panchishkin 2007, p. 246
References
• Faltings, Gerd (1983), "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern", Inventiones Mathematicae, 73 (3): 349–366, Bibcode:1983InMat..73..349F, doi:10.1007/BF01388432, S2CID 121049418
• "Tate module", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002
• Murty, V. Kumar (2000), Introduction to abelian varieties, CRM Monograph Series, vol. 3, American Mathematical Society, ISBN 978-0-8218-1179-5
• Section 13 of Rohrlich, David (1994), "Elliptic curves and the Weil–Deligne group", in Kisilevsky, Hershey; Murty, M. Ram (eds.), Elliptic curves and related topics, CRM Proceedings and Lecture Notes, vol. 4, American Mathematical Society, ISBN 978-0-8218-6994-9
• Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2 (2): 134–144, Bibcode:1966InMat...2..134T, doi:10.1007/bf01404549, MR 0206004, S2CID 245902
| Wikipedia |
Tate topology
In mathematics, the Tate topology is a Grothendieck topology of the space of maximal ideals of a k-affinoid algebra, whose open sets are the admissible open subsets and whose coverings are the admissible open coverings.
References
• Conrad, Brian (2008), "Several approaches to non-Archimedean geometry", in Thakur, Dinesh S.; Savitt, David (eds.), p-adic geometry, Univ. Lecture Ser., vol. 45, Providence, R.I.: American Mathematical Society, pp. 9–63, ISBN 978-0-8218-4468-7, MR 2482345
| Wikipedia |
Tate twist
In number theory and algebraic geometry, the Tate twist,[1] [2] named after John Tate, is an operation on Galois modules.
For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate twist of V, denoted V(1), is the representation on the tensor product V⊗Qp(1), where Qp(1) is the p-adic cyclotomic character (i.e. the Tate module of the group of roots of unity in the separable closure Ks of K). More generally, if m is a positive integer, the mth Tate twist of V, denoted V(m), is the tensor product of V with the m-fold tensor product of Qp(1). Denoting by Qp(−1) the dual representation of Qp(1), the -mth Tate twist of V can be defined as
$V\otimes \mathbf {Q} _{p}(-1)^{\otimes m}.$
References
1. 'The Tate Twist', in Lecture Notes in Mathematics', Vol 1604, 1995, Springer, Berlin p.98-102
2. 'The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist
| Wikipedia |
Tatiana Roque
Tatiana Marins Roque (born April 24, 1970) is a Brazilian historian of mathematics and politician.
Tatiana Roque
Tatiana Roque in 2022
Born
Tatiana Marins Roque
(1970-04-24) April 24, 1970
Rio de Janeiro, Brazil
Occupations
• Mathematician
• professor
• politician
AwardsJabuti Award
Websitetatianaroque.com.br
Academic career
Roque is a professor at the Institute of Mathematics (IM) of the Federal University of Rio de Janeiro (UFRJ),[1] with a Ph.D. in Production Engineering from the Alberto Luiz Coimbra Institute of Graduate Studies and Research in Engineering (Coppe), also at UFRJ.[2] Her research area covers the historiography of mathematics, the relationship between history and mathematics education, and the history of differential equations and celestial mechanics theories at the turn of the 19th to the 20th century. Her book História da matemática: uma visão crítica, desfazendo mitos e lendas (2012) was one of the winners of the 2013 Jabuti Award.[3][4][5]
She was a guest speaker at the 2018 International Congress of Mathematicians, which took place in Rio de Janeiro, and served as the coordinator of UFRJ's Forum of Science and Culture between 2019 and 2022.[6][7]
Political career
In the 2018 elections, Roque ran as a candidate for Federal Deputy for the Socialism and Liberty Party (PSOL) in Rio de Janeiro. She received 15,789 votes but was not elected.[8] In the 2022 elections, she ran for the same position under the Brazilian Socialist Party (PSB).[9] She received 30,764 votes but was not elected, becoming the first alternate to Eduardo Bandeira de Mello.[10][11]
In February 2023, she assumed the position of Secretary of Science and Technology of the city of Rio de Janeiro, being appointed by Mayor Eduardo Paes.[2][12][13]
References
1. "Com comitês de gênero, matemáticas brasileiras ganham força e estreiam em congresso centenário". Gênero e Número (in Portuguese). June 19, 2018. Archived from the original on September 14, 2018. Retrieved September 10, 2022.
2. França, Victor (February 3, 2023). "Professora da UFRJ assumirá Ciência e Tecnologia do Rio". Conexão UFRJ (in Portuguese). Archived from the original on February 4, 2023. Retrieved February 4, 2023.
3. "Premiados do Ano | 64º Prêmio Jabuti". Prêmio Jabuti. Archived from the original on August 11, 2020. Retrieved September 10, 2022.
4. "Anunciados os vencedores do Prêmio Jabuti 2013; confira a lista". GZH (in Portuguese). October 17, 2013. Archived from the original on September 10, 2022. Retrieved September 10, 2022.
5. "Livro da prof.ª Tatiana Roque conquista Prêmio Jabuti 2013". Biblioteca IM. October 18, 2013. Archived from the original on April 11, 2019.
6. "Tatiana Roque, nova coordenadora do FCC, convida para o ciclo de palestras "Desastres e mudanças climáticas"". Fórum de Ciência e Cultura da UFRJ. July 31, 2019. Archived from the original on September 10, 2022. Retrieved September 10, 2022.
7. Neto, Nelson Lima (July 6, 2022). "Coordenadora do Fórum de Ciência e Cultura da UFRJ sai do cargo para ser candidata". O Globo (in Portuguese). Archived from the original on July 16, 2022. Retrieved September 10, 2022.
8. "Tatiana Roque 5010 (PSOL) Deputada Federal | Rio de Janeiro". Eleições 2018 (in Portuguese). Archived from the original on April 7, 2022.
9. Tussini, Gabriel (August 24, 2022). ""Nunca foi tão evidente a relação entre ciência e política", diz Tatiana Roque". ((o))eco (in Portuguese). Archived from the original on August 27, 2022. Retrieved September 10, 2022.
10. "TATIANA ROQUE". Poder360. Archived from the original on September 10, 2022. Retrieved September 10, 2022.
11. "Deputados federais eleitos no Rio de Janeiro: apuração e resultados das Eleições 2022". Folha de S.Paulo (in Portuguese). Archived from the original on October 3, 2022. Retrieved October 8, 2022.
12. Sartori, Caio (January 27, 2023). "Depois do PT, Paes acolhe o PSB com secretaria de Ciência e Tecnologia". Valor Econômico (in Portuguese). Archived from the original on January 28, 2023. Retrieved February 4, 2023.
13. Coelho, Henrique (February 3, 2023). "Prefeito do Rio de Janeiro dá posse a 14 novos secretários". G1 (in Portuguese). Archived from the original on February 3, 2023. Retrieved February 4, 2023.
External links
• Media related to Tatiana Roque at Wikimedia Commons
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Tatiana Shubin
Tatiana Shubin is a Soviet and American mathematician known for her work developing math circles, social structures for the mathematical enrichment of secondary-school students, especially among the Navajo and other Native American people. She is a professor of mathematics at San José State University in California.[1]
Education and career
Shubin is originally from Ukraine, the daughter of a criminologist and a lawyer; she is of Jewish descent on her father's side. When she was ten, her family moved to Almaty in Kazakhstan, where her father had taken a university teaching position. After competing in the All Siberian Mathematics Competition she was invited to a special science boarding school in Akademgorodok, but after spending 8th grade there her parents brought her back to Almaty where she finished high school at age 16. She studied for five years at Moscow State University, earning a bachelor's degree there, but was expelled for non-participation in political activities and instead earned a master's degree at Kazakh State University in Almaty.[2]
After obtaining a letter of invitation from an Israeli, she was allowed to leave the Soviet Union, spent nine months in Austria, and then emigrated to the US in 1978, with support from the Tolstoy Foundation.[2] She completed a Ph.D. at the University of California, Santa Barbara in 1983,[1] and joined the San José State University faculty as a lecturer in 1985.[2][3]
Math circles
Shubin founded the San José Math Circles.[2] She co-founded the first math teachers' circle in 2006, and is a leader of the Math Teachers’ Circle Network that developed out of this circle. She was a co-founder of the Navajo Nation Math Circles project in 2012,[3] and is a director of the Alliance of Indigenous Math Circles.[4]
Publications
Shubin is the coeditor of several books on mathematics:
• Mathematical Adventures for Students and Amateurs (edited with David F. Hayes, Mathematical Association of America, 2004)[5]
• Expeditions in Mathematics (edited with Gerald L. Alexanderson and David F. Hayes, Mathematical Association of America, 2011)[6]
• Inspiring Mathematics: Lessons from the Navajo Nation Math Circles (edited with Dave Auckly, Bob Klein, and Amanda Serenevy, MSRI Mathematical Circles Library 24, Mathematical Sciences Research Institute and American Mathematical Society, 2019)[7]
Her work developing math circles among the Navajo was featured in the documentary film Navajo Math Circles (2016), broadcast on the Public Broadcasting System.[8]
Recognition
Shubin was the 2006 winner of the Award for Distinguished College or University Teaching of Mathematics of the Golden Section (Northern California, Nevada, and Hawaii) of the Mathematical Association of America.[3] She was the 2017 winner of the Mary P. Dolciani Award of the Mathematical Association of America.[9]
She has been named a Sequoyah Fellow by the American Indian Science and Engineering Society. The Navajo Todích’íí’nii (Bitter Water) clan have adopted her as a member.[4]
References
1. "Tatiana Shubin", Faculty, SJSU Department of Mathematics & Statistics, retrieved 2020-10-28; "Tatiana Shubin", People, San José State University, retrieved 2020-10-28; Faculty (alphabetical), SJSU Department of Mathematics & Statistics, retrieved 2020-10-28
2. Tran, Jacqueline My Anh (2004), "A Mathematician Receives a Warm Welcome in a Free Society", 2004 AWM Essay Contest, Association for Women in Mathematics, retrieved 2020-10-28
3. Tatiana Shubin, Tulsa Math Teachers' Circle, retrieved 2020-10-28
4. Our people: Directors, Alliance of Indigenous Math Circles, retrieved 2020-10-28
5. Reviews of Mathematical Adventures for Students and Amateurs:
• Althoen, Steven C., zbMATH, Zbl 1049.00002{{citation}}: CS1 maint: untitled periodical (link)
• Ashbacher, Charles (2003–2004), Journal of Recreational Mathematics, 32 (4): 319–320, ProQuest 89064526{{citation}}: CS1 maint: untitled periodical (link)
• Coupland, Mary (2005), The Australian Mathematics Teacher, 61 (1), Gale A164525443{{citation}}: CS1 maint: untitled periodical (link)
• Glass, Darren (July 2004), "Review", MAA Reviews, Mathematical Association of America
• Rogge, William (August 2005), The Mathematics Teacher, 99 (1): 76, JSTOR 27971867{{citation}}: CS1 maint: untitled periodical (link)
• Rosoff, Jeffrey; Dobler, Carolyn Pillers (November 2005), The American Statistician, 59 (4): 352, doi:10.1198/tas.2005.s35, JSTOR 27643711, S2CID 121040201, ProQuest 228484025{{citation}}: CS1 maint: untitled periodical (link)
6. Reviews of Expeditions in Mathematics:
• Daven, Mike (December 2011), "Review", MAA Reviews, Mathematical Association of America
• Seaberg, Rebecca (December 2012 – January 2013), The Mathematics Teacher, 106 (5): 398, doi:10.5951/mathteacher.106.5.0398, JSTOR 10.5951/mathteacher.106.5.0398{{citation}}: CS1 maint: untitled periodical (link)
7. Review of Inspiring Mathematics: Lessons from the Navajo Nation Math Circles:
• Olszewski, Peter (February 2020), "Review", MAA Reviews, Mathematical Association of America
8. Professor's Project Helps Navajo Students Add Interest in Math, Kansas State University, 14 September 2016, retrieved 2020-10-28 – via Newswise
9. "Shubin Honored with Mary P. Dolciani Award: MTC Network leader honored for bringing Math Circles to new communities", MTCircular, Math Teachers' Circle Network, Spring 2018, retrieved 2020-10-28
External links
• Home page at SJSU
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Tatiana Toro
Tatiana Toro is a Colombian-American mathematician at the University of Washington.[1] Her research is "at the interface of geometric measure theory, harmonic analysis and partial differential equations".[2] Toro was appointed director of the Simons Laufer Mathematical Sciences Institute for 2022–2027.[3]
Tatiana Toro
Toro in 2016
Born1964 (age 58–59)
Colombia
Alma mater
• National University of Colombia (BS)
• Stanford University (Ph.D.)
Awards
• Guggenheim Fellowship (2015)
• Blackwell–Tapia prize (2020)
Scientific career
FieldsMathematics
Institutions
• University of Washington
• Simons Laufer Mathematical Sciences Institute
ThesisFunctions in W2,2(R2) have Lipschitz graphs (1992)
Doctoral advisorLeon Simon
Education and employment
Toro was born in 1964 in Colombia,[2][4] competed for Colombia in the 1981 International Mathematical Olympiad,[5] and earned a bachelor's degree from the National University of Colombia.[6] In 1992, she was awarded her PhD at Stanford University, under the supervision of Leon Simon.[7] After short-term positions at the Institute for Advanced Study, University of California, Berkeley, and University of Chicago, she joined the University of Washington faculty in 1996.[1] Since August 2022, Toro serves as the director of Simons Laufer Mathematical Sciences Institute (formerly MSRI).[4] She will maintain her tenure at the University of Washington throughout her term.[8]
Honors and awards
Toro was an invited speaker at the International Congress of Mathematicians in 2010.[9] She became a Guggenheim Fellow in 2015.[2] She was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to geometric measure theory, potential theory, and free boundary theory".[10] At the University of Washington, she was the Robert R. & Elaine F. Phelps Professor in Mathematics from 2012 to 2016[11] and is currently the Craig McKibben and Sarah Merner Professor. Toro was named MSRI Chancellor's Professor for 2016–17.[12] She was awarded the 2020 Blackwell-Tapia Prize.[13] She was elected fellow of the American Academy of Arts and Sciences (AAAS) in 2020.[14] Toro was honored as the AWM/MAA Falconer Lecturer in 2023.[15]
References
1. Curriculum vitae: Tatiana Toro (PDF), retrieved 2015-10-06.
2. Guggenheim fellows: Tatiana Toro, John Simon Guggenheim Memorial Foundation, retrieved 2015-10-06.
3. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved 2021-07-12.
4. "Tatiana Toro, the Colombian appointed director of the US Mathematical Sciences Research Institute". El Espectador (in Spanish). 2021-06-15.
5. Tatiana Toro, International Mathematical Olympiad, retrieved 2015-10-06.
6. Tatiana Toro, Radcliffe Institute for Advanced Study, Harvard University, retrieved 2015-10-06.
7. Tatiana Toro at the Mathematics Genealogy Project
8. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved 2021-06-15.
9. ICM Plenary and Invited Speakers since 1897, International Mathematical Union, archived from the original on 2017-11-24, retrieved 2015-10-06.
10. 2017 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2016-11-06.
11. Recent faculty awards, University of Washington, retrieved 2016-11-06.
12. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved 2021-06-07.
13. "The Latest", American Mathematical Society, retrieved 2020-07-21
14. "AAAS Fellows Elected" (PDF), Notices of the American Mathematical Society
15. "AWM-MAA Etta Zuber Falconer Lecturer Announced". Association for Women in Mathematics (AWM). 2023-02-07. Retrieved 2023-03-15.
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Tatsujiro Shimizu
Tatsujiro Shimizu (清水 辰次郎, Shimizu Tatsujirō, 7 April 1897 – 8 November 1992) was a Japanese mathematician working in the field of complex analysis. He was the founder of the Japanese Association of Mathematical Sciences.[1]
Tatsujiro Shimizu
Born7 April 1897
Tokyo
Died8 November 1992
Uji, Kyoto
NationalityJapanese
Alma materTokyo Imperial University
Known forAhlfors-Shimizu characteristic, foundation of Japanese Association of Mathematical Sciences
Scientific career
FieldsMathematics, Complex Analysis, Numerical Analysis
InstitutionsTokyo Imperial University, Osaka Imperial University, Kobe University, University of Osaka Prefecture, Tokyo University of Science
Doctoral studentsShizuo Kakutani
Life and career
Shimizu graduated from the Department of Mathematics, School of Science, Tokyo Imperial University in 1924, and stayed there working as a staff member. In 1932 he moved to Osaka Imperial University and became a professor. He made contributions to the establishment of the Department of Mathematics there. In 1949, Shimizu left Osaka and took up a professorship at Kobe University. After two years, he moved again to Osaka Prefectural University. From 1961 he was a professor at the Tokyo University of Science.[2][3]
In 1948, seeing the difficulty in publication of paper in mathematics, Shimizu started a new journal Mathematica Japonicae, for papers of pure and applied mathematics in general, on his own funds. The journal served as the foundation of the Japanese Association of Mathematical Sciences.[2][3]
Shimizu remained active in mathematics into old age. He gave talks at the meeting of the Mathematical Society of Japan until 90 years old. He died in Uji City, Kyoto Prefecture, on November 8, 1992, at the age 95.[2][3]
Works
Function theory
The first works of Shimizu treated topics of function theory, in particular the theory of meromorphic functions. A new form of the Nevanlinna characteristic generalised by him (and separately by Ahlfors) is now known as the Ahlfors-Shimizu characteristic. Additionally, with the idea of function groups, he attained a profound result on the construction of Riemann surface of meromorphic functions. In 1931, as a pioneer in Japan responding to Fatou's study of the theory of iteration of the algebraic functions, Shimizu published two papers introducing the subject in Japanese journals.[4]
Applied mathematics
Since he moved to Osaka in 1932, Shimizu has been interested in the application of mathematical methods into science and technology. He broadly worked on the existence conditions of limit cycles, numerical analysis and applied analysis (including solving ordinary differential equations, numerical solutions and non-linear oscillations), computing machines and devices, as well as artificial intelligence (especially in solving arithmetic problems through electronic computer). His research in these areas was continued through this career. He was also involved in operations research and mathematics in management sciences, as well as probability theory and mathematical statistics.[2][3]
Notable students
Among his students is Shizuo Kakutani, Osaka University, 1941[5]
Books
• Statistical Machine Computing Method (「統計機械計算法」)[6]
• Practical Mathematics (「実用数学」)[6]
• Non-linear Oscillation Theory (「非線形振動論」)[7]
• Applied Mathematics (「応用数学」)[8]
References
1. "Tatsujiro Shimizu - Biography". Maths History. Retrieved 2022-07-02.
2. "Tatsujiro Shimizu". www.jams.jp. Retrieved 2022-07-02.
3. "Japanese Association of Mathematical Sciences". Maths History. Retrieved 2022-06-28.
4. Alexander, Daniel S.; Iavernaro, Felice; Rosa, Alessandro (2012). Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906-1942. American Mathematical Soc. ISBN 978-0-8218-4464-9.
5. "Tatsujiro Shimizu - The Mathematics Genealogy Project". www.mathgenealogy.org. Retrieved 2022-06-28.
6. 20世紀日本人名事典,367日誕生日大事典. "清水 辰次郎とは". コトバンク (in Japanese). Retrieved 2022-06-28.
7. 日本人名大辞典+Plus, デジタル版. "清水辰次郎とは". コトバンク (in Japanese). Retrieved 2022-06-28.
8. "朝倉数学講座 応用数学 （復刊）｜朝倉書店". www.asakura.co.jp. Retrieved 2022-06-28.
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Tatyana Afanasyeva
Tatyana Alexeyevna Afanasyeva (Russian: Татья́на Алексе́евна Афана́сьева) (Kiev, 19 November 1876 – Leiden, 14 April 1964) (also known as Tatiana Ehrenfest-Afanaseva or spelled Afanassjewa) was a Russian/Dutch mathematician and physicist who made contributions to the fields of statistical mechanics and statistical thermodynamics.[1] On 21 December 1904, she married Austrian physicist Paul Ehrenfest (1880–1933). They had two daughters and two sons; one daughter, Tatyana Ehrenfest, also became a mathematician.
Early life
Afanasyeva was born in Kiev, Ukraine, then part of the Russian Empire. Her father was Alexander Afanassjev, a chief engineer on the Imperial Railways, who would bring Tatyana on his travels around the Russian Empire. Her father died while she was still young, so she moved to St Petersburg in Russia to live with her aunt Sonya, and uncle Peter Afanassjev, a professor at the St Petersburg Polytechnic Institute.[2]
Tatyana attended normal school in St Petersburg with a specialty in mathematics and science. At the time, women were not allowed to attend universities in Russian territory, so after graduating from normal school, Tatyana began studying mathematics and physics at the Women's University in St Petersburg under Orest Chvolson. In 1902, she transferred to University of Göttingen in Germany to continue her studies with Felix Klein and David Hilbert.[2]
At the University of Göttingen, Tatyana met Paul Ehrenfest. When Ehrenfest discovered that Tatyana could not attend a mathematics club meeting, he argued with the school to have the rule changed. A friendship developed between the two, and they married in 1904, later returned to St Petersburg in 1907. Under Russian law, marriage was not allowed between two people of different religions. Since Tatyana was a Russian Orthodox and Ehrenfest was Jewish, they both decided to officially renounce their religions in order to remain married.[1]
In 1912 they moved to Leiden in the Netherlands, where Paul Ehrenfest was appointed to succeed Hendrik Lorentz as professor at the University of Leiden,[3] and where the couple lived throughout their career.
Works in mathematics and physics
Initially, Tatyana collaborated closely with her husband, most famously on their classic 1911 review of the statistical mechanics of Boltzmann.[4] The Conceptual Foundations of the Statistical Approach in Mechanics, by Paul and Tatyana Ehrenfest was originally published in 1911 as an article for the German Encyklopädie der mathematischen Wissenschaften (Encyclopedia of Mathematical Sciences), and has since been translated and republished.
She published many papers on various topics such as randomness[5] and entropy,[6] and teaching geometry to children.[7][8]
Contact with Einstein
Albert Einstein was a frequent guest in the 1920s at her home Witte Rozenstraat 57 in Leiden, witness the many signatures on the wall. Later Einstein departed for Princeton University and Afanasyeva corresponded. The archives of Museum Boerhaave in Leiden has three letters to her from Einstein.[9]
Afanasjeva contacted Einstein for his advice on her manuscript on thermodynamics and inquired about a translator. She wanted to give thermodynamics a rigorous mathematical foundation which was lacking and describe pressure, temperature and entropy in changing systems. Einstein responded on 12 August 1947 that he applauded her approach but he also had some criticisms:
"Ich habe den Eindruck gewonnen, dass Sie ein bisschen von logischen Putzteufel besessen sind, und dass daran die Übersichtlichkeit des Buches leide."
(Translation: I have got the impression, that you are possessed somewhat by a logical polishing devil, and that the clarity of the book suffers.)
Einstein did not suggest a translator and sent the manuscript back to Afanasjeva who paid herself for its publication in 1956 as Die Grundlagen der Thermodynamik with Brill Publishers in Leiden with some, but not all of Einstein's corrections.[9]
Legacy
The Dutch Physics Council sponsors the Ehrenfest-Afanassjewa thesis award.[10]
References
1. "Tatyana Ehrenfest-Afanassyeva". www.epigenesys.eu. Retrieved 27 May 2017.
2. "Ehrenfest-Afanassjewa biography". www-history.mcs.st-and.ac.uk. Retrieved 27 May 2017.
3. van Delft, Dirk (April 2006), "Albert Einstein in Leiden" (PDF), Physics Today, vol. 59, no. 4, pp. 57–62, Bibcode:2006PhT....59d..57D, doi:10.1063/1.2207039. See p. 57: "In 1912 Ehrenfest succeeded Hendrik Antoon Lorentz (1853–1928) as professor of theoretical physics at Leiden. ... In Leiden, the Ehrenfests moved into a Russian-style villa designed by Ehrenfest’s Russian wife Tatiana Afanashewa, a mathematician."
4. P. Ehrenfest & T. Ehrenfest (1911) Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, in: Enzyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen. Band IV, 2. Teil ( F. Klein and C. Müller (eds.). Leipzig: Teubner, pp. 3–90. Translated as The conceptual Foundations of the Statistical Approach in Mechanics. New York: Cornell University Press, 1959. ISBN 0-486-49504-3
5. T. Ehrenfest-Afanassjewa, On the Use of the Notion "Probability" in Physics Am. J. Phys. 26: 388 (1958)
6. T. Ehrenfest-Afanassjewa, Die Grundlagen der Thermodynamik (Leiden 1956)
7. Tatjana Ehrenfest-Afanassjewa, "Exercises in Experimental Geometry. 1931" (PDF). Archived from the original (PDF) on 1 April 2013. Retrieved 4 February 2006. (214 KiB).
8. Ed de Moor Van Vormleer naar Realistische Meetkunde, Thesis, Utrecht (1999).
9. Margriet van der Heijden (2019). "Einsteins onbekende brieven aan Afanassjewa". NRC (2–3 February 2019): W4–W5.
10. "Ehrenfest-Afanassjewa thesis award". Leiden: Dutch Physics Council.
Sources
• Klein, Martin J. (1972). The making of a theoretical physicist (1. ed., 2. print. ed.). Amsterdam: North-Holland Publ. ISBN 978-0720401639.
• Pyenson, Lewis (1995). "Ehrenfest, Tatyana Afanaseva". In McMurray, Emily J.; Kosek, Jane Kelly; Valade III, Roger M. (eds.). Notable twentieth-century scientists. Detroit, MI: Gale Research. ISBN 9780810391819.
Further reading
• Vogt, Annette B. (1970–1980). "Ehrenfest-Afanas'eva, Tatiana A.". Dictionary of Scientific Biography. Vol. 20. New York: Charles Scribner's Sons. pp. 356–358. ISBN 978-0-684-10114-9.
External links
Wikimedia Commons has media related to Tatyana Afanasyeva.
• Tatiana Ehrenfest-Afanaseva CWP UCLA biography
• Paul and Tatiana Ehrenfest: The Conceptual Foundations of the Statistical Approach in Mechanics, translation Michael J. Moravesik, Dover publications New York 1990 (reprint of the edition of 1959), with a preface by T. Ehrenfest-Afanassjewa.
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Tatyana Ehrenfest
Tatyana Pavlovna Ehrenfest, later van Aardenne-Ehrenfest, (Vienna, October 28, 1905 – Dordrecht, November 29, 1984) was a Dutch mathematician. She was the daughter of Paul Ehrenfest (1880–1933) and Tatyana Afanasyeva (1876–1964).
Under her married name, Tanja van Aardenne-Ehrenfest, she is known for her contributions to De Bruijn sequences, low-discrepancy sequences, and the BEST theorem.
Education
Tatyana Ehrenfest was born in Vienna and spent her childhood in St Petersburg. In 1912 the Ehrenfests moved to Leiden where her father succeeded Hendrik Lorentz as professor at the University of Leiden. Until 1917 she was home schooled; after that, she attended the Gymnasium in Leiden and passed the final exams in 1922. She studied mathematics and physics at the University of Leiden. In 1928 she went to Göttingen where she took courses from Harald Bohr and Max Born. On December 8, 1931, she obtained her Ph.D. in Leiden. After that, she was never employed and, in particular, never held any academic position.
Contributions
De Bruijn sequences are cyclic sequences of symbols for a given alphabet and parameter $k$ such that every length-$k$ subsequence occurs exactly once within them. They are named after Nicolaas Govert de Bruijn, despite their earlier discovery (for binary alphabets) by Camille Flye Sainte-Marie. De Bruijn and Ehrenfest jointly published the first investigation into de Bruijn sequences for larger alphabets, in 1951.
The BEST theorem, also known as the de Bruijn–van Aardenne-Ehrenfest–Smith–Tutte theorem, relates Euler tours and spanning trees in directed graphs, and gives a product formula for their number. It is a variant of an earlier formula of Smith and Tutte, and was published by de Bruijn and Ehrenfest in the same paper as their work on de Bruijn sequences.
Ehrenfest is also known for her proof of a lower bound on low-discrepancy sequences.
References
1. ^ Oppervlakken met scharen van gesloten geodetische lijnen, Thesis, Leiden, 1931.
2. ^ N.G. de Bruijn, In memoriam T. van Aardenne-Ehrenfest, 1905–1984, Nieuw Archief voor Wiskunde (4), Vol.3, (1985) 235–236.
3. ^ Stanley, Richard P. (2018), Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 160, ISBN 9783319771731
4. ^ Jackson, D. M.; Goulden, I. P. (1979), "Sequence enumeration and the de Bruijn–van Aardenne-Ehrenfest–Smith–Tutte theorem", Canadian Journal of Mathematics, 31 (3): 488–495, doi:10.4153/CJM-1979-054-x, MR 0536359, S2CID 124965207
5. ^ Eric W. Weisstein. Discrepancy Theorem. From MathWorld – A Wolfram Web Resource.
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Tanya Khovanova
Tanya Khovanova (Татьяна Гелиевна Хованова, also spelled Tatyana Hovanova; born 25 January 1959) is a Soviet-American mathematician who became the second female gold medalist at the International Mathematical Olympiads. She is a lecturer in mathematics at the Massachusetts Institute of Technology.
Tanya Khovanova
Татьяна Гелиевна Хованова
NationalitySoviet Union
Other namesTatyana Hovanova
CitizenshipUnited States
Alma materMoscow State University (M.S., Ph.D.)
Children2
Scientific career
FieldsCombinatorics, Recreational mathematics
InstitutionsAdvanced Math and Science Academy Charter School
Research Science Institute
Doctoral advisorIsrael Gelfand
Education
As a high school student, Khovanova became a member of the Soviet team for the International Mathematical Olympiad (IMO). In the summer of 1975, Valery Senderov gave the team a list of difficult mathematical problems used in the entrance exams of Moscow State University to discriminate against Soviet Jews, a topic she later wrote about.[1] Khovanova won the silver medal at the 1975 IMO, and a gold medal at the 1976 Olympiad. Her finish at the 1976 Olympiad was second among all competitors,[2] the highest achievement for female students until 1984, when Karin Gröger from East Germany tied for the first place.[3]
Khovanova graduated with honors from Moscow State University (MSU) with a master's degree in mathematics in 1981. She completed her Ph.D. at MSU in 1988 with Israel Gelfand as her doctoral advisor.[4]
Career
Khovanova left the Soviet Union in 1990, and worked for several years in Israel and the US as a postdoctoral researcher. However, she stopped working as a researcher to raise her children, and then worked in the telecommunications and military contracting industry, before returning to academia as a lecturer at MIT.[5]
Khovanova has been a mathematics competition coach at the Advanced Math and Science Academy Charter School in Marlborough, Massachusetts.[6] In 2010, she helped found the MIT PRIMES program for after school mentoring of local high school students, and she continues to serve as its head mentor. She is also head mentor for mathematics of the Research Science Institute, a summer research program for high school students at MIT.[7]
Research
In Khovanova's earlier mathematical research, she studied representation theory, the theory of integrable systems, quantum group theory, and superstring theory. Her later work explores combinatorics and recreational mathematics.[6]
Online activities
In the mid-1990s Khovanova created a website called Number Gossip, about the special properties of individual numbers.[8] In 2007, she created a mathematics blog, centered on mathematical puzzles and problem solving.[9]
Personal life
When Khovanova emigrated to Boston, she did not know how to drive. A friend gave her a copy of The Boston Driver's Handbook which she studied to learn tips before learning years later that the book was intended to be humorous.[10] She has two sons; her first was born in the Soviet Union.[11]
Recognition
An essay about Khovanova, "To Count the Natural Numbers," by Emily Jia, won the 2016 Essay Contest of the Association for Women in Mathematics.[12][13]
Selected works
Two of Khovanova's papers were included in the annual Best Writing on Mathematics volumes, in 2014 and 2016 respectively.[14][15]
• Khovanova, Tanya (2019). "On the Mathematics of the Fraternal Birth Order Effect and the Genetics of Homosexuality" (PDF). Archives of Sexual Behavior. Springer Science and Business Media LLC. 49 (2): 551–555. ISSN 0004-0002. S2CID 37620479.
References
1. Gonzalez, Robbie (October 11, 2011). "How to solve "Jewish" math problems". Gizmodo. Retrieved 2019-09-08.
2. "Tanya Khovanova". International Mathematical Olympiad results. Retrieved 2019-09-07.
3. "Karin Gröger". International Mathematical Olympiad results. Retrieved 2019-09-07.
4. Tanya Khovanova at the Mathematics Genealogy Project
5. Jia, Emily (2016). "To count the natural numbers". Association for Women in Mathematics.
6. "Tanya Khovanova". MIT Mathematics Department. Retrieved 2019-09-07.
7. Etingof, Pavel; Gerovitch, Slava; Khovanova, Tanya (September 2015). "Mathematical Research in High School: The PRIMES Experience" (PDF). Notices of the American Mathematical Society. 62 (8): 910–918. doi:10.1090/noti1270.
8. Black, Debra (March 9, 2010). "Website explains the mystery of numbers". Toronto Star.
9. Zivkovic, Bora (June 8, 2012). "The Scienceblogging Weekly (June 8th, 2012)". Scientific American.
10. Khovanova, Tanya (2018-12-16). "50 words". Boston Globe. Retrieved 2019-09-08 – via Newspapers.com.{{cite web}}: CS1 maint: url-status (link)
11. Khovanova, Tanya (2017-02-10). "Russian and American Children". Tanya Khovanova's Math Blog. Retrieved 2019-09-08.{{cite web}}: CS1 maint: url-status (link)
12. "Results of 2016 Essay Contest". AWM Programs: Essay Contest. Association for Women in Mathematics. Archived from the original on 2016-09-12. Retrieved 2019-09-07.
13. Jia, Emily. "2016 Student Essay Contest High School Winner". Association for Women in Mathematics (AWM). Retrieved 6 May 2023.
14. Pitici, Mircea (23 November 2014). The best writing on mathematics, 2014. Princeton. ISBN 9781400865307. OCLC 894169899.{{cite book}}: CS1 maint: location missing publisher (link)
15. The best writing on mathematics. 2016. Pitici, Mircea, 1965-. Princeton, New Jersey. 7 March 2017. ISBN 9780691175294. OCLC 958799818.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
External links
• Official website
• Tanya Khovanova publications indexed by Google Scholar
Authority control: Academics
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• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Kendall rank correlation coefficient
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter τ, tau), is a statistic used to measure the ordinal association between two measured quantities. A τ test is a non-parametric hypothesis test for statistical dependence based on the τ coefficient. It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities. It is named after Maurice Kendall, who developed it in 1938,[1] though Gustav Fechner had proposed a similar measure in the context of time series in 1897.[2]
Intuitively, the Kendall correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully different for a correlation of −1) rank between the two variables.
Both Kendall's $\tau $ and Spearman's $\rho $ can be formulated as special cases of a more general correlation coefficient. Its notions of concordance and discordance also appear in other areas of statistics, like the Rand index in cluster analysis.
Definition
Let $(x_{1},y_{1}),...,(x_{n},y_{n})$ be a set of observations of the joint random variables X and Y, such that all the values of ($x_{i}$) and ($y_{i}$) are unique (ties are neglected for simplicity). Any pair of observations $(x_{i},y_{i})$ and $(x_{j},y_{j})$, where $i<j$, are said to be concordant if the sort order of $(x_{i},x_{j})$ and $(y_{i},y_{j})$ agrees: that is, if either both $x_{i}>x_{j}$ and $y_{i}>y_{j}$ holds or both $x_{i}<x_{j}$ and $y_{i}<y_{j}$; otherwise they are said to be discordant.
The Kendall τ coefficient is defined as:
$\tau ={\frac {({\text{number of concordant pairs}})-({\text{number of discordant pairs}})}{({\text{number of pairs}})}}=1-{\frac {2({\text{number of discordant pairs}})}{n \choose 2}}.$[3]
where ${n \choose 2}={n(n-1) \over 2}$ is the binomial coefficient for the number of ways to choose two items from n items.
Properties
The denominator is the total number of pair combinations, so the coefficient must be in the range −1 ≤ τ ≤ 1.
• If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value 1.
• If the disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other) the coefficient has value −1.
• If X and Y are independent and not constant, then the expectation of the coefficient is zero.
• An explicit expression for Kendall's rank coefficient is $\tau ={\frac {2}{n(n-1)}}\sum _{i<j}\operatorname {sgn}(x_{i}-x_{j})\operatorname {sgn}(y_{i}-y_{j})$.
Hypothesis test
The Kendall rank coefficient is often used as a test statistic in a statistical hypothesis test to establish whether two variables may be regarded as statistically dependent. This test is non-parametric, as it does not rely on any assumptions on the distributions of X or Y or the distribution of (X,Y).
Under the null hypothesis of independence of X and Y, the sampling distribution of τ has an expected value of zero. The precise distribution cannot be characterized in terms of common distributions, but may be calculated exactly for small samples; for larger samples, it is common to use an approximation to the normal distribution, with mean zero and variance
${\frac {2(2n+5)}{9n(n-1)}}$.[4]
Accounting for ties
A pair $\{(x_{i},y_{i}),(x_{j},y_{j})\}$ is said to be tied if and only if $x_{i}=x_{j}$ or $y_{i}=y_{j}$; a tied pair is neither concordant nor discordant. When tied pairs arise in the data, the coefficient may be modified in a number of ways to keep it in the range [−1, 1]:
Tau-a
The Tau-a statistic tests the strength of association of the cross tabulations. Both variables have to be ordinal. Tau-a will not make any adjustment for ties. It is defined as:
$\tau _{A}={\frac {n_{c}-n_{d}}{n_{0}}}$
where nc, nd and n0 are defined as in the next section.
Tau-b
The Tau-b statistic, unlike Tau-a, makes adjustments for ties.[5] Values of Tau-b range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.
The Kendall Tau-b coefficient is defined as:
$\tau _{B}={\frac {n_{c}-n_{d}}{\sqrt {(n_{0}-n_{1})(n_{0}-n_{2})}}}$
where
${\begin{aligned}n_{0}&=n(n-1)/2\\n_{1}&=\sum _{i}t_{i}(t_{i}-1)/2\\n_{2}&=\sum _{j}u_{j}(u_{j}-1)/2\\n_{c}&={\text{Number of concordant pairs}}\\n_{d}&={\text{Number of discordant pairs}}\\t_{i}&={\text{Number of tied values in the }}i^{\text{th}}{\text{ group of ties for the first quantity}}\\u_{j}&={\text{Number of tied values in the }}j^{\text{th}}{\text{ group of ties for the second quantity}}\end{aligned}}$
A simple algorithm developed in BASIC computes Tau-b coefficient using an alternative formula. [6]
Be aware that some statistical packages, e.g. SPSS, use alternative formulas for computational efficiency, with double the 'usual' number of concordant and discordant pairs.[7]
Tau-c
Tau-c (also called Stuart-Kendall Tau-c)[8] is more suitable than Tau-b for the analysis of data based on non-square (i.e. rectangular) contingency tables.[8][9] So use Tau-b if the underlying scale of both variables has the same number of possible values (before ranking) and Tau-c if they differ. For instance, one variable might be scored on a 5-point scale (very good, good, average, bad, very bad), whereas the other might be based on a finer 10-point scale.
The Kendall Tau-c coefficient is defined as:[9]
$\tau _{C}={\frac {2(n_{c}-n_{d})}{n^{2}{\frac {(m-1)}{m}}}}$
where
${\begin{aligned}n_{c}&={\text{Number of concordant pairs}}\\n_{d}&={\text{Number of discordant pairs}}\\r&={\text{Number of rows}}\\c&={\text{Number of columns}}\\m&=\min(r,c)\end{aligned}}$
Significance tests
When two quantities are statistically dependent, the distribution of $\tau $ is not easily characterizable in terms of known distributions. However, for $\tau _{A}$ the following statistic, $z_{A}$, is approximately distributed as a standard normal when the variables are statistically independent:
$z_{A}={3(n_{c}-n_{d}) \over {\sqrt {n(n-1)(2n+5)/2}}}$
Thus, to test whether two variables are statistically dependent, one computes $z_{A}$, and finds the cumulative probability for a standard normal distribution at $-|z_{A}|$. For a 2-tailed test, multiply that number by two to obtain the p-value. If the p-value is below a given significance level, one rejects the null hypothesis (at that significance level) that the quantities are statistically independent.
Numerous adjustments should be added to $z_{A}$ when accounting for ties. The following statistic, $z_{B}$, has the same distribution as the $\tau _{B}$ distribution, and is again approximately equal to a standard normal distribution when the quantities are statistically independent:
$z_{B}={n_{c}-n_{d} \over {\sqrt {v}}}$
where
${\begin{array}{ccl}v&=&(v_{0}-v_{t}-v_{u})/18+v_{1}+v_{2}\\v_{0}&=&n(n-1)(2n+5)\\v_{t}&=&\sum _{i}t_{i}(t_{i}-1)(2t_{i}+5)\\v_{u}&=&\sum _{j}u_{j}(u_{j}-1)(2u_{j}+5)\\v_{1}&=&\sum _{i}t_{i}(t_{i}-1)\sum _{j}u_{j}(u_{j}-1)/(2n(n-1))\\v_{2}&=&\sum _{i}t_{i}(t_{i}-1)(t_{i}-2)\sum _{j}u_{j}(u_{j}-1)(u_{j}-2)/(9n(n-1)(n-2))\end{array}}$
This is sometimes referred to as the Mann-Kendall test.[10]
Algorithms
The direct computation of the numerator $n_{c}-n_{d}$, involves two nested iterations, as characterized by the following pseudocode:
numer := 0
for i := 2..N do
for j := 1..(i − 1) do
numer := numer + sign(x[i] − x[j]) × sign(y[i] − y[j])
return numer
Although quick to implement, this algorithm is $O(n^{2})$ in complexity and becomes very slow on large samples. A more sophisticated algorithm[11] built upon the Merge Sort algorithm can be used to compute the numerator in $O(n\cdot \log {n})$ time.
Begin by ordering your data points sorting by the first quantity, $x$, and secondarily (among ties in $x$) by the second quantity, $y$. With this initial ordering, $y$ is not sorted, and the core of the algorithm consists of computing how many steps a Bubble Sort would take to sort this initial $y$. An enhanced Merge Sort algorithm, with $O(n\log n)$ complexity, can be applied to compute the number of swaps, $S(y)$, that would be required by a Bubble Sort to sort $y_{i}$. Then the numerator for $\tau $ is computed as:
$n_{c}-n_{d}=n_{0}-n_{1}-n_{2}+n_{3}-2S(y),$
where $n_{3}$ is computed like $n_{1}$ and $n_{2}$, but with respect to the joint ties in $x$ and $y$.
A Merge Sort partitions the data to be sorted, $y$ into two roughly equal halves, $y_{\mathrm {left} }$ and $y_{\mathrm {right} }$, then sorts each half recursive, and then merges the two sorted halves into a fully sorted vector. The number of Bubble Sort swaps is equal to:
$S(y)=S(y_{\mathrm {left} })+S(y_{\mathrm {right} })+M(Y_{\mathrm {left} },Y_{\mathrm {right} })$
where $Y_{\mathrm {left} }$ and $Y_{\mathrm {right} }$ are the sorted versions of $y_{\mathrm {left} }$ and $y_{\mathrm {right} }$, and $M(\cdot ,\cdot )$ characterizes the Bubble Sort swap-equivalent for a merge operation. $M(\cdot ,\cdot )$ is computed as depicted in the following pseudo-code:
function M(L[1..n], R[1..m]) is
i := 1
j := 1
nSwaps := 0
while i ≤ n and j ≤ m do
if R[j] < L[i] then
nSwaps := nSwaps + n − i + 1
j := j + 1
else
i := i + 1
return nSwaps
A side effect of the above steps is that you end up with both a sorted version of $x$ and a sorted version of $y$. With these, the factors $t_{i}$ and $u_{j}$ used to compute $\tau _{B}$ are easily obtained in a single linear-time pass through the sorted arrays.
Software Implementations
• R's statistics base-package implements the test cor.test(x, y, method = "kendall") in its "stats" package (also cor(x, y, method = "kendall") will work, but the latter does not return the p-value).
• For Python, the SciPy library implements the computation of $\tau $ in scipy.stats.kendalltau
See also
• Correlation
• Kendall tau distance
• Kendall's W
• Spearman's rank correlation coefficient
• Goodman and Kruskal's gamma
• Theil–Sen estimator
• Mann–Whitney U test - it is equivalent to Kendall's tau correlation coefficient if one of the variables is binary.
References
1. Kendall, M. (1938). "A New Measure of Rank Correlation". Biometrika. 30 (1–2): 81–89. doi:10.1093/biomet/30.1-2.81. JSTOR 2332226.
2. Kruskal, W. H. (1958). "Ordinal Measures of Association". Journal of the American Statistical Association. 53 (284): 814–861. doi:10.2307/2281954. JSTOR 2281954. MR 0100941.
3. Nelsen, R.B. (2001) [1994], "Kendall tau metric", Encyclopedia of Mathematics, EMS Press
4. Prokhorov, A.V. (2001) [1994], "Kendall coefficient of rank correlation", Encyclopedia of Mathematics, EMS Press
5. Agresti, A. (2010). Analysis of Ordinal Categorical Data (Second ed.). New York: John Wiley & Sons. ISBN 978-0-470-08289-8.
6. Alfred Brophy (1986). "An algorithm and program for calculation of Kendall's rank correlation coefficient" (PDF). Behavior Research Methods, Instruments, & Computers. 18: 45–46. doi:10.3758/BF03200993. S2CID 62601552.
7. IBM (2016). IBM SPSS Statistics 24 Algorithms. IBM. p. 168. Retrieved 31 August 2017.
8. Berry, K. J.; Johnston, J. E.; Zahran, S.; Mielke, P. W. (2009). "Stuart's tau measure of effect size for ordinal variables: Some methodological considerations". Behavior Research Methods. 41 (4): 1144–1148. doi:10.3758/brm.41.4.1144. PMID 19897822.
9. Stuart, A. (1953). "The Estimation and Comparison of Strengths of Association in Contingency Tables". Biometrika. 40 (1–2): 105–110. doi:10.2307/2333101. JSTOR 2333101.
10. Glen_b. "Relationship between Mann-Kendall and Kendall Tau-b".
11. Knight, W. (1966). "A Computer Method for Calculating Kendall's Tau with Ungrouped Data". Journal of the American Statistical Association. 61 (314): 436–439. doi:10.2307/2282833. JSTOR 2282833.
Further reading
• Abdi, H. (2007). "Kendall rank correlation" (PDF). In Salkind, N.J. (ed.). Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.
• Daniel, Wayne W. (1990). "Kendall's tau". Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 365–377. ISBN 978-0-534-91976-4.
• Kendall, Maurice; Gibbons, Jean Dickinson (1990) [First published 1948]. Rank Correlation Methods. Charles Griffin Book Series (5th ed.). Oxford: Oxford University Press. ISBN 978-0195208375.
• Bonett, Douglas G.; Wright, Thomas A. (2000). "Sample size requirements for estimating Pearson, Kendall, and Spearman correlations". Psychometrika. 65 (1): 23–28. doi:10.1007/BF02294183. S2CID 120558581.
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| Wikipedia |
τ-additivity
In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces.
A measure or set function $\mu $ on a space $X$ whose domain is a sigma-algebra $\Sigma $ is said to be τ-additive if for any upward-directed family ${\mathcal {G}}\subseteq \Sigma $ of nonempty open sets such that its union is in $\Sigma ,$ the measure of the union is the supremum of measures of elements of ${\mathcal {G}};$ that is,:
$\mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G).$
See also
• Net (mathematics) – A generalization of a sequence of points
• Sigma additivity – Mapping functionPages displaying short descriptions of redirect targets
• Valuation (measure theory) – map in measure or domain theoryPages displaying wikidata descriptions as a fallback
References
• Fremlin, D.H. (2003), Measure Theory, Volume 4, Torres Fremlin, ISBN 0-9538129-4-4.
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| Wikipedia |
Ramanujan tau function
The Ramanujan tau function, studied by Ramanujan (1916), is the function $\tau :\mathbb {N} \rightarrow \mathbb {Z} $ :\mathbb {N} \rightarrow \mathbb {Z} } defined by the following identity:
$\sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}\left(1-q^{n}\right)^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),$
where q = exp(2πiz) with Im z > 0, $\phi $ is the Euler function, η is the Dedekind eta function, and the function Δ(z) is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write $\Delta /(2\pi )^{12}$ instead of $\Delta $). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in Dyson (1972).
Values
The first few values of the tau function are given in the following table (sequence A000594 in the OEIS):
n 12345678910111213141516
τ(n) 1−24252−14724830−6048−1674484480−113643−115920534612−370944−5777384018561217160987136
Ramanujan's conjectures
Ramanujan (1916) observed, but did not prove, the following three properties of τ(n):
• τ(mn) = τ(m)τ(n) if gcd(m,n) = 1 (meaning that τ(n) is a multiplicative function)
• τ(pr + 1) = τ(p)τ(pr) − p11 τ(pr − 1) for p prime and r > 0.
• |τ(p)| ≤ 2p11/2 for all primes p.
The first two properties were proved by Mordell (1917) and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by applying them to a Kuga-Sato variety).
Congruences for the tau function
For k ∈ $\mathbb {Z} $ and n ∈ $\mathbb {Z} $>0, define σk(n) as the sum of the kth powers of the divisors of n. The tau function satisfies several congruence relations; many of them can be expressed in terms of σk(n). Here are some:[1]
1. $\tau (n)\equiv \sigma _{11}(n)\ {\bmod {\ }}2^{11}{\text{ for }}n\equiv 1\ {\bmod {\ }}8$[2]
2. $\tau (n)\equiv 1217\sigma _{11}(n)\ {\bmod {\ }}2^{13}{\text{ for }}n\equiv 3\ {\bmod {\ }}8$[2]
3. $\tau (n)\equiv 1537\sigma _{11}(n)\ {\bmod {\ }}2^{12}{\text{ for }}n\equiv 5\ {\bmod {\ }}8$[2]
4. $\tau (n)\equiv 705\sigma _{11}(n)\ {\bmod {\ }}2^{14}{\text{ for }}n\equiv 7\ {\bmod {\ }}8$[2]
5. $\tau (n)\equiv n^{-610}\sigma _{1231}(n)\ {\bmod {\ }}3^{6}{\text{ for }}n\equiv 1\ {\bmod {\ }}3$[3]
6. $\tau (n)\equiv n^{-610}\sigma _{1231}(n)\ {\bmod {\ }}3^{7}{\text{ for }}n\equiv 2\ {\bmod {\ }}3$[3]
7. $\tau (n)\equiv n^{-30}\sigma _{71}(n)\ {\bmod {\ }}5^{3}{\text{ for }}n\not \equiv 0\ {\bmod {\ }}5$[4]
8. $\tau (n)\equiv n\sigma _{9}(n)\ {\bmod {\ }}7{\text{ for }}n\equiv 0,1,2,4\ {\bmod {\ }}7$[5]
9. $\tau (n)\equiv n\sigma _{9}(n)\ {\bmod {\ }}7^{2}{\text{ for }}n\equiv 3,5,6\ {\bmod {\ }}7$[5]
10. $\tau (n)\equiv \sigma _{11}(n)\ {\bmod {\ }}691.$[6]
For p ≠ 23 prime, we have[1][7]
1. $\tau (p)\equiv 0\ {\bmod {\ }}23{\text{ if }}\left({\frac {p}{23}}\right)=-1$
2. $\tau (p)\equiv \sigma _{11}(p)\ {\bmod {\ }}23^{2}{\text{ if }}p{\text{ is of the form }}a^{2}+23b^{2}$[8]
3. $\tau (p)\equiv -1\ {\bmod {\ }}23{\text{ otherwise}}.$
Explicit formula
In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:[9]
$\tau (n)=n^{4}\sigma (n)-24\sum _{i=1}^{n-1}i^{2}(35i^{2}-52in+18n^{2})\sigma (i)\sigma (n-i).$
Conjectures on τ(n)
Suppose that f is a weight-k integer newform and the Fourier coefficients a(n) are integers. Consider the problem:
Given that f does not have complex multiplication, do almost all primes p have the property that a(p) ≢ 0 (mod p)?
Indeed, most primes should have this property, and hence they are called ordinary. Despite the big advances by Deligne and Serre on Galois representations, which determine a(n) (mod p) for n coprime to p, it is unclear how to compute a(p) (mod p). The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes p such that a(p) = 0, which thus are congruent to 0 modulo p. There are no known examples of non-CM f with weight greater than 2 for which a(p) ≢ 0 (mod p) for infinitely many primes p (although it should be true for almost all p). There are also no known examples with a(p) ≡ 0 (mod p) for infinitely many p. Some researchers had begun to doubt whether a(p) ≡ 0 (mod p) for infinitely many p. As evidence, many provided Ramanujan's τ(p) (case of weight 12). The only solutions up to 1010 to the equation τ(p) ≡ 0 (mod p) are 2, 3, 5, 7, 2411, and 7758337633 (sequence A007659 in the OEIS).[10]
Lehmer (1947) conjectured that τ(n) ≠ 0 for all n, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for n up to 214928639999 (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of N for which this condition holds for all n ≤ N.
Nreference
3316799Lehmer (1947)
214928639999Lehmer (1949)
1000000000000000Serre (1973, p. 98), Serre (1985)
1213229187071998Jennings (1993)
22689242781695999Jordan and Kelly (1999)
22798241520242687999Bosman (2007)
982149821766199295999Zeng and Yin (2013)
816212624008487344127999Derickx, van Hoeij, and Zeng (2013)
Ramanujan's L-function
Ramanujan's L-function is defined by
$L(s)=\sum _{n\geq 1}{\frac {\tau (n)}{n^{s}}}$
if $\Re s>6$ and by analytic continuation otherwise. It satisfies the functional equation
${\frac {L(s)\Gamma (s)}{(2\pi )^{s}}}={\frac {L(12-s)\Gamma (12-s)}{(2\pi )^{12-s}}},\quad s\notin \mathbb {Z} _{0}^{-},\,12-s\notin \mathbb {Z} _{0}^{-}$
and has the Euler product
$L(s)=\prod _{p\,{\text{prime}}}{\frac {1}{1-\tau (p)p^{-s}+p^{11-2s}}},\quad \Re s>7.$
Ramanujan conjectured that all nontrivial zeros of $L$ have real part equal to $6$.
Notes
1. Page 4 of Swinnerton-Dyer 1973
2. Due to Kolberg 1962
3. Due to Ashworth 1968
4. Due to Lahivi
5. Due to D. H. Lehmer
6. Due to Ramanujan 1916
7. Due to Wilton 1930
8. Due to J.-P. Serre 1968, Section 4.5
9. Niebur, Douglas (September 1975). "A formula for Ramanujan's $\tau$-function". Illinois Journal of Mathematics. 19 (3): 448–449. doi:10.1215/ijm/1256050746. ISSN 0019-2082.
10. N. Lygeros and O. Rozier (2010). "A new solution for the equation $\tau (p)\equiv 0{\pmod {p}}$" (PDF). Journal of Integer Sequences. 13: Article 10.7.4.
References
• Apostol, T. M. (1997), "Modular Functions and Dirichlet Series in Number Theory", New York: Springer-Verlag 2nd Ed.
• Ashworth, M. H. (1968), Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford)
• Dyson, F. J. (1972), "Missed opportunities", Bull. Amer. Math. Soc., 78 (5): 635–652, doi:10.1090/S0002-9904-1972-12971-9, Zbl 0271.01005
• Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873, Zbl 0168.29502
• Lehmer, D.H. (1947), "The vanishing of Ramanujan's function τ(n)", Duke Math. J., 14 (2): 429–433, doi:10.1215/s0012-7094-47-01436-1, Zbl 0029.34502
• Lygeros, N. (2010), "A New Solution to the Equation τ(p) ≡ 0 (mod p)" (PDF), Journal of Integer Sequences, 13: Article 10.7.4
• Mordell, Louis J. (1917), "On Mr. Ramanujan's empirical expansions of modular functions.", Proceedings of the Cambridge Philosophical Society, 19: 117–124, JFM 46.0605.01
• Newman, M. (1972), A table of τ (p) modulo p, p prime, 3 ≤ p ≤ 16067, National Bureau of Standards
• Rankin, Robert A. (1988), "Ramanujan's tau-function and its generalizations", in Andrews, George E. (ed.), Ramanujan revisited (Urbana-Champaign, Ill., 1987), Boston, MA: Academic Press, pp. 245–268, ISBN 978-0-12-058560-1, MR 0938968
• Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Trans. Camb. Philos. Soc., 22 (9): 159–184, MR 2280861
• Serre, J-P. (1968), "Une interprétation des congruences relatives à la fonction $\tau $ de Ramanujan", Séminaire Delange-Pisot-Poitou, 14
• Swinnerton-Dyer, H. P. F. (1973), "On l-adic representations and congruences for coefficients of modular forms", in Kuyk, Willem; Serre, Jean-Pierre (eds.), Modular functions of one variable, III, Lecture Notes in Mathematics, vol. 350, pp. 1–55, doi:10.1007/978-3-540-37802-0, ISBN 978-3-540-06483-1, MR 0406931
• Wilton, J. R. (1930), "Congruence properties of Ramanujan's function τ(n)", Proceedings of the London Mathematical Society, 31: 1–10, doi:10.1112/plms/s2-31.1.1
| Wikipedia |
Tau function (integrable systems)
Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota[1] in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form.
The term tau function, or $\tau $-function, was first used systematically by Mikio Sato[2] and his students[3][4] in the specific context of the Kadomtsev–Petviashvili (or KP) equation and related integrable hierarchies. It is a central ingredient in the theory of solitons. In this setting, given any $\tau $-function satisfying a Hirota-type system of bilinear equations (see § Hirota bilinear residue relation for KP tau functions below), the corresponding solutions of the equations of the integrable hierarchy are explicitly expressible in terms of it and its logarithmic derivatives up to a finite order. Tau functions also appear as matrix model partition functions in the spectral theory of random matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to moduli spaces of Riemann surfaces, and enumeration of branched coverings, or so-called Hurwitz numbers.
There are two notions of $\tau $-functions, both introduced by the Sato school. The first is isospectral $\tau $-functions of the Sato–Segal–Wilson type[2][5] for integrable hierarchies, such as the KP hierarchy, which are parametrized by linear operators satisfying isospectral deformation equations of Lax type. The second is isomonodromic $\tau $-functions[6].
Depending on the specific application, a $\tau $-function may either be: 1) an analytic function of a finite or infinite number of independent, commuting flow variables, or deformation parameters; 2) a discrete function of a finite or infinite number of denumerable variables; 3) a formal power series expansion in a finite or infinite number of expansion variables, which need have no convergence domain, but serves as generating function for certain enumerative invariants appearing as the coefficients of the series; 4) a finite or infinite (Fredholm) determinant whose entries are either specific polynomial or quasi-polynomial functions, or parametric integrals, and their derivatives; 5) the Pfaffian of a skew symmetric matrix (either finite or infinite dimensional) with entries similarly of polynomial or quasi-polynomial type. Examples of all these types are given below.
In the Hamilton–Jacobi approach to Liouville integrable Hamiltonian systems, Hamilton's principal function, evaluated on the level surfaces of a complete set of Poisson commuting invariants, plays a role similar to the $\tau $-function, serving both as a generating function for the canonical transformation to linearizing canonical coordinates and, when evaluated on simultaneous level sets of a complete set of Poisson commuting invariants, as a complete solution of the Hamilton–Jacobi equation.
Tau functions: isospectral and isomonodromic
A $\tau $-function of isospectral type is defined as a solution of the Hirota bilinear equations (see § Hirota bilinear residue relation for KP tau functions below), from which the linear operator undergoing isospectral evolution can be uniquely reconstructed. Geometrically, in the Sato[2] and Segal-Wilson[5] sense, it is the value of the determinant of a Fredholm integral operator, interpreted as the orthogonal projection of an element of a suitably defined (infinite dimensional) Grassmann manifold onto the origin, as that element evolves under the linear exponential action of a maximal abelian subgroup of the general linear group. It typically arises as a partition function, in the sense of statistical mechanics, many-body quantum mechanics or quantum field theory, as the underlying measure undergoes a linear exponential deformation.
Isomonodromic $\tau $-functions for linear systems of Fuchsian type are defined below in § Fuchsian isomonodromic systems. Schlesinger equations. For the more general case of linear ordinary differential equations with rational coefficients, including irregular singularities, they are developed in reference [6].
Hirota bilinear residue relation for KP tau functions
A KP (Kadomtsev–Petviashvili) $\tau $-function $\tau (\mathbf {t} )$ is a function of an infinite collection $\mathbf {t} =(t_{1},t_{2},\dots )$ of variables (called KP flow variables) that satisfies the bilinear formal residue equation
$\mathrm {res} _{z=0}\left(e^{\sum _{i=1}^{\infty }(\delta t_{i})z^{i}}\tau ({\bf {t}}-[z^{-1}])\tau ({\bf {s}}+[z^{-1}])\right)dz\equiv 0,$
(1)
identically in the $\delta t_{j}$ variables, where $\mathrm {res} _{z=0}$ is the $z^{-1}$ coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series in $z$, and
${\bf {s}}:={\bf {t}}+(\delta t_{1},\delta t_{2},\cdots ),\quad [z^{-1}]:=(z^{-1},{\tfrac {z^{-2}}{2}},\cdots {\tfrac {z^{-j}}{j}},\cdots ).$
As explained below in the section § Formal Baker-Akhiezer function and the KP hierarchy, every such $\tau $-function determines a set of solutions to the equations of the KP hierarchy.
Kadomtsev–Petviashvili equation
If $\tau (t_{1},t_{2},t_{3},\dots \dots )$ is a KP $\tau $-function satisfying the Hirota residue equation (1) and we identify the first three flow variables as
$t_{1}=x,\quad t_{2}=y,\quad t_{3}=t,$
it follows that the function
$u(x,y,t):=2{\frac {\partial ^{2}}{\partial x^{2}}}\log \left(\tau (x,y,t,t_{4},\dots )\right)$
satisfies the $2$ (spatial)$+1$ (time) dimensional nonlinear partial differential equation
$3u_{yy}=\left(4u_{t}-6uu_{x}-u_{xxx}\right)_{x},$
(2)
known as the Kadomtsev-Petviashvili (KP) equation. This equation plays a prominent role in plasma physics and in shallow water ocean waves.
Taking further logarithmic derivatives of $\tau (t_{1},t_{2},t_{3},\dots \dots )$ gives an infinite sequence of functions that satisfy further systems of nonlinear autonomous PDE's, each involving partial derivatives of finite order with respect to a finite number of the KP flow parameters ${\bf {t}}=(t_{1},t_{2},\dots )$. These are collectively known as the KP hierarchy.
Formal Baker–Akhiezer function and the KP hierarchy
If we define the (formal) Baker-Akhiezer function $\psi (z,\mathbf {t} )$ by Sato's formula[2][3]
$\psi (z,\mathbf {t} ):=e^{\sum _{i=1}^{\infty }t_{i}z^{i}}{\frac {\tau (\mathbf {t} -[z^{-1}])}{\tau (\mathbf {t} )}}$
and expand it as a formal series in the powers of the variable $z$
$\psi (z,\mathbf {t} )=e^{\sum _{i=1}^{\infty }t_{i}z^{i}}(1+\sum _{j=1}^{\infty }w_{j}(\mathbf {t} )z^{-j}),$
this satisfies an infinite sequence of compatible evolution equations
${\frac {\partial \psi }{\partial t_{i}}}={\mathcal {D}}_{i}\psi ,\quad i,j,=1,2,\dots ,$
(3)
where ${\mathcal {D}}_{i}$ is a linear ordinary differential operator of degree $i$ in the variable $x:=t_{1}$, with coefficients that are functions of the flow variables $\mathbf {t} =(t_{1},t_{2},\dots )$, defined as follows
${\mathcal {D}}_{i}:={\big (}{\mathcal {L}}^{i}{\big )}_{+}$
where ${\mathcal {L}}$ is the formal pseudo-differential operator
${\mathcal {L}}=\partial +\sum _{j=1}^{\infty }u_{j}(\mathbf {t} )\partial ^{-j}={\mathcal {W}}\circ \partial \circ {\mathcal {W}}^{-1}$
with $\partial :={\frac {\partial }{\partial x}}$ :={\frac {\partial }{\partial x}}} ,
${\mathcal {W}}:=1+\sum _{j=1}^{\infty }w_{j}(\mathbf {t} )\partial ^{-j}$
is the wave operator and ${\big (}{\mathcal {L}}^{i}{\big )}_{+}$ denotes the projection to the part of ${\mathcal {L}}^{i}$ containing purely non-negative powers of $\partial $; i.e. the differential operator part of ${\mathcal {L}}^{i}$ .
The pseudodifferential operator ${\mathcal {L}}$ satisfies the infinite system of isospectral deformation equations
${\frac {\partial {\mathcal {L}}}{\partial t_{i}}}=[{\mathcal {D}}_{i},{\mathcal {L}}],\quad i,=1,2,\dots $
(4)
and the compatibility conditions for both the system (3) and (4) are
${\frac {\partial {\mathcal {D}}_{i}}{\partial t_{j}}}-{\frac {\partial {\mathcal {D}}_{j}}{\partial t_{i}}}+[{\mathcal {D}}_{i},{\mathcal {D}}_{j}]=0,\quad i,j,=1,2,\dots $
(5)
This is a compatible infinite system of nonlinear partial differential equations, known as the KP (Kadomtsev-Petviashvili) hierarchy, for the functions $\{u_{j}(\mathbf {t} )\}_{j\in \mathbf {N} }$, with respect to the set $\mathbf {t} =(t_{1},t_{2},\dots )$ of independent variables, each of which contains only a finite number of $u_{j}$'s, and derivatives only with respect to the three independent variables $(x,t_{i},t_{j})$. The first nontrivial case of these is the Kadomtsev-Petviashvili equation (2).
Thus, every KP $\tau $-function provides a solution, at least in the formal sense, of this infinite system of nonlinear partial differential equations.
Isomonodromic systems. Isomonodromic tau functions
Fuchsian isomonodromic systems. Schlesinger equations
Consider the overdetermined system of first order matrix partial differential equations
${\partial \Psi \over \partial z}-\sum _{i=1}^{n}{N_{i} \over z-\alpha _{i}}\Psi =0,\quad $
(6)
${\partial \Psi \over \partial \alpha _{i}}+{N_{i} \over z-\alpha _{i}}\Psi =0,$
(7)
where $\{N_{i}\}_{i=1,\dots ,n}$ are a set of $n$ $r\times r$ traceless matrices, $\{\alpha _{i}\}_{i=1,\dots ,n}$ a set of $n$ complex parameters, $z$ a complex variable, and $\Psi (z,\alpha _{1},\dots ,\alpha _{m})$ is an invertible $r\times r$ matrix valued function of $z$ and $\{\alpha _{i}\}_{i=1,\dots ,n}$. These are the necessary and sufficient conditions for the based monodromy representation of the fundamental group $\pi _{1}({\bf {P}}^{1}\backslash \{\alpha _{i}\}_{i=1,\dots ,n})$ of the Riemann sphere punctured at the points $\{\alpha _{i}\}_{i=1,\dots ,n}$ corresponding to the rational covariant derivative operator
${\partial \over \partial z}-\sum _{i=1}^{n}{N_{i} \over z-\alpha _{i}}$
to be independent of the parameters $\{\alpha _{i}\}_{i=1,\dots ,n}$; i.e. that changes in these parameters induce an isomonodromic deformation. The compatibility conditions for this system are the Schlesinger equations[6]
${\partial N_{i} \over \partial \alpha _{j}}={[N_{i},N_{j}] \over \alpha _{i}-\alpha _{j}}\quad {\text{for }}i\neq j,\quad {\partial N_{i} \over \partial \alpha _{i}}=-\sum _{1\leq j\leq n,j\neq i}{[N_{i},N_{j}] \over \alpha _{i}-\alpha _{j}}.$
(8)
Isomonodromic $\tau $-function
Defining $n$ functions
$H_{i}:={\frac {1}{2}}\sum _{1\leq j\leq n,j\neq i}{{\rm {Tr}}(N_{i}N_{j}) \over \alpha _{i}-\alpha _{j}},\quad i=1,\dots ,n,$
(9)
the Schlesinger equations (8) imply that the differential form
$\omega :=\sum _{i=1}^{n}H_{i}d\alpha _{i}$ :=\sum _{i=1}^{n}H_{i}d\alpha _{i}}
on the space of parameters is closed:
$d\omega =0$
and hence, locally exact. Therefore, at least locally, there exists a function $\tau (\alpha _{1},\dots ,\alpha _{n})$ of the parameters, defined within a multiplicative constant, such that
$\omega =d\mathrm {ln} \tau $
The function $\tau (\alpha _{1},\dots ,\alpha _{n})$ is called the isomonodromic $\tau $-function associated to the fundamental solution $\Psi $ of the system (6), (7).
Hamiltonian structure of the Schlesinger equations
Defining the Lie Poisson brackets on the space of $n$-tuples $\{N_{i}\}_{i=1,\dots ,n}$ of $r\times r$ matrices:
$\{(N_{i})_{ab},(N_{j})_{c,d}\}=\delta _{ij}\left((N_{i})_{ad}\delta _{bc}-(N_{i})_{cb}\delta _{ad}\right)$
$1\leq i,j\leq n,\quad 1\leq a,b,c,d\leq r,$
and viewing the $n$ functions $\{H_{i}\}_{i=1,\dots ,n}$ defined in (9) as Hamiltonian functions on this Poisson space, the Schlesinger equations (8) may be expressed in Hamiltonian form as [7] [8]
${\frac {\partial f(N_{1},\dots ,N_{n})}{\partial \alpha _{i}}}=\{f,H_{i}\},\quad 1\leq i\leq n$
for any differentiable function $f(N_{1},\dots ,N_{n})$.
Reduction of $r=2$, $n=3$ case to $P_{VI}$
The simplest nontrivial case of the Schlesinger equations is when $r=2$ and $n=3$. By applying a Möbius transformation to the variable $z$, two of the finite poles may be chosen to be at $0$ and $1$, and the third viewed as the independent variable. Setting the sum $\sum _{i=1}^{3}N_{i}$ of the matrices appearing in (6), which is an invariant of the Schlesinger equations, equal to a constant, and quotienting by its stabilizer under $Gl(2)$ conjugation, we obtain a system equivalent to the most generic case $P_{VI}$ of the six Painlevé transcendent equations, for which many detailed classes of explicit solutions are known [9] [10] [11].
Non-Fuchsian isomonodromic systems
For non-Fuchsian systems, with higher order poles, the generalized monodromy data include Stokes matrices and connection matrices, and there are further isomonodromic deformation parameters associated with the local asymptotics, but the isomonodromic $\tau $-functions may be defined in a similar way, using differentials on the extended parameter space [6]. There is similarly a Poisson bracket structure on the space of rational matrix values functions of the spectral parameter $z$ and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics [7] [8].
Taking all possible confluences of the poles appearing in (6) for the $r=2$ and $n=3$ case, including the one at $z=\infty $, and making the corresponding reductions, we obtain all other instances $P_{I}\cdots P_{V}$ of the Painlevé transcendents, for which numerous special solutions are also known [9] [10].
Fermionic VEV (vacuum expectation value) representations
The fermionic Fock space ${\mathcal {F}}$, is a semi-infinite exterior product space [12]
${\mathcal {F}}=\Lambda ^{\infty /2}{\mathcal {H}}=\oplus _{n\in \mathbf {Z} }{\mathcal {F}}_{n}$
defined on a (separable) Hilbert space ${\mathcal {H}}$ with basis elements $\{e_{i}\}_{i\in \mathbf {Z} }$ and dual basis elements $\{e^{i}\}_{i\in \mathbf {Z} }$ for ${\mathcal {H}}^{*}$.
The free fermionic creation and annihilation operators $\{\psi _{j},\psi _{j}^{\dagger }\}_{j\in \mathbf {Z} }$ act as endomorphisms on ${\mathcal {F}}$ via exterior and interior multiplication by the basis elements
$\psi _{i}:=e_{i}\wedge ,\quad \psi _{i}^{\dagger }:=i_{e^{i}},\quad i\in \mathbf {Z} ,$
and satisfy the canonical anti-commutation relations
$[\psi _{i},\psi _{k}]_{+}=[\psi _{i}^{\dagger },\psi _{k}^{\dagger }]_{+}=0,\quad [\psi _{i},\psi _{k}^{\dagger }]_{+}=\delta _{ij}.$
These generate the standard fermionic representation of the Clifford algebra on the direct sum ${\mathcal {H}}+{\mathcal {H}}^{*}$, corresponding to the scalar product
$Q(u+\mu ,w+\nu ):=\nu (u)+\mu (v),\quad u,v\in {\mathcal {H}},\ \mu ,\nu \in {\mathcal {H}}^{*}$
with the Fock space ${\mathcal {F}}$ as irreducible module. Denote the vacuum state, in the zero fermionic charge sector ${\mathcal {F}}_{0}$, as
$|0\rangle :=e_{-1}\wedge e_{-2}\wedge \cdots $ :=e_{-1}\wedge e_{-2}\wedge \cdots } ,
which corresponds to the Dirac sea of states along the real integer lattice in which all negative integer locations are occupied and all non-negative ones are empty.
This is annihilated by the following operators
$\psi _{-j}|0\rangle =0,\quad \psi _{j-1}^{\dagger }|0\rangle =0,\quad j=0,1,\dots $
The dual fermionic Fock space vacuum state, denoted $\langle 0|$, is annihilated by the adjoint operators, acting to the left
$\langle 0|\psi _{-j}^{\dagger }=0,\quad \langle 0|\psi _{j-1}|0=0,\quad j=0,1,\dots $
Normal ordering $:L_{1},\cdots L_{m}:$ of a product of linear operators (i.e., finite or infinite linear combinations of creation and annihilation operators) is defined so that its vacuum expectation value (VEV) vanishes
$\langle 0|:L_{1},\cdots L_{m}:|0\rangle =0.$
In particular, for a product $L_{1}L_{2}$ of a pair $(L_{1},L_{2})$ of linear operators, one has
${:L_{1}L_{2}:}=L_{1}L_{2}-\langle 0|L_{1}L_{2}|0\rangle .$
The fermionic charge operator $C$ is defined as
$C=\sum _{i\in \mathbf {Z} }:\psi _{i}\psi _{i}^{\dagger }:$
The subspace ${\mathcal {F}}_{n}\subset {\mathcal {F}}$ is the eigenspace of $C$ consisting of all eigenvectors with eigenvalue $n$
$C|v;n\rangle =n|v;n\rangle ,\quad \forall |v;n\rangle \in {\mathcal {F}}_{n}$.
The standard orthonormal basis $\{|\lambda \rangle \}$ for the zero fermionic charge sector ${\mathcal {F}}_{0}$ is labelled by integer partitions $\lambda =(\lambda _{1},\dots ,\lambda _{\ell (\lambda )})$, where $\lambda _{1}\geq \cdots \geq \lambda _{\ell (\lambda )}$ is a weakly decreasing sequence of $\ell (\lambda )$ positive integers, which can equivalently be represented by a Young diagram, as depicted here for the partition $(5,4,1)$.
An alternative notation for a partition $\lambda $ consists of the Frobenius indices $(\alpha _{1},\dots \alpha _{r}|\beta _{1},\dots \beta _{r})$, where $\alpha _{i}$ denotes the arm length; i.e. the number $\lambda _{i}-i$ of boxes in the Young diagram to the right of the $i$'th diagonal box, $\beta _{i}$ denotes the leg length, i.e. the number of boxes in the Young diagram below the $i$'th diagonal box, for $i=1,\dots ,r$, where $r$ is the Frobenius rank, which is the number of elements along the principal diagonal.
The basis element $|\lambda \rangle $ is then given by acting on the vacuum with a product of $r$ pairs of creation and annihilation operators, labelled by the Frobenius indices
$|\lambda \rangle =(-1)^{\sum _{j=1}^{r}\beta _{j}}\prod _{k=1}^{r}{\big (}\psi _{\alpha _{k}}\psi _{-\beta _{k}-1}^{\dagger }{\big )}|0\rangle .$
The integers $\{\alpha _{i}\}_{i=1,\dots ,r}$ indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while $\{-\beta _{i}-1\}_{i=1,\dots ,r}$ indicate the unoccupied negative integer sites. The corresponding diagram, consisting of infinitely many occupied and unoccupied sites on the integer lattice that are a finite perturbation of the Dirac sea are referred to as a Maya diagram.[2]
The case of the null (emptyset) partition $|\emptyset \rangle =|0\rangle $ gives the vacuum state, and the dual basis $\{\langle \mu |\}$ is defined by
$\langle \mu |\lambda \rangle =\delta _{\lambda ,\mu }$
Then any KP $\tau $-function can be expressed as a sum
$\tau _{w}(\mathbf {t} )=\sum _{\lambda }\pi _{\lambda }(w)s_{\lambda }(\mathbf {t} )$
(10)
where $\mathbf {t} =(t_{1},t_{2},\dots ,\dots )$ are the KP flow variables, $s_{\lambda }(\mathbf {t} )$ is the Schur function corresponding to the partition $\lambda $, viewed as a function of the normalized power sum variables
$t_{i}:=[\mathbf {x} ]_{i}:={\frac {1}{i}}\sum _{a=1}^{n}x_{a}^{i}\quad i=1,2,\dots $
in terms of an auxiliary (finite or infinite) sequence of variables $\mathbf {x} :=(x_{1},\dots ,x_{N})$ :=(x_{1},\dots ,x_{N})} and the constant coefficients $\pi _{\lambda }(w)$ may be viewed as the Plücker coordinates of an element $w\in \mathrm {Gr} _{{\mathcal {H}}_{+}}({\mathcal {H}})$ of the infinite dimensional Grassmannian consisting of the orbit, under the action of the general linear group $\mathrm {Gl} ({\mathcal {H}})$, of the subspace ${\mathcal {H}}_{+}=\mathrm {span} \{e_{-i}\}_{i\in \mathbf {N} }\subset {\mathcal {H}}$ of the Hilbert space ${\mathcal {H}}$.
This corresponds, under the Bose-Fermi correspondence, to a decomposable element
$|\tau _{w}\rangle =\sum _{\lambda }\pi _{\lambda }(w)|\lambda \rangle $
of the Fock space ${\mathcal {F}}_{0}$ which, up to projectivization is the image of the Grassmannian element $w\in \mathrm {Gr} _{{\mathcal {H}}_{+}}({\mathcal {H}})$ under the Plücker map
${\mathcal {Pl}}:\mathrm {span} (w_{1},w_{2},\dots )\longrightarrow [w_{1}\wedge w_{2}\wedge \cdots ]=[|\tau _{w}\rangle ],$
where $(w_{1},w_{2},\dots )$ is a basis for the subspace $w\subset {\mathcal {H}}$ and $[\cdots ]$ denotes projectivization of an element of ${\mathcal {F}}$.
The Plücker coordinates $\{\pi _{\lambda }(w)\}$ satisfy an infinite set of bilinear relations, the Plücker relations, defining the image of the Plücker embedding into the projectivization $\mathbf {P} ({\mathcal {F}})$ of the fermionic Fock space, which are equivalent to the Hirota bilinear residue relation (1).
If $w=g({\mathcal {H}}_{+})$ for a group element $g\in \mathrm {Gl} ({\mathcal {H}})$ with fermionic representation ${\hat {g}}$, then the $\tau $-function $\tau _{w}(\mathbf {t} )$ can be expressed as the fermionic vacuum state expectation value (VEV):
$\tau _{w}(\mathbf {t} )=\langle 0|{\hat {\gamma }}_{+}(\mathbf {t} ){\hat {g}}|0\rangle ,$
where
$\Gamma _{+}=\{{\hat {\gamma }}_{+}(\mathbf {t} )=e^{\sum _{i=1}^{\infty }t_{i}J_{i}}\}\subset \mathrm {Gl} ({\mathcal {H}})$
is the abelian subgroup of $\mathrm {Gl} ({\mathcal {H}})$ that generates the KP flows, and
$J_{i}:=\sum _{j\in \mathbf {Z} }\psi _{j}\psi _{j+i}^{\dagger },\quad i=1,2\dots $
are the ""current"" components.
Examples of solutions to the equations of the KP hierarchy
Schur functions
As seen in equation (9), every KP $\tau $-function can be represented (at least formally) as a linear combination of Schur functions, in which the coefficients $\pi _{\lambda }(w)$ satisfy the bilinear set of Plucker relations corresponding to an element $w$ of an infinite (or finite) Grassmann manifold. In fact, the simplest class of (polynomial) tau functions consists of the Schur functions $s_{\lambda }(\mathbf {t} )$ themselves, which correspond to the special element of the Grassmann manifold whose image under the Plücker map is $|\lambda >$.
Multisoliton solutions
If we choose $3N$ complex constants $\{\alpha _{k},\beta _{k},\gamma _{k}\}_{k=1,\dots ,N}$ with $\alpha _{k},\beta _{k}$'s all distinct, $\gamma _{k}\neq 0$, and define the functions
$y_{k}({\bf {t}}):=e^{\sum _{i=1}^{\infty }t_{i}\alpha _{k}^{i}}+\gamma _{k}e^{\sum _{i=1}^{\infty }t_{i}\beta _{k}^{i}}\quad k=1,\dots ,N,$
we arrive at the Wronskian determinant formula
$\tau _{{\vec {\alpha }},{\vec {\beta }},{\vec {\gamma }}}^{(N)}({\bf {t}}):={\begin{vmatrix}y_{1}({\bf {t}})&y_{2}({\bf {t}})&\cdots &y_{N}({\bf {t}})\\y_{1}'({\bf {t}})&y_{2}'({\bf {t}})&\cdots &y_{N}'({\bf {t}})\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(N-1)}({\bf {t}})&y_{2}^{(N-1)}({\bf {t}})&\cdots &y_{N}^{(N-1)}({\bf {t}})\\\end{vmatrix}}.$
which gives the general $N$-soliton $\tau $-function.[3][4]
Theta function solutions associated to algebraic curves
Let $X$ be a compact Riemann surface of genus $g$ and fix a canonical homology basis $a_{1},\dots ,a_{g},b_{1},\dots ,b_{g}$ of $H_{1}(X,\mathbf {Z} )$ with intersection numbers
$a_{i}\circ a_{j}=b_{i}\circ b_{j}=0,\quad a_{i}\circ b_{j}=\delta _{ij},\quad 1\leq i,j\leq g.$
Let $\{\omega _{i}\}_{i=1,\dots ,g}$ be a basis for the space $H^{1}(X)$ of holomorphic differentials satisfying the standard normalization conditions
$\oint _{a_{i}}\omega _{j}=\delta _{ij},\quad \oint _{b_{j}}\omega _{j}=B_{ij},$
where $B$ is the Riemann matrix of periods. The matrix $B$ belongs to the Siegel upper half space
$\mathbf {S} _{g}=\left\{B\in \mathrm {Mat} _{g\times g}(\mathbf {C} )\ \colon \ B^{T}=B,\ {\text{Im}}(B){\text{ is positive definite}}\right\}.$
The Riemann $\theta $ function on $\mathbf {C} ^{g}$ corresponding to the period matrix $B$ is defined to be
$\theta (Z|B):=\sum _{N\in \mathbb {Z} ^{g}}e^{i\pi (N,BN)+2i\pi (N,Z)}.$
Choose a point $p_{\infty }\in X$, a local parameter $\zeta $ in a neighbourhood of $p_{\infty }$ with $\zeta (p_{\infty })=0$ and a positive divisor of degree $g$
${\mathcal {D}}:=\sum _{i=1}^{g}p_{i},\quad p_{i}\in X.$
For any positive integer $k\in \mathbf {N} ^{+}$ let $\Omega _{k}$ be the unique meromorphic differential of the second kind characterized by the following conditions:
• The only singularity of $\Omega _{k}$ is a pole of order $k+1$ at $p=p_{\infty }$ with vanishing residue.
• The expansion of $\Omega _{k}$ around $p=p_{\infty }$ is
$\Omega _{k}=d(\zeta ^{-k})+\sum _{j=1}^{\infty }Q_{ij}\zeta ^{j}d\zeta $.
• $\Omega _{k}$ is normalized to have vanishing $a$-cycles:
$\oint _{a_{i}}\Omega _{j}=0.$
Denote by $\mathbf {U} _{k}\in \mathbf {C} ^{g}$ the vector of $b$-cycles of $\Omega _{k}$:
$(\mathbf {U} _{k})_{j}:=\oint _{b_{j}}\Omega _{k}.$
Denote the image of ${\mathcal {D}}$ under the Abel map ${\mathcal {A}}:{\mathcal {S}}^{g}(X)\to \mathbf {C} ^{g}$
$\mathbf {E} :={\mathcal {A}}({\mathcal {D}})\in \mathbf {C} ^{g},\quad \mathbf {E} _{j}={\mathcal {A}}_{j}({\mathcal {D}}):=\sum _{j=1}^{g}\int _{p_{0}}^{p_{i}}\omega _{j}$ :={\mathcal {A}}({\mathcal {D}})\in \mathbf {C} ^{g},\quad \mathbf {E} _{j}={\mathcal {A}}_{j}({\mathcal {D}}):=\sum _{j=1}^{g}\int _{p_{0}}^{p_{i}}\omega _{j}}
with arbitrary base point $p_{0}$.
Then the following is a KP $\tau $-function: [13]
$\tau _{(X,{\mathcal {D}},p_{\infty },\zeta )}(\mathbf {t} ):=e^{-{1 \over 2}\sum _{ij}Q_{ij}t_{i}t_{j}}\theta \left(\mathbf {E} +\sum _{k=1}^{\infty }t_{k}\mathbf {U} _{k}{\Big |}B\right).$
Matrix model partition functions as KP $\tau $-functions
Let $d\mu _{0}(M)$ be the Lebesgue measure on the $N^{2}$ dimensional space ${\mathbf {H} }^{N\times N}$ of $N\times N$ complex Hermitian matrices. Let $\rho (M)$ be a conjugation invariant integrable density function
$\rho (UMU^{\dagger })=\rho (M),\quad U\in U(N).$
Define a deformation family of measures
$d\mu _{N,\rho }(\mathbf {t} ):=e^{{\text{ Tr }}(\sum _{i=1}^{\infty }t_{i}M^{i})}\rho (M)d\mu _{0}(M)$
for small $\mathbf {t} =(t_{1},t_{2},\cdots )$ and let
$\tau _{N,\rho }({\bf {t}}):=\int _{{\mathbf {H} }^{N\times N}}d\mu _{N,\rho }({\bf {t}}).$
be the partition function for this random matrix model.[14] Then $\tau _{N,\rho }(\mathbf {t} )$ satisfies the bilinear Hirota residue equation (1), and hence is a $\tau $-function of the KP hierarchy.[15]
$\tau $-functions of hypergeometric type. Generating function for Hurwitz numbers
Let $\{r_{i}\}_{i\in \mathbf {Z} }$ be a (doubly) infinite sequence of complex numbers. For any integer partition $\lambda =(\lambda _{1},\dots ,\lambda _{\ell (\lambda )})$ define the content product coefficient
$r_{\lambda }:=\prod _{(i,j)\in \lambda }r_{j-i}$,
where the product is over all pairs $(i,j)$ of positive integers that correspond to boxes of the Young diagram of the partition $\lambda $, viewed as positions of matrix elements of the corresponding $\ell (\lambda )\times \lambda _{1}$ matrix. Then, for every pair of infinite sequences $\mathbf {t} =(t_{1},t_{2},\dots )$ and $\mathbf {s} =(s_{1},s_{2},\dots )$ of complex vaiables, viewed as (normalized) power sums $\mathbf {t} =[\mathbf {x} ],\ \mathbf {s} =[\mathbf {y} ]$ of the infinite sequence of auxiliary variables $\mathbf {x} =(x_{1},x_{2},\dots )$ and $\mathbf {y} =(y_{1},y_{2},\dots )$, defined by: $t_{j}:={\tfrac {1}{j}}\sum _{a=1}^{\infty }x_{a}^{j},\quad s_{j}:={\tfrac {1}{j}}\sum _{j=1}^{\infty }y_{a}^{j}$, the function
$\tau ^{r}(\mathbf {t} ,\mathbf {s} ):=\sum _{\lambda }r_{\lambda }s_{\lambda }(\mathbf {t} )s_{\lambda }(\mathbf {s} )$
(11)
is a double KP $\tau $-function, both in the $\mathbf {t} $ and the $\mathbf {s} $ variables, known as a $\tau $-function of hypergeometric type.[16]
In particular, choosing
$r_{j}=r_{j}^{\beta }:=e^{j\beta }$
for some small parameter $\beta $, denoting the corresponding content product coefficient as $r_{\lambda }^{\beta }$ and setting $\mathbf {s} =(1,0,\dots )=:\mathbf {t} _{0}$, the resulting $\tau $-function can be equivalently expanded as
$\tau ^{r^{\beta }}(\mathbf {t} ,\mathbf {t} _{0})=\sum _{\lambda }\sum _{d=0}^{\infty }{\frac {\beta ^{d}}{d!}}H_{d}(\lambda )p_{\lambda }(\mathbf {t} ),$
(12)
where $\{H_{d}(\lambda )\}$ are the simple Hurwitz numbers, which are ${\frac {1}{n!}}$ times the number of ways in which an element $k_{\lambda }\in {\mathcal {S}}_{n}$ of the symmetric group ${\mathcal {S}}_{n}$ in $n=|\lambda |$ elements, with cycle lengths equal to the parts of the partition $\lambda $, can be factorized as a product of $d$ $2$-cycles
$k_{\lambda }=(a_{1}b_{1})\dots (a_{d}b_{d})$,
and
$p_{\lambda }(\mathbf {t} )=\prod _{i=1}^{\ell (\lambda )}p_{\lambda _{i}}(\mathbf {t} ),\ {\text{with}}\ p_{i}(\mathbf {t} ):=\sum _{a=1}^{\infty }x_{a}^{i}=it_{i}$
is the power sum symmetric function. Equation (12) thus shows that the (formal) KP hypergeometric $\tau $-function (11) corresponding to the content product coefficients $r_{\lambda }^{\beta }$ is a generating function, in the combinatorial sense, for simple Hurwitz numbers. [17] [18]
References
1. Hirota, Ryogo (1986). "Reduction of soliton equations in bilinear form". Physica D: Nonlinear Phenomena. Elsevier BV. 18 (1–3): 161–170. Bibcode:1986PhyD...18..161H. doi:10.1016/0167-2789(86)90173-9. ISSN 0167-2789.
2. Sato, Mikio, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds", Kokyuroku, RIMS, Kyoto Univ., 30–46 (1981).
3. Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1981). "Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–". Journal of the Physical Society of Japan. Physical Society of Japan. 50 (11): 3806–3812. Bibcode:1981JPSJ...50.3806D. doi:10.1143/jpsj.50.3806. ISSN 0031-9015.
4. Jimbo, Michio; Miwa, Tetsuji (1983). "Solitons and infinite-dimensional Lie algebras". Publications of the Research Institute for Mathematical Sciences. European Mathematical Society Publishing House. 19 (3): 943–1001. doi:10.2977/prims/1195182017. ISSN 0034-5318.
5. Segal, Graeme; Wilson, George (1985). "Loop groups and equations of KdV type". Publications mathématiques de l'IHÉS. Springer Science and Business Media LLC. 61 (1): 5–65. doi:10.1007/bf02698802. ISSN 0073-8301. S2CID 54967353.
6. Jimbo, Michio; Miwa, Tetsuji; Ueno, Kimio (1981). "Monodromy preserving deformation of linear ordinary differential equations with rational coefficients". Physica D: Nonlinear Phenomena. Elsevier BV. 2 (2): 306–352. doi:10.1016/0167-2789(81)90013-0. ISSN 0167-2789.
7. Harnad, J. (1994). "Dual Isomonodromic Deformations and Moment Maps into Loop Algebras". Communications in Mathematical Physics. Springer. 166 (11): 337–365. arXiv:hep-th/9301076. doi:10.1007/BF02112319.
8. Bertola, M.; Harnad, J.; Hurtubise, J. (2023). "Hamiltonian structure of rational isomonodromic deformation systems". Journal of Mathematical Physics. American Institute of Physics. 64: 083502. arXiv:2212.06880. doi:10.1063/5.0142532.
9. Fokas, Athanassios S.; Its, Alexander R.; Kapaev, Andrei A.; Novokshenov, Victor Yu. (2006), Painlevé transcendents: The Riemann–Hilbert approach, Mathematical Surveys and Monographs, vol. 128, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3651-4, MR 2264522
10. Conte, R.; Musette, M. (2020), The Painlevé handbook, second edition, Mathematical physics studies, Switzerland: Springer Nature, ISBN 978-3-030-53339-7
11. Lisovyy, Oleg; Tykhyy, Yuriy (2014). "Algebraic solutions of the sixth Painlevé equation". Journal of Geometry and Physics. 85: 124–163.
12. Kac, V.; Peterson, D.H. (1981). "Spin and wedge representations of infinite-dimensional Lie Algebras and groups". Proc. Natl. Acad. Sci. U.S.A. 58 (6): 3308–3312. doi:10.1073/pnas.78.6.3308. PMC 319557.
13. Dubrovin, B.A. (1981). "Theta Functions and Nonlinear Equations". Russ. Math. Surv. 36 (1): 11–92. doi:10.1070/RM1981v036n02ABEH002596. S2CID 54967353.
14. M.L. Mehta, "Random Matrices", 3rd ed.,vol. 142 of Pure and Applied Mathematics, Elsevier, Academic Press, ISBN 9780120884094 (2004).
15. Kharchev, S.; Marshakov, A.; Mironov, A.; Orlov, A.; Zabrodin, A. (1991). "Matrix models among integrable theories: Forced hierarchies and operator formalism". Nuclear Physics B. Elsevier BV. 366 (3): 569–601. Bibcode:1991NuPhB.366..569K. doi:10.1016/0550-3213(91)90030-2. ISSN 0550-3213.
16. Orlov, A. Yu. (2006). "Hypergeometric Functions as Infinite-Soliton Tau Functions". Theoretical and Mathematical Physics. Springer Science and Business Media LLC. 146 (2): 183–206. Bibcode:2006TMP...146..183O. doi:10.1007/s11232-006-0018-4. ISSN 0040-5779. S2CID 122017484.
17. Pandharipande, R. (2000). "The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere". Letters in Mathematical Physics. Springer Science and Business Media LLC. 53 (1): 59–74. doi:10.1023/a:1026571018707. ISSN 0377-9017. S2CID 17477158.
18. Okounkov, Andrei (2000). "Toda equations for Hurwitz numbers". Mathematical Research Letters. International Press of Boston. 7 (4): 447–453. arXiv:math/0004128. doi:10.4310/mrl.2000.v7.n4.a10. ISSN 1073-2780. S2CID 55141973.
Bibliography
• Dickey, L.A. (2003), "Soliton Equations and Hamiltonian Systems", Vol. 26 of Advanced Series in Mathematical Physics. World Scientific Publishing Co., Inc., River Edge, NJ, 2nd Ed.
• Harnad, J.; Balogh, F. (2021), "Tau functions and Their Applications", Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K.
• Hirota, R. (2004), "The Direct Method in Soliton Theory", Cambridge University Press, Cambridge, U.K.
• Jimbo, M.; Miwa, T. (1999), "Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras", Cambridge University Press, Cambridge, U.K., Cambridge Tracts in Mathematics, 135
• Kodama, Y. (2017), KP Solitons and the Grassmannians: Combinatorics and Geometry of Two-Dimensional Wave Patterns, Springer Briefs in Mathematical Physics, vol. Springer Nature
| Wikipedia |
Tau-leaping
In probability theory, tau-leaping, or τ-leaping, is an approximate method for the simulation of a stochastic system.[1] It is based on the Gillespie algorithm, performing all reactions for an interval of length tau before updating the propensity functions.[2] By updating the rates less often this sometimes allows for more efficient simulation and thus the consideration of larger systems.
Many variants of the basic algorithm have been considered.[3][4][5][6][7]
Algorithm
The algorithm is analogous to the Euler method for deterministic systems, but instead of making a fixed change
$x(t+\tau )=x(t)+\tau x'(t)$
the change is
$x(t+\tau )=x(t)+P(\tau x'(t))$
where $P(\tau x'(t))$ is a Poisson distributed random variable with mean $\tau x'(t)$.
Given a state $\mathbf {x} (t)=\{X_{i}(t)\}$ with events $E_{j}$ occurring at rate $R_{j}(\mathbf {x} (t))$ and with state change vectors $\mathbf {v} _{ij}$ (where $i$ indexes the state variables, and $j$ indexes the events), the method is as follows:
1. Initialise the model with initial conditions $\mathbf {x} (t_{0})=\{X_{i}(t_{0})\}$.
2. Calculate the event rates $R_{j}(\mathbf {x} (t))$.
3. Choose a time step $\tau $. This may be fixed, or by some algorithm dependent on the various event rates.
4. For each event $E_{j}$ generate $K_{j}\sim {\text{Poisson}}(R_{j}\tau )$, which is the number of times each event occurs during the time interval $[t,t+\tau )$.
5. Update the state by
$\mathbf {x} (t+\tau )=\mathbf {x} (t)+\sum _{j}K_{j}v_{ij}$
where $v_{ij}$ is the change on state variable $X_{i}$ due to event $E_{j}$. At this point it may be necessary to check that no populations have reached unrealistic values (such as a population becoming negative due to the unbounded nature of the Poisson variable $K_{j}$).
6. Repeat from Step 2 onwards until some desired condition is met (e.g. a particular state variable reaches 0, or time $t_{1}$ is reached).
Algorithm for efficient step size selection
This algorithm is described by Cao et al.[4] The idea is to bound the relative change in each event rate $R_{j}$ by a specified tolerance $\epsilon $ (Cao et al. recommend $\epsilon =0.03$, although it may depend on model specifics). This is achieved by bounding the relative change in each state variable $X_{i}$ by $\epsilon /g_{i}$, where $g_{i}$ depends on the rate that changes the most for a given change in $X_{i}$. Typically $g_{i}$ is equal the highest order event rate, but this may be more complex in different situations (especially epidemiological models with non-linear event rates).
This algorithm typically requires computing $2N$ auxiliary values (where $N$ is the number of state variables $X_{i}$), and should only require reusing previously calculated values $R_{j}(\mathbf {x} )$. An important factor in this since $X_{i}$ is an integer value, then there is a minimum value by which it can change, preventing the relative change in $R_{j}$ being bounded by 0, which would result in $\tau $ also tending to 0.
1. For each state variable $X_{i}$, calculate the auxiliary values
$\mu _{i}(\mathbf {x} )=\sum _{j}v_{ij}R_{j}(\mathbf {x} )$
$\sigma _{i}^{2}(\mathbf {x} )=\sum _{j}v_{ij}^{2}R_{j}(\mathbf {x} )$
2. For each state variable $X_{i}$, determine the highest order event in which it is involved, and obtain $g_{i}$
3. Calculate time step $\tau $ as
$\tau =\min _{i}{\left\{{\frac {\max {\{\epsilon X_{i}/g_{i},1\}}}{|\mu _{i}(\mathbf {x} )|}},{\frac {\max {\{\epsilon X_{i}/g_{i},1\}}^{2}}{\sigma _{i}^{2}(\mathbf {x} )}}\right\}}$
This computed $\tau $ is then used in Step 3 of the $\tau $ leaping algorithm.
References
1. Gillespie, D. T. (2001). "Approximate accelerated stochastic simulation of chemically reacting systems" (PDF). The Journal of Chemical Physics. 115 (4): 1716–1733. Bibcode:2001JChPh.115.1716G. doi:10.1063/1.1378322.
2. Erhard, F.; Friedel, C. C.; Zimmer, R. (2010). "FERN – Stochastic Simulation and Evaluation of Reaction Networks". Systems Biology for Signaling Networks. p. 751. doi:10.1007/978-1-4419-5797-9_30. ISBN 978-1-4419-5796-2.
3. Cao, Y.; Gillespie, D. T.; Petzold, L. R. (2005). "Avoiding negative populations in explicit Poisson tau-leaping". The Journal of Chemical Physics. 123 (5): 054104. Bibcode:2005JChPh.123e4104C. CiteSeerX 10.1.1.123.3650. doi:10.1063/1.1992473. PMID 16108628. S2CID 1652735.
4. Cao, Y.; Gillespie, D. T.; Petzold, L. R. (2006). "Efficient step size selection for the tau-leaping simulation method" (PDF). The Journal of Chemical Physics. 124 (4): 044109. Bibcode:2006JChPh.124d4109C. doi:10.1063/1.2159468. PMID 16460151.
5. Anderson, David F. (2008-02-07). "Incorporating postleap checks in tau-leaping". The Journal of Chemical Physics. 128 (5): 054103. arXiv:0708.0377. Bibcode:2008JChPh.128e4103A. doi:10.1063/1.2819665. ISSN 0021-9606. PMID 18266441. S2CID 1166923.
6. Chatterjee, Abhijit; Vlachos, Dionisios G.; Katsoulakis, Markos A. (2005-01-08). "Binomial distribution based τ-leap accelerated stochastic simulation". The Journal of Chemical Physics. 122 (2): 024112. Bibcode:2005JChPh.122b4112C. doi:10.1063/1.1833357. ISSN 0021-9606. PMID 15638577.
7. Moraes, Alvaro; Tempone, Raul; Vilanova, Pedro (2014-04-24). "Hybrid Chernoff Tau-Leap". Multiscale Modeling & Simulation. 12 (2): 581–615. CiteSeerX 10.1.1.756.9799. doi:10.1137/130925657. ISSN 1540-3467.
| Wikipedia |
Refactorable number
A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that $\tau (n)\mid n$. The first few refactorable numbers are listed in (sequence A033950 in the OEIS) as
1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ...
For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.
Properties
Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable.[1] Colton proved that no refactorable number is perfect. The equation $\gcd(n,x)=\tau (n)$ has solutions only if $n$ is a refactorable number, where $\gcd $ is the greatest common divisor function.
Let $T(x)$ be the number of refactorable numbers which are at most $x$. The problem of determining an asymptotic for $T(x)$ is open. Spiro has proven that $T(x)={\frac {x}{{\sqrt {\log x}}(\log \log x)^{1+o(1)}}}$[2]
There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large $n$ such that both $n$ and $n+1$ are refactorable. Zelinsky wondered if there exists a refactorable number $n_{0}\equiv a\mod m$, does there necessarily exist $n>n_{0}$ such that $n$ is refactorable and $n\equiv a\mod m$.
History
First defined by Curtis Cooper and Robert E. Kennedy[3] where they showed that the tau numbers have natural density zero, they were later rediscovered by Simon Colton using a computer program he had made which invents and judges definitions from a variety of areas of mathematics such as number theory and graph theory.[4] Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.
See also
• Divisor function
References
1. J. Zelinsky, "Tau Numbers: A Partial Proof of a Conjecture and Other Results," Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8
2. Spiro, Claudia (1985). "How often is the number of divisors of n a divisor of n?". Journal of Number Theory. 21 (1): 81–100. doi:10.1016/0022-314X(85)90012-5.
3. Cooper, C.N. and Kennedy, R. E. "Tau Numbers, Natural Density, and Hardy and Wright's Theorem 437." Internat. J. Math. Math. Sci. 13, 383-386, 1990
4. S. Colton, "Refactorable Numbers - A Machine Invention," Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.2
Classes of natural numbers
Powers and related numbers
• Achilles
• Power of 2
• Power of 3
• Power of 10
• Square
• Cube
• Fourth power
• Fifth power
• Sixth power
• Seventh power
• Eighth power
• Perfect power
• Powerful
• Prime power
Of the form a × 2b ± 1
• Cullen
• Double Mersenne
• Fermat
• Mersenne
• Proth
• Thabit
• Woodall
Other polynomial numbers
• Hilbert
• Idoneal
• Leyland
• Loeschian
• Lucky numbers of Euler
Recursively defined numbers
• Fibonacci
• Jacobsthal
• Leonardo
• Lucas
• Padovan
• Pell
• Perrin
Possessing a specific set of other numbers
• Amenable
• Congruent
• Knödel
• Riesel
• Sierpiński
Expressible via specific sums
• Nonhypotenuse
• Polite
• Practical
• Primary pseudoperfect
• Ulam
• Wolstenholme
Figurate numbers
2-dimensional
centered
• Centered triangular
• Centered square
• Centered pentagonal
• Centered hexagonal
• Centered heptagonal
• Centered octagonal
• Centered nonagonal
• Centered decagonal
• Star
non-centered
• Triangular
• Square
• Square triangular
• Pentagonal
• Hexagonal
• Heptagonal
• Octagonal
• Nonagonal
• Decagonal
• Dodecagonal
3-dimensional
centered
• Centered tetrahedral
• Centered cube
• Centered octahedral
• Centered dodecahedral
• Centered icosahedral
non-centered
• Tetrahedral
• Cubic
• Octahedral
• Dodecahedral
• Icosahedral
• Stella octangula
pyramidal
• Square pyramidal
4-dimensional
non-centered
• Pentatope
• Squared triangular
• Tesseractic
Combinatorial numbers
• Bell
• Cake
• Catalan
• Dedekind
• Delannoy
• Euler
• Eulerian
• Fuss–Catalan
• Lah
• Lazy caterer's sequence
• Lobb
• Motzkin
• Narayana
• Ordered Bell
• Schröder
• Schröder–Hipparchus
• Stirling first
• Stirling second
• Telephone number
• Wedderburn–Etherington
Primes
• Wieferich
• Wall–Sun–Sun
• Wolstenholme prime
• Wilson
Pseudoprimes
• Carmichael number
• Catalan pseudoprime
• Elliptic pseudoprime
• Euler pseudoprime
• Euler–Jacobi pseudoprime
• Fermat pseudoprime
• Frobenius pseudoprime
• Lucas pseudoprime
• Lucas–Carmichael number
• Somer–Lucas pseudoprime
• Strong pseudoprime
Arithmetic functions and dynamics
Divisor functions
• Abundant
• Almost perfect
• Arithmetic
• Betrothed
• Colossally abundant
• Deficient
• Descartes
• Hemiperfect
• Highly abundant
• Highly composite
• Hyperperfect
• Multiply perfect
• Perfect
• Practical
• Primitive abundant
• Quasiperfect
• Refactorable
• Semiperfect
• Sublime
• Superabundant
• Superior highly composite
• Superperfect
Prime omega functions
• Almost prime
• Semiprime
Euler's totient function
• Highly cototient
• Highly totient
• Noncototient
• Nontotient
• Perfect totient
• Sparsely totient
Aliquot sequences
• Amicable
• Perfect
• Sociable
• Untouchable
Primorial
• Euclid
• Fortunate
Other prime factor or divisor related numbers
• Blum
• Cyclic
• Erdős–Nicolas
• Erdős–Woods
• Friendly
• Giuga
• Harmonic divisor
• Jordan–Pólya
• Lucas–Carmichael
• Pronic
• Regular
• Rough
• Smooth
• Sphenic
• Størmer
• Super-Poulet
• Zeisel
Numeral system-dependent numbers
Arithmetic functions
and dynamics
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| Wikipedia |
Abelian and Tauberian theorems
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods.
There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense.
In the theory of integral transforms, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function.[1]
Abelian theorems
For any summation method L, its Abelian theorem is the result that if c = (cn) is a convergent sequence, with limit C, then L(c) = C.
An example is given by the Cesàro method, in which L is defined as the limit of the arithmetic means of the first N terms of c, as N tends to infinity. One can prove that if c does converge to C, then so does the sequence (dN) where
$d_{N}={\frac {c_{1}+c_{2}+\cdots +c_{N}}{N}}.$
To see that, subtract C everywhere to reduce to the case C = 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take N large enough to make the initial segment of terms up to cN average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also necessarily bounded.
The name derives from Abel's theorem on power series. In that case L is the radial limit (thought of within the complex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with term
anzn
and set z = r ·eiθ. That theorem has its main interest in the case that the power series has radius of convergence exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] so that the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum of the an. When the radius is 1 the power series will have some singularity on |z| = 1; the assertion is that, nonetheless, if the sum of the an exists, it is equal to the limit over r. This therefore fits exactly into the abstract picture.
Tauberian theorems
Partial converses to Abelian theorems are called Tauberian theorems. The original result of Alfred Tauber (1897)[2] stated that if we assume also
an = o(1/n)
(see Little o notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent. This was strengthened by John Edensor Littlewood: we need only assume O(1/n). A sweeping generalization is the Hardy–Littlewood Tauberian theorem.
In the abstract setting, therefore, an Abelian theorem states that the domain of L contains the convergent sequences, and its values there are equal to those of the Lim functional. A Tauberian theorem states, under some growth condition, that the domain of L is exactly the convergent sequences and no more.
If one thinks of L as some generalised type of weighted average, taken to the limit, a Tauberian theorem allows one to discard the weighting, under the correct hypotheses. There are many applications of this kind of result in number theory, in particular in handling Dirichlet series.
The development of the field of Tauberian theorems received a fresh turn with Norbert Wiener's very general results, namely Wiener's Tauberian theorem and its large collection of corollaries.[3] The central theorem can now be proved by Banach algebra methods, and contains much, though not all, of the previous theory.
See also
• Wiener's Tauberian theorem
• Hardy–Littlewood Tauberian theorem
• Haar's Tauberian theorem
References
1. Froese Fischer, Charlotte (1954). A method for finding the asymptotic behavior of a function from its Laplace transform (Thesis). University of British Columbia. doi:10.14288/1.0080631.
2. Tauber, Alfred (1897). "Ein Satz aus der Theorie der unendlichen Reihen" [A theorem about infinite series]. Monatshefte für Mathematik und Physik (in German). 8: 273–277. doi:10.1007/BF01696278. JFM 28.0221.02. S2CID 120692627.
3. Wiener, Norbert (1932). "Tauberian theorems". Annals of Mathematics. 33 (1): 1–100. doi:10.2307/1968102. JFM 58.0226.02. JSTOR 1968102. MR 1503035. Zbl 0004.05905.
External links
• "Tauberian theorems", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Korevaar, Jacob (2004). Tauberian theory. A century of developments. Grundlehren der Mathematischen Wissenschaften. Vol. 329. Springer-Verlag. pp. xvi+483. doi:10.1007/978-3-662-10225-1. ISBN 978-3-540-21058-0. MR 2073637. Zbl 1056.40002.
• Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cambridge Studies in Advanced Mathematics. Vol. 97. Cambridge: Cambridge University Press. pp. 147–167. ISBN 978-0-521-84903-6. MR 2378655. Zbl 1142.11001.
| Wikipedia |
Taubes's Gromov invariant
In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold, where the curves are holomorphic with respect to an auxiliary compatible almost complex structure. (Multiple covers of 2-tori with self-intersection 0 are also counted.)
Taubes proved the information contained in this invariant is equivalent to invariants derived from the Seiberg–Witten equations in a series of four long papers. Much of the analytical complexity connected to this invariant comes from properly counting multiply covered pseudoholomorphic curves so that the result is invariant of the choice of almost complex structure. The crux is a topologically defined index for pseudoholomorphic curves which controls embeddedness and bounds the Fredholm index.
Embedded contact homology is an extension due to Michael Hutchings of this work to noncompact four-manifolds of the form $Y\times \mathbb {R} $, where Y is a compact contact 3-manifold. ECH is a symplectic field theory-like invariant; namely, it is the homology of a chain complex generated by certain combinations of Reeb orbits of a contact form on Y, and whose differential counts certain embedded pseudoholomorphic curves and multiply covered pseudoholomorphic cylinders with "ECH index" 1 in $Y\times \mathbb {R} $. The ECH index is a version of Taubes's index for the cylindrical case, and again, the curves are pseudoholomorphic with respect to a suitable almost complex structure. The result is a topological invariant of Y, which Taubes proved is isomorphic to monopole Floer homology, a version of Seiberg–Witten homology for Y.
References
• Taubes, Clifford (2000). Wentworth, Richard (ed.). Seiberg Witten and Gromov invariants for symplectic 4-manifolds. First International Press Lecture Series. Vol. 2. Somerville, MA: International Press. ISBN 1-57146-061-6. MR 1798809.
• Taubes, Clifford (2010). "Embedded contact homology and Seiberg-Witten Floer cohomology I.". Geometry & Topology. 14 (5): 2497–2581. arXiv:0811.3985. doi:10.2140/gt.2010.14.2497. MR 2746723.
| Wikipedia |
Tautness (topology)
In mathematics, particularly in algebraic topology, a taut pair is a topological pair whose direct limit of cohomology module of open neighborhood of that pair which is directed downward by inclusion is isomorphic to the cohomology module of original pair.
Definition
For a topological pair $(A,B)$ in a topological space $X$, a neighborhood $(U,V)$ of such a pair is defined to be a pair such that $U$ and $V$ are neighborhoods of $A$ and $B$ respectively.
If we collect all neighborhoods of $(A,B)$, then we can form a directed set which is directed downward by inclusion. Hence its cohomology module $H^{q}(U,V;G)$ is a direct system where $G$ is a module over a ring with unity. If we denote its direct limit by
${\bar {H}}^{q}(A,B;G)=\varinjlim H^{q}(U,V;G)$
the restriction maps $H^{q}(U,V;G)\to H^{q}(A,B;G)$ define a natural homomorphism $i:{\bar {H}}^{q}(A,B;G)\to H^{q}(A,B;G)$.
The pair $(A,B)$ is said to be tautly embedded in $X$ (or a taut pair in $X$) if $i$ is an isomorphism for all $q$ and $G$.[1]
Basic properties
• For pair $(A,B)$ of $X$, if two of the three pairs $(B,\emptyset ),(A,\emptyset )$, and $(A,B)$ are taut in $X$, so is the third.
• For pair $(A,B)$ of $X$, if $A,B$ and $X$ have compact triangulation, then $(A,B)$ in $X$ is taut.
• If $U$ varies over the neighborhoods of $A$, there is an isomorphism $\varinjlim {\bar {H}}^{q}(U;G)\simeq {\bar {H}}^{q}(A;G)$.
• If $(A,B)$ and $(A',B')$ are closed pairs in a normal space $X$, there is an exact relative Mayer-Vietoris sequence for any coefficient module $G$[2]
$\cdots \to {\bar {H}}^{q}(A\cup A',B\cup B')\to {\bar {H}}^{q}(A,B)\oplus {\bar {H}}^{q}(A',B')\to {\bar {H}}^{q}(A\cap A',B\cap B')\to \cdots $
Properties related to cohomology theory
• Let $A$ be any subspace of a topological space $X$ which is a neighborhood retract of $X$. Then $A$ is a taut subspace of $X$ with respect to Alexander-Spanier cohomology.
• every retract of an arbitrary topological space is a taut subspace of $X$ with respect to Alexander-Spanier cohomology.
• A closed subspace of a paracompactt Hausdorff space is a taut subspace of relative to the Alexander cohomology theory[3]
Note
Since the Čech cohomology and the Alexander-Spanier cohomology are naturally isomorphic on the category of all topological pairs,[4] all of the above properties are valid for Čech cohomology. However, it's not true for singular cohomology (see Example)
Dependence of cohomology theory
Example[5]
Let $X$ be the subspace of $\mathbb {R} ^{2}\subset S^{2}$ which is the union of four sets
$A_{1}=\{(x,y)\mid x=0,-2\leq y\leq 1\}$
$A_{2}=\{(x,y)\mid 0\leq x\leq 1,y=-2\}$
$A_{3}=\{(x,y)\mid x=1,-2\leq y\leq 0\}$
$A_{4}=\{(x,y)\mid 0<x\leq 1,y=\sin 2\pi /x\}$
The first singular cohomology of $X$ is $H^{1}(X;Z)=0$ and using the Alexander duality theorem on $S^{2}-X$, $\varinjlim \{H^{q}(U;\mathbb {Z} )\}=\mathbb {Z} $ as $U$ varies over neighborhoods of $X$.
Therefore, $\varinjlim \{H^{q}(U;\mathbb {Z} )\}\to H^{1}(X;\mathbb {Z} )$ is not a monomorphism so that $X$ is not a taut subspace of $\mathbb {R} ^{2}$ with respect to singular cohomology. However, since $X$ is closed in $\mathbb {R} ^{2}$, it's taut subspace with respect to Alexander cohomology.[6]
See also
• Alexander-Spanier cohomology
• Čech cohomology
References
1. Spanier, Edwin H. (1966). Algebraic topology. p. 289. ISBN 978-0387944265.
2. Spanier, Edwin H. (1966). Algebraic topology. p. 290-291. ISBN 978-0387944265.
3. Deo, Satya (197). "On the tautness property of Alexander-Spanier cohomology". American Mathematical Society. 52: 441-442. doi:10.2307/2040179.
4. Dowker, C. H. (1952). "Homology groups of relations". Annals of Mathematics. (2) 56 (1): 84–95. doi:10.2307/1969768. JSTOR 1969768.
5. Spanier, Edwin H. (1966). Algebraic topology. p. 317. ISBN 978-0387944265.
6. Spanier, Edwin H. (1978). "Tautness for Alexander-Spanier cohomology". Pacific Journal of Mathematics. 75 (2): 562. doi:10.2140/pjm.1978.75.561. S2CID 122337937.
| Wikipedia |
Tautological ring
In algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained from 1 by pushforward along various morphisms described below. The tautological cohomology ring is the image of the tautological ring under the cycle map (from the Chow ring to the cohomology ring).
Definition
Let ${\overline {\mathcal {M}}}_{g,n}$ be the moduli stack of stable marked curves $(C;x_{1},\ldots ,x_{n})$, such that
• C is a complex curve of arithmetic genus g whose only singularities are nodes,
• the n points x1, ..., xn are distinct smooth points of C,
• the marked curve is stable, namely its automorphism group (leaving marked points invariant) is finite.
The last condition requires $2g-2+n>0$ in other words (g,n) is not among (0,0), (0,1), (0,2), (1,0). The stack ${\overline {\mathcal {M}}}_{g,n}$ then has dimension $3g-3+n$. Besides permutations of the marked points, the following morphisms between these moduli stacks play an important role in defining tautological classes:
• Forgetful maps ${\overline {\mathcal {M}}}_{g,n}\to {\overline {\mathcal {M}}}_{g,n-1}$ which act by removing a given point xk from the set of marked points, then restabilizing the marked curved if it is not stable anymore.
• Gluing maps ${\overline {\mathcal {M}}}_{g,n+1}\times {\overline {\mathcal {M}}}_{g',n'+1}\to {\overline {\mathcal {M}}}_{g+g',n+n'}$ that identify the k-th marked point of a curve to the l-th marked point of the other. Another set of gluing maps is ${\overline {\mathcal {M}}}_{g,n+2}\to {\overline {\mathcal {M}}}_{g+1,n}$ that identify the k-th and l-th marked points, thus increasing the genus by creating a closed loop.
The tautological rings $R^{\bullet }({\overline {\mathcal {M}}}_{g,n})$ are simultaneously defined as the smallest subrings of the Chow rings closed under pushforward by forgetful and gluing maps.[1]
The tautological cohomology ring $RH^{\bullet }({\overline {\mathcal {M}}}_{g,n})$ is the image of $R^{\bullet }({\overline {\mathcal {M}}}_{g,n})$ under the cycle map. As of 2016, it is not known whether the tautological and tautological cohomology rings are isomorphic.
Generating set
For $1\leq k\leq n$ we define the class $\psi _{k}\in R^{\bullet }({\overline {\mathcal {M}}}_{g,n})$ as follows. Let $\delta _{k}$ be the pushforward of 1 along the gluing map ${\overline {\mathcal {M}}}_{g,n}\times {\overline {\mathcal {M}}}_{0,3}\to {\overline {\mathcal {M}}}_{g,n+1}$ which identifies the marked point xk of the first curve to one of the three marked points yi on the sphere (the latter choice is unimportant thanks to automorphisms). For definiteness order the resulting points as x1, ..., xk−1, y1, y2, xk+1, ..., xn. Then $\psi _{k}$ is defined as the pushforward of $-\delta _{k}^{2}$ along the forgetful map that forgets the point y2. This class coincides with the first Chern class of a certain line bundle.[1]
For $i\geq 1$ we also define $\kappa _{i}\in R^{\bullet }({\overline {\mathcal {M}}}_{g,n})$ be the pushforward of $(\psi _{k})^{i+1}$ along the forgetful map ${\overline {\mathcal {M}}}_{g,n+1}\to {\overline {\mathcal {M}}}_{g,n}$ that forgets the k-th point. This is independent of k (simply permute points).
Theorem. $R^{\bullet }({\overline {\mathcal {M}}}_{g,n})$ is additively generated by pushforwards along (any number of) gluing maps of monomials in $\psi $ and $\kappa $ classes.
These pushforwards of monomials (hereafter called basic classes) do not form a basis. The set of relations is not fully known.
Theorem. The tautological rings are invariant under pullback along gluing and forgetful maps. There exist universal combinatorial formulae expressing pushforwards, pullbacks, and products of basic classes as linear combinations of basic classes.
Faber conjectures
The tautological ring $R^{\bullet }({\mathcal {M}}_{g,n})$ on the moduli space of smooth n-pointed genus g curves simply consists of restrictions of classes in $R^{\bullet }({\overline {\mathcal {M}}}_{g,n})$. We omit n when it is zero (when there is no marked point).
In the case $n=0$ of curves with no marked point, Mumford conjectured, and Madsen and Weiss proved, that for any $d>0$ the map $\mathbb {Q} [\kappa _{1},\kappa _{2},\ldots ]\to H^{\bullet }({\mathcal {M}}_{g})$ is an isomorphism in degree d for large enough g. In this case all classes are tautological.
Conjecture (Faber). (1) Large-degree tautological rings vanish: $R^{d}({\mathcal {M}}_{g})=0$ for $d>g-2.$ (2) $R^{g-2}({\mathcal {M}}_{g})\cong \mathbb {Q} $ and there is an explicit combinatorial formula for this isomorphism. (3) The product (coming from the Chow ring) of classes defines a perfect pairing $R^{d}({\mathcal {M}}_{g})\times R^{g-d-2}({\mathcal {M}}_{g})\to R^{g-2}({\mathcal {M}}_{g})\cong \mathbb {Q} .$
Although $R^{d}({\mathcal {M}}_{g})$ trivially vanishes for $d>3g-3$ because of the dimension of ${\mathcal {M}}_{g}$, the conjectured bound is much lower. The conjecture would completely determine the structure of the ring: a polynomial in the $\kappa _{j}$ of cohomological degree d vanishes if and only if its pairing with all polynomials of cohomological degree $g-d-2$ vanishes.
Parts (1) and (2) of the conjecture were proven. Part (3), also called the Gorenstein conjecture, was only checked for $g<24$. For $g=24$ and higher genus, several methods of constructing relations between $\kappa $ classes find the same set of relations which suggest that the dimensions of $R^{d}({\mathcal {M}}_{g})$ and $R^{g-d-2}({\mathcal {M}}_{g})$ are different. If the set of relations found by these methods is complete then the Gorenstein conjecture is wrong. Besides Faber's original non-systematic computer search based on classical maps between vector bundles over ${\mathcal {C}}_{g}^{d}$, the d-th fiber power of the universal curve ${\mathcal {C}}_{g}={\mathcal {M}}_{g,1}\twoheadrightarrow {\mathcal {M}}_{g}$, the following methods have been used to find relations:
• Virtual classes of the moduli space of stable quotients (over $\mathbb {P} ^{1}$) by Pandharipande and Pixton.[2]
• Witten's r-spin class and Givental-Telemann's classification of cohomological field theories, used by Pandharipande, Pixton, Zvonkine.[3]
• Geometry of the universal Jacobian over ${\mathcal {M}}_{g,1}$, by Yin.
• Powers of theta-divisor on the universal abelian variety, by Grushevsky and Zakharov.[4]
These four methods are proven to give the same set of relations.
Similar conjectures were formulated for moduli spaces ${\overline {\mathcal {M}}}_{g,n}$ of stable curves and ${\mathcal {M}}_{g,n}^{\text{c.t.}}$ of compact-type stable curves. However, Petersen-Tommasi[5] proved that $R^{\bullet }({\overline {\mathcal {M}}}_{2,20})$ and $R^{\bullet }({\mathcal {M}}_{2,8}^{\text{c.t.}})$ fail to obey the (analogous) Gorenstein conjecture. On the other hand, Tavakol[6] proved that for genus 2 the moduli space of rational-tails stable curves ${\mathcal {M}}_{2,n}^{\text{r.t.}}$ obeys the Gorenstein condition for every n.
See also
• ELSV formula
• Hodge bundle
• Witten's conjecture
References
1. Faber, C.; Pandharipande, R. (2011). "Tautological and non-tautological cohomology of the moduli space of curves". arXiv:1101.5489 [math.AG].
2. Pandharipande, R.; Pixton, A. (2013). "Relations in the tautological ring of the moduli space of curves". arXiv:1301.4561 [math.AG].
3. Pandharipande, R.; Pixton, A.; Zvonkine, D. (2016). "Tautological relations via r-spin structures". arXiv:1607.00978 [math.AG].
4. Grushevsky, Samuel; Zakharov, Dmitry (2012). "The zero section of the universal semiabelian variety, and the double ramification cycle". Duke Mathematical Journal. 163 (5): 953–982. arXiv:1206.3534. doi:10.1215/00127094-26444575.
5. Petersen, Dan; Tommasi, Orsola (2012). "The Gorenstein conjecture fails for the tautological ring of $\mathcal{\bar M}_{2,n}$". Inventiones mathematicae. 196 (2014): 139. arXiv:1210.5761. Bibcode:2014InMat.196..139P. doi:10.1007/s00222-013-0466-z.
6. Tavakol, Mehdi (2011). "The tautological ring of the moduli space M_{2,n}^rt". arXiv:1101.5242 [math.AG].
• Vakil, Ravi (2003), "The moduli space of curves and its tautological ring" (PDF), Notices of the American Mathematical Society, 50 (6): 647–658, MR 1988577
• Graber, Tom; Vakil, Ravi (2001), "On the tautological ring of ${\overline {\mathcal {M}}}_{g,n}$" (PDF), Turkish Journal of Mathematics, 25 (1): 237–243, MR 1829089
| Wikipedia |
Tav (number)
In his work on set theory, Georg Cantor denoted the collection of all cardinal numbers by the last letter of the Hebrew alphabet, ת (transliterated as Tav, Taw, or Sav.) As Cantor realized, this collection could not itself have a cardinality, as this would lead to a paradox of the Burali-Forti type. Cantor instead said that it was an "inconsistent" collection which was absolutely infinite.[3][4]
See also
• Taw (letter)
• Aleph number
• Absolute Infinite
References
1. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Georg Cantor, ed. Ernst Zermelo, with biography by Adolf Fraenkel; orig. pub. Berlin: Verlag von Julius Springer, 1932; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, ISBN 3-540-09849-6.
2. The Rediscovery of the Cantor-Dedekind Correspondence, I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 76 (1974/75), pp. 104–139, at p. 126 ff.
3. Gesammelte Abhandlungen,[1] Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962, pp. 443–447; translated into English in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as Ivor Grattan-Guinness has discovered,[2] this is in fact an amalgamation by Cantor's editor, Ernst Zermelo, of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August 3.
4. The Correspondence between Georg Cantor and Philip Jourdain, I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 73 (1971/72), pp. 111–130, at pp. 116–117.
| Wikipedia |
Taxicab number
In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), also called the nth Ramanujan–Hardy number, is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103.
The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy:
I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways."[1][2]
History and definition
The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, who published the Hardy–Ramanujan number Ta(2) = 1729. This particular example of 1729 was made famous in the early 20th century by a story involving Srinivasa Ramanujan. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are the smallest possible and so it cannot be used to find the actual value of Ta(n).
The taxicab numbers subsequent to 1729 were found with the help of computers. John Leech obtained Ta(3) in 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1989.[3] J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999.[4][5] Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008,[6] following a 2003 paper by Calude et al. that gave a 99% probability that the number was actually Ta(6).[7] Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006.[8]
The restriction of the summands to positive numbers is necessary, because allowing negative numbers allows for more (and smaller) instances of numbers that can be expressed as sums of cubes in n distinct ways. The concept of a cabtaxi number has been introduced to allow for alternative, less restrictive definitions of this nature. In a sense, the specification of two summands and powers of three is also restrictive; a generalized taxicab number allows for these values to be other than two and three, respectively.
Known taxicab numbers
So far, the following 6 taxicab numbers are known:
${\begin{aligned}\operatorname {Ta} (1)=2&=1^{3}+1^{3}\end{aligned}}$
${\begin{aligned}\operatorname {Ta} (2)=1729&=1^{3}+12^{3}\\&=9^{3}+10^{3}\end{aligned}}$
${\begin{aligned}\operatorname {Ta} (3)=87539319&=167^{3}+436^{3}\\&=228^{3}+423^{3}\\&=255^{3}+414^{3}\end{aligned}}$
${\begin{aligned}\operatorname {Ta} (4)=6963472309248&=2421^{3}+19083^{3}\\&=5436^{3}+18948^{3}\\&=10200^{3}+18072^{3}\\&=13322^{3}+16630^{3}\end{aligned}}$
${\begin{aligned}\operatorname {Ta} (5)=48988659276962496&=38787^{3}+365757^{3}\\&=107839^{3}+362753^{3}\\&=205292^{3}+342952^{3}\\&=221424^{3}+336588^{3}\\&=231518^{3}+331954^{3}\end{aligned}}$
${\begin{aligned}\operatorname {Ta} (6)=24153319581254312065344&=582162^{3}+28906206^{3}\\&=3064173^{3}+28894803^{3}\\&=8519281^{3}+28657487^{3}\\&=16218068^{3}+27093208^{3}\\&=17492496^{3}+26590452^{3}\\&=18289922^{3}+26224366^{3}\end{aligned}}$
Upper bounds for taxicab numbers
For the following taxicab numbers upper bounds are known:
${\begin{matrix}\operatorname {Ta} (7)&\leq &24885189317885898975235988544&=&2648660966^{3}+1847282122^{3}\\&&&=&2685635652^{3}+1766742096^{3}\\&&&=&2736414008^{3}+1638024868^{3}\\&&&=&2894406187^{3}+860447381^{3}\\&&&=&2915734948^{3}+459531128^{3}\\&&&=&2918375103^{3}+309481473^{3}\\&&&=&2919526806^{3}+58798362^{3}\end{matrix}}$
${\begin{matrix}\operatorname {Ta} (8)&\leq &50974398750539071400590819921724352&=&299512063576^{3}+288873662876^{3}\\&&&=&336379942682^{3}+234604829494^{3}\\&&&=&341075727804^{3}+224376246192^{3}\\&&&=&347524579016^{3}+208029158236^{3}\\&&&=&367589585749^{3}+109276817387^{3}\\&&&=&370298338396^{3}+58360453256^{3}\\&&&=&370633638081^{3}+39304147071^{3}\\&&&=&370779904362^{3}+7467391974^{3}\end{matrix}}$
${\begin{matrix}\operatorname {Ta} (9)&\leq &136897813798023990395783317207361432493888&=&41632176837064^{3}+40153439139764^{3}\\&&&=&46756812032798^{3}+32610071299666^{3}\\&&&=&47409526164756^{3}+31188298220688^{3}\\&&&=&48305916483224^{3}+28916052994804^{3}\\&&&=&51094952419111^{3}+15189477616793^{3}\\&&&=&51471469037044^{3}+8112103002584^{3}\\&&&=&51518075693259^{3}+5463276442869^{3}\\&&&=&51530042142656^{3}+4076877805588^{3}\\&&&=&51538406706318^{3}+1037967484386^{3}\end{matrix}}$
${\begin{matrix}\operatorname {Ta} (10)&\leq &7335345315241855602572782233444632535674275447104&=&15695330667573128^{3}+15137846555691028^{3}\\&&&=&17627318136364846^{3}+12293996879974082^{3}\\&&&=&17873391364113012^{3}+11757988429199376^{3}\\&&&=&18211330514175448^{3}+10901351979041108^{3}\\&&&=&19262797062004847^{3}+5726433061530961^{3}\\&&&=&19404743826965588^{3}+3058262831974168^{3}\\&&&=&19422314536358643^{3}+2059655218961613^{3}\\&&&=&19426825887781312^{3}+1536982932706676^{3}\\&&&=&19429379778270560^{3}+904069333568884^{3}\\&&&=&19429979328281886^{3}+391313741613522^{3}\end{matrix}}$
${\begin{matrix}\operatorname {Ta} (11)&\leq &2818537360434849382734382145310807703728251895897826621632&=&11410505395325664056^{3}+11005214445987377356^{3}\\&&&=&12815060285137243042^{3}+8937735731741157614^{3}\\&&&=&12993955521710159724^{3}+8548057588027946352^{3}\\&&&=&13239637283805550696^{3}+7925282888762885516^{3}\\&&&=&13600192974314732786^{3}+6716379921779399326^{3}\\&&&=&14004053464077523769^{3}+4163116835733008647^{3}\\&&&=&14107248762203982476^{3}+2223357078845220136^{3}\\&&&=&14120022667932733461^{3}+1497369344185092651^{3}\\&&&=&14123302420417013824^{3}+1117386592077753452^{3}\\&&&=&14125159098802697120^{3}+657258405504578668^{3}\\&&&=&14125594971660931122^{3}+284485090153030494^{3}\end{matrix}}$
${\begin{aligned}\operatorname {Ta} (12)\leq &73914858746493893996583617733225161086864012865017882136931801625152\\&=33900611529512547910376^{3}+32696492119028498124676^{3}\\&=38073544107142749077782^{3}+26554012859002979271194^{3}\\&=38605041855000884540004^{3}+25396279094031028611792^{3}\\&=39334962370186291117816^{3}+23546015462514532868036^{3}\\&=40406173326689071107206^{3}+19954364747606595397546^{3}\\&=41606042841774323117699^{3}+12368620118962768690237^{3}\\&=41912636072508031936196^{3}+6605593881249149024056^{3}\\&=41950587346428151112631^{3}+4448684321573910266121^{3}\\&=41960331491058948071104^{3}+3319755565063005505892^{3}\\&=41965847682542813143520^{3}+1952714722754103222628^{3}\\&=41965889731136229476526^{3}+1933097542618122241026^{3}\\&=41967142660804626363462^{3}+845205202844653597674^{3}\end{aligned}}$
Cubefree taxicab numbers
A more restrictive taxicab problem requires that the taxicab number be cubefree, which means that it is not divisible by any cube other than 13. When a cubefree taxicab number T is written as T = x3 + y3, the numbers x and y must be relatively prime. Among the taxicab numbers Ta(n) listed above, only Ta(1) and Ta(2) are cubefree taxicab numbers. The smallest cubefree taxicab number with three representations was discovered by Paul Vojta (unpublished) in 1981 while he was a graduate student. It is
15170835645
= 5173 + 24683
= 7093 + 24563
= 17333 + 21523.
The smallest cubefree taxicab number with four representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003. It is
1801049058342701083
= 922273 + 12165003
= 1366353 + 12161023
= 3419953 + 12076023
= 6002593 + 11658843
(sequence A080642 in the OEIS).
See also
• 1729 (number) – Hardy-Ramanujan number
• Diophantine equation – Polynomial equation whose integer solutions are sought
• Euler's sum of powers conjecture – Disproved conjecture in number theory
• Generalized taxicab number
• Beal's conjecture – Mathematical conjecturePages displaying short descriptions of redirect targets
• Jacobi–Madden equation
• Prouhet–Tarry–Escott problem
• Pythagorean quadruple – Four integers where the sum of the squares of three equals the square of the fourth
• Sums of three cubes – Problem in number theory
• Sums of powers, a list of related conjectures and theorems
Notes
1. Quotations by G. H. Hardy, MacTutor History of Mathematics Archived 2012-07-16 at the Wayback Machine
2. Silverman, Joseph H. (1993). "Taxicabs and sums of two cubes". Amer. Math. Monthly. 100 (4): 331–340. doi:10.2307/2324954. JSTOR 2324954.
3. Numbers Count column, Personal Computer World, page 234, November 1989
4. Numbers Count column of Personal Computer World, page 610, Feb 1995
5. "The Fifth Taxicab Number is 48988659276962496" by David W. Wilson
6. NMBRTHRY Archives – March 2008 (#10) "The sixth taxicab number is 24153319581254312065344" by Uwe Hollerbach
7. C. S. Calude, E. Calude and M. J. Dinneen: What is the value of Taxicab(6)?, Journal of Universal Computer Science, Vol. 9 (2003), pp. 1196–1203
8. "'New Upper Bounds for Taxicab and Cabtaxi Numbers" Christian Boyer, France, 2006–2008
References
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford University Press, London & NY, 1954, Thm. 412.
• J. Leech, Some Solutions of Diophantine Equations, Proc. Camb. Phil. Soc. 53, 778–780, 1957.
• E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel, The four least solutions in distinct positive integers of the Diophantine equations = x3 + y3 = z3 + w3 = u3 + v3 = m3 + n3, Bull. Inst. Math. Appl., 27(1991) 155–157; MR1125858, online.
• David W. Wilson, The Fifth Taxicab Number is 48988659276962496, Journal of Integer Sequences, Vol. 2 (1999), online. (Wilson was unaware of J. A. Dardis' prior discovery of Ta(5) in 1994 when he wrote this.)
• D. J. Bernstein, Enumerating solutions to $p(a)+q(b)=r(c)+s(d)$, Mathematics of Computation 70, 233 (2000), 389–394.
• C. S. Calude, E. Calude and M. J. Dinneen: What is the value of Taxicab(6)?, Journal of Universal Computer Science, Vol. 9 (2003), p. 1196–1203
External links
• A 2002 post to the Number Theory mailing list by Randall L. Rathbun
• Grime, James; Bowley, Roger. Haran, Brady (ed.). 1729: Taxi Cab Number or Hardy-Ramanujan Number. Numberphile.
• Taxicab and other maths at Euler
• Singh, Simon. Haran, Brady (ed.). "Taxicab Numbers in Futurama". Numberphile.
| Wikipedia |
Socolar–Taylor tile
The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and reflections of the tile allowed.[1] It is the first known example of a single aperiodic tile, or "einstein".[2] The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed.[3]
Smith et al. in 2023 proposed a connected set tile implementing the rule geometrically in two dimensions.
This is, however, confirmed to be possible in three dimensions, and, in their original paper, Socolar and Taylor suggest a three-dimensional analogue to the monotile.[1] Taylor and Socolar remark that the 3D monotile aperiodically tiles three-dimensional space. However the tile does allow tilings with a period, shifting one (non-periodic) two dimensional layer to the next, and so the tile is only "weakly aperiodic".
Physical copies of the three-dimensional tile could not be fitted together without allowing reflections, which would require access to four-dimensional space.[2][4]
Gallery
• The monotile implemented geometrically. Black lines are included to show how the structure is enforced.
• A three-dimensional analogue of the Socolar-Taylor tile (all matching rules implemented geometrically)
• A three-dimensional analogue of the monotile, with matching rules implemented geometrically. Red lines are included only to illuminate the structure of the tiling. Note that this shape remains a connected set.
• A partial tiling of three-dimensional space with the 3D monotile.
• A tiling of 3D space with one tile removed to demonstrate the structure.
References
1. Socolar, Joshua E. S.; Taylor, Joan M. (2011), "An aperiodic hexagonal tile", Journal of Combinatorial Theory, Series A, 118 (8): 2207–2231, arXiv:1003.4279, doi:10.1016/j.jcta.2011.05.001, MR 2834173.
2. Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144
3. Frettlöh, Dirk. "Hexagonal aperiodic monotile". Tilings Encyclopedia. Retrieved 3 June 2013.
4. Harriss, Edmund. "Socolar and Taylor's Aperiodic Tile". Maxwell's Demon. Retrieved 3 June 2013.
External links
• Previewable digital models of the three-dimensional tile, suitable for 3D printing, at Thingiverse
• Original diagrams and further information on Joan Taylor's personal website
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
Modularity theorem
FieldNumber theory
Conjectured byYutaka Taniyama
Goro Shimura
Conjectured in1957
First proof byChristophe Breuil
Brian Conrad
Fred Diamond
Richard Taylor
First proof in2001
ConsequencesFermat's Last Theorem
Statement
The theorem states that any elliptic curve over $\mathbf {Q} $ can be obtained via a rational map with integer coefficients from the classical modular curve $X_{0}(N)$ for some integer $N$; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level $N$. If $N$ is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level $N$, a normalized newform with integer $q$-expansion, followed if need be by an isogeny.
Related statements
The modularity theorem implies a closely related analytic statement:
To each elliptic curve E over $\mathbf {Q} $ we may attach a corresponding L-series. The $L$-series is a Dirichlet series, commonly written
$L(E,s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.$
The generating function of the coefficients $a_{n}$ is then
$f(E,q)=\sum _{n=1}^{\infty }a_{n}q^{n}.$
If we make the substitution
$q=e^{2\pi i\tau }$
we see that we have written the Fourier expansion of a function $f(E,\tau )$ of the complex variable $\tau $, so the coefficients of the $q$-series are also thought of as the Fourier coefficients of $f$. The function obtained in this way is, remarkably, a cusp form of weight two and level $N$ and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem.
Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it).
History
Yutaka Taniyama[1] stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō. Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil[2] rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted $L$-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the Langlands program.[3][4]
The conjecture attracted considerable interest when Gerhard Frey[5] suggested in 1986 that it implies Fermat's Last Theorem. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre[6] identified a missing link (now known as the epsilon conjecture or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet[7]'s completion of a proof of the epsilon conjecture.
Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof.[8] For example, Wiles's Ph.D. supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".
In 1995 Andrew Wiles, with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves, which he used to prove Fermat's Last Theorem,[9] and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond,[10] Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999.[11][12]
Further information: Fermat's Last Theorem and Wiles's proof of Fermat's Last Theorem
Once fully proven, the conjecture became known as the modularity theorem.
Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime $n$-th powers, $n\geq 3$. (The case $n=3$ was already known by Euler.)
Generalizations
The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. However, Freitas, Le Hung & Siksek[13] proved that elliptic curves defined over real quadratic fields are modular.
Example
For example,[14][15][16] the elliptic curve $y^{2}-y=x^{3}-x$, with discriminant (and conductor) 37, is associated to the form
$f(z)=q-2q^{2}-3q^{3}+2q^{4}-2q^{5}+6q^{6}+\cdots ,\qquad q=e^{2\pi iz}$
For prime numbers ℓ not equal to 37, one can verify the property about the coefficients. Thus, for ℓ = 3, there are 6 solutions of the equation modulo 3: (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1); thus a(3) = 3 − 6 = −3.
The conjecture, going back to the 1950s, was completely proven by 1999 using ideas of Andrew Wiles, who proved it in 1994 for a large family of elliptic curves.[17]
There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over Q, from the modular curve X0(N) to E. In particular, the points of E can be parametrized by modular functions.
For example, a modular parametrization of the curve $y^{2}-y=x^{3}-x$ is given by[18]
${\begin{aligned}x(z)&=q^{-2}+2q^{-1}+5+9q+18q^{2}+29q^{3}+51q^{4}+\cdots \\y(z)&=q^{-3}+3q^{-2}+9q^{-1}+21+46q+92q^{2}+180q^{3}+\cdots \end{aligned}}$
where, as above, q = exp(2πiz). The functions x(z) and y(z) are modular of weight 0 and level 37; in other words they are meromorphic, defined on the upper half-plane Im(z) > 0 and satisfy
$x\!\left({\frac {az+b}{cz+d}}\right)=x(z)$
and likewise for y(z), for all integers a, b, c, d with ad − bc = 1 and 37|c.
Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.
The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5, the Fermat equation
$a^{p}+b^{p}=c^{p}$
has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch was the first to notice,[19] the elliptic curve
$y^{2}=x(x-a^{p})(x+b^{p})$
of discriminant
$\Delta ={\frac {1}{256}}(abc)^{2p}$
cannot be modular.[7] Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987.[20]
See also
Serre's modularity conjecture
Notes
1. Taniyama 1956.
2. Weil 1967.
3. Harris, Michael (2020). "Virtues of Priority". arXiv:2003.08242 [math.HO].
4. Lang, Serge (November 1995). "Some History of the Shimura-Taniyama Conjecture" (PDF). Notices of the American Mathematical Society. 42 (11): 1301–1307. Retrieved 2022-11-08.
5. Frey 1986.
6. Serre 1987.
7. Ribet 1990.
8. Singh 1997, pp. 203–205, 223, 226.
9. Wiles 1995a; Wiles 1995b.
10. Diamond 1996.
11. Conrad, Diamond & Taylor 1999.
12. Breuil et al. 2001.
13. Freitas, Le Hung & Siksek 2015.
14. For the calculations, see for example Zagier 1985, pp. 225–248
15. LMFDB: http://www.lmfdb.org/EllipticCurve/Q/37/a/1
16. OEIS: https://oeis.org/A007653
17. A synthetic presentation (in French) of the main ideas can be found in this Bourbaki article of Jean-Pierre Serre. For more details see Hellegouarch 2001
18. Zagier, D. (1985). "Modular points, modular curves, modular surfaces and modular forms". Arbeitstagung Bonn 1984. Lecture Notes in Mathematics. Vol. 1111. Springer. pp. 225–248. doi:10.1007/BFb0084592. ISBN 978-3-540-39298-9.
19. Hellegouarch, Yves (1974). "Points d'ordre 2ph sur les courbes elliptiques" (PDF). Acta Arithmetica. 26 (3): 253–263. doi:10.4064/aa-26-3-253-263. ISSN 0065-1036. MR 0379507.
20. See the survey of Ribet, K. (1990b). "From the Taniyama–Shimura conjecture to Fermat's Last Theorem". Annales de la Faculté des Sciences de Toulouse. 11: 116–139. doi:10.5802/afst.698.
References
• Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", Journal of the American Mathematical Society, 14 (4): 843–939, doi:10.1090/S0894-0347-01-00370-8, ISSN 0894-0347, MR 1839918
• Conrad, Brian; Diamond, Fred; Taylor, Richard (1999), "Modularity of certain potentially Barsotti–Tate Galois representations", Journal of the American Mathematical Society, 12 (2): 521–567, doi:10.1090/S0894-0347-99-00287-8, ISSN 0894-0347, MR 1639612
• Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn, eds. (1997), Modular forms and Fermat's last theorem, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94609-2, MR 1638473
• Darmon, Henri (1999), "A proof of the full Shimura–Taniyama–Weil conjecture is announced" (PDF), Notices of the American Mathematical Society, 46 (11): 1397–1401, ISSN 0002-9920, MR 1723249Contains a gentle introduction to the theorem and an outline of the proof.
• Diamond, Fred (1996), "On deformation rings and Hecke rings", Annals of Mathematics, Second Series, 144 (1): 137–166, doi:10.2307/2118586, ISSN 0003-486X, JSTOR 2118586, MR 1405946
• Freitas, Nuno; Le Hung, Bao V.; Siksek, Samir (2015), "Elliptic curves over real quadratic fields are modular", Inventiones Mathematicae, 201 (1): 159–206, arXiv:1310.7088, Bibcode:2015InMat.201..159F, doi:10.1007/s00222-014-0550-z, ISSN 0020-9910, MR 3359051, S2CID 119132800
• Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, ISSN 0933-8268, MR 0853387
• Mazur, Barry (1991), "Number theory as gadfly", The American Mathematical Monthly, 98 (7): 593–610, doi:10.2307/2324924, ISSN 0002-9890, JSTOR 2324924, MR 1121312 Discusses the Taniyama–Shimura–Weil conjecture 3 years before it was proven for infinitely many cases.
• Ribet, Kenneth A. (1990), "On modular representations of Gal(Q/Q) arising from modular forms", Inventiones Mathematicae, 100 (2): 431–476, Bibcode:1990InMat.100..431R, doi:10.1007/BF01231195, hdl:10338.dmlcz/147454, ISSN 0020-9910, MR 1047143, S2CID 120614740
• Serre, Jean-Pierre (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", Duke Mathematical Journal, 54 (1): 179–230, doi:10.1215/S0012-7094-87-05413-5, ISSN 0012-7094, MR 0885783
• Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", The Bulletin of the London Mathematical Society, 21 (2): 186–196, doi:10.1112/blms/21.2.186, ISSN 0024-6093, MR 0976064
• Singh, Simon (1997), Fermat's Last Theorem, Fourth Estate, ISBN 978-1-85702-521-7
• Taniyama, Yutaka (1956), "Problem 12", Sugaku (in Japanese), 7: 269 English translation in (Shimura 1989, p. 194)
• Taylor, Richard; Wiles, Andrew (1995), "Ring-theoretic properties of certain Hecke algebras", Annals of Mathematics, Second Series, 141 (3): 553–572, CiteSeerX 10.1.1.128.531, doi:10.2307/2118560, ISSN 0003-486X, JSTOR 2118560, MR 1333036
• Weil, André (1967), "Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen, 168: 149–156, doi:10.1007/BF01361551, ISSN 0025-5831, MR 0207658, S2CID 120553723
• Wiles, Andrew (1995a), "Modular elliptic curves and Fermat's last theorem", Annals of Mathematics, Second Series, 141 (3): 443–551, CiteSeerX 10.1.1.169.9076, doi:10.2307/2118559, ISSN 0003-486X, JSTOR 2118559, MR 1333035
• Wiles, Andrew (1995b), "Modular forms, elliptic curves, and Fermat's last theorem", Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel, Boston, Berlin: Birkhäuser, pp. 243–245, MR 1403925
External links
• Darmon, H. (2001) [1994], "Shimura–Taniyama conjecture", Encyclopedia of Mathematics, EMS Press
• Weisstein, Eric W. "Taniyama–Shimura Conjecture". MathWorld.
Topics in algebraic curves
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| Wikipedia |
Taylor Booth (mathematician)
Taylor Lockwood Booth (September 22, 1933 – October 20, 1986) was a mathematician known for his work in automata theory.
Taylor Booth
Born
Taylor Lockwood Booth
(1933-09-22)September 22, 1933
Manchester, Connecticut, U.S.
DiedOctober 20, 1986(1986-10-20) (aged 53)
Alma materUniversity of Connecticut
Known forSequential Machines and Automata Theory (1967)
AwardsIEEE Centennial Medal (1984)
Scientific career
FieldsMathematics, computer science, computer engineering
One of his fundamental works is Sequential Machines and Automata Theory (1967). It is a wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. It deals with state minimization techniques, Finite state machines, Turing machines, Markov processes, and undecidability.
Education
Booth studied at the University of Connecticut, where he received his B.S., M.S. and Ph.D. degrees.[1]
Professional career
At his alma mater Booth was professor at the Computer Science and Engineering department.[1]
He was the founder and director of the Computer Applications & Research Center (CARC) at the University of Connecticut's School of Engineering. In 1981 the center was created to support the school's growing need for centralized computing research and development services. After his death the center was renamed to "Taylor L. Booth Center for Computer Applications and Research" or in its shorter form the "Booth Research Center". In 2002 this merged with the Advanced Technology Institute (ATI), another center at the School of Engineering, to form the "Booth Engineering Center for Advanced Technology" (BECAT).[1][2][3]
Booth was the first president of the Computing Sciences Accreditation Board, founded in 1984 and since renamed to CSAB.[1][4]
Awards and honors
Professor Booth received following awards and honors:[1]
• The Frederick Emmons Terman Award from the American Society for Engineering Education in 1972, to recognize the outstanding young electrical engineering educator.[5]
• The IEEE Centennial Medal from the Institute of Electrical and Electronics Engineers (IEEE) in 1984.
• The Distinguished Service Award from the IEEE Computer Society in 1985, for his accreditation work.
Taylor L. Booth Education Award
After Booth's death, the IEEE Computer Society established the Taylor L. Booth Education Award, to keep his name in memory. The award is given annually for individuals with an "outstanding record in computer science and engineering education".[1][6]
References
1. "Tribute to Taylor L. Booth". IEEE-CS. Archived from the original on 2010-06-17. Retrieved October 23, 2010.
2. "BECAT Overview and History". University of Connecticut. Archived from the original on 2010-06-22. Retrieved October 28, 2010.
3. "School of Engineering Annual report 2001-2002" (PDF). University of Connecticut. Retrieved September 17, 2020.
4. "IEEE Computer Society Marks 60th Anniversary". IEEE-CS. August 7, 2007. Archived from the original on 2011-06-29. Retrieved October 23, 2010.
5. "Past Frederick Emmons Terman Award Winners". American Society for Engineering Education. Archived from the original on 2013-04-02. Retrieved November 3, 2010.
6. "Taylor L. Booth Education Award". IEEE-CS. Retrieved July 9, 2022.
External links
• Taylor Booth (1967) Sequential Machines and Automata Theory, John Wiley and Sons, New York. Library of Congress Catalog Card Number: 67–25924.
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| Wikipedia |
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev (Russian: Пафну́тий Льво́вич Чебышёв, IPA: [pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof]) (16 May [O.S. 4 May] 1821 – 8 December [O.S. 26 November] 1894)[2] was a Russian mathematician and considered to be the founding father of Russian mathematics.
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev
Born(1821-05-16)16 May 1821[1]
Akatovo, Kaluga Governorate, Russian Empire[1]
Died8 December 1894(1894-12-08) (aged 73)[1]
St. Petersburg, Russian Empire[1]
NationalityRussian
Other namesChebysheff, Chebyshov, Tschebyscheff, Tschebycheff, Tchebycheff
Alma materMoscow University
Known forWork on probability, statistics, mechanics, analytical geometry and number theory
AwardsDemidov Prize (1849)
Scientific career
FieldsMathematician
InstitutionsSt. Petersburg University
Academic advisorsNikolai Brashman
Notable studentsDmitry Grave
Aleksandr Korkin
Aleksandr Lyapunov
Andrey Markov
Vladimir Andreevich Markov
Konstantin Posse
Yegor Ivanovich Zolotarev
Signature
Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias.
Transcription
The surname Chebyshev has been transliterated in several different ways, like Tchebichef, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Čebyčev, Čebyšev, Chebysheff, Chebychov, Chebyshov (according to native Russian speakers, this one provides the closest pronunciation in English to the correct pronunciation in old Russian), and Chebychev, a mixture between English and French transliterations considered erroneous. It is one of the most well known data-retrieval nightmares in mathematical literature. Currently, the English transliteration Chebyshev has gained widespread acceptance, except by the French, who prefer Tchebychev. The correct transliteration according to ISO 9 is Čebyšëv. The American Mathematical Society adopted the transcription Chebyshev in its Mathematical Reviews.[3]
His first name comes from the Greek Paphnutius (Παφνούτιος), which in turn takes its origin in the Coptic Paphnuty (Ⲡⲁⲫⲛⲟⲩϯ), meaning "that who belongs to God" or simply "the man of God".
Biography
Early years
One of nine children,[4] Chebyshev was born in the village of Okatovo in the district of Borovsk, province of Kaluga. His father, Lev Pavlovich, was a Russian nobleman and wealthy landowner. Pafnuty Lvovich was first educated at home by his mother Agrafena Ivanovna Pozniakova (in reading and writing) and by his cousin Avdotya Kvintillianovna Sukhareva (in French and arithmetic). Chebyshev mentioned that his music teacher also played an important role in his education, for she "raised his mind to exactness and analysis."
Trendelenburg's gait affected Chebyshev's adolescence and development. From childhood, he limped and walked with a stick and so his parents abandoned the idea of his becoming an officer in the family tradition. His disability prevented his playing many children's games and he devoted himself instead to mathematics.
In 1832, the family moved to Moscow, mainly to attend to the education of their eldest sons (Pafnuty and Pavel, who would become lawyers). Education continued at home and his parents engaged teachers of excellent reputation, including (for mathematics and physics) P.N. Pogorelski, held to be one of the best teachers in Moscow and who had taught (for example) the writer Ivan Sergeevich Turgenev.
University studies
In summer 1837, Chebyshev passed the registration examinations and, in September of that year, began his mathematical studies at the second philosophical department of Moscow University. His teachers included N.D. Brashman, N.E. Zernov and D.M. Perevoshchikov of whom it seems clear that Brashman had the greatest influence on Chebyshev. Brashman instructed him in practical mechanics and probably showed him the work of French engineer J.V. Poncelet. In 1841 Chebyshev was awarded the silver medal for his work "calculation of the roots of equations" which he had finished in 1838. In this, Chebyshev derived an approximating algorithm for the solution of algebraic equations of nth degree based on Newton's method. In the same year, he finished his studies as "most outstanding candidate".
In 1841, Chebyshev's financial situation changed drastically. There was famine in Russia, and his parents were forced to leave Moscow. Although they could no longer support their son, he decided to continue his mathematical studies and prepared for the master examinations, which lasted six months. Chebyshev passed the final examination in October 1843 and, in 1846, defended his master thesis "An Essay on the Elementary Analysis of the Theory of Probability." His biographer Prudnikov suggests that Chebyshev was directed to this subject after learning of recently published books on probability theory or on the revenue of the Russian insurance industry.
Adult years
In 1847, Chebyshev promoted his thesis pro venia legendi "On integration with the help of logarithms" at St Petersburg University and thus obtained the right to teach there as a lecturer. At that time some of Leonhard Euler's works were rediscovered by P. N. Fuss and were being edited by V. Ya. Bunyakovsky, who encouraged Chebyshev to study them. This would come to influence Chebyshev's work. In 1848, he submitted his work The Theory of Congruences for a doctorate, which he defended in May 1849.[1] He was elected an extraordinary professor at St Petersburg University in 1850, ordinary professor in 1860 and, after 25 years of lectureship, he became merited professor in 1872. In 1882 he left the university and devoted his life to research.
During his lectureship at the university (1852–1858), Chebyshev also taught practical mechanics at the Alexander Lyceum in Tsarskoe Selo (now Pushkin), a southern suburb of St Petersburg.
His scientific achievements were the reason for his election as junior academician (adjunkt) in 1856. Later, he became an extraordinary (1856) and in 1858 an ordinary member of the Imperial Academy of Sciences. In the same year he became an honorary member of Moscow University. He accepted other honorary appointments and was decorated several times. In 1856, Chebyshev became a member of the scientific committee of the ministry of national education. In 1859, he became an ordinary member of the ordnance department of the academy with the adoption of the headship of the commission for mathematical questions according to ordnance and experiments related to ballistics. The Paris academy elected him corresponding member in 1860 and full foreign member in 1874. In 1893, he was elected honorable member of the St. Petersburg Mathematical Society, which had been founded three years earlier.
Chebyshev died in St Petersburg on 26 November 1894.
Mathematical contributions
Chebyshev is known for his work in the fields of probability, statistics, mechanics, and number theory. The Chebyshev inequality states that if $X$ is a random variable with standard deviation σ > 0, then the probability that the outcome of $X$ is no less than $a\sigma $ away from its mean is no more than $1/a^{2}$:
$\Pr(|X-{\mathbf {E} }(X)|\geq a\ )\leq {\frac {\sigma ^{2}}{a^{2}}}.$
The Chebyshev inequality is used to prove the weak law of large numbers.
The Bertrand–Chebyshev theorem (1845, 1852) states that for any $n>3$, there exists a prime number $p$ such that $n<p<2n$. This is a consequence of the Chebyshev inequalities for the number $\pi (n)$ of prime numbers less than $n$, which state that $\pi (n)$ is of the order of $n/\log(n)$. A more precise form is given by the celebrated prime number theorem: the quotient of the two expressions approaches 1.0 as $n$ tends to infinity.
Chebyshev is also known for the Chebyshev polynomials and the Chebyshev bias – the difference between the number of primes that are congruent to 3 (modulo 4) and 1 (modulo 4).
Chebyshev was the first person to think systematically in terms of random variables and their moments and expectations.[5]
Legacy
Chebyshev is considered to be a founding father of Russian mathematics.[1] Among his well-known students were the mathematicians Dmitry Grave, Aleksandr Korkin, Aleksandr Lyapunov, and Andrei Markov. According to the Mathematics Genealogy Project, Chebyshev has 13,709 mathematical "descendants" as of January 2020.[6]
The lunar crater Chebyshev and the asteroid 2010 Chebyshev were named to honor his major achievements in the mathematical realm.[7]
Publications
• Tchebychef, P. L. (1899), Markov, Andrey Andreevich; Sonin, N. (eds.), Oeuvres, vol. I, New York: Commissionaires de l'Académie impériale des sciences, MR 0147353, Reprinted by Chelsea 1962
• Tchebychef, P. L. (1907), Markov, Andrey Andreevich; Sonin, N. (eds.), Oeuvres, vol. II, New York: Commissionaires de l'Académie impériale des sciences, MR 0147353, Reprinted by Chelsea 1962
• Butzer (1999), "P. L. Chebyshev (1821–1894): A Guide to his Life and Work", Journal of Approximation Theory, 96: 111–138, doi:10.1006/jath.1998.3289
See also
• List of things named after Pafnuty Chebyshev
References
1. Pafnuty Chebyshev. Encyclopaedia Britannica
2. Pafnuty Lvovich Chebyshev – Britannica Online Encyclopedia
3. Chebyshev, Pafnutiĭ L'vovich, on MathSciNet.
4. Biography in MacTutor Archive
5. Mackey, George (July 1980). "Harmonic analysis as the exploitation of symmetry-a historical survey". Bulletin of the American Mathematical Society. New Series. 3 (1): 549. doi:10.1090/S0273-0979-1980-14783-7.
6. Pafnuty Chebyshev at the Mathematics Genealogy Project
7. Schmadel, Lutz D. (2007). "(2010) Chebyshev". Dictionary of Minor Planet Names – (2010) Chebyshev. Springer Berlin Heidelberg. p. 163. doi:10.1007/978-3-540-29925-7_2011. ISBN 978-3-540-00238-3.
External links
• Media related to Pafnuty Chebyshev at Wikimedia Commons
Wikisource has the text of the 1911 Encyclopædia Britannica article "Chebichev, Pafnutiy Lvovich".
• Mechanisms by Chebyshev – short 3d films – embodiment of Tchebishev's inventions
• Pafnuty Chebyshev at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Pafnuty Chebyshev", MacTutor History of Mathematics Archive, University of St Andrews
• Biography, another one, and yet another (all in Russian).
• Biography in French.
• Œuvres de P.L. Tchebychef Vol. I, Vol. II (in French)
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| Wikipedia |
Chebotarev's density theorem
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field $\mathbb {Q} $ of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in (Tschebotareff 1926).
A special case that is easier to state says that if K is an algebraic number field which is a Galois extension of $\mathbb {Q} $ of degree n, then the prime numbers that completely split in K have density
1/n
among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its Frobenius element, which is a representative of a well-defined conjugacy class in the Galois group
Gal(K/Q).
Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with k elements occurs with frequency asymptotic to
k/n.
History and motivation
When Carl Friedrich Gauss first introduced the notion of complex integers Z[i], he observed that the ordinary prime numbers may factor further in this new set of integers. In fact, if a prime p is congruent to 1 mod 4, then it factors into a product of two distinct prime gaussian integers, or "splits completely"; if p is congruent to 3 mod 4, then it remains prime, or is "inert"; and if p is 2 then it becomes a product of the square of the prime (1+i) and the invertible gaussian integer -i; we say that 2 "ramifies". For instance,
$5=(1+2i)(1-2i)$ splits completely;
$3$ is inert;
$2=-i(1+i)^{2}$ ramifies.
From this description, it appears that as one considers larger and larger primes, the frequency of a prime splitting completely approaches 1/2, and likewise for the primes that remain primes in Z[i]. Dirichlet's theorem on arithmetic progressions demonstrates that this is indeed the case. Even though the prime numbers themselves appear rather erratically, splitting of the primes in the extension
$\mathbb {Z} \subset \mathbb {Z} [i]$
follows a simple statistical law.
Similar statistical laws also hold for splitting of primes in the cyclotomic extensions, obtained from the field of rational numbers by adjoining a primitive root of unity of a given order. For example, the ordinary integer primes group into four classes, each with probability 1/4, according to their pattern of splitting in the ring of integers corresponding to the 8th roots of unity. In this case, the field extension has degree 4 and is abelian, with the Galois group isomorphic to the Klein four-group. It turned out that the Galois group of the extension plays a key role in the pattern of splitting of primes. Georg Frobenius established the framework for investigating this pattern and proved a special case of the theorem. The general statement was proved by Nikolai Grigoryevich Chebotaryov in 1922.
Relation with Dirichlet's theorem
The Chebotarev density theorem may be viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that if N≥2 is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1/n, where n=φ(N) is the Euler totient function. This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K. Indeed, the Galois group of K/Q is abelian and can be canonically identified with the group of invertible residue classes mod N. The splitting invariant of a prime p not dividing N is simply its residue class because the number of distinct primes into which p splits is φ(N)/m, where m is multiplicative order of p modulo N; hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to N.
Formulation
In their survey article, Lenstra & Stevenhagen (1996) give an earlier result of Frobenius in this area. Suppose K is a Galois extension of the rational number field Q, and P(t) a monic integer polynomial such that K is a splitting field of P. It makes sense to factorise P modulo a prime number p. Its 'splitting type' is the list of degrees of irreducible factors of P mod p, i.e. P factorizes in some fashion over the prime field Fp. If n is the degree of P, then the splitting type is a partition Π of n. Considering also the Galois group G of K over Q, each g in G is a permutation of the roots of P in K; in other words by choosing an ordering of α and its algebraic conjugates, G is faithfully represented as a subgroup of the symmetric group Sn. We can write g by means of its cycle representation, which gives a 'cycle type' c(g), again a partition of n.
The theorem of Frobenius states that for any given choice of Π the primes p for which the splitting type of P mod p is Π has a natural density δ, with δ equal to the proportion of g in G that have cycle type Π.
The statement of the more general Chebotarev theorem is in terms of the Frobenius element of a prime (ideal), which is in fact an associated conjugacy class C of elements of the Galois group G. If we fix C then the theorem says that asymptotically a proportion |C|/|G| of primes have associated Frobenius element as C. When G is abelian the classes of course each have size 1. For the case of a non-abelian group of order 6 they have size 1, 2 and 3, and there are correspondingly (for example) 50% of primes p that have an order 2 element as their Frobenius. So these primes have residue degree 2, so they split into exactly three prime ideals in a degree 6 extension of Q with it as Galois group.[1]
Statement
Let L be a finite Galois extension of a number field K with Galois group G. Let X be a subset of G that is stable under conjugation. The set of primes v of K that are unramified in L and whose associated Frobenius conjugacy class Fv is contained in X has density
${\frac {\#X}{\#G}}.$[2]
The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.[3]
Effective Version
The Generalized Riemann hypothesis implies an effective version[4] of the Chebotarev density theorem: if L/K is a finite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramified primes of K of norm below x with Frobenius conjugacy class in C is
${\frac {|C|}{|G|}}{\Bigl (}\mathrm {li} (x)+O{\bigl (}{\sqrt {x}}(n\log x+\log |\Delta |){\bigr )}{\Bigr )},$
where the constant implied in the big-O notation is absolute, n is the degree of L over Q, and Δ its discriminant.
The effective form of Chebotarev's density theory becomes much weaker without GRH. Take L to be a finite Galois extension of Q with Galois group G and degree d. Take $\rho $ to be a nontrivial irreducible representation of G of degree n, and take ${\mathfrak {f}}(\rho )$ to be the Artin conductor of this representation. Suppose that, for $\rho _{0}$ a subrepresentation of $\rho \otimes \rho $ or $\rho \otimes {\bar {\rho }}$, $L(\rho _{0},s)$ is entire; that is, the Artin conjecture is satisfied for all $\rho _{0}$. Take $\chi _{\rho }$ to be the character associated to $\rho $. Then there is an absolute positive $c$ such that, for $x\geq 2$,
$\sum _{p\leq x,p\not \mid {\mathfrak {f}}(\rho )}\chi _{\rho }({\text{Fr}}_{p})\log p=rx+O{\biggl (}{\frac {x^{\beta }}{\beta }}+x\exp {\biggl (}{\frac {-c(dn)^{-4}\log x}{3\log {\mathfrak {f}}(\rho )+{\sqrt {\log x}}}}{\biggr )}(dn\log(x{\mathfrak {f}}(\rho )){\biggr )},$
where $r$ is 1 if $\rho $ is trivial and is otherwise 0, and where $\beta $ is an exceptional real zero of $L(\rho ,s)$; if there is no such zero, the $x^{\beta }/\beta $ term can be ignored. The implicit constant of this expression is absolute. [5]
Infinite extensions
The statement of the Chebotarev density theorem can be generalized to the case of an infinite Galois extension L / K that is unramified outside a finite set S of primes of K (i.e. if there is a finite set S of primes of K such that any prime of K not in S is unramified in the extension L / K). In this case, the Galois group G of L / K is a profinite group equipped with the Krull topology. Since G is compact in this topology, there is a unique Haar measure μ on G. For every prime v of K not in S there is an associated Frobenius conjugacy class Fv. The Chebotarev density theorem in this situation can be stated as follows:[2]
Let X be a subset of G that is stable under conjugation and whose boundary has Haar measure zero. Then, the set of primes v of K not in S such that Fv ⊆ X has density
${\frac {\mu (X)}{\mu (G)}}.$
This reduces to the finite case when L / K is finite (the Haar measure is then just the counting measure).
A consequence of this version of the theorem is that the Frobenius elements of the unramified primes of L are dense in G.
Important consequences
The Chebotarev density theorem reduces the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions. Specifically, it implies that as a Galois extension of K, L is uniquely determined by the set of primes of K that split completely in it.[6] A related corollary is that if almost all prime ideals of K split completely in L, then in fact L = K.[7]
See also
• Splitting of prime ideals in Galois extensions
• Grothendieck–Katz p-curvature conjecture
Notes
1. This particular example already follows from the Frobenius result, because G is a symmetric group. In general, conjugacy in G is more demanding than having the same cycle type.
2. Section I.2.2 of Serre
3. Lenstra, Hendrik (2006). "The Chebotarev Density Theorem" (PDF). Retrieved 7 June 2018.
4. Lagarias, J.C.; Odlyzko, A.M. (1977). "Effective Versions of the Chebotarev Theorem". Algebraic Number Fields: 409–464.
5. Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic Number Theory. Providence, RI: American Mathematical Society. p. 111.
6. Corollary VII.13.10 of Neukirch
7. Corollary VII.13.7 of Neukirch
References
• Lenstra, H. W.; Stevenhagen, P. (1996), "Chebotarëv and his density theorem" (PDF), The Mathematical Intelligencer, 18 (2): 26–37, CiteSeerX 10.1.1.116.9409, doi:10.1007/BF03027290, MR 1395088
• Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
• Serre, Jean-Pierre (1998) [1968], Abelian l-adic representations and elliptic curves (Revised reprint of the 1968 original ed.), Wellesley, MA: A K Peters, Ltd., ISBN 1-56881-077-6, MR 1484415
• Tschebotareff, N. (1926), "Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören", Mathematische Annalen, 95 (1): 191–228, doi:10.1007/BF01206606
| Wikipedia |
Paper bag problem
In geometry, the paper bag problem or teabag problem is to calculate the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch.
According to Anthony C. Robin, an approximate formula for the capacity of a sealed expanded bag is:[1]
$V=w^{3}\left(h/\left(\pi w\right)-0.142\left(1-10^{\left(-h/w\right)}\right)\right),$
where w is the width of the bag (the shorter dimension), h is the height (the longer dimension), and V is the maximum volume. The approximation ignores the crimping round the equator of the bag.
A very rough approximation to the capacity of a bag that is open at one edge is:
$V=w^{3}\left(h/\left(\pi w\right)-0.071\left(1-10^{\left(-2h/w\right)}\right)\right)$
(This latter formula assumes that the corners at the bottom of the bag are linked by a single edge, and that the base of the bag is not a more complex shape such as a lens).
The square teabag
For the special case where the bag is sealed on all edges and is square with unit sides, h = w = 1, the first formula estimates a volume of roughly
$V={\frac {1}{\pi }}-0.142\cdot 0.9$
or roughly 0.19. According to Andrew Kepert at the University of Newcastle, Australia, an upper bound for this version of the teabag problem is 0.217+, and he has made a construction that appears to give a volume of 0.2055+.
Robin also found a more complicated formula for the general paper bag,[1] which gives 0.2017, below the bounds given by Kepert (i.e., 0.2055+ ≤ maximum volume ≤ 0.217+).
See also
• Biscornu, a shape formed by attaching two squares in a different way, with the corner of one at the midpoint of the other
• Mylar balloon (geometry)
Notes
1. Robin 2004.
References
• Robin, Anthony C (2004). "Paper Bag Problem". Mathematics Today. Institute of Mathematics and its Applications. June: 104–107. ISSN 1361-2042.
• Weisstein, Eric W. "Paper Bag". MathWorld. Archived from the original on 2011-06-29.
External links
• The original statement of the teabag problem
• Andrew Kepert's work on the teabag problem (mirror)
• Curved folds for the teabag problem
• A numerical approach to the teabag problem by Andreas Gammel
• Weisstein, Eric W. "Paper Bag Surface". MathWorld.
Mathematics of paper folding
Flat folding
• Big-little-big lemma
• Crease pattern
• Huzita–Hatori axioms
• Kawasaki's theorem
• Maekawa's theorem
• Map folding
• Napkin folding problem
• Pureland origami
• Yoshizawa–Randlett system
Strip folding
• Dragon curve
• Flexagon
• Möbius strip
• Regular paperfolding sequence
3d structures
• Miura fold
• Modular origami
• Paper bag problem
• Rigid origami
• Schwarz lantern
• Sonobe
• Yoshimura buckling
Polyhedra
• Alexandrov's uniqueness theorem
• Blooming
• Flexible polyhedron (Bricard octahedron, Steffen's polyhedron)
• Net
• Source unfolding
• Star unfolding
Miscellaneous
• Fold-and-cut theorem
• Lill's method
Publications
• Geometric Exercises in Paper Folding
• Geometric Folding Algorithms
• Geometric Origami
• A History of Folding in Mathematics
• Origami Polyhedra Design
• Origamics
People
• Roger C. Alperin
• Margherita Piazzola Beloch
• Robert Connelly
• Erik Demaine
• Martin Demaine
• Rona Gurkewitz
• David A. Huffman
• Tom Hull
• Kôdi Husimi
• Humiaki Huzita
• Toshikazu Kawasaki
• Robert J. Lang
• Anna Lubiw
• Jun Maekawa
• Kōryō Miura
• Joseph O'Rourke
• Tomohiro Tachi
• Eve Torrence
| Wikipedia |
Mathematics education
In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge.
Educational research
Disciplines
• Evaluation
• History
• Organization
• Philosophy
• Psychology (school)
• Technology (electronic marking)
• International education
• School counseling
• Special education
• Female education
• Teacher education
Core ideas
• Free education
• Right to education
Curricular domains
• Arts
• Business
• Computer science
• Early childhood
• Engineering
• Language
• Literacy
• Mathematics
• Performing arts
• Science
• Social science
• Technology
• Vocational
Methods
• Case method
• Conversation analysis
• Discourse analysis
• Factor analysis
• Factorial experiment
• Focus group
• Meta-analysis
• Multivariate statistics
• Participant observation
Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education.
History
Ancient
Elementary mathematics were a core part of education in many ancient civilisations, including ancient Egypt, ancient Babylonia, ancient Greece, ancient Rome, and Vedic India. In most cases, formal education was only available to male children with sufficiently high status, wealth, or caste. The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE.[1]
Pythagorean theorem
Historians of Mesopotamia have confirmed that use of the Pythagorean rule dates back to the Old Babylonian Empire (20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth of Pythagoras.[2][3][4][5][6]
In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. The teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants, and moneylenders could expect to learn such practical mathematics as was relevant to their profession.
Medieval and early modern
In the Middle Ages, the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian.[7] Although it continued to be taught in European universities, it was seen as subservient to the study of natural, metaphysical, and moral philosophy. The first modern arithmetic curriculum (starting with addition, then subtraction, multiplication, and division) arose at reckoning schools in Italy in the 1300s.[8] Spreading along trade routes, these methods were designed to be used in commerce. They contrasted with Platonic math taught at universities, which was more philosophical and concerned numbers as concepts rather than calculating methods.[8] They also contrasted with mathematical methods learned by artisan apprentices, which were specific to the tasks and tools at hand. For example, the division of a board into thirds can be accomplished with a piece of string, instead of measuring the length and using the arithmetic operation of division.[7]
The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes in 1543. However, there are many different writings on mathematics and mathematics methodology that date back to 1800 BCE. These were mostly located in Mesopotamia, where the Sumerians were practicing multiplication and division. There are also artifacts demonstrating their methodology for solving equations like the quadratic equation. After the Sumerians, some of the most famous ancient works on mathematics came from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. The more famous Rhind Papyrus has been dated back to approximately 1650 BCE, but it is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students.
The social status of mathematical study was improving by the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics being established by the University of Cambridge in 1662.
Modern
In the 18th and 19th centuries, the Industrial Revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money, and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age.
By the twentieth century, mathematics was part of the core curriculum in all developed countries.
During the twentieth century, mathematics education was established as an independent field of research. Main events in this development include the following:
• In 1893, a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein.
• The International Commission on Mathematical Instruction (ICMI) was founded in 1908, and Felix Klein became the first president of the organisation.
• The professional periodical literature on mathematics education in the United States had generated more than 4,000 articles after 1920, so in 1941 William L. Schaaf published a classified index, sorting them into their various subjects.[9]
• A renewed interest in mathematics education emerged in the 1960s, and the International Commission was revitalized.
• In 1968, the Shell Centre for Mathematical Education was established in Nottingham.
• The first International Congress on Mathematical Education (ICME) was held in Lyon in 1969. The second congress was in Exeter in 1972, and after that, it has been held every four years.
In the 20th century, the cultural impact of the "electronic age" (McLuhan) was also taken up by educational theory and the teaching of mathematics. While previous approach focused on "working with specialized 'problems' in arithmetic", the emerging structural approach to knowledge had "small children meditating about number theory and 'sets'."[10]
Objectives
At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included:
• The teaching and learning of basic numeracy skills to all students[11]
• The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid geometry, trigonometry, probability, statistics) to most students, to equip them to follow a trade or craft and to understand mathematics commonly used in news and Internet (such as percentages, charts, probability, and statistics)
• The teaching of abstract mathematical concepts (such as set and function) at an early age
• The teaching of selected areas of mathematics (such as Euclidean geometry)[12] as an example of an axiomatic system[13] and a model of deductive reasoning
• The teaching of selected areas of mathematics (such as calculus) as an example of the intellectual achievements of the modern world
• The teaching of advanced mathematics to those students who wish to follow a career in science, technology, engineering, and mathematics (STEM) fields
• The teaching of heuristics[14] and other problem-solving strategies to solve non-routine problems
• The teaching of mathematics in social sciences and actuarial sciences, as well as in some selected arts under liberal arts education in liberal arts colleges or universities
Methods
The method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve. Methods of teaching mathematics include the following:
• Computer-based math: an approach based on the use of mathematical software as the primary tool of computation.
• Computer-based mathematics education: involves the use of computers to teach mathematics. Mobile applications have also been developed to help students learn mathematics.[15][16][17]
• Classical education: the teaching of mathematics within the quadrivium, part of the classical education curriculum of the Middle Ages, which was typically based on Euclid's Elements taught as a paradigm of deductive reasoning.[18]
• Conventional approach: the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with arithmetic and is followed by Euclidean geometry and elementary algebra taught concurrently. Requires the instructor to be well informed about elementary mathematics since didactic and curriculum decisions are often dictated by the logic of the subject rather than pedagogical considerations. Other methods emerge by emphasizing some aspects of this approach.
• Relational approach: uses class topics to solve everyday problems and relates the topic to current events.[19] This approach focuses on the many uses of mathematics and helps students understand why they need to know it as well as helps them to apply mathematics to real-world situations outside of the classroom.
• Historical method: teaching the development of mathematics within a historical, social, and cultural context. Proponents argue it provides more human interest than the conventional approach.[20]
• Discovery math: a constructivist method of teaching (discovery learning) mathematics which centres around problem-based or inquiry-based learning, with the use of open-ended questions and manipulative tools.[21] This type of mathematics education was implemented in various parts of Canada beginning in 2005.[22] Discovery-based mathematics is at the forefront of the Canadian "math wars" debate with many criticizing it for declining math scores.
• New Math: a method of teaching mathematics which focuses on abstract concepts such as set theory, functions, and bases other than ten. Adopted in the US as a response to the challenge of early Soviet technical superiority in space, it began to be challenged in the late 1960s. One of the most influential critiques of the New Math was Morris Kline's 1973 book Why Johnny Can't Add. The New Math method was the topic of one of Tom Lehrer's most popular parody songs, with his introductory remarks to the song: "...in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer."
• Recreational mathematics: mathematical problems that are fun can motivate students to learn mathematics and can increase their enjoyment of mathematics.[23]
• Standards-based mathematics: a vision for pre-college mathematics education in the United States and Canada, focused on deepening student understanding of mathematical ideas and procedures, and formalized by the National Council of Teachers of Mathematics which created the Principles and Standards for School Mathematics.
• Mastery: an approach in which most students are expected to achieve a high level of competence before progressing.
• Problem solving: the cultivation of mathematical ingenuity, creativity, and heuristic thinking by setting students open-ended, unusual, and sometimes unsolved problems. The problems can range from simple word problems to problems from international mathematics competitions such as the International Mathematical Olympiad. Problem-solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings.
• Exercises: the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding simple fractions or solving quadratic equations.
• Rote learning: the teaching of mathematical results, definitions and concepts by repetition and memorisation typically without meaning or supported by mathematical reasoning. A derisory term is drill and kill. In traditional education, rote learning is used to teach multiplication tables, definitions, formulas, and other aspects of mathematics.
• Math walk: a walk where experience of perceived objects and scenes is translated into mathematical language.
Content and age levels
Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries. Sometimes a class may be taught at an earlier age than typical as a special or honors class.
Elementary mathematics in most countries is taught similarly, though there are differences. Most countries tend to cover fewer topics in greater depth than in the United States.[24] During the primary school years, children learn about whole numbers and arithmetic, including addition, subtraction, multiplication, and division.[25] Comparisons and measurement are taught, in both numeric and pictorial form, as well as fractions and proportionality, patterns, and various topics related to geometry.[26]
At high school level in most of the US, algebra, geometry, and analysis (pre-calculus and calculus) are taught as separate courses in different years. On the other hand, in most other countries (and in a few US states), mathematics is taught as an integrated subject, with topics from all branches of mathematics studied every year; students thus undertake a pre-defined course - entailing several topics - rather than choosing courses à la carte as in the United States. Even in these cases, however, several "mathematics" options may be offered, selected based on the student's intended studies post high school. (In South Africa, for example, the options are Mathematics, Mathematical Literacy and Technical Mathematics.) Thus, a science-oriented curriculum typically overlaps the first year of university mathematics, and includes differential calculus and trigonometry at age 16–17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions, and infinite series in their final year of secondary school; Probability and statistics are similarly often taught.
At college and university level, science and engineering students will be required to take multivariable calculus, differential equations, and linear algebra; at several US colleges, the minor or AS in mathematics substantively comprises these courses. Mathematics majors study additional other areas within pure mathematics—and often in applied mathematics—with the requirement of specified advanced courses in analysis and modern algebra. Applied mathematics may be taken as a major subject in its own right, while specific topics are taught within other courses: for example, civil engineers may be required to study fluid mechanics,[27] and "math for computer science" might include graph theory, permutation, probability, and formal mathematical proofs.[28] Pure and applied math degrees often include modules in probability theory or mathematical statistics, while a course in numerical methods is a common requirement for applied math. (Theoretical) physics is mathematics-intensive, often overlapping substantively with the pure or applied math degree. Business mathematics is usually limited to introductory calculus and (sometimes) matrix calculations; economics programs additionally cover optimization, often differential equations and linear algebra, and sometimes analysis.
Standards
Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to, realistic for, and considered socially appropriate for their pupils.
In modern times, there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England, for example, standards for mathematics education are set as part of the National Curriculum for England,[29] while Scotland maintains its own educational system. Many other countries have centralized ministries which set national standards or curricula, and sometimes even textbooks.
Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized mathematics tests had taken more mathematics courses in high school. This led some states to require three years of mathematics instead of two. But because this requirement was often met by taking another lower-level mathematics course, the additional courses had a “diluted” effect in raising achievement levels.[30]
In North America, the National Council of Teachers of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000 for the United States and Canada, which boosted the trend towards reform mathematics. In 2006, the NCTM released Curriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards were guidelines to implement as American states and Canadian provinces chose. In 2010, the National Governors Association Center for Best Practices and the Council of Chief State School Officers published the Common Core State Standards for US states, which were subsequently adopted by most states. Adoption of the Common Core State Standards in mathematics is at the discretion of each state, and is not mandated by the federal government.[31] "States routinely review their academic standards and may choose to change or add onto the standards to best meet the needs of their students."[32] The NCTM has state affiliates that have different education standards at the state level. For example, Missouri has the Missouri Council of Teachers of Mathematics (MCTM) which has its pillars and standards of education listed on its website. The MCTM also offers membership opportunities to teachers and future teachers so that they can stay up to date on the changes in math educational standards.[33]
The Programme for International Student Assessment (PISA), created by the Organisation for the Economic Co-operation and Development (OECD), is a global program studying the reading, science, and mathematics abilities of 15-year-old students.[34] The first assessment was conducted in the year 2000 with 43 countries participating.[35] PISA has repeated this assessment every three years to provide comparable data, helping to guide global education to better prepare youth for future economies. There have been many ramifications following the results of triennial PISA assessments due to implicit and explicit responses of stakeholders, which have led to education reform and policy change.[35][36][21]
Research
According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist."[37] However, there are useful theories on how children learn mathematics, and much research has been conducted in recent decades to explore how these theories can be applied to teaching. The following results are examples of some of the current findings in the field of mathematics education.
Important results[37]
One of the strongest results in recent research is that the most important feature of effective teaching is giving students "the opportunity to learn". Teachers can set expectations, times, kinds of tasks, questions, acceptable answers, and types of discussions that will influence students' opportunities to learn. This must involve both skill efficiency and conceptual understanding.
Conceptual understanding[37]
Two of the most important features of teaching in the promotion of conceptual understanding times are attending explicitly to concepts and allowing students to struggle with important mathematics. Both of these features have been confirmed through a wide variety of studies. Explicit attention to concepts involves making connections between facts, procedures, and ideas. (This is often seen as one of the strong points in mathematics teaching in East Asian countries, where teachers typically devote about half of their time to making connections. At the other extreme is the US, where essentially no connections are made in school classrooms.[38]) These connections can be made through explanation of the meaning of a procedure, questions comparing strategies and solutions of problems, noticing how one problem is a special case of another, reminding students of the main point, discussing how lessons connect, and so on.
Deliberate, productive struggle with mathematical ideas refers to the fact that when students exert effort with important mathematical ideas, even if this struggle initially involves confusion and errors, the result is greater learning. This is true whether the struggle is due to intentionally challenging, well-implemented teaching, or unintentionally confusing, faulty teaching.
Formative assessment[39]
Formative assessment is both the best and cheapest way to boost student achievement, student engagement, and teacher professional satisfaction. Results surpass those of reducing class size or increasing teachers' content knowledge. Effective assessment is based on clarifying what students should know, creating appropriate activities to obtain the evidence needed, giving good feedback, encouraging students to take control of their learning and letting students be resources for one another.
Homework[40]
Homework which leads students to practice past lessons or prepare future lessons is more effective than those going over today's lesson. Students benefit from feedback. Students with learning disabilities or low motivation may profit from rewards. For younger children, homework helps simple skills, but not broader measures of achievement.
Students with difficulties[40]
Students with genuine difficulties (unrelated to motivation or past instruction) struggle with basic facts, answer impulsively, struggle with mental representations, have poor number sense, and have poor short-term memory. Techniques that have been found productive for helping such students include peer-assisted learning, explicit teaching with visual aids, instruction informed by formative assessment, and encouraging students to think aloud.
Algebraic reasoning[40]
Elementary school children need to spend a long time learning to express algebraic properties without symbols before learning algebraic notation. When learning symbols, many students believe letters always represent unknowns and struggle with the concept of variable. They prefer arithmetic reasoning to algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with the minus sign and understand the equals sign to mean "the answer is...".
Methodology
As with other educational research (and the social sciences in general), mathematics education research depends on both quantitative and qualitative studies. Quantitative research includes studies that use inferential statistics to answer specific questions, such as whether a certain teaching method gives significantly better results than the status quo. The best quantitative studies involve randomized trials where students or classes are randomly assigned different methods to test their effects. They depend on large samples to obtain statistically significant results.
Qualitative research, such as case studies, action research, discourse analysis, and clinical interviews, depend on small but focused samples in an attempt to understand student learning and to look at how and why a given method gives the results it does. Such studies cannot conclusively establish that one method is better than another, as randomized trials can, but unless it is understood why treatment X is better than treatment Y, application of results of quantitative studies will often lead to "lethal mutations"[37] of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting new hypotheses, which can eventually be tested by randomized experiments. Both qualitative and quantitative studies, therefore, are considered essential in education—just as in the other social sciences.[41] Many studies are “mixed”, simultaneously combining aspects of both quantitative and qualitative research, as appropriate.
Randomized trials
There has been some controversy over the relative strengths of different types of research. Because randomized trials provide clear, objective evidence on “what works”, policymakers often consider only those studies. Some scholars have pushed for more random experiments in which teaching methods are randomly assigned to classes.[42][43] In other disciplines concerned with human subjects—like biomedicine, psychology, and policy evaluation—controlled, randomized experiments remain the preferred method of evaluating treatments.[44][45] Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.[43] On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections, such as the ethical difficulty of randomly assigning students to various treatments when the effects of such treatments are not yet known to be effective,[46] or the difficulty of assuring rigid control of the independent variable in fluid, real school settings.[47]
In the United States, the National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies, some of which used randomized assignment of treatments to experimental units, such as classrooms or students. The NMAP report's preference for randomized experiments received criticism from some scholars.[48] In 2010, the What Works Clearinghouse (essentially the research arm for the Department of Education) responded to ongoing controversy by extending its research base to include non-experimental studies, including regression discontinuity designs and single-case studies.[49]
Organizations
• Advisory Committee on Mathematics Education
• American Mathematical Association of Two-Year Colleges
• Association of Teachers of Mathematics
• Canadian Mathematical Society
• C.D. Howe Institute
• Mathematical Association
• National Council of Teachers of Mathematics
• OECD
See also
Aspects of mathematics education
• Cognitively Guided Instruction
• Critical mathematics pedagogy
• Ethnomathematics
• Number sentence, primary level mathematics education
• Pre-math skills
• Sir Cumference, children's mathematics educational book series
• Statistics education
North American issues
• Mathematics education in the United States
Mathematical difficulties
• Dyscalculia
• Mathematical anxiety
References
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33. "MoCTM - Home". www.moctm.org. Archived from the original on 2018-02-12. Retrieved 2018-02-11.
34. "What is PISA?". OECD. 2018. Archived from the original on 2018-03-04. Retrieved 2019-10-14.
35. Lockheed, Marlaine (2015). The Experience of Middle-Income Countries Participating in PISA 2000. PISA. France: OECD Publishing. p. 30. ISBN 978-92-64-24618-8.
36. Sellar, S., & Lingard, B., Sam; Lingard, Bob (April 2018). "International large-scale assessments, affective worlds and policy impacts in education" (PDF). International Journal of Qualitative Studies in Education. 31 (5): 367–381. doi:10.1080/09518398.2018.1449982. S2CID 149999527. Archived (PDF) from the original on 2020-03-07. Retrieved 2019-11-30.{{cite journal}}: CS1 maint: multiple names: authors list (link)
37. Hiebert, James; Grouws, Douglas (2007), "9", The Effects of Classroom Mathematics Teaching on Students' Learning, vol. 1, Reston VA: National Council of Teachers of Mathematics, pp. 371–404
38. Institute of Education Sciences, ed. (2003), "Highlights From the TIMSS 1999 Video Study of Eighth-Grade Mathematics Teaching", Trends in International Mathematics and Science Study (TIMSS) - Overview, U.S. Department of Education, archived from the original on 2012-05-08, retrieved 2012-05-08
39. Black, P.; Wiliam, Dylan (1998). "Assessment and Classroom Learning" (PDF). Assessment in Education. 5 (1): 7–74. doi:10.1080/0969595980050102. S2CID 143347721. Archived (PDF) from the original on 2018-07-26. Retrieved 2018-07-25.
40. "Research clips and briefs". Archived from the original on 2014-10-02. Retrieved 2009-11-15.
41. Raudenbush, Stephen (2005). "Learning from Attempts to Improve Schooling: The Contribution of Methodological Diversity". Educational Researcher. 34 (5): 25–31. CiteSeerX 10.1.1.649.7042. doi:10.3102/0013189X034005025. S2CID 145667765.
42. Cook, Thomas D. (2002). "Randomized Experiments in Educational Policy Research: A Critical Examination of the Reasons the Educational Evaluation Community has Offered for Not Doing Them". Educational Evaluation and Policy Analysis. 24 (3): 175–199. doi:10.3102/01623737024003175. S2CID 144583638.
43. Working Group on Statistics in Mathematics Education Research (2007). "Using Statistics Effectively in Mathematics Education Research: A report from a series of workshops organized by the American Statistical Association with funding from the National Science Foundation" (PDF). The American Statistical Association. Archived from the original (PDF) on 2007-02-02. Retrieved 2013-03-25.
44. Shadish, William R.; Cook, Thomas D.; Campbell, Donald T. (2002). Experimental and quasi-experimental designs for generalized causal inference (2nd ed.). Boston: Houghton Mifflin. ISBN 978-0-395-61556-0.
45. See articles on NCLB, National Mathematics Advisory Panel, Scientifically based research and What Works Clearinghouse
46. Mosteller, Frederick; Boruch, Robert (2002), Evidence Matters: Randomized Trials in Education Research, Brookings Institution Press
47. Chatterji, Madhabi (December 2004). "Evidence on "What Works": An Argument for Extended-Term Mixed-Method (ETMM) Evaluation Designs". Educational Researcher. 33 (9): 3–13. doi:10.3102/0013189x033009003. S2CID 14742527. Archived from the original on 2023-01-15. Retrieved 2019-11-30.
48. Kelly, Anthony (2008). "Reflections on the National Mathematics Advisory Panel Final Report". Educational Researcher. 37 (9): 561–4. doi:10.3102/0013189X08329353. S2CID 143471869. This is the introductory article to an issue devoted to this debate on report of the National Mathematics Advisory Panel, particularly on its use of randomized experiments.
49. Sparks, Sarah (October 20, 2010). "Federal Criteria For Studies Grow". Education Week. p. 1.
Further reading
• Anderson, John R.; Reder, Lynne M.; Simon, Herbert A.; Ericsson, K. Anders; Glaser, Robert (1998). "Radical Constructivism and Cognitive Psychology" (PDF). Brookings Papers on Education Policy (1): 227–278. Archived from the original (PDF) on 2010-06-26. Retrieved 2011-09-25.
• Auslander, Maurice; et al. (2004). "Goals for School Mathematics: The Report of the Cambridge Conference on School Mathematics 1963" (PDF). Cambridge MA: Center for the Study of Mathematics Curriculum. Archived (PDF) from the original on 2010-07-15. Retrieved 2009-08-06.
• Ball, Lynda, et al. Uses of Technology in Primary and Secondary Mathematics Education (Cham, Switzerland: Springer, 2018).
• Dreher, Anika, et al. "What kind of content knowledge do secondary mathematics teachers need?." Journal für Mathematik-Didaktik 39.2 (2018): 319-341 online Archived 2021-04-18 at the Wayback Machine.
• Drijvers, Paul, et al. Uses of technology in lower secondary mathematics education: A concise topical survey (Springer Nature, 2016).
• Gosztonyi, Katalin. "Mathematical culture and mathematics education in Hungary in the XXth century." in Mathematical cultures (Birkhäuser, Cham, 2016) pp. 71–89. online
• Paul Lockhart (2009). A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. Bellevue Literary Press. ISBN 978-1934137178.
• Losano, Leticia, and Márcia Cristina de Costa Trindade Cyrino. "Current research on prospective secondary mathematics teachers’ professional identity." in The mathematics education of prospective secondary teachers around the world (Springer, Cham, 2017) pp. 25-32.
• Sriraman, Bharath; English, Lyn (2010). Theories of Mathematics Education. Springer. ISBN 978-3-642-00774-3.
• Strogatz, Steven Henry; Joffray, Don (2009). The Calculus of Friendship: What a Teacher and a Student Learned about Life While Corresponding about Math. Princeton University Press. ISBN 978-0-691-13493-2.
• Strutchens, Marilyn E., et al. The mathematics education of prospective secondary teachers around the world (Springer Nature, 2017) online Archived 2021-04-18 at the Wayback Machine.
• Wong, Khoon Yoong. "Enriching secondary mathematics education with 21st century competencies." in Developing 21st Century Competencies In The Mathematics Classroom: Yearbook 2016 (Association Of Mathematics Educators. 2016) pp. 33–50.
External links
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| Wikipedia |
Technology readiness
Technology readiness refers to people's propensity to embrace and use new technologies for accomplishing goals in home life and at work. The construct can be viewed as an overall state of mind resulting from a gestalt of mental enablers and inhibitors that collectively determine a person's predisposition to use new technologies.[1]
The Technology Readiness Index (TRI), introduced by A. Parasuraman in 2000, consists of 36 attributes that measure the construct and its components. A streamlined and updated version with 16 attributes, "TRI 2.0," was introduced by Parasuraman and Colby in 2015.[2] The Technology Readiness model differs from well-known acceptance models such as the Technology acceptance model (TAM) in that TRI measures beliefs an individual has about cutting-edge technology in general while the TAM model measures acceptance towards a specific technology.[3][4] Technology Readiness is a multidimensional psychographic construct, offering a way to segment consumers based upon their underlying positive and negative technology beliefs.
Technology readiness has four underlying dimensions:[5]
• optimism, a positive view of technology and a belief that it offers people increased control, flexibility, and efficiency
• innovativeness, the tendency to be a technology pioneer and thought leader
• discomfort, perceived lack of control over technology and a feeling of being overwhelmed by it
• insecurity, distrust of technology and skepticism about its ability to work properly.
While optimism and innovativeness are contributors to technology readiness, discomfort and insecurity are inhibitors. The model captures the paradox that individuals may simultaneously hold both positive and negative beliefs.
The Technology Readiness Index has been validated as being a predictor of adoption of innovative technologies, and the findings it provides in a certain context equate to different strategies that would apply to a cutting-edge product or service.[6] It is frequently used in research to identify the general innovativeness of a population and/or as moderating variable in a more complete model that explains acceptance of a technology. The TRI and TRI 2.0 instruments are copyrighted, and academic researchers may license them at no cost by contacting the authors for permission.
References
1. Parasuraman, A. (2000). "Technology Readiness Index (TRI) a multiple-item scale to measure readiness to embrace new technologies". Journal of Service Research. 2 (4): 307–320. doi:10.1177/109467050024001.
2. "An Updated and Streamlined Technology Readiness: TRI 2.0, A. Parasuraman and Charles L. Colby, Journal of Service Research, Volume 18:1, pages 59-74
3. A Technology Readiness Index Primer A Technology Index Primer
4. "What is Technology Readiness Index". igi-global.com. Retrieved 31 January 2016.
5. Techno-Ready Marketing: How and Why Your Customers Adopt Technology, A. Parasuraman and Charles L. Colby, The Free Press, 2001
6. E-Service: New Directions in Theory and Practice, Roland Rust and P.K. Kannan, Pages 25-44, M.E. Sharpe, 2002.
| Wikipedia |
Theodore Frankel
Theodore Frankel (June 17, 1929 – August 5, 2017)[1] was a mathematician who introduced the Andreotti–Frankel theorem and the Frankel conjecture.
Frankel received his Ph.D. from the University of California, Berkeley in 1955. His doctoral advisor was Harley Flanders.[2] A Professor Emeritus of Mathematics at University of California, San Diego, Frankel was a longtime member of the Institute for Advanced Study in Princeton, New Jersey. He is known for his work in global differential geometry, Morse theory, and relativity theory. He joined the UC San Diego mathematics department in 1965, after serving on the faculties at Stanford University and Brown University.
Research
In the 1930s, John Synge established what is now known as Synge's theorem, by applying the second variation formula for arclength to a minimal loop. Frankel adapted Synge's method to higher-dimensional objects. As a consequence, he was able to prove that, when given a positively curved Riemannian metric on a closed manifold, any two totally geodesic compact submanifolds must intersect if their dimensions are large enough. The idea is to apply Synge's method to a minimizing geodesic between the two submanifolds. By the same approach, Frankel proved that complex submanifolds of positively curved Kähler manifolds must intersect if their dimensions are sufficiently large. These results were later extended by Samuel Goldberg and Shoshichi Kobayashi to allow positivity of the holomorphic bisectional curvature.[3]
Inspired by work of René Thom, Frankel and Aldo Andreotti gave a new proof of the Lefschetz hyperplane theorem using Morse theory. The crux of the argument is the algebraic fact that the eigenvalues of the real part of a complex quadratic form must occur in pairs of the form ±z. This becomes relevant in the context of Lefschetz's theorem, by considering a Morse function given by the distance to a fixed point. The second-order analysis at critical points is immediately aided by the above algebraic analysis, and the homology vanishing phenomena follows via the Morse inequalities.[4]
Given a Killing vector field for which the corresponding one-parameter group of isometries acts by holomorphic mappings, Frankel used the Cartan formula to show that the interior product of the vector field with the Kähler form is closed. Assuming that the first Betti number is zero, the de Rham theorem applies to construct a function whose critical points coincide with the zeros of the vector field. A second-order analysis at the critical points shows that the set of zeros of the vector field is a nondegenerate critical manifold for the function. Following Raoul Bott's development of Morse theory for critical manifolds, Frankel was able to establish that the Betti numbers of the manifold are fully encoded by the Betti numbers of the critical manifolds, together with the index of his Morse function along these manifolds. These ideas of Frankel were later important for works of Michael Atiyah and Nigel Hitchin, among others.[5][6]
Major Publications
Articles
• Andreotti, Aldo; Frankel, Theodore (1959). "The Lefschetz theorem on hyperplane sections". Annals of Mathematics. Second Series. 69 (3): 713–717. doi:10.2307/1970034. MR 0177422. Zbl 0115.38405.
• Frankel, Theodore (1959). "Fixed points and torsion on Kähler manifolds". Annals of Mathematics. Second Series. 70 (1): 1–8. doi:10.2307/1969889. MR 0131883. Zbl 0088.38002.
• Frankel, Theodore (1961). "Manifolds with positive curvature". Pacific Journal of Mathematics. 11 (1): 165–174. doi:10.2140/pjm.1961.11.165. MR 0123272. Zbl 0107.39002.
Textbooks
• Frankel, Theodore (1979). Gravitational Curvature. An Introduction to Einstein's Theory. San Francisco: W. H. Freeman and Co. ISBN 0-7167-1062-5. MR 0518868. Zbl 0427.53009.[7]
• Frankel, Theodore (2011). The Geometry of Physics: An Introduction (Third edition of 1997 original ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9781139061377. ISBN 978-1-107-60260-1. MR 2884939. Zbl 1250.58001.
References
1. UC San Diego Campus Notice: Passing of Professor Emeritus Ted Frankel
2. Theodore Frankel at the Mathematics Genealogy Project
3. Goldberg, Samuel I.; Kobayashi, Shoshichi (1967). "Holomorphic bisectional curvature". Journal of Differential Geometry. 1 (3–4): 225–233. doi:10.4310/jdg/1214428090. MR 0227901. Zbl 0169.53202.
4. John Milnor, Morse theory (1963), section 7
5. Atiyah, M. F. (1982). "Convexity and commuting Hamiltonians". Bulletin of the London Mathematical Society. 14 (1): 1–15. doi:10.1112/blms/14.1.1. MR 0642416. Zbl 0482.58013.
6. Hitchin, N. J. (1987). "The self-duality equations on a Riemann surface". Proceedings of the London Mathematical Society. 3. 55 (1): 59–126. doi:10.1112/plms/s3-55.1.59. MR 0887284. Zbl 0634.53045.
7. Trautman, Andrzej (1986). "Review: Gravitational Curvature, by Theodore Frankel". Bull. Amer. Math. Soc. (N.S.). 14 (1): 152–158. doi:10.1090/s0273-0979-1986-15425-x.
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| Wikipedia |
John William Theodore Youngs
John William Theodore Youngs (usually cited as J. W. T. Youngs, known as Ted Youngs; 21 August 1910 Bilaspur, Chhattisgarh, India – 20 July 1970 Santa Cruz, California) was an American mathematician.
John William Theodore Youngs
Born21 August 1910
Bilaspur
Died20 July 1970 (aged 59)
Alma mater
• Ohio State University
• Wheaton College
Employer
• Indiana University
• Purdue University
• University of California, Santa Cruz
Awards
• Guggenheim Fellowship (1946)
Youngs was the son of a missionary. He completed his undergraduate study at Wheaton College and received his PhD from Ohio State University in 1934 under Tibor Radó. He then taught for 18 years at Indiana University, where for eight years he was chair of the mathematics department. From 1964 he was a professor at the University of California, Santa Cruz, where he developed the mathematics faculty and was chair of the academic senate of the university.
Youngs worked in geometric topology, for example, questions on the Frechét-equivalence of topological maps.[1] He is famous for the Ringel–Youngs theorem (i.e. Ringel and Youngs's 1968 proof of the Heawood conjecture),[2] which is closely related to the analogue of the four-color theorem for surfaces of higher genus.
John Youngs was a consultant for Sandia National Laboratories, the Rand Corporation and the Institute for Defense Analyses as well as a trustee for Carver Research Foundation Institute in Tuskegee. In 1946–1947 he was a Guggenheim Fellow. At the University of Santa Cruz a mathematics prize for undergraduates in named after him.
Sources
• Obituary in Journal of Combinatorial Theory, vol 13, 1972
References
1. Youngs The representation problem for Frechét Surfaces, Memoirs American Mathematical Society 1951
2. Ringel, Gerhard; Youngs, J.W.T. (1968). "Solution of the Heawood map-coloring problem". Proc. Natl. Acad. Sci. USA. 60 (2): 438–445. Bibcode:1968PNAS...60..438R. doi:10.1073/pnas.60.2.438. MR 0228378. PMC 225066. PMID 16591648.
External links
• Obituary from the University of California
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| Wikipedia |
Teichmüller space
In mathematics, the Teichmüller space $T(S)$ of a (real) topological (or differential) surface $S$, is a space that parametrizes complex structures on $S$ up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.
Each point in a Teichmüller space $T(S)$ may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from $S$ to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension $6g-6$ for a surface of genus $g\geq 2$. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space.
The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research.
The sub-field of mathematics that studies the Teichmüller space is called Teichmüller theory.
History
Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann (1826-1866), who knew that $6g-6$ parameters were needed to describe the variations of complex structures on a surface of genus $g\geq 2$. The early study of Teichmüller space, in the late nineteenth–early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were Felix Klein, Henri Poincaré, Paul Koebe, Jakob Nielsen, Robert Fricke and Werner Fenchel.
The main contribution of Teichmüller to the study of moduli was the introduction of quasiconformal mappings to the subject. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by Lars Ahlfors and Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers).
The geometric vein in the study of Teichmüller space was revived following the work of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmüller space, and this is a very active subject of research in geometric group theory.
Definitions
Teichmüller space from complex structures
Let $S$ be an orientable smooth surface (a differentiable manifold of dimension 2). Informally the Teichmüller space $T(S)$ of $S$ is the space of Riemann surface structures on $S$ up to isotopy.
Formally it can be defined as follows. Two complex structures $X,Y$ on $S$ are said to be equivalent if there is a diffeomorphism $f\in \operatorname {Diff} (S)$ such that:
• It is holomorphic (the differential is complex linear at each point, for the structures $X$ at the source and $Y$ at the target) ;
• it is isotopic to the identity of $S$ (there is a continuous map $\gamma :[0,1]\to \operatorname {Diff} (S)$ :[0,1]\to \operatorname {Diff} (S)} such that $\gamma (0)=f,\gamma (1)=\mathrm {Id} $.
Then $T(S)$ is the space of equivalence classes of complex structures on $S$ for this relation.
Another equivalent definition is as follows: $T(S)$ is the space of pairs $(X,g)$ where $X$ is a Riemann surface and $g:S\to X$ a diffeomorphism, and two pairs $(X,g),(Y,h)$ are regarded as equivalent if $h\circ g^{-1}:X\to Y$ is isotopic to a holomorphic diffeomorphism. Such a pair is called a marked Riemann surface; the marking being the diffeomorphism; another definition of markings is by systems of curves.[1]
There are two simple examples that are immediately computed from the Uniformization theorem: there is a unique complex structure on the sphere $\mathbb {S} ^{2}$ (see Riemann sphere) and there are two on $\mathbb {R} ^{2}$ (the complex plane and the unit disk) and in each case the group of positive diffeomorphisms is contractible. Thus the Teichmüller space of $\mathbb {S} ^{2}$ is a single point and that of $\mathbb {R} ^{2}$ contains exactly two points.
A slightly more involved example is the open annulus, for which the Teichmüller space is the interval $[0,1)$ (the complex structure associated to $\lambda $ is the Riemann surface $\{z\in \mathbb {C} :\lambda <|z|<\lambda ^{-1}\}$ :\lambda <|z|<\lambda ^{-1}\}} ).
The Teichmüller space of the torus and flat metrics
The next example is the torus $\mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}.$ In this case any complex structure can be realised by a Riemann surface of the form $\mathbb {C} /(\mathbb {Z} +\tau \mathbb {Z} )$ (a complex elliptic curve) for a complex number $\tau \in \mathbb {H} $ where
$\mathbb {H} =\{z\in \mathbb {C} :\operatorname {Im} (z)>0\},$ :\operatorname {Im} (z)>0\},}
is the complex upper half-plane. Then we have a bijection:[2]
$\mathbb {H} \longrightarrow T(\mathbb {T} ^{2})$
$\tau \longmapsto (\mathbb {C} /(\mathbb {Z} +\tau \mathbb {Z} ),(x,y)\mapsto x+\tau y)$
and thus the Teichmüller space of $\mathbb {T} ^{2}$ is $\mathbb {H} .$
If we identify $\mathbb {C} $ with the Euclidean plane then each point in Teichmüller space can also be viewed as a marked flat structure on $\mathbb {T} ^{2}.$ Thus the Teichmüller space is in bijection with the set of pairs $(M,f)$ where $M$ is a flat surface and $f:\mathbb {T} ^{2}\to M$ is a diffeomorphism up to isotopy on $f$.
Finite type surfaces
These are the surfaces for which Teichmüller space is most often studied, which include closed surfaces. A surface is of finite type if it is diffeomorphic to a compact surface minus a finite set. If $S$ is a closed surface of genus $g$ then the surface obtained by removing $k$ points from $S$ is usually denoted $S_{g,k}$ and its Teichmüller space by $T_{g,k}.$
Teichmüller spaces and hyperbolic metrics
Every finite type orientable surface other than the ones above admits complete Riemannian metrics of constant curvature $-1$. For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the uniformisation theorem. Thus if $2g-2+k>0$ the Teichmüller space $T_{g,k}$ can be realised as the set of marked hyperbolic surfaces of genus $g$ with $k$ cusps, that is the set of pairs $(M,f)$ where $M$ is a hyperbolic surface and $f:S\to M$ is a diffeomorphism, modulo the equivalence relation where $(M,f)$ and $(N,g)$ are identified if $f\circ g^{-1}$ is isotopic to an isometry.
The topology on Teichmüller space
In all cases computed above there is an obvious topology on Teichmüller space. In the general case there are many natural ways to topologise $T(S)$, perhaps the simplest is via hyperbolic metrics and length functions.
If $\alpha $ is a closed curve on $S$ and $x=(M,f)$ a marked hyperbolic surface then one $f_{*}\alpha $ is homotopic to a unique closed geodesic $\alpha _{x}$ on $M$ (up to parametrisation). The value at $x$ of the length function associated to (the homotopy class of) $\alpha $ is then:
$\ell _{\alpha }(x)=\operatorname {Length} (\alpha _{x}).$
Let ${\mathcal {S}}$ be the set of simple closed curves on $S$. Then the map
$T(S)\to \mathbb {R} ^{\mathcal {S}}$
$x\mapsto \left(\ell _{\alpha }(x)\right)_{\alpha \in {\mathcal {S}}}$
is an embedding. The space $\mathbb {R} ^{\mathcal {S}}$ has the product topology and $T(S)$ is endowed with the induced topology. With this topology $T(S_{g,k})$ is homeomorphic to $\mathbb {R} ^{6g-6+2k}.$
In fact one can obtain an embedding with $9g-9$ curves,[3] and even $6g-5+2k$.[4] In both case one can use the embedding to give a geometric proof of the homeomorphism above.
More examples of small Teichmüller spaces
There is a unique complete hyperbolic metric of finite volume on the three-holed sphere[5] and so the Teichmüller space of finite-volume complete metrics of constant curvature $T(S_{0,3})$ is a point (this also follows from the dimension formula of the previous paragraph).
The Teichmüller spaces $T(S_{0,4})$ and $T(S_{1,1})$ are naturally realised as the upper half-plane, as can be seen using Fenchel–Nielsen coordinates.
Teichmüller space and conformal structures
Instead of complex structures of hyperbolic metrics one can define Teichmüller space using conformal structures. Indeed, conformal structures are the same as complex structures in two (real) dimensions.[6] Moreover, the Uniformisation Theorem also implies that in each conformal class of Riemannian metrics on a surface there is a unique metric of constant curvature.
Teichmüller spaces as representation spaces
Yet another interpretation of Teichmüller space is as a representation space for surface groups. If $S$ is hyperbolic, of finite type and $\Gamma =\pi _{1}(S)$ is the fundamental group of $S$ then Teichmüller space is in natural bijection with:
• The set of injective representations $\Gamma \to \mathrm {PSL} _{2}(\mathbb {R} )$ with discrete image, up to conjugation by an element of $\mathrm {PSL} _{2}(\mathbb {R} )$, if $S$ is compact ;
• In general, the set of such representations, with the added condition that those elements of $\Gamma $ which are represented by curves freely homotopic to a puncture are sent to parabolic elements of $\mathrm {PSL} _{2}(\mathbb {R} )$, again up to conjugation by an element of $\mathrm {PSL} _{2}(\mathbb {R} )$.
The map sends a marked hyperbolic structure $(M,f)$ to the composition $\rho \circ f_{*}$ where $\rho :\pi _{1}(M)\to \mathrm {PSL} _{2}(\mathbb {R} )$ :\pi _{1}(M)\to \mathrm {PSL} _{2}(\mathbb {R} )} is the monodromy of the hyperbolic structure and $f_{*}:\pi _{1}(S)\to \pi _{1}(M)$ is the isomorphism induced by $f$.
Note that this realises $T(S)$ as a closed subset of $\mathrm {PSL} _{2}(\mathbb {R} )^{2g+k-1}$ which endows it with a topology. This can be used to see the homeomorphism $T(S)\cong \mathbb {R} ^{6g-6+2k}$ directly.[7]
This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group $\mathrm {PSL} _{2}(\mathbb {R} )$ is replaced by an arbitrary semisimple Lie group.
A remark on categories
All definitions above can be made in the topological category instead of the category of differentiable manifolds, and this does not change the objects.
Infinite-dimensional Teichmüller spaces
Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to $\mathbb {R} ^{\mathbb {N} }$). Another example of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces.[8][9]
Action of the mapping class group and relation to moduli space
The map to moduli space
There is a map from Teichmüller space to the moduli space of Riemann surfaces diffeomorphic to $S$, defined by $(X,f)\mapsto X$. It is a covering map, and since $T(S)$ is simply connected it is the orbifold universal cover for the moduli space.
Action of the mapping class group
The mapping class group of $S$ is the coset group $MCG(S)$ of the diffeomorphism group of $S$ by the normal subgroup of those that are isotopic to the identity (the same definition can be made with homeomorphisms instead of diffeomorphisms and, for surfaces, this does not change the resulting group). The group of diffeomorphisms acts naturally on Teichmüller space by
$g\cdot (X,f)\mapsto (X,f\circ g^{-1}).$
If $\gamma \in MCG(S)$ is a mapping class and $g,h$ two diffeomorphisms representing it then they are isotopic. Thus the classes of $(X,f\circ g^{-1})$ and $(X,f\circ h^{-1})$ are the same in Teichmüller space, and the action above factorises through the mapping class group.
The action of the mapping class group $MCG(S)$ on the Teichmüller space is properly discontinuous, and the quotient is the moduli space.
Fixed points
Main article: Nielsen realization problem
The Nielsen realisation problem asks whether any finite subgroup of the mapping class group has a global fixed point (a point fixed by all group elements) in Teichmüller space. In more classical terms the question is: can every finite subgroup of $MCG(S)$ be realised as a group of isometries of some complete hyperbolic metric on $S$ (or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by Steven Kerckhoff.[10]
Coordinates
Fenchel–Nielsen coordinates
Main article: Fenchel–Nielsen coordinates
The Fenchel–Nielsen coordinates (so named after Werner Fenchel and Jakob Nielsen) on the Teichmüller space $T(S)$ are associated to a pants decomposition of the surface $S$. This is a decomposition of $S$ into pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in Teichmüller space, and another real parameter called the twist which is more involved to define.[11]
In case of a closed surface of genus $g$ there are $3g-3$ curves in a pants decomposition and we get $6g-6$ parameters, which is the dimension of $T(S_{g})$. The Fenchel–Nielsen coordinates in fact define a homeomorphism $T(S_{g})\to ]0,+\infty [^{3g-3}\times \mathbb {R} ^{3g-3}$.[12]
In the case of a surface with punctures some pairs of pants are "degenerate" (they have a cusp) and give only two length and twist parameters. Again in this case the Fenchel–Nielsen coordinates define a homeomorphism $T(S_{g,k})\to ]0,+\infty [^{3g-3+k}\times \mathbb {R} ^{3g-3+k}$.
Shear coordinates
If $k>0$ the surface $S=S_{g,k}$ admits ideal triangulations (whose vertices are exactly the punctures). By the formula for the Euler characteristic such a triangulation has $4g-4+2k$ triangles. An hyperbolic structure $M$ on $S$ determines a (unique up to isotopy) diffeomorphism $S\to M$ sending every triangle to an hyperbolic ideal triangle, thus a point in $T(S)$. The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation.[13] There are $6g-6+3k$ such parameters which can each take any value in $\mathbb {R} $, and the completeness of the structure corresponds to a linear equation and thus we get the right dimension $6g-6+2k$. These coordinates are called shear coordinates.
For closed surfaces, a pair of pants can be decomposed as the union of two ideal triangles (it can be seen as an incomplete hyperbolic metric on the three-holed sphere[14]). Thus we also get $3g-3$ shear coordinates on $T(S_{g})$.
Earthquakes
Main article: Earthquake map
A simple earthquake path in Teichmüller space is a path determined by varying a single shear or length Fenchel–Nielsen coordinate (for a fixed ideal triangulation of a surface). The name comes from seeing the ideal triangles or the pants as tectonic plates and the shear as plate motion.
More generally one can do earthquakes along geodesic laminations. A theorem of Thurston then states that two points in Teichmüller space are joined by a unique earthquake path.
Analytic theory
Quasiconformal mappings
Main article: Quasiconformal mapping
A quasiconformal mapping between two Riemann surfaces is a homeomorphism which deforms the conformal structure in a bounded manner over the surface. More precisely it is differentiable almost everywhere and there is a constant $K\geq 1$, called the dilatation, such that
${\frac {|f_{z}|+|f_{\bar {z}}|}{|f_{z}|-|f_{\bar {z}}|}}\leq K$
where $f_{z},f_{\bar {z}}$ are the derivatives in a conformal coordinate $z$ and its conjugate ${\bar {z}}$.
There are quasi-conformal mappings in every isotopy class and so an alternative definition for The Teichmüller space is as follows. Fix a Riemann surface $X$ diffeomorphic to $S$, and Teichmüller space is in natural bijection with the marked surfaces $(Y,g)$ where $g:X\to Y$ is a quasiconformal mapping, up to the same equivalence relation as above.
Quadratic differentials and the Bers embedding
Main article: Schwarzian derivative
Main article: Bers slice
With the definition above, if $X=\Gamma \setminus \mathbb {H} ^{2}$ there is a natural map from Teichmüller space to the space of $\Gamma $-equivariant solutions to the Beltrami differential equation.[15] These give rise, via the Schwarzian derivative, to quadratic differentials on $X$.[16] The space of those is a complex space of complex dimension $3g-3$, and the image of Teichmüller space is an open set.[17] This map is called the Bers embedding.
A quadratic differential on $X$ can be represented by a translation surface conformal to $X$.
Teichmüller mappings
Teichmüller's theorem[18] states that between two marked Riemann surfaces $(X,g)$ and $(Y,h)$ there is always a unique quasiconformal mapping $X\to Y$ in the isotopy class of $h\circ g^{-1}$ which has minimal dilatation. This map is called a Teichmüller mapping.
In the geometric picture this means that for every two diffeomorphic Riemann surfaces $X,Y$ and diffeomorphism $f:X\to Y$ there exists two polygons representing $X,Y$ and an affine map sending one to the other, which has smallest dilatation among all quasiconformal maps $X\to Y$.
Metrics
The Teichmüller metric
If $x,y\in T(S)$ and the Teichmüller mapping between them has dilatation $K$ then the Teichmüller distance between them is by definition ${\frac {1}{2}}\log K$. This indeed defines a distance on $T(S)$ which induces its topology, and for which it is complete. This is the metric most commonly used for the study of the metric geometry of Teichmüller space. In particular it is of interest to geometric group theorists.
There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on $T(S)\times T(S)$, which is not symmetric.[19]
The Weil–Petersson metric
Main article: Weil–Petersson metric
Quadratic differentials on a Riemann surface $X$ are identified with the cotangent space at $(X,f)$ to Teichmüller space.[20] The Weil–Petersson metric is the Riemannian metric defined by the $L^{2}$ inner product on quadratic differentials.
Compactifications
There are several inequivalent compactifications of Teichmüller spaces that have been studied. Several of the earlier compactifications depend on the choice of a point in Teichmüller space so are not invariant under the modular group, which can be inconvenient. William Thurston later found a compactification without this disadvantage, which has become the most widely used compactification.
Thurston compactification
By looking at the hyperbolic lengths of simple closed curves for each point in Teichmüller space and taking the closure in the (infinite-dimensional) projective space, Thurston (1988) introduced a compactification whose points at infinity correspond to projective measured laminations. The compactified space is homeomorphic to a closed ball. This Thurston compactification is acted on continuously by the modular group. In particular any element of the modular group has a fixed point in Thurston's compactification, which Thurston used in his classification of elements of the modular group.
Bers compactification
The Bers compactification is given by taking the closure of the image of the Bers embedding of Teichmüller space, studied by Bers (1970). The Bers embedding depends on the choice of a point in Teichmüller space so is not invariant under the modular group, and in fact the modular group does not act continuously on the Bers compactification.
Teichmüller compactification
The "points at infinity" in the Teichmüller compactification consist of geodesic rays (for the Teichmüller metric) starting at a fixed basepoint. This compactification depends on the choice of basepoint so is not acted on by the modular group, and in fact Kerckhoff showed that the action of the modular group on Teichmüller space does not extend to a continuous action on this compactification.
Gardiner–Masur compactification
Gardiner & Masur (1991) considered a compactification similar to the Thurston compactification, but using extremal length rather than hyperbolic length. The modular group acts continuously on this compactification, but they showed that their compactification has strictly more points at infinity.
Large-scale geometry
There has been an extensive study of the geometric properties of Teichmüller space endowed with the Teichmüller metric. Known large-scale properties include:
• Teichmüller space $T(S_{g,k})$ contains flat subspaces of dimension $3g-3+k$, and there are no higher-dimensional quasi-isometrically embedded flats.[21]
• In particular, if $g>1$ or $g=1,k>1$ or $g=0,k>4$ then $T(S_{g,k})$ is not hyperbolic.
On the other hand, Teichmüller space exhibits several properties characteristic of hyperbolic spaces, such as:
• Some geodesics behave like they do in hyperbolic space.[22]
• Random walks on Teichmüller space converge almost surely to a point on the Thurston boundary.[23]
Some of these features can be explained by the study of maps from Teichmüller space to the curve complex, which is known to be hyperbolic.
Complex geometry
The Bers embedding gives $T(S)$ a complex structure as an open subset of $\mathbb {C} ^{3g-3}.$
Metrics coming from the complex structure
Since Teichmüller space is a complex manifold it carries a Carathéodory metric. Teichmüller space is Kobayashi hyperbolic and its Kobayashi metric coincides with the Teichmüller metric.[24] This latter result is used in Royden's proof that the mapping class group is the full group of isometries for the Teichmüller metric.
The Bers embedding realises Teichmüller space as a domain of holomorphy and hence it also carries a Bergman metric.
Kähler metrics on Teichmüller space
The Weil–Petersson metric is Kähler but it is not complete.
Cheng and Yau showed that there is a unique complete Kähler–Einstein metric on Teichmüller space.[25] It has constant negative scalar curvature.
Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by McMullen (2000) that is Kähler-hyperbolic.
Equivalence of metrics
With the exception of the incomplete Weil–Petersson metric, all metrics on Teichmüller space introduced here are quasi-isometric to each other.[26]
See also
• Moduli of algebraic curves
• p-adic Teichmüller theory
• Inter-universal Teichmüller theory
• Teichmüller modular form
References
1. Imayoshi & Taniguchi 1992, p. 14.
2. Imayoshi & Taniguchi 1992, p. 13.
3. Imayoshi & Taniguchi 1992, Theorem 3.12.
4. Hamenstädt, Ursula (2003). "Length functions and parameterizations of Teichmüller space for surfaces with cusps". Annales Acad. Scient. Fenn. 28: 75–88.
5. Ratcliffe 2006, Theorem 9.8.8.
6. Imayoshi & Taniguchi 1992, Theorem 1.7.
7. Imayoshi & Taniguchi 1992, Theorem 2.25.
8. Ghys, Etienne (1999). "Laminations par surfaces de Riemann". Panor. Synthèses. 8: 49–95. MR 1760843.
9. Deroin, Bertrand (2007). "Nonrigidity of hyperbolic surfaces laminations". Proceedings of the American Mathematical Society. 135 (3): 873–881. doi:10.1090/s0002-9939-06-08579-0. MR 2262885.
10. Kerckhoff 1983.
11. Imayoshi & Taniguchi 1992, p. 61.
12. Imayoshi & Taniguchi 1992, Theorem 3.10.
13. Thurston 1988, p. 40.
14. Thurston 1988, p. 42.
15. Ahlfors 2006, p. 69.
16. Ahlfors 2006, p. 71.
17. Ahlfors 2006, Chapter VI.C.
18. Ahlfors 2006, p. 96.
19. Thurston, William (1998) [1986], Minimal stretch maps between hyperbolic surfaces, arXiv:math/9801039, Bibcode:1998math......1039T
20. Ahlfors 2006, Chapter VI.D
21. Eskin, Alex; Masur, Howard; Rafi, Kasra (2017). "Large scale rank of Teichmüller space". Duke Mathematical Journal. 166 (8): 1517–1572. arXiv:1307.3733. doi:10.1215/00127094-0000006X. S2CID 15393033.
22. Rafi, Kasra (2014). "Hyperbolicity in Teichmüller space". Geometry & Topology. 18 (5): 3025–3053. arXiv:1011.6004. doi:10.2140/gt.2014.18.3025. S2CID 73575721.
23. Duchin, Moon (2005). Thin triangles and a multiplicative ergodic theorem for Teichmüller geometry (Ph.D.). University of Chicago. arXiv:math/0508046.
24. Royden, Halsey L. (1970). "Report on the Teichmüller metric". Proc. Natl. Acad. Sci. U.S.A. 65 (3): 497–499. Bibcode:1970PNAS...65..497R. doi:10.1073/pnas.65.3.497. MR 0259115. PMC 282934. PMID 16591819.
25. Cheng, Shiu Yuen; Yau, Shing Tung (1980). "On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation". Comm. Pure Appl. Math. 33 (4): 507–544. doi:10.1002/cpa.3160330404. MR 0575736.
26. Yeung, Sai-Kee (2005). "Quasi-isometry of metrics on Teichmüller spaces". Int. Math. Res. Not. 2005 (4): 239–255. doi:10.1155/IMRN.2005.239. MR 2128436.
Sources
• Ahlfors, Lars V. (2006). Lectures on quasiconformal mappings. Second edition. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. American Math. Soc. pp. viii+162. ISBN 978-0-8218-3644-6.
• Bers, Lipman (1970), "On boundaries of Teichmüller spaces and on Kleinian groups. I", Annals of Mathematics, Second Series, 91 (3): 570–600, doi:10.2307/1970638, JSTOR 1970638, MR 0297992
• Fathi, Albert; Laudenbach, François; Poenaru, Valentin (2012). Thurston's work on surfaces. Princeton University Press. pp. xvi+254. ISBN 978-0-691-14735-2. MR 3053012.
• Gardiner, Frederic P.; Masur, Howard (1991), "Extremal length geometry of Teichmüller space", Complex Variables Theory Appl., 16 (2–3): 209–237, doi:10.1080/17476939108814480, MR 1099913
• Imayoshi, Yôichi; Taniguchi, Masahiko (1992). An introduction to Teichmüller spaces. Springer. pp. xiv+279. ISBN 978-4-431-70088-3.
• Kerckhoff, Steven P. (1983). "The Nielsen realization problem". Annals of Mathematics. Second Series. 117 (2): 235–265. CiteSeerX 10.1.1.353.3593. doi:10.2307/2007076. JSTOR 2007076. MR 0690845.
• McMullen, Curtis T. (2000), "The moduli space of Riemann surfaces is Kähler hyperbolic", Annals of Mathematics, Second Series, 151 (1): 327–357, arXiv:math/0010022, doi:10.2307/121120, JSTOR 121120, MR 1745010, S2CID 8032847
• Ratcliffe, John (2006). Foundations of hyperbolic manifolds, Second edition. Springer. pp. xii+779. ISBN 978-0387-33197-3.
• Thurston, William P. (1988), "On the geometry and dynamics of diffeomorphisms of surfaces", Bulletin of the American Mathematical Society, New Series, 19 (2): 417–431, doi:10.1090/S0273-0979-1988-15685-6, MR 0956596
Further reading
• Bers, Lipman (1981), "Finite-dimensional Teichmüller spaces and generalizations", Bulletin of the American Mathematical Society, New Series, 5 (2): 131–172, doi:10.1090/S0273-0979-1981-14933-8, MR 0621883
• Gardiner, Frederick P. (1987), Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), New York: John Wiley & Sons, ISBN 978-0-471-84539-3, MR 0903027
• Hubbard, John Hamal (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, ISBN 978-0-9715766-2-9, MR 2245223
• Papadopoulos, Athanase, ed. (2007–2016), Handbook of Teichmüller theory. Vols. I-V (PDF), IRMA Lectures in Mathematics and Theoretical Physics, vol. 11, 13, 17, 19, 26, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 978-3-03719-029-6, MR 2284826, S2CID 9203341 The last volume contains translations of several of Teichmüller's papers.
• Teichmüller, Oswald (1939), "Extremale quasikonforme Abbildungen und quadratische Differentiale", Abh. Preuss. Akad. Wiss. Math.-Nat. Kl., 1939 (22): 197, JFM 66.1252.01, MR 0003242
• Teichmüller, Oswald (1982), Ahlfors, Lars V.; Gehring, Frederick W. (eds.), Gesammelte Abhandlungen, Berlin, New York: Springer-Verlag, ISBN 978-3-540-10899-3, MR 0649778
• Voitsekhovskii, M.I. (2001) [1994], "Teichmüller space", Encyclopedia of Mathematics, EMS Press
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| Wikipedia |
Teichmüller–Tukey lemma
In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion. Over Zermelo–Fraenkel set theory, the Teichmüller–Tukey lemma is equivalent to the axiom of choice, and therefore to the well-ordering theorem, Zorn's lemma, and the Hausdorff maximal principle.[1]
Definitions
A family of sets ${\mathcal {F}}$ is of finite character provided it has the following properties:
1. For each $A\in {\mathcal {F}}$, every finite subset of $A$ belongs to ${\mathcal {F}}$.
2. If every finite subset of a given set $A$ belongs to ${\mathcal {F}}$, then $A$ belongs to ${\mathcal {F}}$.
Statement of the lemma
Let $Z$ be a set and let ${\mathcal {F}}\subseteq {\mathcal {P}}(Z)$. If ${\mathcal {F}}$ is of finite character and $X\in {\mathcal {F}}$, then there is a maximal $Y\in {\mathcal {F}}$ (according to the inclusion relation) such that $X\subseteq Y$.[2]
Applications
In linear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider the collection ${\mathcal {F}}$ of linearly independent sets of vectors. This is a collection of finite character. Thus, a maximal set exists, which must then span V and be a basis for V.
Notes
1. Jech, Thomas J. (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 978-0-486-46624-8.
2. Kunen, Kenneth (2009). The Foundations of Mathematics. College Publications. ISBN 978-1-904987-14-7.
References
• Brillinger, David R. "John Wilder Tukey"
| Wikipedia |
Mapping class group of a surface
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.
The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory.
The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups.
History
The mapping class group appeared in the first half of the twentieth century. Its origins lie in the study of the topology of hyperbolic surfaces, and especially in the study of the intersections of closed curves on these surfaces. The earliest contributors were Max Dehn and Jakob Nielsen: Dehn proved finite generation of the group,[1] and Nielsen gave a classification of mapping classes and proved that all automorphisms of the fundamental group of a surface can be represented by homeomorphisms (the Dehn–Nielsen–Baer theorem).
The Dehn–Nielsen theory was reinterpreted in the mid-seventies by Thurston who gave the subject a more geometric flavour[2] and used this work to great effect in his program for the study of three-manifolds.
More recently the mapping class group has been by itself a central topic in geometric group theory, where it provides a testing ground for various conjectures and techniques.
Definition and examples
Mapping class group of orientable surfaces
Let $S$ be a connected, closed, orientable surface and $\operatorname {Homeo} ^{+}(S)$ the group of orientation-preserving, or positive, homeomorphisms of $S$. This group has a natural topology, the compact-open topology. It can be defined easily by a distance function: if we are given a metric $d$ on $S$ inducing its topology then the function defined by
$\delta (f,g)=\sup _{x\in S}\left(d(f(x),g(x))\right)$
is a distance inducing the compact-open topology on $\operatorname {Homeo} ^{+}(S)$. The connected component of the identity for this topology is denoted $\operatorname {Homeo} _{0}(S)$. By definition it is equal to the homeomorphisms of $S$ which are isotopic to the identity. It is a normal subgroup of the group of positive homeomorphisms, and the mapping class group of $S$ is the group
$\operatorname {Mod} (S)=\operatorname {Homeo} ^{+}(S)/\operatorname {Homeo} _{0}(S)$.
This is a countable group.
If we modify the definition to include all homeomorphisms we obtain the extended mapping class group $\operatorname {Mod} ^{\pm }(S)$, which contains the mapping class group as a subgroup of index 2.
This definition can also be made in the differentiable category: if we replace all instances of "homeomorphism" above with "diffeomorphism" we obtain the same group, that is the inclusion $\operatorname {Diff} ^{+}(S)\subset \operatorname {Homeo} ^{+}(S)$ induces an isomorphism between the quotients by their respective identity components.
The mapping class groups of the sphere and the torus
Suppose that $S$ is the unit sphere in $\mathbb {R} ^{3}$. Then any homeomorphism of $S$ is isotopic to either the identity or to the restriction to $S$ of the symmetry in the plane $z=0$. The latter is not orientation-preserving and we see that the mapping class group of the sphere is trivial, and its extended mapping class group is $\mathbb {Z} /2\mathbb {Z} $, the cyclic group of order 2.
The mapping class group of the torus $\mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}$ is naturally identified with the modular group $\operatorname {SL} _{2}(\mathbb {Z} )$. It is easy to construct a morphism $\Phi :\operatorname {SL} _{2}(\mathbb {Z} )\to \operatorname {Mod} (\mathbb {T} ^{2})$ :\operatorname {SL} _{2}(\mathbb {Z} )\to \operatorname {Mod} (\mathbb {T} ^{2})} : every $A\in \operatorname {SL} _{2}(\mathbb {Z} )$ induces a diffeomorphism of $\mathbb {T} ^{2}$ via $x+\mathbb {Z} ^{2}\mapsto Ax+\mathbb {Z} ^{2}$. The action of diffeomorphisms on the first homology group of $\mathbb {T} ^{2}$ gives a left-inverse $\Pi $ to the morphism $\Phi $ (proving in particular that it is injective) and it can be checked that $\Pi $ is injective, so that $\Pi ,\Phi $ are inverse isomorphisms between $\operatorname {Mod} (\mathbb {T} ^{2})$ and $\operatorname {SL} _{2}(\mathbb {Z} )$.[3] In the same way, the extended mapping class group of $\mathbb {T} ^{2}$ is $\operatorname {GL} _{2}(\mathbb {Z} )$.
Mapping class group of surfaces with boundary and punctures
In the case where $S$ is a compact surface with a non-empty boundary $\partial S$ then the definition of the mapping class group needs to be more precise. The group $\operatorname {Homeo} ^{+}(S,\partial S)$ of homeomorphisms relative to the boundary is the subgroup of $\operatorname {Homeo} ^{+}(S)$ which restrict to the identity on the boundary, and the subgroup $\operatorname {Homeo} _{0}(S,\partial S)$ is the connected component of the identity. The mapping class group is then defined as
$\operatorname {Mod} (S)=\operatorname {Homeo} ^{+}(S,\partial S)/\operatorname {Homeo} _{0}(S,\partial S)$.
A surface with punctures is a compact surface with a finite number of points removed ("punctures"). The mapping class group of such a surface is defined as above (note that the mapping classes are allowed to permute the punctures, but not the boundary components).
Mapping class group of an annulus
Any annulus is homeomorphic to the subset $A_{0}=\{1\leq |z|\leq 2\}$ of $\mathbb {C} $. One can define a diffeomorphism $\tau _{0}$ by the following formula:
$\tau _{0}(z)=e^{2i\pi |z|}z$
which is the identity on both boundary components $\{|z|=1\},\{|z|=2\}$. The mapping class group of $A$ is then generated by the class of $\tau _{0}$.
Braid groups and mapping class groups
Main article: Braid groups
Braid groups can be defined as the mapping class groups of a disc with punctures. More precisely, the braid group on n strands is naturally isomorphic to the mapping class group of a disc with n punctures.[4]
The Dehn–Nielsen–Baer theorem
If $S$ is closed and $f$ is a homeomorphism of $S$ then we can define an automorphism $f_{*}$ of the fundamental group $\pi _{1}(S,x_{0})$ as follows: fix a path $\gamma $ between $x_{0}$ and $f(x_{0})$ and for a loop $\alpha $ based at $x_{0}$ representing an element $[\alpha ]\in \pi _{1}(S,x_{0})$ define $f_{*}([\alpha ])$ to be the element of the fundamental group associated to the loop ${\bar {\gamma }}*f(\alpha )*\gamma $. This automorphism depends on the choice of $\gamma $, but only up to conjugation. Thus we get a well-defined map from $\operatorname {Homeo} (S)$ to the outer automorphism group $\operatorname {Out} (\pi _{1}(S,x_{0}))$. This map is a morphism and its kernel is exactly the subgroup $\operatorname {Homeo} _{0}(S)$. The Dehn–Nielsen–Baer theorem states that it is in addition surjective.[5] In particular, it implies that:
The extended mapping class group $\operatorname {Mod} ^{\pm }(S)$ is isomorphic to the outer automorphism group $\operatorname {Out} (\pi _{1}(S))$.
The image of the mapping class group is an index 2 subgroup of the outer automorphism group, which can be characterised by its action on homology.
The conclusion of the theorem does not hold when $S$ has a non-empty boundary (except in a finite number of cases). In this case the fundamental group is a free group and the outer automorphism group Out(Fn) is strictly larger than the image of the mapping class group via the morphism defined in the previous paragraph. The image is exactly those outer automorphisms which preserve each conjugacy class in the fundamental group corresponding to a boundary component.
The Birman exact sequence
This is an exact sequence relating the mapping class group of surfaces with the same genus and boundary but a different number of punctures. It is a fundamental tool which allows to use recursive arguments in the study of mapping class groups. It was proven by Joan Birman in 1969.[6] The exact statement is as follows.[7]
Let $S$ be a compact surface and $x\in S$. There is an exact sequence
$1\to \pi _{1}(S,x)\to \operatorname {Mod} (S\setminus \{x\})\to \operatorname {Mod} (S)\to 1$.
In the case where $S$ itself has punctures the mapping class group $\operatorname {Mod} (S\setminus \{x\})$ must be replaced by the finite-index subgroup of mapping classes fixing $x$.
Elements of the mapping class group
Dehn twists
Main article: Dehn twist
If $c$ is an oriented simple closed curve on $S$ and one chooses a closed tubular neighbourhood $A$ then there is a homeomorphism $f$ from $A$ to the canonical annulus $A_{0}$ defined above, sending $c$ to a circle with the counterclockwise orientation. This is used to define a homeomorphism $\tau _{c}$ of $S$ as follows: on $S\setminus A$ it is the identity, and on $A$ it is equal to $f^{-1}\circ \tau _{0}\circ f$. The class of $\tau _{c}$ in the mapping class group $\operatorname {Mod} (S)$ does not depend on the choice of $f$ made above, and the resulting element is called the Dehn twist about $c$. If $c$ is not null-homotopic this mapping class is nontrivial, and more generally the Dehn twists defined by two non-homotopic curves are distinct elements in the mapping class group.
In the mapping class group of the torus identified with $\operatorname {SL} _{2}(\mathbb {Z} )$ the Dehn twists correspond to unipotent matrices. For example, the matrix
${\begin{pmatrix}1&1\\0&1\end{pmatrix}}$
corresponds to the Dehn twist about a horizontal curve in the torus.
The Nielsen–Thurston classification
There is a classification of the mapping classes on a surface, originally due to Nielsen and rediscovered by Thurston, which can be stated as follows. An element $g\in \operatorname {Mod} (S)$ is either:
• of finite order (i.e. there exists $n>0$ such that $g^{n}$ is the identity),
• reducible: there exists a set of disjoint closed curves on $S$ which is preserved by the action of $g$;
• or pseudo-Anosov.
The main content of the theorem is that a mapping class which is neither of finite order nor reducible must be pseudo-Anosov, which can be defined explicitly by dynamical properties.[8]
Pseudo-Anosov diffeomorphisms
Main article: Pseudo-Anosov map
The study of pseudo-Anosov diffeomorphisms of a surface is fundamental. They are the most interesting diffeomorphisms, since finite-order mapping classes are isotopic to isometries and thus well understood, and the study of reducible classes indeed essentially reduces to the study of mapping classes on smaller surfaces which may themselves be either finite order or pseudo-Anosov.
Pseudo-Anosov mapping classes are "generic" in the mapping class group in various ways. For example, a random walk on the mapping class group will end on a pseudo-Anosov element with a probability tending to 1 as the number of steps grows.
Actions of the mapping class group
Action on Teichmüller space
Given a punctured surface $S$ (usually without boundary) the Teichmüller space $T(S)$ is the space of marked complex (equivalently, conformal or complete hyperbolic) structures on $S$. These are represented by pairs $(X,f)$ where $X$ is a Riemann surface and $f:S\to X$ a homeomorphism, modulo a suitable equivalence relation. There is an obvious action of the group $\operatorname {Homeo} ^{+}(S)$ on such pairs, which descends to an action of $\operatorname {Mod} (S)$ on Teichmüller space.
This action has many interesting properties; for example it is properly discontinuous (though not free). It is compatible with various geometric structures (metric or complex) with which $T(S)$ can be endowed. In particular, the Teichmüller metric can be used to establish some large-scale properties of the mapping class group, for example that the maximal quasi-isometrically embedded flats in $\operatorname {Mod} (S)$ are of dimension $3g-3+k$.[9]
The action extends to the Thurston boundary of Teichmüller space, and the Nielsen-Thurston classification of mapping classes can be seen in the dynamical properties of the action on Teichmüller space together with its Thurston boundary. Namely:[10]
• Finite-order elements fix a point inside Teichmüller space (more concretely this means that any mapping class of finite order in $\operatorname {Mod} (S)$ can be realised as an isometry for some hyperbolic metric on $S$);
• Pseudo-Anosov classes fix the two points on the boundary corresponding to their stable and unstable foliation and the action is minimal (has a dense orbit) on the boundary;
• Reducible classes do not act minimally on the boundary.
Action on the curve complex
The curve complex of a surface $S$ is a complex whose vertices are isotopy classes of simple closed curves on $S$. The action of the mapping class groups $\operatorname {Mod} (S)$ on the vertices carries over to the full complex. The action is not properly discontinuous (the stabiliser of a simple closed curve is an infinite group).
This action, together with combinatorial and geometric properties of the curve complex, can be used to prove various properties of the mapping class group.[11] In particular, it explains some of the hyperbolic properties of the mapping class group: while as mentioned in the previous section the mapping class group is not a hyperbolic group it has some properties reminiscent of those.
Pants complex
The pants complex of a compact surface $S$ is a complex whose vertices are the pants decompositions of $S$ (isotopy classes of maximal systems of disjoint simple closed curves). The action of $\operatorname {Mod} (S)$ extends to an action on this complex. This complex is quasi-isometric to Teichmüller space endowed with the Weil–Petersson metric.[12]
Markings complex
The stabilisers of the mapping class group's action on the curve and pants complexes are quite large. The markings complex is a complex whose vertices are markings of $S$, which are acted upon by, and have trivial stabilisers in, the mapping class group $\operatorname {Mod} (S)$. It is (in opposition to the curve or pants complex) a locally finite complex which is quasi-isometric to the mapping class group.[13]
A marking[lower-alpha 1] is determined by a pants decomposition $\alpha _{1},\ldots ,\alpha _{\xi }$ and a collection of transverse curves $\beta _{1},\ldots ,\beta _{\xi }$ such that every one of the $\beta _{i}$ intersects at most one of the $\alpha _{i}$, and this "minimally" (this is a technical condition which can be stated as follows: if $\alpha _{i},\beta _{i}$ are contained in a subsurface homeomorphic to a torus then they intersect once, and if the surface is a four-holed sphere they intersect twice). Two distinct markings are joined by an edge if they differ by an "elementary move", and the full complex is obtained by adding all possible higher-dimensional simplices.
Generators and relations for mapping class groups
The Dehn–Lickorish theorem
The mapping class group is generated by the subset of Dehn twists about all simple closed curves on the surface. The Dehn–Lickorish theorem states that it is sufficient to select a finite number of those to generate the mapping class group.[14] This generalises the fact that $\operatorname {SL} _{2}(\mathbb {Z} )$ is generated by the matrices
${\begin{pmatrix}1&1\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\1&1\end{pmatrix}}$.
In particular, the mapping class group of a surface is a finitely generated group.
The smallest number of Dehn twists that can generate the mapping class group of a closed surface of genus $g\geq 2$ is $2g+1$; this was proven later by Humphries.
Finite presentability
It is possible to prove that all relations between the Dehn twists in a generating set for the mapping class group can be written as combinations of a finite number among them. This means that the mapping class group of a surface is a finitely presented group.
One way to prove this theorem is to deduce it from the properties of the action of the mapping class group on the pants complex: the stabiliser of a vertex is seen to be finitely presented, and the action is cofinite. Since the complex is connected and simply connected it follows that the mapping class group must be finitely generated. There are other ways of getting finite presentations, but in practice the only one to yield explicit relations for all geni is that described in this paragraph with a slightly different complex instead of the curve complex, called the cut system complex.[15]
An example of a relation between Dehn twists occurring in this presentation is the lantern relation.
Other systems of generators
There are other interesting systems of generators for the mapping class group besides Dehn twists. For example, $\operatorname {Mod} (S)$ can be generated by two elements[16] or by involutions.[17]
Cohomology of the mapping class group
If $S$ is a surface of genus $g$ with $b$ boundary components and $k$ punctures then the virtual cohomological dimension of $\operatorname {Mod} (S)$ is equal to $4g-4+b+k$.
The first homology of the mapping class group is finite[18] and it follows that the first cohomology group is finite as well.
Subgroups of the mapping class groups
The Torelli subgroup
As singular homology is functorial, the mapping class group $\operatorname {Mod} (S)$ acts by automorphisms on the first homology group $H_{1}(S)$. This is a free abelian group of rank $2g$ if $S$ is closed of genus $g$. This action thus gives a linear representation $\operatorname {Mod} (S)\to \operatorname {GL} _{2g}(\mathbb {Z} )$.
This map is in fact a surjection with image equal to the integer points $\operatorname {Sp} _{2g}(\mathbb {Z} )$ of the symplectic group. This comes from the fact that the intersection number of closed curves induces a symplectic form on the first homology, which is preserved by the action of the mapping class group. The surjectivity is proven by showing that the images of Dehn twists generate $\operatorname {Sp} _{2g}(\mathbb {Z} )$.[19]
The kernel of the morphism $\operatorname {Mod} (S)\to \operatorname {Sp} _{2g}(\mathbb {Z} )$ is called the Torelli group of $S$. It is a finitely generated, torsion-free subgroup[20] and its study is of fundamental importance for its bearing on both the structure of the mapping class group itself (since the arithmetic group $\operatorname {Sp} _{2g}(\mathbb {Z} )$ is comparatively very well understood, a lot of facts about $\operatorname {Mod} (S)$ boil down to a statement about its Torelli subgroup) and applications to 3-dimensional topology and algebraic geometry.
Residual finiteness and finite-index subgroups
An example of application of the Torelli subgroup is the following result:
The mapping class group is residually finite.
The proof proceeds first by using residual finiteness of the linear group $\operatorname {Sp} _{2g}(\mathbb {Z} )$, and then, for any nontrivial element of the Torelli group, constructing by geometric means subgroups of finite index which does not contain it.[21]
An interesting class of finite-index subgroups is given by the kernels of the morphisms:
$\Phi _{n}:\operatorname {Mod} (S)\to \operatorname {Sp} _{2g}(\mathbb {Z} )\to \operatorname {Sp} _{2g}(\mathbb {Z} /n\mathbb {Z} )$
The kernel of $\Phi _{n}$ is usually called a congruence subgroup of $\operatorname {Mod} (S)$. It is a torsion-free group for all $n\geq 3$ (this follows easily from a classical result of Minkowski on linear groups and the fact that the Torelli group is torsion-free).
Finite subgroups
The mapping class group has only finitely many classes of finite groups, as follows from the fact that the finite-index subgroup $\ker(\Phi _{3})$ is torsion-free, as discussed in the previous paragraph. Moreover, this also implies that any finite subgroup of $\operatorname {Mod} (S)$ is a subgroup of the finite group $\operatorname {Mod} (S)/\ker(\Phi _{3})\cong \operatorname {Sp} _{2g}(\mathbb {Z} /3)$.
A bound on the order of finite subgroups can also be obtained through geometric means. The solution to the Nielsen realisation problem implies that any such group is realised as the group of isometries of an hyperbolic surface of genus $g$. Hurwitz's bound then implies that the maximal order is equal to $84(g-1)$.
General facts on subgroups
The mapping class groups satisfy the Tits alternative: that is, any subgroup of it either contains a non-abelian free subgroup or it is virtually solvable (in fact abelian).[22]
Any subgroup which is not reducible (that is it does not preserve a set of isotopy class of disjoint simple closed curves) must contain a pseudo-Anosov element.[23]
Linear representations
It is an open question whether the mapping class group is a linear group or not. Besides the symplectic representation on homology explained above there are other interesting finite-dimensional linear representations arising from topological quantum field theory. The images of these representations are contained in arithmetic groups which are not symplectic, and this allows to construct many more finite quotients of $\operatorname {Mod} (S)$.[24]
In the other direction there is a lower bound for the dimension of a (putative) faithful representation, which has to be at least $2{\sqrt {g-1}}$.[25]
Notes
1. We describe here only "clean, complete" (in the terminology of Masur & Minsky (2000)) markings.
Citations
1. Acta Math. 1938, pp. 135–206.
2. Bull. Amer. Math. Soc. 1988, pp. 417–431.
3. Farb & Margalit 2012, Theorem 2.5.
4. Birman 1974.
5. Farb & Margalit 2012, Theorem 8.1.
6. Birman 1969, pp. 213–238.
7. Farb & Margalit 2012, Theorem 4.6.
8. Fathi, Laudenbach & Poénaru 2012, Chapter 9.
9. Eskin, Masur & Rafi. sfn error: no target: CITEREFEskinMasurRafi (help)
10. Fathi, Laudenbach & Poénaru 2012.
11. Invent. Math. 1999, pp. 103–149.
12. Brock 2002.
13. Masur & Minsky 2000.
14. Farb & Margalit 2012, Theorem 4.1.
15. Hatcher & Thurston 1980.
16. Topology 1996, pp. 377–383.
17. J. Algebra 2004.
18. Proc. Amer. Math. Soc. 2010, pp. 753–758.
19. Farb & Margalit 2012, Theorem 6.4.
20. Farb & Margalit 2012, Theorem 6.15 and Theorem 6.12.
21. Farb & Margalit 2012, Theorem 6.11.
22. Ivanov 1992, Theorem 4.
23. Ivanov 1992, Theorem 1.
24. Geom. Topol. 2012, pp. 1393–1411.
25. Duke Math. J. 2001, pp. 581–597.
Sources
• Birman, Joan (1969). "Mapping class groups and their relationship to braid groups". Comm. Pure Appl. Math. 22 (2): 213–238. doi:10.1002/cpa.3160220206. MR 0243519.
• Birman, Joan S. (1974). Braids, links, and mapping class groups. Annals of Mathematics Studies. Vol. 82. Princeton University Press.
• Brendle, Tara E.; Farb, Benson (2004). "Every mapping class group is generated by 3 torsion elements and by 6 involutions". J. Algebra. 278. arXiv:math/0307039. doi:10.1016/j.jalgebra.2004.02.019. S2CID 14784932.
• Brock, Jeff (2002). "Pants decompositions and the Weil–Petersson metric". Complex Manifolds and Hyperbolic Geometry. American Mathematical Society. MR 1940162.
• Dehn, Max (1938). "Die Gruppe der Abbildungsklassen: Das arithmetische Feld auf Flächen". Acta Math. (in German). 69: 135–206. doi:10.1007/bf02547712, translated in Dehn 1987.
• Dehn, Max (1987). Papers on group theory and topology. translated and introduced by John Stillwell. Springer-Verlag. ISBN 978-038796416-4.
• Eskin, Alex; Masur, Howard; Rafi, Kasra (2017). "Large-scale rank of Teichmüller space". Duke Mathematical Journal. 166 (8). arXiv:1307.3733. doi:10.1215/00127094-0000006X. S2CID 15393033.
• Farb, Benson; Lubotzky, Alexander; Minsky, Yair (2001). "Rank-1 phenomena for mapping class groups". Duke Math. J. 106 (3): 581–597. doi:10.1215/s0012-7094-01-10636-4. MR 1813237.
• Farb, Benson; Margalit, Dan (2012). A primer on mapping class groups. Princeton University press.
• Fathi, Albert; Laudenbach, François; Poénaru, Valentin (2012). Thurston's work on surfaces. Mathematical Notes. Vol. 48. translated from the 1979 French original by Djun M. Kim and Dan Margalit. Princeton University Press. pp. xvi+254. ISBN 978-0-691-14735-2.
• Hatcher, Allen; Thurston, William (1980). "A presentation for the mapping class group of a closed orientable surface". Topology. 19 (3): 221–237. doi:10.1016/0040-9383(80)90009-9.
• Ivanov, Nikolai (1992). Subgroups of Teichmüller Modular Groups. American Math. Soc.
• Masbaum, Gregor; Reid, Alan W. (2012). "All finite groups are involved in the mapping class group". Geom. Topol. 16 (3): 1393–1411. arXiv:1106.4261. doi:10.2140/gt.2012.16.1393. MR 2967055. S2CID 17330187.
• Masur, Howard A.; Minsky, Yair N. (1999). "Geometry of the complex of curves. I. Hyperbolicity". Invent. Math. 138: 103–149. arXiv:math/9804098. Bibcode:1999InMat.138..103M. doi:10.1007/s002220050343. MR 1714338. S2CID 16199015.
• Masur, Howard A.; Minsky, Yair N. (2000). "Geometry of the complex of curves II: Hierarchical structure". Geom. Funct. Anal. 10 (4): 902–974. arXiv:math/9807150. doi:10.1007/pl00001643. S2CID 14834205.
• Putman, Andy (2010). "A note on the abelianizations of finite-index subgroups of the mapping class group". Proc. Amer. Math. Soc. 138 (2): 753–758. arXiv:0812.0017. doi:10.1090/s0002-9939-09-10124-7. MR 2557192. S2CID 2047111.
• Thurston, William P. (1988). "On the geometry and dynamics of diffeomorphisms of surfaces". Bull. Amer. Math. Soc. 19 (2): 417–431. doi:10.1090/s0273-0979-1988-15685-6. MR 0956596.
• Wajnryb, B. (1996). "Mapping class group of a surface is generated by two elements". Topology. 35 (2): 377–383. doi:10.1016/0040-9383(95)00037-2.
| Wikipedia |
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors $W(\mathbb {F} _{p})$ over the finite field of order $p$ is isomorphic to $\mathbb {Z} _{p}$, the ring of $p$-adic integers. They have a highly non-intuitive structure[1] upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers.
The main idea[1] behind Witt vectors is instead of using the standard $p$-adic expansion
$a=a_{0}+a_{1}p+a_{2}p^{2}+\cdots $
to represent an element in $\mathbb {Z} _{p}$, we can instead consider an expansion using the Teichmüller character
$\omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}$ :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}}
which sends each element in the solution set of $x^{p-1}-1$ in $\mathbb {F} _{p}$ to an element in the solution set of $x^{p-1}-1$ in $\mathbb {Z} _{p}$. That is, we expand out elements in $\mathbb {Z} _{p}$ in terms of roots of unity instead of as profinite elements in $\prod \mathbb {F} _{p}$. We can then express a $p$-adic integer as an infinite sum
$\omega (a)=\omega (a_{0})+\omega (a_{1})p+\omega (a_{2})p^{2}+\cdots $
which gives a Witt vector
$(\omega (a_{0}),\omega (a_{1}),\omega (a_{2}),\ldots )$
Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give $W(\mathbb {F} _{p})$ an additive and multiplicative structure such that $\omega $ induces a commutative ring morphism.
History
In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject now known as Kummer theory. Let $k$ be a field containing a primitive $n$-th root of unity. Kummer theory classifies degree $n$ cyclic field extensions $K$ of $k$. Such fields are in bijection with order $n$ cyclic groups $\Delta \subseteq k^{\times }/(k^{\times })^{n}$, where $\Delta $ corresponds to $K=k({\sqrt[{n}]{\Delta }})$.
But suppose that $k$ has characteristic $p$. The problem of studying degree $p$ extensions of $k$, or more generally degree $p^{n}$ extensions, may appear superficially similar to Kummer theory. However, in this situation, $k$ cannot contain a primitive $p$-th root of unity. Indeed, if $x$ is a $p$-th root of unity in $k$, then it satisfies $x^{p}=1$. But consider the expression $(x-1)^{p}=0$. By expanding using binomial coefficients we see that the operation of raising to the $p$-th power, known here as the Frobenius homomorphism, introduces the factor $p$ to every coefficient except the first and the last, and so modulo $p$ these equations are the same. Therefore $x=1$. Consequently, Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.
The case where the characteristic divides the degree is today called Artin–Schreier theory because the first progress was made by Artin and Schreier. Their initial motivation was the Artin–Schreier theorem, which characterizes the real closed fields as those whose absolute Galois group has order two.[2] This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving that no other such fields exist, they proved that degree $p$ extensions of a field $k$ of characteristic $p$ were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form $x^{p}-x-a.$ By repeating their construction, they described degree $p^{2}$ extensions. Abraham Adrian Albert used this idea to describe degree $p^{n}$ extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.[3]
Schmid[4] generalized further to non-commutative cyclic algebras of degree $p^{n}$. In the process of doing so, certain polynomials related to the addition of $p$-adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree $p^{n}$ field extensions and cyclic algebras. Specifically, he introduced a ring now called $W_{n}(k)$, the ring of $n$-truncated $p$-typical Witt vectors. This ring has $k$ as a quotient, and it comes with an operator $F$ which is called the Frobenius operator because it reduces to the Frobenius operator on $k$. Witt observes that the degree $p^{n}$ analog of Artin–Schreier polynomials is
$F(x)-x-a,$
where $a\in W_{n}(k)$. To complete the analogy with Kummer theory, define $\wp $ to be the operator $x\mapsto F(x)-x.$ Then the degree $p^{n}$ extensions of $k$ are in bijective correspondence with cyclic subgroups $\Delta \subseteq W_{n}(k)/\wp (W_{n}(k))$ of order $p^{n}$, where $\Delta $ corresponds to the field $k(\wp ^{-1}(\Delta ))$.
Motivation
Any $p$-adic integer (an element of $\mathbb {Z} _{p}$, not to be confused with $\mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}$) can be written as a power series $a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots $, where the $a_{i}$ are usually taken from the integer interval $[0,p-1]=\{0,1,2,\ldots ,p-1\}$. It is hard to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits. However, taking representative coefficients $a_{i}\in [0,p-1]$ is only one of many choices, and Hensel himself (the creator of $p$-adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number $0$ together with the $(p-1)^{\text{th}}$ roots of unity; that is, the solutions of $x^{p}-x=0$ in $\mathbb {Z} _{p}$, so that $a_{i}=a_{i}^{p}$. This choice extends naturally to ring extensions of $\mathbb {Z} _{p}$ in which the residue field is enlarged to $\mathbb {F} _{q}$ with $q=p^{f}$, some power of $p$. Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the $q$ solutions in the field to $x^{q}-x=0$. Call the field $\mathbb {Z} _{p}(\eta )$, with $\eta $ an appropriate primitive $(q-1)^{\text{th}}$ root of unity (over $\mathbb {Z} _{p}$). The representatives are then $0$ and $\eta ^{i}$ for $0\leq i\leq q-2$. Since these representatives form a multiplicative set they can be thought of as characters. Some thirty years after Hensel's works Teichmüller studied these characters, which now bear his name, and this led him to a characterisation of the structure of the whole field in terms of the residue field. These Teichmüller representatives can be identified with the elements of the finite field $\mathbb {F} _{q}$ of order $q$ by taking residues modulo $p$ in $\mathbb {Z} _{p}(\eta )$, and elements of $\mathbb {F} _{q}^{\times }$ are taken to their representatives by the Teichmüller character $\omega :\mathbb {F} _{q}^{\times }\to \mathbb {Z} _{p}(\eta )^{\times }$ :\mathbb {F} _{q}^{\times }\to \mathbb {Z} _{p}(\eta )^{\times }} . This operation identifies the set of integers in $\mathbb {Z} _{p}(\eta )$ with infinite sequences of elements of $\omega (\mathbb {F} _{q}^{\times })\cup \{0\}$.
Taking those representatives the expressions for addition and multiplication can be written in closed form. We now have the following problem (stated for the simplest case: $q=p$): given two infinite sequences of elements of $\omega (\mathbb {F} _{p}^{\times })\cup \{0\},$ describe their sum and product as $p$-adic integers explicitly. This problem was solved by Witt using Witt vectors.
Detailed motivational sketch
We derive the ring of $p$-adic integers $\mathbb {Z} _{p}$ from the finite field $\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} $ using a construction which naturally generalizes to the Witt vector construction.
The ring $\mathbb {Z} _{p}$ of $p$-adic integers can be understood as the inverse limit of the rings $\mathbb {Z} /p^{i}\mathbb {Z} $ taken along the obvious projections. Specifically, it consists of the sequences $(n_{0},n_{1},\ldots )$ with $n_{i}\in \mathbb {Z} /p^{i+1}\mathbb {Z} ,$ such that $n_{j}\equiv n_{i}{\bmod {p}}^{i+1}$ for $j\geq i.$ That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections $\mathbb {Z} /p^{i+1}\mathbb {Z} \to \mathbb {Z} /p^{i}\mathbb {Z} .$
The elements of $\mathbb {Z} _{p}$ can be expanded as (formal) power series in $p$
$a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots ,$
where the coefficients $a_{i}$ are taken from the integer interval $[0,p-1]=\{0,1,\ldots ,p-1\}.$ Of course, this power series usually will not converge in $\mathbb {R} $ using the standard metric on the reals, but it will converge in $\mathbb {Z} _{p},$ with the $p$-adic metric. We will sketch a method of defining ring operations for such power series.
Letting $a+b$ be denoted by $c$, one might consider the following definition for addition:
${\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}+c_{1}p&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p&&{\bmod {p}}^{2}\\c_{0}+c_{1}p+c_{2}p^{2}&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p+(a_{2}+b_{2})p^{2}&&{\bmod {p}}^{3}\end{aligned}}$
and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set $[0,p-1].$
Representing elements in Fp as elements in the ring of Witt vectors W(Fp)
There is a better coefficient subset of $\mathbb {Z} _{p}$ which does yield closed formulas, the Teichmüller representatives: zero together with the $(p-1)^{\text{th}}$ roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives $[0,p-1]$) as roots of $x^{p-1}-1=0$ through Hensel lifting, the $p$-adic version of Newton's method. For example, in $\mathbb {Z} _{5},$ to calculate the representative of $2,$ one starts by finding the unique solution of $x^{4}-1=0$ in $\mathbb {Z} /25\mathbb {Z} $ with $x\equiv 2{\bmod {5}}$; one gets $7.$ Repeating this in $\mathbb {Z} /125\mathbb {Z} ,$ with the conditions $x^{4}-1=0$ and $x\equiv 7{\bmod {2}}5$, gives $57,$ and so on; the resulting Teichmüller representative of $2$, denoted $\omega (2)$, is the sequence
$\omega (2)=(2,7,57,\ldots )\in W(\mathbb {F} _{5}).$
The existence of a lift in each step is guaranteed by the greatest common divisor $(x^{p-1}-1,(p-1)x^{p-2})=1$ in every $\mathbb {Z} /p^{n}\mathbb {Z} .$
This algorithm shows that for every $j\in [0,p-1]$, there is exactly one Teichmüller representative with $a_{0}=j$, which we denote $\omega (j).$ Indeed, this defines the Teichmüller character $\omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}$ :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}} as a (multiplicative) group homomorphism, which moreover satisfies $m\circ \omega =\mathrm {id} _{\mathbb {F} _{p}}$ if we let $m:\mathbb {Z} _{p}\to \mathbb {Z} _{p}/p\mathbb {Z} _{p}\cong \mathbb {F} _{p}$ denote the canonical projection. Note however that $\omega $ is not additive, as the sum need not be a representative. Despite this, if $\omega (k)\equiv \omega (i)+\omega (j){\bmod {p}}$ in $\mathbb {Z} _{p},$ then $i+j=k$ in $\mathbb {F} _{p}.$
Representing elements in Zp as elements in the ring of Witt vectors W(Fp)
Because of this one-to-one correspondence given by $\omega $, one can expand every $p$-adic integer as a power series in $p$ with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as $\omega (t_{0})=t_{0}+t_{1}p^{1}+t_{2}p^{2}+\cdots .$ Then, if one has some arbitrary $p$-adic integer of the form $x=x_{0}+x_{1}p^{1}+x_{2}p^{2}+\cdots ,$ one takes the difference $x-\omega (x_{0})=x'_{1}p^{1}+x'_{2}p^{2}+\cdots ,$ leaving a value divisible by $p$. Hence, $x-\omega (x_{0})=0{\bmod {p}}$. The process is then repeated, subtracting $\omega (x'_{1})p$ and proceed likewise. This yields a sequence of congruences
${\begin{aligned}x&\equiv \omega (x_{0})&&{\bmod {p}}\\x&\equiv \omega (x_{0})+\omega (x'_{1})p&&{\bmod {p}}^{2}\\&\cdots \end{aligned}}$
So that
$x\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}$
and $i'>i$ implies:
$\sum _{j=0}^{i'}\omega ({\bar {x}}_{j})p^{j}\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}$
for
${\bar {x}}_{i}:=m\left({\frac {x-\sum _{j=0}^{i-1}\omega ({\bar {x}}_{j})p^{j}}{p^{i}}}\right).$
Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than $\{0,\ldots ,p-1\}$. It is clear that
$\sum _{j=0}^{\infty }\omega ({\bar {x}}_{j})p^{j}=x,$
since
$p^{i+1}\mid x-\sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}$
for all $i$ as $i\to \infty ,$ so the difference tends to 0 with respect to the $p$-adic metric. The resulting coefficients will typically differ from the $a_{i}$ modulo $p^{i}$ except the first one.
Additional properties of elements in the ring of Witt vectors motivating general definition
The Teichmüller coefficients have the key additional property that $\omega ({\bar {x}}_{i})^{p}=\omega ({\bar {x}}_{i}),$ which is missing for the numbers in $[0,p-1]$. This can be used to describe addition, as follows. Consider the equation $ c=a+b$ in $ \mathbb {Z} _{p}$ and let the coefficients $ a_{i},b_{i},c_{i}\in \mathbb {Z} _{p}$ now be as in the Teichmüller expansion. Since the Teichmüller character is not additive, $c_{0}=a_{0}+b_{0}$ is not true in $\mathbb {Z} _{p}$. But it holds in $\mathbb {F} _{p},$ as the first congruence implies. In particular,
$c_{0}^{p}\equiv (a_{0}+b_{0})^{p}{\bmod {p}}^{2},$
and thus
$c_{0}-a_{0}-b_{0}\equiv (a_{0}+b_{0})^{p}-a_{0}-b_{0}\equiv {\binom {p}{1}}a_{0}^{p-1}b_{0}+\cdots +{\binom {p}{p-1}}a_{0}b_{0}^{p-1}{\bmod {p}}^{2}.$
Since the binomial coefficient ${\binom {p}{i}}$ is divisible by $p$, this gives
$c_{1}\equiv a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}{\bmod {p}}.$
This completely determines $c_{1}$ by the lift. Moreover, the congruence modulo $p$ indicates that the calculation can actually be done in $\mathbb {F} _{p},$ satisfying the basic aim of defining a simple additive structure.
For $c_{2}$ this step is already very cumbersome. Write
$c_{1}=c_{1}^{p}\equiv \left(a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}\right)^{p}{\bmod {p}}^{2}.$
Just as for $c_{0},$ a single $p$th power is not enough: one must take
$c_{0}=c_{0}^{p^{2}}\equiv (a_{0}+b_{0})^{p^{2}}{\bmod {p}}^{3}.$
However, ${\binom {p^{2}}{i}}$ is not in general divisible by $p^{2},$ but it is divisible when $i=pd,$ in which case $a^{i}b^{p^{2}-i}=a^{d}b^{p-d}$ combined with similar monomials in $c_{1}^{p}$ will make a multiple of $p^{2}$.
At this step, it becomes clear that one is actually working with addition of the form
${\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}^{p}+c_{1}p&\equiv a_{0}^{p}+a_{1}p+b_{0}^{p}+b_{1}p&&{\bmod {p}}^{2}\\c_{0}^{p^{2}}+c_{1}^{p}p+c_{2}p^{2}&\equiv a_{0}^{p^{2}}+a_{1}^{p}p+a_{2}p^{2}+b_{0}^{p^{2}}+b_{1}^{p}p+b_{2}p^{2}&&{\bmod {p}}^{3}\end{aligned}}$
This motivates the definition of Witt vectors.
Construction of Witt rings
Fix a prime number p. A Witt vector[5] over a commutative ring $R$ (relative to the prime $p$) is a sequence $(X_{0},X_{1},X_{2},\ldots )$ of elements of $R$. Define the Witt polynomials $W_{i}$ by
1. $W_{0}=X_{0}$
2. $W_{1}=X_{0}^{p}+pX_{1}$
3. $W_{2}=X_{0}^{p^{2}}+pX_{1}^{p}+p^{2}X_{2}$
and in general
$W_{n}=\sum _{i=0}^{n}p^{i}X_{i}^{p^{n-i}}.$
The $W_{n}$ are called the ghost components of the Witt vector $(X_{0},X_{1},X_{2},\ldots )$, and are usually denoted by $X^{(n)}$; taken together, the $W_{n}$ define the ghost map to $ \prod _{i=0}^{\infty }R$. If $ R$ is p-torsionfree, then the ghost map is injective and the ghost components can be thought of as an alternative coordinate system for the $R$-module of sequences (though note that the ghost map is not surjective unless $ R$ is p-divisible).
The ring of (p-typical) Witt vectors $W(R)$ is defined by componentwise addition and multiplication of the ghost components. That is, that there is a unique way to make the set of Witt vectors over any commutative ring $R$ into a ring such that:
1. the sum and product are given by polynomials with integral coefficients that do not depend on $R$, and
2. projection to each ghost component is a ring homomorphism from the Witt vectors over $R$, to $R$.
In other words,
• $(X+Y)_{i}$ and $(XY)_{i}$ are given by polynomials with integral coefficients that do not depend on R, and
• $X^{(i)}+Y^{(i)}=(X+Y)^{(i)}$ and $X^{(i)}Y^{(i)}=(XY)^{(i)}.$
The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,
$(X_{0},X_{1},\ldots )+(Y_{0},Y_{1},\ldots )=(X_{0}+Y_{0},X_{1}+Y_{1}-((X_{0}+Y_{0})^{p}-X_{0}^{p}-Y_{0}^{p})/p,\ldots )$
$(X_{0},X_{1},\ldots )\times (Y_{0},Y_{1},\ldots )=(X_{0}Y_{0},X_{0}^{p}Y_{1}+X_{1}Y_{0}^{p}+pX_{1}Y_{1},\ldots )$
These are to be understood as shortcuts for the actual formulas. If for example the ring $R$ has characteristic $p$, the division by $p$ in the first formula above, the one by $p^{2}$ that would appear in the next component and so forth, do not make sense. However, if the $p$-power of the sum is developed, the terms $X_{0}^{p}+Y_{0}^{p}$ are cancelled with the previous ones and the remaining ones are simplified by $p$, no division by $p$ remains and the formula makes sense. The same consideration applies to the ensuing components.
Examples of addition and multiplication
As would be expected, the unit in the ring of Witt vectors $W(A)$ is the element
${\underline {1}}=(1,0,0,\ldots )$
Adding this element to itself gives a non-trivial sequence, for example in $W(\mathbb {F} _{5})$,
${\underline {1}}+{\underline {1}}=(2,4,\ldots )$
since
${\begin{aligned}2&=1+1\\4&=-{\frac {32-1-1}{5}}\mod 5\\&\cdots \end{aligned}}$
which is not the expected behavior, since it doesn't equal ${\underline {2}}$. But, when we reduce with the map $m:W(\mathbb {F} _{5})\to \mathbb {F} _{5}$, we get $m(\omega (1)+\omega (1))=m(\omega (2))$. Note if we have an element $x\in A$ and an element $a\in W(A)$ then
${\underline {x}}a=(xa_{0},x^{p}a_{1},\ldots ,x^{p^{n}}a_{n},\ldots )$
showing multiplication also behaves in a highly non-trivial manner.
Examples
• The Witt ring of any commutative ring $R$ in which $p$ is invertible is just isomorphic to $R^{\mathbb {N} }$ (the product of a countable number of copies of $R$). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to $R^{\mathbb {N} }$, and if $p$ is invertible this homomorphism is an isomorphism.
• The Witt ring $W(\mathbb {F} _{p})\cong \mathbb {Z} _{p}$ of the finite field of order $p$ is the ring of $p$-adic integers written in terms of the Teichmüller representatives, as demonstrated above.
• The Witt ring $W(\mathbb {F} _{q})\cong {\mathcal {O}}_{K}$ of a finite field of order $p^{n}$ is the ring of integers of the unique unramified extension of degree $n$ of the ring of $p$-adic numbers $K/\mathbb {Q} _{p}$. Note $K\cong \mathbb {Q} _{p}(\mu _{q-1})$ for $\mu _{q-1}$ the $(q-1)$-th root of unity, hence $W(\mathbb {F} _{q})\cong \mathbb {Z} _{p}[\mu _{q-1}]$.
Universal Witt vectors
The Witt polynomials for different primes $p$ are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime $p$). Define the universal Witt polynomials $W_{n}$ for $n\geq 1$ by
1. $W_{1}=X_{1}$
2. $W_{2}=X_{1}^{2}+2X_{2}$
3. $W_{3}=X_{1}^{3}+3X_{3}$
4. $W_{4}=X_{1}^{4}+2X_{2}^{2}+4X_{4}$
and in general
$W_{n}=\sum _{d|n}dX_{d}^{n/d}.$
Again, $(W_{1},W_{2},W_{3},\ldots )$ is called the vector of ghost components of the Witt vector $(X_{1},X_{2},X_{3},\ldots )$, and is usually denoted by $(X^{(1)},X^{(2)},X^{(3)},\ldots )$.
We can use these polynomials to define the ring of universal Witt vectors or big Witt ring of any commutative ring $R$ in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring $R$).
Generating Functions
Witt also provided another approach using generating functions.[6]
Definition
Let $X$ be a Witt vector and define
$f_{X}(t)=\prod _{n\geq 1}(1-X_{n}t^{n})=\sum _{n\geq 0}A_{n}t^{n}$
For $n\geq 1$ let ${\mathcal {I}}_{n}$ denote the collection of subsets of $\{1,2,\ldots ,n\}$ whose elements add up to $n$. Then
$A_{n}=\sum _{I\in {\mathcal {I}}_{n}}(-1)^{|I|}\prod _{i\in I}{X_{i}}.$
We can get the ghost components by taking the logarithmic derivative:
${\begin{aligned}-t{\frac {d}{dt}}\log f_{X}(t)&=-t{\frac {d}{dt}}\sum _{n\geq 1}\log(1-X_{n}t^{n})\\&=t{\frac {d}{dt}}\sum _{n\geq 1}\sum _{d\geq 1}{\frac {X_{n}^{d}t^{nd}}{d}}\\&=\sum _{n\geq 1}\sum _{d\geq 1}nX_{n}^{d}t^{nd}\\&=\sum _{m\geq 1}\sum _{d|m}dX_{d}^{m/d}t^{m}\\&=\sum _{m\geq 1}X^{(m)}t^{m}\end{aligned}}$
Sum
Now we can see $f_{Z}(t)=f_{X}(t)f_{Y}(t)$ if $Z=X+Y$. So that
$C_{n}=\sum _{0\leq i\leq n}A_{n}B_{n-i},$
if $A_{n},B_{n},C_{n}$ are the respective coefficients in the power series $f_{X}(t),f_{Y}(t),f_{Z}(t)$. Then
$Z_{n}=\sum _{0\leq i\leq n}A_{n}B_{n-i}-\sum _{I\in {\mathcal {I}}_{n},I\neq \{n\}}(-1)^{|I|}\prod _{i\in I}{Z_{i}}.$
Since $A_{n}$ is a polynomial in $X_{1},\ldots ,X_{n}$ and likewise for $B_{n}$, we can show by induction that $Z_{n}$ is a polynomial in $X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.$
Product
If we set $W=XY$ then
$-t{\frac {d}{dt}}\log f_{W}(t)=-\sum _{m\geq 1}X^{(m)}Y^{(m)}t^{m}.$
But
$\sum _{m\geq 1}X^{(m)}Y^{(m)}t^{m}=\sum _{m\geq 1}\sum _{d|m}dX_{d}^{m/d}\sum _{e|m}eY_{e}^{m/e}t^{m}$.
Now 3-tuples ${m,d,e}$ with $m\in \mathbb {Z} ^{+},d|m,e|m$ are in bijection with 3-tuples ${d,e,n}$ with $d,e,n\in \mathbb {Z} ^{+}$, via $n=m/[d,e]$ ($[d,e]$ is the least common multiple), our series becomes
$\sum _{d,e\geq 1}de\sum _{n\geq 1}\left(X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{n}=-t{\frac {d}{dt}}\log \prod _{d,e\geq 1}\left(1-X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{\frac {de}{[d,e]}}$
So that
$f_{W}(t)=\prod _{d,e\geq 1}\left(1-X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{\frac {de}{[d,e]}}=\sum _{n\geq 0}D_{n}t^{n},$
where $D_{n}$ are polynomials of $X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.$ So by similar induction, suppose
$f_{W}(t)=\prod _{n\geq 1}(1-W_{n}t^{n}),$
then $W_{n}$ can be solved as polynomials of $X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.$
Ring schemes
The map taking a commutative ring $R$ to the ring of Witt vectors over $R$ (for a fixed prime $p$) is a functor from commutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, called the Witt scheme, over $\operatorname {Spec} (\mathbb {Z} ).$ The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions.
Similarly, the rings of truncated Witt vectors, and the rings of universal Witt vectors correspond to ring schemes, called the truncated Witt schemes and the universal Witt scheme.
Moreover, the functor taking the commutative ring $R$ to the set $R^{n}$ is represented by the affine space $\mathbb {A} _{\mathbb {Z} }^{n}$, and the ring structure on $R^{n}$ makes $\mathbb {A} _{\mathbb {Z} }^{n}$ into a ring scheme denoted ${\underline {\mathcal {O}}}^{n}$. From the construction of truncated Witt vectors, it follows that their associated ring scheme $\mathbb {W} _{n}$ is the scheme $\mathbb {A} _{\mathbb {Z} }^{n}$ with the unique ring structure such that the morphism $\mathbb {W} _{n}\to {\underline {\mathcal {O}}}^{n}$ given by the Witt polynomials is a morphism of ring schemes.
Commutative unipotent algebraic groups
Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group $G_{a}$. The analogue of this for fields of characteristic $p$ is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic $p$, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.
Universal property
André Joyal explicated the universal property of the (p-typical) Witt vectors.[7] The basic intuition is that the formation of Witt vectors is the universal way to deform a characteristic $p$ ring to characteristic 0 together with a lift of its Frobenius endomorphism.[8] To make this precise, define a $\delta $-ring $ (R,\delta )$ to consist of a commutative ring $ R$ together with a map of sets $ \delta :R\to R$ that is a $p$-derivation, so that $ \delta $ satisfies the relations
• $\delta (0)=\delta (1)=0$;
• $\delta (xy)=x^{p}\delta (y)+y^{p}\delta (x)+p\delta (x)\delta (y)$;
• $\delta (x+y)=\delta (x)+\delta (y)+{\frac {x^{p}+y^{p}-(x+y)^{p}}{p}}$.
The definition is such that given a $\delta $-ring $ (R,\delta )$, if one defines the map $ \phi :R\to R$ by the formula $ \phi (x)=x^{p}+p\delta (x)$, then $ \phi $ is a ring homomorphism lifting Frobenius on $R/p$. Conversely, if $ R$ is $p$-torsionfree, then this formula uniquely defines the structure of a $\delta $-ring on $ R$ from that of a Frobenius lift. One may thus regard the notion of $\delta $-ring as a suitable replacement for a Frobenius lift in the non $p$-torsionfree case.
The collection of $\delta $-rings and ring homomorphisms thereof respecting the $\delta $-structure assembles to a category $ \mathrm {CRing} _{\delta }$. One then has a forgetful functor
$U:\mathrm {CRing} _{\delta }\to \mathrm {CRing} $
whose right adjoint identifies with the functor $ W$ of Witt vectors. In fact, the functor $ U$ creates limits and colimits and admits an explicitly describable left adjoint as a type of free functor; from this, it is not hard to show that $ \mathrm {CRing} _{\delta }$ inherits local presentability from $\mathrm {CRing} $ so that one can construct the functor $ W$ by appealing to the adjoint functor theorem.
One further has that $ W$ restricts to a fully faithful functor on the full subcategory of perfect rings of characteristic p. Its essential image then consists of those $\delta $-rings that are perfect (in the sense that the associated map $ \phi $ is an isomorphism) and whose underlying ring is $p$-adically complete.[9]
See also
• p-derivation
• Formal group
• Artin–Hasse exponential
• Necklace ring
References
1. Fisher, Benji (1999). "Notes on Witt Vectors: a motivated approach" (PDF). Archived (PDF) from the original on 12 January 2019.
2. Artin, Emil and Schreier, Otto, Über eine Kennzeichnung der reell abgeschlossenen Körper, Abh. Math. Sem. Hamburg 3 (1924).
3. A. A. Albert, Cyclic fields of degree $p^{n}$ over $F$ of characteristic $p$, Bull. Amer. Math. Soc. 40 (1934).
4. Schmid, H. L., Zyklische algebraische Funktionenkörper vom Grad pn über endlichen Konstantenkörper der Charakteristik p, Crelle 175 (1936).
5. Illusie, Luc (1979). "Complexe de de Rham-Witt et cohomologie cristalline". Annales scientifiques de l'École Normale Supérieure (in French). 12 (4): 501–661. doi:10.24033/asens.1374.
6. Lang, Serge (September 19, 2005). "Chapter VI: Galois Theory". Algebra (3rd ed.). Springer. pp. 330. ISBN 978-0-387-95385-4.
7. Joyal, André (1985). "δ-anneaux et vecteurs de Witt". C.R. Math. Rep. Acad. Sci. Canada. 7 (3): 177–182.
8. "Is there a universal property for Witt vectors?". MathOverflow. Retrieved 2022-09-06.
9. Bhatt, Bhargav (October 8, 2018). "Lecture II: Delta rings" (PDF). Archived (PDF) from the original on September 6, 2022.
Introductory
• Notes on Witt vectors: a motivated approach - Basic notes giving the main ideas and intuition. Best to start here!
• The Theory of Witt Vectors - Elementary introduction to the theory.
• Complexe de de Rham-Witt et cohomologie cristalline - Note he uses a different but equivalent convention as in this article. Also, the main points in the introduction are still valid.
Applications
• Mumford, David (1966-08-21), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, vol. 59, Princeton, NJ: Princeton University Press, ISBN 978-0-691-07993-6
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0554237, section II.6
• Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1035-1, ISBN 978-0-387-96648-9, MR 0918564
• Greenberg, Marvin J. (1969). Lectures on Forms in Many Variables. New York and Amsterdam: Benjamin. ASIN B0006BX17M. MR 0241358.
References
• Dolgachev, Igor V. (2001) [1994], "Witt vector", Encyclopedia of Mathematics, EMS Press
• Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, doi:10.1016/S1570-7954(08)00207-6, ISBN 978-0-444-53257-2, MR 2553661
• Witt, Ernst (1936), "Zyklische Körper und Algebren der Characteristik p vom Grad pn. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pn", Journal für die Reine und Angewandte Mathematik (in German), 1937 (176): 126–140, doi:10.1515/crll.1937.176.126
| Wikipedia |
Teichmüller character
In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where $q=p$ if $p$ is odd and $q=4$ if $p=2$, taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller. Identifying the roots of unity in the p-adic integers with the corresponding ones in the complex numbers, ω can be considered as a usual Dirichlet character of conductor q. More generally, given a complete discrete valuation ring O whose residue field k is perfect of characteristic p, there is a unique multiplicative section ω : k → O of the natural surjection O → k. The image of an element under this map is called its Teichmüller representative. The restriction of ω to k× is called the Teichmüller character.
Definition
If x is a p-adic integer, then $\omega (x)$ is the unique solution of $\omega (x)^{p}=\omega (x)$ that is congruent to x mod p. It can also be defined by
$\omega (x)=\lim _{n\rightarrow \infty }x^{p^{n}}$
The multiplicative group of p-adic units is a product of the finite group of roots of unity and a group isomorphic to the p-adic integers. The finite group is cyclic of order p – 1 or 2, as p is odd or even, respectively, and so it is isomorphic to (Z/qZ)×. The Teichmüller character gives a canonical isomorphism between these two groups.
A detailed exposition of the construction of Teichmüller representatives for the p-adic integers, by means of Hensel lifting, is given in the article on Witt vectors, where they provide an important role in providing a ring structure.
See also
• Witt vector
References
• Section 4.3 of Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337
• Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96017-3, MR 0754003
| Wikipedia |
Monique Teillaud
Monique Teillaud is a French researcher in computational geometry at the French Institute for Research in Computer Science and Automation (INRIA) in Nancy, France. She moved to Nancy in 2014 from a different INRIA center in Sophia Antipolis,[1] where she was one of the developers of CGAL, a software library of computational geometry algorithms.[2]
Monique Teillaud
Born (1961-06-14) 14 June 1961
Paris
NationalityFrench
Alma materÉcole normale supérieure de jeunes filles
Scientific career
FieldsComputer science
InstitutionsInria, École nationale supérieure d'informatique pour l'industrie et l'entreprise
Doctoral advisorJean-Daniel Boissonnat
Teillaud graduated from the École Normale Supérieure de Jeunes Filles in 1985, she then got a position at École nationale supérieure d'informatique pour l'industrie et l'entreprise before moving to Inria in 1989. She completed her Ph.D. in 1991 at Paris-Sud University under the supervision of Jean-Daniel Boissonnat.[3] She was the 2008 program chair of the Symposium on Computational Geometry.[4] She is also the author or editor of two books in computational geometry:
• Towards Dynamic Randomized Algorithms in Computational Geometry (Lecture Notes in Computer Science 758, Springer, 1993)[5]
• Effective Computational Geometry for Curves and Surfaces (edited with Boissonat, Springer, 2007)
References
1. Blanchard, Marie (August 2015), Portrait of Monique Teillaud
2. Monique Teillaud, INRIA, retrieved 2018-05-25
3. Monique Teillaud at the Mathematics Genealogy Project
4. SoCG program committees, retrieved 2018-05-25
5. Review of Towards Dynamic Randomized Algorithms in Computational Geometry: Gritzmann, Peter (1996), Mathematical Reviews, 758, MR 1290122{{citation}}: CS1 maint: untitled periodical (link)
External links
• Monique Teillaud publications indexed by Google Scholar
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| Wikipedia |
Teiresias algorithm
The Teiresias algorithm is a combinatorial algorithm for the discovery of rigid patterns (motifs) in biological sequences. It is named after the Greek prophet Teiresias and was created in 1997 by Isidore Rigoutsos and Aris Floratos.[1]
The problem of finding sequence similarities in the primary structure of related proteins or genes arises in the analysis of biological sequences. It can be shown that pattern discovery in its general form is NP-hard.[2] The Teiresias algorithm is based on the observation that if a pattern spans many positions and appears exactly k times in the input then all fragments (sub patterns) of the pattern have to appear at least k times in the input. The algorithm is able to produce all patterns that have a user-defined number of copies in the given input, and manages to be very efficient by avoiding the enumeration of the entire space. Finally, the algorithm reports motifs that are maximal in both length and composition.
A new implementation of the Teiresias algorithm was recently made available by the Computational Medicine Center at Thomas Jefferson University. Teiresias is also accessible through an interactive web-based user interface by the same center. See external links for both.
Pattern description
The Teiresias algorithm uses regular expressions to define the patterns. This allows the patterns reported to consist not only from the characters that appear in each position (literals) but from a specific group of characters (bracketed literals) or even from any character (wild card). The patterns created by the algorithm are <L,W> patterns that have at least k instances in the input, where L ≤ W and L, W, k positive integers. A pattern is called an <L,W> pattern if and only if any L consecutive literals or bracketed literals span at most W positions (i.e. there can be no more than W-L wild cards).
The algorithm reports only maximal patterns. Given a set of sequences S, a pattern P that appears k times in S is called maximal if and only if there exists no pattern P' which is more specific than P and also appears exactly k times in S. If there exists such a pattern P' then we say that P cannot be maximal and P is considered to be subsumed by P'. A pattern P' is said to be more specific than a pattern P if and only if P' can be obtained from P by (a) dereferencing a wild card or (b) instantiating a bracketed literal to a literal, or (c) appending a string of literals, bracketed literals or/and wild cards to the right of P, or (d) prepending a string of literals, bracketed literals or/and wild cards to the left of P.[3]
Algorithm description
Teiresias consists of two phases, Scanning and Convolution. During the first phase the input is scanned for the patterns that satisfy the minimum requirements, the elementary patterns. The elementary patterns consist of exactly L literals and/or bracketed literals and includes at most W-L wild cards. During convolution, the elementary patterns are recursively combined and maximal patterns are created. The order in which the convolutions are performed is very important since it guarantees that all patterns will be generated and all maximal patterns are generated before all the patterns that are subsumed by them. The order is dictated by the following rules
• The priority of each pattern is defined by its contents from left to right.
• A literal has higher priority than a bracketed literal and both have higher priority than wild cards (the more specific first).
• Longer patterns have higher priority than shorter ones.
• Ties are resolved alphabetically.
Given the assurance that all maximal patterns will be created first, the maximality of a newly created pattern can be easily checked. If the new pattern is subsumed by any maximal patterns found so far, it is discarded. Otherwise, a new maximal pattern is found. To avoid checking against all maximal patterns found so far, maximal patterns are placed in a hash map. The hashing scheme is designed so that a maximal pattern P and all non maximal patterns subsumed by P hash to the same entry, but two different maximal patterns are unlikely to hash to the same entry. The algorithm terminates when no more patterns can be combined to form new maximal patterns. The length of any maximal pattern is bounded from above by the length of the longest input sequence.
Time complexity
The algorithm is "output-sensitive." The time complexity of the TEIRESIAS algorithm is[3]
$O\left(W^{L}m\log m+W(Cm+t_{H})\sum _{P_{max}}{rc(P)}\right)$
where L and W are user-specified parameters that define the "minimum density" of a pattern (any L literals or brackets cannot span more than W positions), m is the number of characters the input includes, C is the average number of patterns found in a hash entry, tH is the time needed for locating the hash entry corresponding to any given hash value, rc(P) is number of literals in P, and it can be shown at most rc(P) patterns can be placed on the stack while building P. And the summation Σ is the maximum number of patterns that will ever be placed in the stack that keeps the patterns for extension during convolution.
External links
• A C++ based implementation of the algorithm can be found here.
• The interactive web-based user interface of Teiresias can be found here.
References
1. Rigoutsos, I, Floratos, A (1998) Combinatorial pattern discovery in biological sequences: The TEIRESIAS algorithm. Bioinformatics 14: 55-67
2. Maier, D., "The Complexity of Some Problems on Subsequences and Supersequences", Journal of the ACM, 322-336, 1978
3. Floratos A., and Rigoutsos, I., "On the time complexity of the Teiresias algorithm", IBM technical report RC 21161 (94582), IBM TJ Watson Research Center, 1998
| Wikipedia |
Telephone number (mathematics)
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on n vertices, the number of permutations on n elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with n cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated,[1] giving the values (starting from n = 0)
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence A000085 in the OEIS).
Applications
John Riordan provides the following explanation for these numbers: suppose that n people subscribe to a telephone service that can connect any two of them by a call, but cannot make a single call connecting more than two people. How many different patterns of connection are possible? For instance, with three subscribers, there are three ways of forming a single telephone call, and one additional pattern in which no calls are being made, for a total of four patterns.[2] For this reason, the numbers counting how many patterns are possible are sometimes called the telephone numbers.[3][4]
Every pattern of pairwise connections between n people defines an involution, a permutation of the people that is its own inverse. In this permutation, each two people who call each other are swapped, and the people not involved in calls remain fixed in place. Conversely, every possible involution has the form of a set of pairwise swaps of this type. Therefore, the telephone numbers also count involutions. The problem of counting involutions was the original combinatorial enumeration problem studied by Rothe in 1800[1] and these numbers have also been called involution numbers.[5][6]
In graph theory, a subset of the edges of a graph that touches each vertex at most once is called a matching. Counting the matchings of a given graph is important in chemical graph theory, where the graphs model molecules and the number of matchings is the Hosoya index. The largest possible Hosoya index of an n-vertex graph is given by the complete graphs, for which any pattern of pairwise connections is possible; thus, the Hosoya index of a complete graph on n vertices is the same as the nth telephone number.[7]
A Ferrers diagram is a geometric shape formed by a collection of n squares in the plane, grouped into a polyomino with a horizontal top edge, a vertical left edge, and a single monotonic chain of edges from top right to bottom left. A standard Young tableau is formed by placing the numbers from 1 to n into these squares in such a way that the numbers increase from left to right and from top to bottom throughout the tableau. According to the Robinson–Schensted correspondence, permutations correspond one-for-one with ordered pairs of standard Young tableaux. Inverting a permutation corresponds to swapping the two tableaux, and so the self-inverse permutations correspond to single tableaux, paired with themselves.[8] Thus, the telephone numbers also count the number of Young tableaux with n squares.[1] In representation theory, the Ferrers diagrams correspond to the irreducible representations of the symmetric group of permutations, and the Young tableaux with a given shape form a basis of the irreducible representation with that shape. Therefore, the telephone numbers give the sum of the degrees of the irreducible representations.[9]
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
A diagonally symmetric non-attacking placement of eight rooks on a chessboard
In the mathematics of chess, the telephone numbers count the number of ways to place n rooks on an n × n chessboard in such a way that no two rooks attack each other (the so-called eight rooks puzzle), and in such a way that the configuration of the rooks is symmetric under a diagonal reflection of the board. Via the Pólya enumeration theorem, these numbers form one of the key components of a formula for the overall number of "essentially different" configurations of n mutually non-attacking rooks, where two configurations are counted as essentially different if there is no symmetry of the board that takes one into the other.[10]
Mathematical properties
Recurrence
The telephone numbers satisfy the recurrence relation
$T(n)=T(n-1)+(n-1)T(n-2),$
first published in 1800 by Heinrich August Rothe, by which they may easily be calculated.[1] One way to explain this recurrence is to partition the T(n) connection patterns of the n subscribers to a telephone system into the patterns in which the first person is not calling anyone else, and the patterns in which the first person is making a call. There are T(n − 1) connection patterns in which the first person is disconnected, explaining the first term of the recurrence. If the first person is connected to someone, there are n − 1 choices for that person, and T(n − 2) patterns of connection for the remaining n − 2 people, explaining the second term of the recurrence.[11]
Summation formula and approximation
The telephone numbers may be expressed exactly as a summation
$T(n)=\sum _{k=0}^{\lfloor n/2\rfloor }{\binom {n}{2k}}(2k-1)!!=\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n!}{2^{k}(n-2k)!k!}}.$
In each term of the first sum, $k$ gives the number of matched pairs, the binomial coefficient ${\tbinom {n}{2k}}$ counts the number of ways of choosing the $2k$ elements to be matched, and the double factorial
$(2k-1)!!={\frac {(2k)!}{2^{k}\,k!}}$
is the product of the odd integers up to its argument and counts the number of ways of completely matching the 2k selected elements.[1][11] It follows from the summation formula and Stirling's approximation that, asymptotically,[1][11][12]
$T(n)\sim \left({\frac {n}{e}}\right)^{n/2}{\frac {e^{\sqrt {n}}}{(4e)^{1/4}}}\,.$
Generating function
The exponential generating function of the telephone numbers is[11][13]
$\sum _{n=0}^{\infty }{\frac {T(n)x^{n}}{n!}}=\exp \left({\frac {x^{2}}{2}}+x\right).$
In other words, the telephone numbers may be read off as the coefficients of the Taylor series of exp(x2/2 + x), and the nth telephone number is the value at zero of the nth derivative of this function. This function is closely related to the exponential generating function of the Hermite polynomials, which are the matching polynomials of the complete graphs.[13] The sum of absolute values of the coefficients of the nth (probabilist's) Hermite polynomial is the nth telephone number, and the telephone numbers can also be realized as certain special values of the Hermite polynomials:[5][13]
$T(n)={\frac {{\mathit {He}}_{n}(i)}{i^{n}}}.$
Prime factors
For large values of n, the nth telephone number is divisible by a large power of two, 2n/4 + O(1). More precisely, the 2-adic order (the number of factors of two in the prime factorization) of T(4k) and of T(4k + 1) is k; for T(4k + 2) it is k + 1, and for T(4k + 3) it is k + 2.[14]
For any prime number p, one can test whether there exists a telephone number divisible by p by computing the recurrence for the sequence of telephone numbers, modulo p, until either reaching zero or detecting a cycle. The primes that divide at least one telephone number are[15]
2, 5, 13, 19, 23, 29, 31, 43, 53, 59, ... (sequence A264737 in the OEIS)
The odd primes in this sequence have been called inefficient. Each of them divides infinitely many telephone numbers.[16]
References
1. Knuth, Donald E. (1973), The Art of Computer Programming, Volume 3: Sorting and Searching, Reading, Mass.: Addison-Wesley, pp. 65–67, MR 0445948
2. Riordan, John (2002), Introduction to Combinatorial Analysis, Dover, pp. 85–86
3. Peart, Paul; Woan, Wen-Jin (2000), "Generating functions via Hankel and Stieltjes matrices" (PDF), Journal of Integer Sequences, 3 (2), Article 00.2.1, Bibcode:2000JIntS...3...21P, MR 1778992
4. Getu, Seyoum (1991), "Evaluating determinants via generating functions", Mathematics Magazine, 64 (1): 45–53, doi:10.2307/2690455, JSTOR 2690455, MR 1092195
5. Solomon, A. I.; Blasiak, P.; Duchamp, G.; Horzela, A.; Penson, K.A. (2005), "Combinatorial physics, normal order and model Feynman graphs", in Gruber, Bruno J.; Marmo, Giuseppe; Yoshinaga, Naotaka (eds.), Symmetries in Science XI, Kluwer Academic Publishers, pp. 527–536, arXiv:quant-ph/0310174, doi:10.1007/1-4020-2634-X_25, S2CID 5702844
6. Blasiak, P.; Dattoli, G.; Horzela, A.; Penson, K. A.; Zhukovsky, K. (2008), "Motzkin numbers, central trinomial coefficients and hybrid polynomials", Journal of Integer Sequences, 11 (1), Article 08.1.1, arXiv:0802.0075, Bibcode:2008JIntS..11...11B, MR 2377567
7. Tichy, Robert F.; Wagner, Stephan (2005), "Extremal problems for topological indices in combinatorial chemistry" (PDF), Journal of Computational Biology, 12 (7): 1004–1013, doi:10.1089/cmb.2005.12.1004, PMID 16201918
8. A direct bijection between involutions and tableaux, inspired by the recurrence relation for the telephone numbers, is given by Beissinger, Janet Simpson (1987), "Similar constructions for Young tableaux and involutions, and their application to shiftable tableaux", Discrete Mathematics, 67 (2): 149–163, doi:10.1016/0012-365X(87)90024-0, MR 0913181
9. Halverson, Tom; Reeks, Mike (2015), "Gelfand models for diagram algebras", Journal of Algebraic Combinatorics, 41 (2): 229–255, doi:10.1007/s10801-014-0534-5, MR 3306071, S2CID 7419411
10. Holt, D. F. (1974), "Rooks inviolate", The Mathematical Gazette, 58 (404): 131–134, doi:10.2307/3617799, JSTOR 3617799, S2CID 250441965
11. Chowla, S.; Herstein, I. N.; Moore, W. K. (1951), "On recursions connected with symmetric groups. I", Canadian Journal of Mathematics, 3: 328–334, doi:10.4153/CJM-1951-038-3, MR 0041849, S2CID 123802787
12. Moser, Leo; Wyman, Max (1955), "On solutions of xd = 1 in symmetric groups", Canadian Journal of Mathematics, 7: 159–168, doi:10.4153/CJM-1955-021-8, MR 0068564
13. Banderier, Cyril; Bousquet-Mélou, Mireille; Denise, Alain; Flajolet, Philippe; Gardy, Danièle; Gouyou-Beauchamps, Dominique (2002), "Generating functions for generating trees", Discrete Mathematics, 246 (1–3): 29–55, arXiv:math/0411250, doi:10.1016/S0012-365X(01)00250-3, MR 1884885, S2CID 14804110
14. Kim, Dongsu; Kim, Jang Soo (2010), "A combinatorial approach to the power of 2 in the number of involutions", Journal of Combinatorial Theory, Series A, 117 (8): 1082–1094, arXiv:0902.4311, doi:10.1016/j.jcta.2009.08.002, MR 2677675, S2CID 17457503
15. Sloane, N. J. A. (ed.), "Sequence A264737", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
16. Amdeberhan, Tewodros; Moll, Victor (2015), "Involutions and their progenies", Journal of Combinatorics, 6 (4): 483–508, arXiv:1406.2356, doi:10.4310/JOC.2015.v6.n4.a5, MR 3382606, S2CID 119708272
| Wikipedia |
Telescoping series
In mathematics, a telescoping series is a series whose general term $t_{n}$ is of the form $t_{n}=a_{n+1}-a_{n}$, i.e. the difference of two consecutive terms of a sequence $(a_{n})$.[1]
As a consequence the partial sums only consists of two terms of $(a_{n})$ after cancellation.[2][3] The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.
For example, the series
$\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}$
(the series of reciprocals of pronic numbers) simplifies as
${\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}&{}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\sum _{n=1}^{N}\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\left\lbrack {\left(1-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{N}}-{\frac {1}{N+1}}\right)}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{N}}+{\frac {1}{N}}\right)-{\frac {1}{N+1}}}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1-{\frac {1}{N+1}}}\right\rbrack =1.\end{aligned}}$
An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, De dimensione parabolae.[4]
In general
Telescoping sums are finite sums in which pairs of consecutive terms cancel each other, leaving only the initial and final terms.[5]
Let $a_{n}$ be a sequence of numbers. Then,
$\sum _{n=1}^{N}\left(a_{n}-a_{n-1}\right)=a_{N}-a_{0}$
If $a_{n}\rightarrow 0$
$\sum _{n=1}^{\infty }\left(a_{n}-a_{n-1}\right)=-a_{0}$
Telescoping products are finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms.
Let $a_{n}$ be a sequence of numbers. Then,
$\prod _{n=1}^{N}{\frac {a_{n-1}}{a_{n}}}={\frac {a_{0}}{a_{N}}}$
If $a_{n}\rightarrow 1$
$\prod _{n=1}^{\infty }{\frac {a_{n-1}}{a_{n}}}=a_{0}$
More examples
• Many trigonometric functions also admit representation as a difference, which allows telescopic canceling between the consecutive terms.
${\begin{aligned}\sum _{n=1}^{N}\sin \left(n\right)&{}=\sum _{n=1}^{N}{\frac {1}{2}}\csc \left({\frac {1}{2}}\right)\left(2\sin \left({\frac {1}{2}}\right)\sin \left(n\right)\right)\\&{}={\frac {1}{2}}\csc \left({\frac {1}{2}}\right)\sum _{n=1}^{N}\left(\cos \left({\frac {2n-1}{2}}\right)-\cos \left({\frac {2n+1}{2}}\right)\right)\\&{}={\frac {1}{2}}\csc \left({\frac {1}{2}}\right)\left(\cos \left({\frac {1}{2}}\right)-\cos \left({\frac {2N+1}{2}}\right)\right).\end{aligned}}$
• Some sums of the form
$\sum _{n=1}^{N}{f(n) \over g(n)}$
where f and g are polynomial functions whose quotient may be broken up into partial fractions, will fail to admit summation by this method. In particular, one has
${\begin{aligned}\sum _{n=0}^{\infty }{\frac {2n+3}{(n+1)(n+2)}}={}&\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}+{\frac {1}{n+2}}\right)\\={}&\left({\frac {1}{1}}+{\frac {1}{2}}\right)+\left({\frac {1}{2}}+{\frac {1}{3}}\right)+\left({\frac {1}{3}}+{\frac {1}{4}}\right)+\cdots \\&{}\cdots +\left({\frac {1}{n-1}}+{\frac {1}{n}}\right)+\left({\frac {1}{n}}+{\frac {1}{n+1}}\right)+\left({\frac {1}{n+1}}+{\frac {1}{n+2}}\right)+\cdots \\={}&\infty .\end{aligned}}$
The problem is that the terms do not cancel.
• Let k be a positive integer. Then
$\sum _{n=1}^{\infty }{\frac {1}{n(n+k)}}={\frac {H_{k}}{k}}$
where Hk is the kth harmonic number. All of the terms after 1/(k − 1) cancel.
• Let k,m with k $\neq $ m be positive integers. Then
$\sum _{n=1}^{\infty }{\frac {1}{(n+k)(n+k+1)\dots (n+m-1)(n+m)}}={\frac {1}{m-k}}\cdot {\frac {k!}{m!}}$
An application in probability theory
In probability theory, a Poisson process is a stochastic process of which the simplest case involves "occurrences" at random times, the waiting time until the next occurrence having a memoryless exponential distribution, and the number of "occurrences" in any time interval having a Poisson distribution whose expected value is proportional to the length of the time interval. Let Xt be the number of "occurrences" before time t, and let Tx be the waiting time until the xth "occurrence". We seek the probability density function of the random variable Tx. We use the probability mass function for the Poisson distribution, which tells us that
$\Pr(X_{t}=x)={\frac {(\lambda t)^{x}e^{-\lambda t}}{x!}},$
where λ is the average number of occurrences in any time interval of length 1. Observe that the event {Xt ≥ x} is the same as the event {Tx ≤ t}, and thus they have the same probability. Intuitively, if something occurs at least $x$ times before time $t$, we have to wait at most $t$ for the $xth$ occurrence. The density function we seek is therefore
${\begin{aligned}f(t)&{}={\frac {d}{dt}}\Pr(T_{x}\leq t)={\frac {d}{dt}}\Pr(X_{t}\geq x)={\frac {d}{dt}}(1-\Pr(X_{t}\leq x-1))\\\\&{}={\frac {d}{dt}}\left(1-\sum _{u=0}^{x-1}\Pr(X_{t}=u)\right)={\frac {d}{dt}}\left(1-\sum _{u=0}^{x-1}{\frac {(\lambda t)^{u}e^{-\lambda t}}{u!}}\right)\\\\&{}=\lambda e^{-\lambda t}-e^{-\lambda t}\sum _{u=1}^{x-1}\left({\frac {\lambda ^{u}t^{u-1}}{(u-1)!}}-{\frac {\lambda ^{u+1}t^{u}}{u!}}\right)\end{aligned}}$
The sum telescopes, leaving
$f(t)={\frac {\lambda ^{x}t^{x-1}e^{-\lambda t}}{(x-1)!}}.$
Similar concepts
Telescoping product
A telescoping product is a finite product (or the partial product of an infinite product) that can be cancelled by method of quotients to be eventually only a finite number of factors.[6][7]
For example, the infinite product[6]
$\prod _{n=2}^{\infty }\left(1-{\frac {1}{n^{2}}}\right)$
simplifies as
${\begin{aligned}\prod _{n=2}^{\infty }\left(1-{\frac {1}{n^{2}}}\right)&=\prod _{n=2}^{\infty }{\frac {(n-1)(n+1)}{n^{2}}}\\&=\lim _{N\to \infty }\prod _{n=2}^{N}{\frac {n-1}{n}}\times \prod _{n=2}^{N}{\frac {n+1}{n}}\\&=\lim _{N\to \infty }\left\lbrack {{\frac {1}{2}}\times {\frac {2}{3}}\times {\frac {3}{4}}\times \cdots \times {\frac {N-1}{N}}}\right\rbrack \times \left\lbrack {{\frac {3}{2}}\times {\frac {4}{3}}\times {\frac {5}{4}}\times \cdots \times {\frac {N}{N-1}}\times {\frac {N+1}{N}}}\right\rbrack \\&=\lim _{N\to \infty }\left\lbrack {\frac {1}{2}}\right\rbrack \times \left\lbrack {\frac {N+1}{N}}\right\rbrack \\&={\frac {1}{2}}\times \lim _{N\to \infty }\left\lbrack {\frac {N+1}{N}}\right\rbrack \\&={\frac {1}{2}}\times \lim _{N\to \infty }\left\lbrack {\frac {N}{N}}+{\frac {1}{N}}\right\rbrack \\&={\frac {1}{2}}.\end{aligned}}$
Other applications
For other applications, see:
• Grandi's series;
• Proof that the sum of the reciprocals of the primes diverges, where one of the proofs uses a telescoping sum;
• Fundamental theorem of calculus, a continuous analog of telescoping series;
• Order statistic, where a telescoping sum occurs in the derivation of a probability density function;
• Lefschetz fixed-point theorem, where a telescoping sum arises in algebraic topology;
• Homology theory, again in algebraic topology;
• Eilenberg–Mazur swindle, where a telescoping sum of knots occurs;
• Faddeev–LeVerrier algorithm.
References
1. Apostol, Tom (1967). Calculus, Volume 1 (Second ed.). John Wiley & Sons. p. 386.
2. Tom M. Apostol, Calculus, Volume 1, Blaisdell Publishing Company, 1962, pages 422–3
3. Brian S. Thomson and Andrew M. Bruckner, Elementary Real Analysis, Second Edition, CreateSpace, 2008, page 85
4. Weil, André (1989). "Prehistory of the zeta-function". In Aubert, Karl Egil; Bombieri, Enrico; Goldfeld, Dorian (eds.). Number Theory, Trace Formulas and Discrete Groups: Symposium in Honor of Atle Selberg, Oslo, Norway, July 14–21, 1987. Boston, Massachusetts: Academic Press. pp. 1–9. doi:10.1016/B978-0-12-067570-8.50009-3. MR 0993308.
5. Weisstein, Eric W. "Telescoping Sum". MathWorld. Wolfram.
6. "Telescoping Series - Product". Brilliant Math & Science Wiki. Brilliant.org. Retrieved 9 February 2020.
7. Bogomolny, Alexander. "Telescoping Sums, Series and Products". Cut the Knot. Retrieved 9 February 2020.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
Telescoping Markov chain
In probability theory, a telescoping Markov chain (TMC) is a vector-valued stochastic process that satisfies a Markov property and admits a hierarchical format through a network of transition matrices with cascading dependence.
For any $N>1$ consider the set of spaces $\{{\mathcal {S}}^{\ell }\}_{\ell =1}^{N}$. The hierarchical process $\theta _{k}$ defined in the product-space
$\theta _{k}=(\theta _{k}^{1},\ldots ,\theta _{k}^{N})\in {\mathcal {S}}^{1}\times \cdots \times {\mathcal {S}}^{N}$
is said to be a TMC if there is a set of transition probability kernels $\{\Lambda ^{n}\}_{n=1}^{N}$ such that
1. $\theta _{k}^{1}$ is a Markov chain with transition probability matrix $\Lambda ^{1}$
$\mathbb {P} (\theta _{k}^{1}=s\mid \theta _{k-1}^{1}=r)=\Lambda ^{1}(s\mid r)$
2. there is a cascading dependence in every level of the hierarchy,
$\mathbb {P} (\theta _{k}^{n}=s\mid \theta _{k-1}^{n}=r,\theta _{k}^{n-1}=t)=\Lambda ^{n}(s\mid r,t)$ for all $n\geq 2.$
3. $\theta _{k}$ satisfies a Markov property with a transition kernel that can be written in terms of the $\Lambda $'s,
$\mathbb {P} (\theta _{k+1}={\vec {s}}\mid \theta _{k}={\vec {r}})=\Lambda ^{1}(s_{1}\mid r_{1})\prod _{\ell =2}^{N}\Lambda ^{\ell }(s_{\ell }\mid r_{\ell },s_{\ell -1})$
where ${\vec {s}}=(s_{1},\ldots ,s_{N})\in {\mathcal {S}}^{1}\times \cdots \times {\mathcal {S}}^{N}$ and ${\vec {r}}=(r_{1},\ldots ,r_{N})\in {\mathcal {S}}^{1}\times \cdots \times {\mathcal {S}}^{N}.$
| Wikipedia |
Tellegen's theorem
Tellegen's theorem is one of the most powerful theorems in network theory. Most of the energy distribution theorems and extremum principles in network theory can be derived from it. It was published in 1952 by Bernard Tellegen.[1] Fundamentally, Tellegen's theorem gives a simple relation between magnitudes that satisfy Kirchhoff's laws of electrical circuit theory.
The Tellegen theorem is applicable to a multitude of network systems. The basic assumptions for the systems are the conservation of flow of extensive quantities (Kirchhoff's current law, KCL) and the uniqueness of the potentials at the network nodes (Kirchhoff's voltage law, KVL). The Tellegen theorem provides a useful tool to analyze complex network systems including electrical circuits, biological and metabolic networks, pipeline transport networks, and chemical process networks.
The theorem
Consider an arbitrary lumped network that has $b$ branches and $n$ nodes. In an electrical network, the branches are two-terminal components and the nodes are points of interconnection. Suppose that to each branch we assign arbitrarily a branch potential difference $W_{k}$ and a branch current $F_{k}$ for $k=1,2,\dots ,b$, and suppose that they are measured with respect to arbitrarily picked associated reference directions. If the branch potential differences $W_{1},W_{2},\dots ,W_{b}$ satisfy all the constraints imposed by KVL and if the branch currents $F_{1},F_{2},\dots ,F_{b}$ satisfy all the constraints imposed by KCL, then
$\sum _{k=1}^{b}W_{k}F_{k}=0.$
Tellegen's theorem is extremely general; it is valid for any lumped network that contains any elements, linear or nonlinear, passive or active, time-varying or time-invariant. The generality is extended when $W_{k}$ and $F_{k}$ are linear operations on the set of potential differences and on the set of branch currents (respectively) since linear operations don't affect KVL and KCL. For instance, the linear operation may be the average or the Laplace transform. More generally, operators that preserve KVL are called Kirchhoff voltage operators, operators that preserve KCL are called Kirchhoff current operators, and operators that preserve both are simply called Kirchhoff operators. These operators need not necessarily be linear for Tellegen's theorem to hold.[2]
The set of currents can also be sampled at a different time from the set of potential differences since KVL and KCL are true at all instants of time. Another extension is when the set of potential differences $W_{k}$ is from one network and the set of currents $F_{k}$ is from an entirely different network, so long as the two networks have the same topology (same incidence matrix) Tellegen's theorem remains true. This extension of Tellegen's Theorem leads to many theorems relating to two-port networks.[3]
Definitions
We need to introduce a few necessary network definitions to provide a compact proof.
Incidence matrix: The $n\times b$ matrix $\mathbf {A_{a}} $ is called node-to-branch incidence matrix for the matrix elements $a_{ij}$ being
$a_{ij}={\begin{cases}1,&{\text{if current in branch }}j{\text{ leaves node }}i\\-1,&{\text{if current in branch }}j{\text{ enters node }}i\\0,&{\text{otherwise}}\end{cases}}$
A reference or datum node $P_{0}$ is introduced to represent the environment and connected to all dynamic nodes and terminals. The $(n-1)\times b$ matrix $\mathbf {A} $, where the row that contains the elements $a_{0j}$ of the reference node $P_{0}$ is eliminated, is called reduced incidence matrix.
The conservation laws (KCL) in vector-matrix form:
$\mathbf {A} \mathbf {F} =\mathbf {0} $
The uniqueness condition for the potentials (KVL) in vector-matrix form:
$\mathbf {W} =\mathbf {A^{T}} \mathbf {w} $
where $w_{k}$ are the absolute potentials at the nodes to the reference node $P_{0}$.
Proof
Using KVL:
${\begin{aligned}\mathbf {W^{T}} \mathbf {F} =\mathbf {(A^{T}w)^{T}} \mathbf {F} =\mathbf {(w^{T}A)} \mathbf {F} =\mathbf {w^{T}AF} =\mathbf {0} \end{aligned}}$
because $\mathbf {AF} =\mathbf {0} $ by KCL. So:
$\sum _{k=1}^{b}W_{k}F_{k}=\mathbf {W^{T}} \mathbf {F} =0$
Applications
Network analogs have been constructed for a wide variety of physical systems, and have proven extremely useful in analyzing their dynamic behavior. The classical application area for network theory and Tellegen's theorem is electrical circuit theory. It is mainly in use to design filters in signal processing applications.
A more recent application of Tellegen's theorem is in the area of chemical and biological processes. The assumptions for electrical circuits (Kirchhoff laws) are generalized for dynamic systems obeying the laws of irreversible thermodynamics. Topology and structure of reaction networks (reaction mechanisms, metabolic networks) can be analyzed using the Tellegen theorem.
Another application of Tellegen's theorem is to determine stability and optimality of complex process systems such as chemical plants or oil production systems. The Tellegen theorem can be formulated for process systems using process nodes, terminals, flow connections and allowing sinks and sources for production or destruction of extensive quantities.
A formulation for Tellegen's theorem of process systems:
$\sum _{j=1}^{n_{P}}W_{j}{\frac {\operatorname {d} Z_{j}}{\operatorname {d} t}}=\sum _{k=1}^{b}W_{k}f_{k}+\sum _{j=1}^{n_{P}}w_{j}p_{j}+\sum _{j=1}^{n_{t}}w_{j}t_{j},\quad j=1,\dots ,n_{p}+n_{t}$
where $p_{j}$ are the production terms, $t_{j}$ are the terminal connections, and ${\frac {\operatorname {d} Z_{j}}{\operatorname {d} t}}$ are the dynamic storage terms for the extensive variables.
References
In-line references
1. Tellegen, B. D. H. (1952). "A general network theorem with applications". Philips Research Reports. 7: 259–269.
2. Penfield, P. (1970). "A Generalized Form of Tellegen's Theorem" (PDF). IEEE Transactions on Circuit Theory. CT-17: 302–305. Retrieved November 8, 2016.
3. Tellegen's Theorem and Electrical Networks by Paul Penfield, Jr., Robert Spence, and Simon Duinker, The MIT Press, Cambridge, MA, 1970
General references
• Basic Circuit Theory by C.A. Desoer and E.S. Kuh, McGraw-Hill, New York, 1969
• "Tellegen's Theorem and Thermodynamic Inequalities", G.F. Oster and C.A. Desoer, J. Theor. Biol 32 (1971), 219–241
• "Network Methods in Models of Production", Donald Watson, Networks, 10 (1980), 1–15
External links
• Circuit example for Tellegen's theorem
• G.F. Oster and C.A. Desoer, Tellegen's Theorem and Thermodynamic Inequalities
• Network thermodynamics
| Wikipedia |
Telman Malikov
Telman Malikov (born January 5, 1950, in Baku) is an Azerbaijani scientist. He is a professor at Azerbaijan National Academy of Sciences Institute of Mathematics and Mechanics.[1]
Early life
In 1972, Malikov graduated in Mechanics and Mathematics at Azerbaijan State University (now Baku State University (BSU)) with an honors diploma. That year, he went to Ganja State University (GSU) as a teacher. In December 1972 – 1975 he became a postgraduate at BSU. From 1976 to 1977 he worked at GSU. From 1977 to 2013 he worked at Azerbaijan Technology University in Ganja. From 1990 to 2005 he was a manager of the Higher Mathematics Department. From 2000 to 2013, he was rector of Azerbaijan Technology University.[2] Starting in 2014 he began work at the University of Mathematics and Mechanics of Azerbaijan National Academy of Sciences.
Research
In 1972 Malikov entered postgraduate study at Azerbaijan State University. In 1976 he defended his dissertation on "The research of intrinsic processes in optimum systems" on "Differential and Integral equations". He earned the degree of physical-mathematics sciences.
In 2005, he defended his thesis on a "Discrete Mathematics and Mathematical Cybernetics", on the topic of "Necessary conditions for optimality in some of optimal management processes".
Malikov studied optimal management following the work of Q. T. Ahmadov, associate member of Azerbaijan National Academy of Sciences.
Malikov suggested new methods to obtain the necessary conditions for optimality in described processes with simple equations, integrodifferential equations, Goursat-Darboux and acted equations. His methods give an opportunity for optimality in some problems that were impossible to explore (for example, in processes with neutral type equations) and to obtain necessary conditions for optimality of special management in necessary conditions and different meanings. He authored more than 80 articles, 2 textbooks, 3 monographs, and crafted more than 10 inventions and patents. His scientific works were published in Russia, US, UK and in scientific journals.
Malikov led scientific investigators and advised doctoral candidates. In 2002, he was awarded with the "Gold Medal" of the French Association for industry for his achievements in education. He was a member of the defence council of doctors and candidates of sciences on Discrete Mathematics and Mathematical Cybernetics of Cybernetic Institute of Azerbaijan National Academy of Sciences.
References
1. "MƏLİKOV Telman Qulu oğlu | Şamaxı Ensiklopediyası". shamakhi-encyclopedia.az. Retrieved 2016-05-16.
2. "Azərbaycan Texnologiya Universiteti". atu.edu.az. Retrieved 2016-05-16.
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| Wikipedia |
Temperley–Lieb algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras.
Structure
Generators and relations
Let $R$ be a commutative ring and fix $\delta \in R$. The Temperley–Lieb algebra $TL_{n}(\delta )$ is the $R$-algebra generated by the elements $e_{1},e_{2},\ldots ,e_{n-1}$, subject to the Jones relations:
• $e_{i}^{2}=\delta e_{i}$ for all $1\leq i\leq n-1$
• $e_{i}e_{i+1}e_{i}=e_{i}$ for all $1\leq i\leq n-2$
• $e_{i}e_{i-1}e_{i}=e_{i}$ for all $2\leq i\leq n-1$
• $e_{i}e_{j}=e_{j}e_{i}$ for all $1\leq i,j\leq n-1$ such that $|i-j|\neq 1$
Using these relations, any product of generators $e_{i}$ can be brought to Jones' normal form:
$E={\big (}e_{i_{1}}e_{i_{1}-1}\cdots e_{j_{1}}{\big )}{\big (}e_{i_{2}}e_{i_{2}-1}\cdots e_{j_{2}}{\big )}\cdots {\big (}e_{i_{r}}e_{i_{r}-1}\cdots e_{j_{r}}{\big )}$
where $(i_{1},i_{2},\dots ,i_{r})$ and $(j_{1},j_{2},\dots ,j_{r})$ are two strictly increasing sequences in $\{1,2,\dots ,n-1\}$. Elements of this type form a basis of the Temperley-Lieb algebra.[1]
The dimensions of Temperley-Lieb algebras are Catalan numbers:[2]
$\dim(TL_{n}(\delta ))={\frac {(2n)!}{n!(n+1)!}}$
The Temperley–Lieb algebra $TL_{n}(\delta )$ is a subalgebra of the Brauer algebra ${\mathfrak {B}}_{n}(\delta )$,[3] and therefore also of the partition algebra $P_{n}(\delta )$. The Temperley–Lieb algebra $TL_{n}(\delta )$ is semisimple for $\delta \in \mathbb {C} -F_{n}$ where $F_{n}$ is a known, finite set.[4] For a given $n$, all semisimple Temperley-Lieb algebras are isomorphic.[3]
Diagram algebra
$TL_{n}(\delta )$ may be represented diagrammatically as the vector space over noncrossing pairings of $2n$ points on two opposite sides of a rectangle with n points on each of the two sides.
The identity element is the diagram in which each point is connected to the one directly across the rectangle from it. The generator $e_{i}$ is the diagram in which the $i$-th and $(i+1)$-th point on the left side are connected to each other, similarly the two points opposite to these on the right side, and all other points are connected to the point directly across the rectangle.
The generators of $TL_{5}(\delta )$ are:
From left to right, the unit 1 and the generators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): e_{1} , $e_{2}$, $e_{3}$, $e_{4}$.
Multiplication on basis elements can be performed by concatenation: placing two rectangles side by side, and replacing any closed loops by a factor $\delta $, for example $e_{1}e_{4}e_{3}e_{2}\times e_{2}e_{4}e_{3}=\delta \,e_{1}e_{4}e_{3}e_{2}e_{4}e_{3}$:
× = = $\delta $ .
The Jones relations can be seen graphically:
= $\delta $
=
=
The five basis elements of $TL_{3}(\delta )$ are the following:
.
From left to right, the unit 1, the generators $e_{2}$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): e_{1} , and $e_{1}e_{2}$, $e_{2}e_{1}$.
Representations
Structure
For $\delta $ such that $TL_{n}(\delta )$ is semisimple, a complete set $\{W_{\ell }\}$ of simple modules is parametrized by integers $0\leq \ell \leq n$ with $\ell \equiv n{\bmod {2}}$. The dimension of a simple module is written in terms of binomial coefficients as[4]
$\dim(W_{\ell })={\binom {n}{\frac {n-\ell }{2}}}-{\binom {n}{{\frac {n-\ell }{2}}-1}}$
A basis of the simple module $W_{\ell }$ is the set $M_{n,\ell }$ of monic noncrossing pairings from $n$ points on the left to $\ell $ points on the right. (Monic means that each point on the right is connected to a point on the left.) There is a natural bijection between $\cup _{\begin{array}{c}0\leq \ell \leq n\\\ell \equiv n{\bmod {2}}\end{array}}M_{n,\ell }\times M_{n,\ell }$, and the set of diagrams that generate $TL_{n}(\delta )$: any such diagram can be cut into two elements of $M_{n,\ell }$ for some $\ell $.
Then $TL_{n}(\delta )$ acts on $W_{\ell }$ by diagram concatenation from the left.[3] (Concatenation can produce non-monic pairings, which have to be modded out.) The module $W_{\ell }$ may be called a standard module or link module.[1]
If $\delta =q+q^{-1}$ with $q$ a root of unity, $TL_{n}(\delta )$ may not be semisimple, and $W_{\ell }$ may not be irreducible:
$W_{\ell }{\text{ reducible }}\iff \exists j\in \{1,2,\dots ,\ell \},\ q^{2n-4\ell +2+2j}=1$
If $W_{\ell }$ is reducible, then its quotient by its maximal proper submodule is irreducible.[1]
Branching rules from the Brauer algebra
Simple modules of the Brauer algebra ${\mathfrak {B}}_{n}(\delta )$ can be decomposed into simple modules of the Temperley-Lieb algebra. The decomposition is called a branching rule, and it is a direct sum with positive integer coefficients:
$W_{\lambda }\left({\mathfrak {B}}_{n}(\delta )\right)=\bigoplus _{\begin{array}{c}|\lambda |\leq \ell \leq n\\\ell \equiv |\lambda |{\bmod {2}}\end{array}}c_{\ell }^{\lambda }W_{\ell }\left(TL_{n}(\delta )\right)$
The coefficients $c_{\ell }^{\lambda }$ do not depend on $n,\delta $, and are given by[4]
$c_{\ell }^{\lambda }=f^{\lambda }\sum _{r=0}^{\frac {\ell -|\lambda |}{2}}(-1)^{r}{\binom {\ell -r}{r}}{\binom {\ell -2r}{\ell -|\lambda |-2r}}(\ell -|\lambda |-2r)!!$
where $f^{\lambda }$ is the number of standard Young tableaux of shape $\lambda $, given by the hook length formula.
Affine Temperley-Lieb algebra
The affine Temperley-Lieb algebra $aTL_{n}(\delta )$ is an infinite-dimensional algebra such that $TL_{n}(\delta )\subset aTL_{n}(\delta )$. It is obtained by adding generators $e_{n},\tau ,\tau ^{-1}$ such that[5]
• $\tau e_{i}=e_{i+1}\tau $ for all $1\leq i\leq n$,
• $e_{1}\tau ^{2}=e_{1}e_{2}\cdots e_{n-1}$,
• $\tau \tau ^{-1}=\tau ^{-1}\tau ={\text{id}}$.
The indices are supposed to be periodic i.e. $e_{n+1}=e_{1},e_{n}=e_{0}$, and the Temperley-Lieb relations are supposed to hold for all $1\leq i\leq n$. Then $\tau ^{n}$ is central. A finite-dimensional quotient of the algebra $aTL_{n}(\delta )$, sometimes called the unoriented Jones-Temperley-Lieb algebra,[6] is obtained by assuming $\tau ^{n}={\text{id}}$, and replacing non-contractible lines with the same factor $\delta $ as contractible lines (for example, in the case $n=4$, this implies $e_{1}e_{3}e_{2}e_{4}e_{1}e_{3}=\delta ^{2}e_{1}e_{3}$).
The diagram algebra for $aTL_{n}(\delta )$ is deduced from the diagram algebra for $TL_{n}(\delta )$ by turning rectangles into cylinders. The algebra $aTL_{n}(\delta )$ is infinite-dimensional because lines can wind around the cylinder. If $n$ is even, there can even exist closed winding lines, which are non-contractible.
The Temperley-Lieb algebra is a quotient of the corresponding affine Temperley-Lieb algebra.[5]
The cell module $W_{\ell ,z}$ of $aTL_{n}(\delta )$ is generated by the set of monic pairings from $n$ points to $\ell $ points, just like the module $W_{\ell }$ of $TL_{n}(\delta )$. However, the pairings are now on a cylinder, and the right-multiplication with $\tau $ is identified with $z\cdot {\text{id}}$ for some $z\in \mathbb {C} ^{*}$. If $\ell =0$, there is no right-multiplication by $\tau $, and it is the addition of a non-contractible loop on the right which is identified with $z+z^{-1}$. Cell modules are finite-dimensional, with
$\dim(W_{\ell ,z})={\binom {n}{\frac {n-\ell }{2}}}$
The cell module $W_{\ell ,z}$ is irreducible for all $z\in \mathbb {C} ^{*}-R(\delta )$, where the set $R(\delta )$ is countable. For $z\in R(\delta )$, $W_{\ell ,z}$ has an irreducible quotient. The irreducible cell modules and quotients thereof form a complete set of irreducible modules of $aTL_{n}(\delta )$.[5] Cell modules of the unoriented Jones-Temperley-Lieb algebra must obey $z^{\ell }=1$ if $\ell \neq 0$, and $z+z^{-1}=\delta $ if $\ell =0$.
Applications
Temperley–Lieb Hamiltonian
Consider an interaction-round-a-face model e.g. a square lattice model and let $n$ be the number of sites on the lattice. Following Temperley and Lieb[7] we define the Temperley–Lieb Hamiltonian (the TL Hamiltonian) as
${\mathcal {H}}=\sum _{j=1}^{n-1}(\delta -e_{j})$
In what follows we consider the special case $\delta =1$.
We will firstly consider the case $n=3$. The TL Hamiltonian is ${\mathcal {H}}=2-e_{1}-e_{2}$, namely
${\mathcal {H}}$ = 2 - - .
We have two possible states,
and .
In acting by ${\mathcal {H}}$ on these states, we find
${\mathcal {H}}$ = 2 - - = - ,
and
${\mathcal {H}}$ = 2 - - = - + .
Writing ${\mathcal {H}}$ as a matrix in the basis of possible states we have,
${\mathcal {H}}=\left({\begin{array}{rr}1&-1\\-1&1\end{array}}\right)$
The eigenvector of ${\mathcal {H}}$ with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue $\lambda _{0}$ for ${\mathcal {H}}$ is $\lambda _{0}=0$. The corresponding eigenvector is $\psi _{0}=(1,1)$. As we vary the number of sites $n$ we find the following table[8]
$n$ $\psi _{0}$ $n$ $\psi _{0}$
2 (1) 3 (1, 1)
4 (2, 1) 5 $(3_{3},1_{2})$
6 $(11,5_{2},4,1)$ 7 $(26_{4},10_{2},9_{2},8_{2},5_{2},1_{2})$
8 $(170,75_{2},71,56_{2},50,30,14_{4},6,1)$ 9 $(646,\ldots )$
$\vdots $ $\vdots $ $\vdots $ $\vdots $
where we have used the notation $m_{j}=(m,\ldots ,m)$ $j$-times e.g., $5_{2}=(5,5)$.
An interesting observation is that the largest components of the ground state of ${\mathcal {H}}$ have a combinatorial enumeration as we vary the number of sites,[9] as was first observed by Murray Batchelor, Jan de Gier and Bernard Nienhuis.[8] Using the resources of the on-line encyclopedia of integer sequences, Batchelor et al. found, for an even numbers of sites
$1,2,11,170,\ldots =\prod _{j=0}^{\frac {n-2}{2}}\left(3j+1\right){\frac {(2j)!(6j)!}{(4j)!(4j+1)!}}\qquad (n=2,4,6,\dots )$
and for an odd numbers of sites
$1,3,26,646,\ldots =\prod _{j=0}^{\frac {n-3}{2}}(3j+2){\frac {(2j+1)!(6j+3)!}{(4j+2)!(4j+3)!}}\qquad (n=3,5,7,\dots )$
Surprisingly, these sequences corresponded to well known combinatorial objects. For $n$ even, this (sequence A051255 in the OEIS) corresponds to cyclically symmetric transpose complement plane partitions and for $n$ odd, (sequence A005156 in the OEIS), these correspond to alternating sign matrices symmetric about the vertical axis.
References
1. Ridout, David; Saint-Aubin, Yvan (2012-04-20). "Standard Modules, Induction and the Temperley-Lieb Algebra". arXiv:1204.4505v4 [math-ph].
2. Kassel, Christian; Turaev, Vladimir (2008). "Braid Groups". Graduate Texts in Mathematics. New York, NY: Springer New York. doi:10.1007/978-0-387-68548-9. ISBN 978-0-387-33841-5. ISSN 0072-5285.
3. Halverson, Tom; Jacobson, Theodore N. (2018-08-24). "Set-partition tableaux and representations of diagram algebras". arXiv:1808.08118v2 [math.RT].
4. Benkart, Georgia; Moon, Dongho (2005-04-26), "Tensor product representations of Temperley-Lieb algebras and Chebyshev polynomials", Representations of Algebras and Related Topics, Providence, Rhode Island: American Mathematical Society, pp. 57–80, doi:10.1090/fic/045/05, ISBN 9780821834152
5. Belletête, Jonathan; Saint-Aubin, Yvan (2018-02-10). "On the computation of fusion over the affine Temperley-Lieb algebra". Nuclear Physics B. 937: 333–370. arXiv:1802.03575v1. Bibcode:2018NuPhB.937..333B. doi:10.1016/j.nuclphysb.2018.10.016. S2CID 119131017.
6. Read, N.; Saleur, H. (2007-01-11). "Enlarged symmetry algebras of spin chains, loop models, and S-matrices". Nuclear Physics B. 777 (3): 263–315. arXiv:cond-mat/0701259. Bibcode:2007NuPhB.777..263R. doi:10.1016/j.nuclphysb.2007.03.007. S2CID 119152756.
7. Temperley, Neville; Lieb, Elliott (1971). "Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 322 (1549): 251–280. Bibcode:1971RSPSA.322..251T. doi:10.1098/rspa.1971.0067. JSTOR 77727. MR 0498284. S2CID 122770421.
8. Batchelor, Murray; de Gier, Jan; Nienhuis, Bernard (2001). "The quantum symmetric $XXZ$ chain at $\Delta =-1/2$, alternating-sign matrices and plane partitions". Journal of Physics A. 34 (19): L265–L270. arXiv:cond-mat/0101385. doi:10.1088/0305-4470/34/19/101. MR 1836155. S2CID 118048447.
9. de Gier, Jan (2005). "Loops, matchings and alternating-sign matrices". Discrete Mathematics. 298 (1–3): 365–388. arXiv:math/0211285. doi:10.1016/j.disc.2003.11.060. MR 2163456. S2CID 2129159.
Further reading
• Kauffman, Louis H. (1991). Knots and Physics. World Scientific. ISBN 978-981-02-0343-6.
• Kauffman, Louis H. (1987). "State Models and the Jones Polynomial". Topology. 26 (3): 395–407. doi:10.1016/0040-9383(87)90009-7. MR 0899057.
• Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. London: Academic Press Inc. ISBN 0-12-083180-5. MR 0690578.
| Wikipedia |
Ten Computational Canons
The Ten Computational Canons was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (602–670), as the official mathematical texts for imperial examinations in mathematics.
The Ten Computational Canons includes:
1. Zhoubi Suanjing (Zhou Shadow Mathematical Classic)
2. Jiuzhang Suanshu (The Nine Chapters on the Mathematical Art)
3. Haidao Suanjing (The Sea Island Mathematical Classic)
4. Sunzi Suanjing (The Mathematical Classic of Sun Zi)
5. Zhang Qiujian Suanjing (The Mathematical Classic of Zhang Qiujian)
6. Wucao Suanjing (Computational Canon of the Five Administrative Sections)
7. Xiahou Yang Suanjing (The Mathematical Classic of Xiahou Yang)
8. Wujing Suanshu (Computational Prescriptions of the Five Classics)
9. Jigu Suanjing (Continuation of Ancient Mathematical Classic)
10. Zhui Shu (Method of Interpolation)
It was specified in Tang dynasty laws on examination that Sunzi Suanjing and the Computational Canon of the Five Administrative Sections together required one year of study; The Nine Chapters on the Mathematical Art plus Haidao Suanjing three years; Jigu Suanjing three years; Zhui Shu four years; and Zhang Qiujian and Xia Houyang one year each.
The government of the Song dynasty actively promoted the study of mathematics. There were two government xylograph editions of The Ten Computational Canons in the years 1084 and 1213. The wide availability of these mathematical texts contributed to the flourishing of mathematics in the Song and Yuan dynasties, inspiring mathematicians such as Jia Xian, Qin Jiushao, Yang Hui, Li Zhi and Zhu Shijie.
In the Ming dynasty during the reign of the Yongle Emperor, some of the Ten Canons were copied into the Yongle Encyclopedia. During the reign of the Qianlong Emperor in the Qing dynasty, scholar Dai Zhen made copies of the Zhoubi Suanjing, The Nine Chapters on the Mathematical Art, Haidao Suanjing, Sunzi Suanjing, Zhang Qiujian Suanjing, Computational Canon of the Five Administrative Sections, Xiahou Yang Suanjing, Computational Prescriptions of the Five Classics, Jigu Suanjing, and Shushu Jiyi from the Yongle Encyclopedia and transferred them into another encyclopedia, the Siku Quanshu.
• Zhoubi Suanjing
• The Nine Chapters on the Mathematical Art
• Haidao Suanjing
• Sunzi Suanjing
• Computational Canon of the Five Administrative Sections
• Jigu Suanjing
• Shushu Jiyi
References
• Jean Claude Martzloff, A History of Chinese Mathematics, pp. 123–126. ISBN 3-540-33782-2.
| Wikipedia |
99 Bottles of Beer
"99 Bottles of Beer" or "100 Bottles of Pop on the Wall" is a song dating to the mid-20th century. It is a traditional reverse counting song in both the United States and Canada. It is popular to sing on road trips, as it has a very repetitive format which is easy to memorize and can take a long time when families sing. In particular, the song is often sung by children on long school bus trips, such as class field trips, or on Scout or Girl Guide outings.
"99 bottles"
Song
GenreFolk
Lyrics
The song's lyrics are as follows, with mathematical values substituted:[1][2]
(n) bottles of beer on the wall. (n) bottles of beer. Take one down, pass it around, (n-1) bottles of beer on the wall. (caution: this mathematical formula ends with n=1, the song does not use negative numbers).
Alternative line:[3]
If one of those bottles should happen to fall, 98 bottles of beer on the wall...
The same verse is repeated, each time with one bottle fewer, until there is none left. Variations on the last verse following the last bottle going down include lines such as:
No more bottles of beer on the wall, no more bottles of beer.
Go to the store and buy some more, 99 bottles of beer on the wall...
Or:
No more bottles of beer on the wall, no more bottles of beer.
We've taken them down and passed them around; now we're drunk and passed out!
Other alternate lines read:
If that one bottle should happen to fall, what a waste of alcohol!
Or:
No more bottles of beer on the wall, no more bottles of beer.
There's nothing else to fall, because there's no more bottles of beer on the wall.
Or:
The song does not stop at the last "1" or "0" bottles of beer but continues counting with −1 (Negative one) Bottles of beer on the wall Take one down, pass it around, −2 (negative 2) bottles of beer on the wall... continuing onward through the negative numbers
Andy Kaufman routine
The boring and time-consuming nature of the "99 Bottles of Beer" song means that probably only a minority of renditions are done to the final verse. The American comedian Andy Kaufman exploited this fact in the routine early in his career when he would actually sing all 100 verses.[4]
Atticus
Atticus, a band from Knoxville, Tennessee recorded a thirteen and a half minute live version of the song in its entirety at a club in Glasgow, Scotland called The Cathouse. It was included in the 2001 album Figment. Rich Stewart aka Homebrew Stew listed it the number one drinking song out of 86 in an article for Modern Drunkard Magazine the following year.[5]
Mathematically inspired variants
Donald Byrd has collected dozens of variants inspired by mathematical concepts and written by himself and others.[6] (A subset of his collection has been published.[7]) Byrd argues that the collection has pedagogic as well as amusement value. Among his variants are:
• "Infinity bottles of beer on the wall". If one bottle is taken down, there are still infinite bottles of beer on the wall (thus creating an unending sequence much like "The Song That Never Ends").
• "Aleph-null bottles of beer on the wall". Aleph-null is the size of the set of all natural numbers, and is the smallest infinity and the only countable one; therefore, even if an infinite aleph-null of bottles fall, the same amount remains.
• "Aleph-one/two/three/etc. bottles of beer on the wall". Aleph-one, two, three, etc. are uncountable infinite sets, which are larger than countable ones; therefore, if only a countable infinity of bottles fall, an uncountable number remains.
Other versions in Byrd's collection involve concepts including geometric progressions, differentials, Euler's identity, complex numbers, summation notation, the Cantor set, the Fibonacci sequence, and the continuum hypothesis, among others.
References in computer science
The computer scientist Donald Knuth proved that the song has a complexity of $O(\log N)$ in his in-joke-article "The Complexity of Songs".[8]
Numerous computer programs exist to output the lyrics to the song. This is analogous to "Hello, World!" programs, with the addition of a loop. As with "Hello World!", this can be a practice exercise for those studying computer programming, and a demonstration of different programming paradigms dealing with looping constructs and syntactic differences between programming languages within a paradigm.
The program has been written in over 1,500 different programming languages.[9]
Example
C
#include <stdio.h>
int main(void) {
for( size_t i = 99; i > 0; i-- ) {
printf("%zu bottle%s of beer on the wall, %zu bottle%s of beer.\nTake one down & pass it around, now there's ",
i, (i==1? "" : "s"), i, (i==1? "" : "s"));
printf(( i > 1 )? "%zu bottles of beer on the wall\n" : "no more bottles of beer on the wall!\n", i-1);
}
}
Rust
fn main() {
let mut i = 99;
loop {
if i == 1 {
println!("{} bottle of beer on the wall, {} bottle of beer.\nTake one down and pass it around, there's no more bottles of beer on the wall!", i, i);
break;
} else {
println!("{} bottles of beer on the wall, {} bottles of beer.\nTake one down and pass it around, now there's {} more bottles of beer on the wall!", i, i, (i - 1));
}
i -= 1;
}
}
See also
• "Potje met vet" – a traditional Dutch song sung in the same style
• "Ten Green Bottles" – a similar song which is popular in the United Kingdom
References
1. Nyberg, Tim (2006). 99 Bottles of Beer on the Wall: The Complete Lyrics. Andrews McMeel Publishing. p. 112. ISBN 978-0-7407-6074-7.
2. Baird, Kevin C. (2007). Ruby by example: concepts and code. No Starch Press. p. 25. ISBN 978-1-59327-148-0.
3. Cohen, Norm (2005). Folk Music: A Regional Exploration. Greenwood Press. p. 60. ISBN 0-313-32872-2.
4. Patton, Charlie (December 23, 1999). "Ever-annoying Andy Kaufman gets last laugh | Jacksonville.com". Archived from the original on 2018-02-01. Retrieved 15 Sep 2012.
5. Stewart, Rich. "Rhythm and Booze: The Top 86 Drinking Songs". Modern Drunkard Magazine. Retrieved 2018-12-13.
6. Byrd, Donald (2015-11-30). "Infinite Bottles of Beer: Mathematical Concepts with Epsilon Pain, Or: A Cantorial Approach to Cantorian Arithmetic and Other Mathematical Melodies" (PDF). Indiana University, School of Informatics. Retrieved 2020-03-26.
7. Donald Byrd (2010). "Infinite Bottles of Beer: A cantorial approach to Cantorian arithmetic and other mathematical melodies". Math Horizons: 16–17.
8. Knuth, Donald. "The Complexity of Songs" (PDF). Retrieved 2020-09-02.
9. Team, 99 Bottles of Beer. "99 Bottles of Beer - Start". www.99-bottles-of-beer.net.
| Wikipedia |
Almost Mathieu operator
In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by
$[H_{\omega }^{\lambda ,\alpha }u](n)=u(n+1)+u(n-1)+2\lambda \cos(2\pi (\omega +n\alpha ))u(n),\,$
acting as a self-adjoint operator on the Hilbert space $\ell ^{2}(\mathbb {Z} )$. Here $\alpha ,\omega \in \mathbb {T} ,\lambda >0$ are parameters. In pure mathematics, its importance comes from the fact of being one of the best-understood examples of an ergodic Schrödinger operator. For example, three problems (now all solved) of Barry Simon's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.[1] In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the Aubry–André model.
For $\lambda =1$, the almost Mathieu operator is sometimes called Harper's equation.
The spectral type
If $\alpha $ is a rational number, then $H_{\omega }^{\lambda ,\alpha }$ is a periodic operator and by Floquet theory its spectrum is purely absolutely continuous.
Now to the case when $\alpha $ is irrational. Since the transformation $\omega \mapsto \omega +\alpha $ is minimal, it follows that the spectrum of $H_{\omega }^{\lambda ,\alpha }$ does not depend on $\omega $. On the other hand, by ergodicity, the supports of absolutely continuous, singular continuous, and pure point parts of the spectrum are almost surely independent of $\omega $. It is now known, that
• For $0<\lambda <1$, $H_{\omega }^{\lambda ,\alpha }$ has surely purely absolutely continuous spectrum.[2] (This was one of Simon's problems.)
• For $\lambda =1$, $H_{\omega }^{\lambda ,\alpha }$ has surely purely singular continuous spectrum for any irrational $\alpha $.[3]
• For $\lambda >1$, $H_{\omega }^{\lambda ,\alpha }$ has almost surely pure point spectrum and exhibits Anderson localization.[4] (It is known that almost surely can not be replaced by surely.)[5][6]
That the spectral measures are singular when $\lambda \geq 1$ follows (through the work of Yoram Last and Simon) [7] from the lower bound on the Lyapunov exponent $\gamma (E)$ given by
$\gamma (E)\geq \max\{0,\log(\lambda )\}.\,$
This lower bound was proved independently by Joseph Avron, Simon and Michael Herman, after an earlier almost rigorous argument of Serge Aubry and Gilles André. In fact, when $E$ belongs to the spectrum, the inequality becomes an equality (the Aubry–André formula), proved by Jean Bourgain and Svetlana Jitomirskaya.[8]
The structure of the spectrum
Another striking characteristic of the almost Mathieu operator is that its spectrum is a Cantor set for all irrational $\alpha $ and $\lambda >0$. This was shown by Avila and Jitomirskaya solving the by-then famous "ten martini problem"[9] (also one of Simon's problems) after several earlier results (including generically[10] and almost surely[11] with respect to the parameters).
Furthermore, the Lebesgue measure of the spectrum of the almost Mathieu operator is known to be
$\operatorname {Leb} (\sigma (H_{\omega }^{\lambda ,\alpha }))=|4-4\lambda |\,$
for all $\lambda >0$. For $\lambda =1$ this means that the spectrum has zero measure (this was first proposed by Douglas Hofstadter and later became one of Simon's problems).[12] For $\lambda \neq 1$, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky. Earlier Last [13][14] had proven this formula for most values of the parameters.
The study of the spectrum for $\lambda =1$ leads to the Hofstadter's butterfly, where the spectrum is shown as a set.
References
1. Simon, Barry (2000). "Schrödinger operators in the twenty-first century". Mathematical Physics 2000. London: Imp. Coll. Press. pp. 283–288. ISBN 978-1860942303.
2. Avila, A. (2008). "The absolutely continuous spectrum of the almost Mathieu operator". arXiv:0810.2965 [math.DS].
3. Jitomirskaya, S. "On point spectrum of critical almost Mathieu operators" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
4. Jitomirskaya, Svetlana Ya. (1999). "Metal-insulator transition for the almost Mathieu operator". Ann. of Math. 150 (3): 1159–1175. arXiv:math/9911265. Bibcode:1999math.....11265J. doi:10.2307/121066. JSTOR 121066. S2CID 10641385.
5. Avron, J.; Simon, B. (1982). "Singular continuous spectrum for a class of almost periodic Jacobi matrices". Bull. Amer. Math. Soc. 6 (1): 81–85. doi:10.1090/s0273-0979-1982-14971-0. Zbl 0491.47014.
6. Jitomirskaya, S.; Simon, B. (1994). "Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators" (PDF). Comm. Math. Phys. 165 (1): 201–205. Bibcode:1994CMaPh.165..201J. CiteSeerX 10.1.1.31.4995. doi:10.1007/bf02099743. S2CID 16267690. Zbl 0830.34074.
7. Last, Y.; Simon, B. (1999). "Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators". Invent. Math. 135 (2): 329–367. arXiv:math-ph/9907023. Bibcode:1999InMat.135..329L. doi:10.1007/s002220050288. S2CID 9429122.
8. Bourgain, J.; Jitomirskaya, S. (2002). "Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential". Journal of Statistical Physics. 108 (5–6): 1203–1218. doi:10.1023/A:1019751801035. S2CID 14062549.
9. Avila, A.; Jitomirskaya, S. (2005). "Solving the Ten Martini Problem". The Ten Martini problem. Lecture Notes in Physics. Vol. 690. pp. 5–16. arXiv:math/0503363. Bibcode:2006LNP...690....5A. doi:10.1007/3-540-34273-7_2. ISBN 978-3-540-31026-6. S2CID 55259301.
10. Bellissard, J.; Simon, B. (1982). "Cantor spectrum for the almost Mathieu equation". J. Funct. Anal. 48 (3): 408–419. doi:10.1016/0022-1236(82)90094-5.
11. Puig, Joaquim (2004). "Cantor spectrum for the almost Mathieu operator". Comm. Math. Phys. 244 (2): 297–309. arXiv:math-ph/0309004. Bibcode:2004CMaPh.244..297P. doi:10.1007/s00220-003-0977-3. S2CID 120589515.
12. Avila, A.; Krikorian, R. (2006). "Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles". Annals of Mathematics. 164 (3): 911–940. arXiv:math/0306382. doi:10.4007/annals.2006.164.911. S2CID 14625584.
13. Last, Y. (1993). "A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants". Comm. Math. Phys. 151 (1): 183–192. Bibcode:1993CMaPh.151..183L. doi:10.1007/BF02096752. S2CID 189834787.
14. Last, Y. (1994). "Zero measure spectrum for the almost Mathieu operator". Comm. Math. Phys. 164 (2): 421–432. Bibcode:1993CMaPh.151..183L. doi:10.1007/BF02096752. S2CID 189834787.
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| Wikipedia |
Ten-of-diamonds decahedron
In geometry, the ten-of-diamonds decahedron is a space-filling polyhedron with 10 faces, 2 opposite rhombi with orthogonal major axes, connected by 8 identical isosceles triangle faces. Although it is convex, it is not a Johnson solid because its faces are not composed entirely of regular polygons. Michael Goldberg named it after a playing card, as a 10-faced polyhedron with two opposite rhombic (diamond-shaped) faces. He catalogued it in a 1982 paper as 10-II, the second in a list of 26 known space-filling decahedra.[1]
Ten-of-diamonds decahedron
Faces8 triangles
2 rhombi
Edges16
Vertices8
Symmetry groupD2d, order 8
Dual polyhedronSkew-truncated tetragonal disphenoid
Propertiesspace-filling
Coordinates
If the space-filling polyhedron is placed in a 3-D coordinate grid, the coordinates for the 8 vertices can be given as: (0, ±2, −1), (±2, 0, 1), (±1, 0, −1), (0, ±1, 1).
Symmetry
The ten-of-diamonds has D2d symmetry, which projects as order-4 dihedral (square) symmetry in two dimensions. It can be seen as a triakis tetrahedron, with two pairs of coplanar triangles merged into rhombic faces. The dual is similar to a truncated tetrahedron, except two edges from the original tetrahedron are reduced to zero length making pentagonal faces. The dual polyhedra can be called a skew-truncated tetragonal disphenoid, where 2 edges along the symmetry axis completely truncated down to the edge midpoint.
Symmetric projection
Ten of diamonds Related Dual Related
Solid faces
Edges
triakis tetrahedron
Solid faces
Edges
Truncated tetrahedron
v=8, e=16, f=10 v=8, e=18, f=12 v=10, e=16, f=8 v=12, e=18, f=8
Honeycomb
Ten-of-diamonds honeycomb
Schläfli symboldht1,2{4,3,4}
Coxeter diagram
CellTen-of-diamonds
Vertex figuresdodecahedron
tetrahedron
Space
Fibrifold
Coxeter
I3 (204)
8−o
[[4,3+,4]]
DualAlternated bitruncated cubic honeycomb
PropertiesCell-transitive
The ten-of-diamonds is used in the honeycomb with Coxeter diagram , being the dual of an alternated bitruncated cubic honeycomb, . Since the alternated bitruncated cubic honeycomb fills space by pyritohedral icosahedra, , and tetragonal disphenoidal tetrahedra, vertex figures of this honeycomb are their duals – pyritohedra, and tetragonal disphenoids.
Cells can be seen as the cells of the tetragonal disphenoid honeycomb, , with alternate cells removed and augmented into neighboring cells by a center vertex. The rhombic faces in the honeycomb are aligned along 3 orthogonal planes.
Uniform Dual Alternated Dual alternated
t1,2{4,3,4}
dt1,2{4,3,4}
ht1,2{4,3,4}
dht1,2{4,3,4}
Bitruncated cubic honeycomb of truncated octahedral cells
tetragonal disphenoid honeycomb
Dual honeycomb of icosahedra and tetrahedra
Ten-of-diamonds honeycomb
Honeycomb structure orthogonally viewed along cubic plane
Related space-filling polyhedra
The ten-of-diamonds can be dissected in an octagonal cross-section between the two rhombic faces. It is a decahedron with 12 vertices, 20 edges, and 10 faces (4 triangles, 4 trapezoids, 1 rhombus, and 1 isotoxal octagon). Michael Goldberg labels this polyhedron 10-XXV, the 25th in a list of space-filling decahedra.[2]
The ten-of-diamonds can be dissected as a half-model on a symmetry plane into a space-filling heptahedron with 6 vertices, 11 edges, and 7 faces (6 triangles and 1 trapezoid). Michael Goldberg identifies this polyhedron as a triply truncated quadrilateral prism, type 7-XXIV, the 24th in a list of heptagonal space-fillers.[3]
It can be further dissected as a quarter-model by another symmetry plane into a space-filling hexahedron with 6 vertices, 10 edges, and 6 faces (4 triangles, 2 right trapezoids). Michael Goldberg identifies this polyhedron as an ungulated quadrilateral pyramid, type 6-X, the 10th in a list of space-filling hexahedron.[4]
Dissected models in symmetric projections
Relation Decahedral
half model
Heptahedral
half model
Hexahedral
quarter model
Symmetry C2v, order 4 Cs, order 2 C2, order 2
Edges
Net
Elements v=12, e=20, f=10 v=6, e=11, f=7 v=6, e=10, f=6
Rhombic bowtie
Rhombic bowtie
Faces16 triangles
2 rhombi
Edges28
Vertices12
Symmetry groupD2h, order 8
Propertiesspace-filling
Net
Pairs of ten-of-diamonds can be attached as a nonconvex bow-tie space-filler, called a rhombic bowtie for its cross-sectional appearance. The two right-most symmetric projections below show the rhombi edge-on on the top, bottom and a middle neck where the two halves are connected. The 2D projections can look convex or concave.
It has 12 vertices, 28 edges, and 18 faces (16 triangles and 2 rhombi) within D2h symmetry. These paired-cells stack more easily as inter-locking elements. Long sequences of these can be stacked together in 3 axes to fill space.[5]
The 12 vertex coordinates in a 2-unit cube. (further augmentations on the rhombi can be done with 2 unit translation in z.)
(0, ±1, −1), (±1, 0, 0), (0, ±1, 1),
(±1/2, 0, −1), (0, ±1/2, 0), (±1/2, 0, 1)
Bow-tie model (two ten-of-diamonds)
SkewSymmetric
See also
• Elongated gyrobifastigium
References
1. Goldberg, Michael. On the Space-filling Decahedra. Structural Topology, 1982, num. Type 10-II
2. On Space-filling Decahedra, type 10-XXV.
3. Goldberg, Michael On the space-filling heptahedra Geometriae Dedicata, June 1978, Volume 7, Issue 2, pp 175–184 PDF type 7-XXIV
4. Goldberg, Michael On the space-filling hexahedra Geom. Dedicata, June 1977, Volume 6, Issue 1, pp 99–108 PDF type 6-X
5. Robert Reid, Anthony Steed Bowties: A Novel Class of Space Filling Polyhedron 2003
• Koch 1972 Koch, Elke, Wirkungsbereichspolyeder und Wirkungsbereichsteilunger zukubischen Gitterkomplexen mit weniger als drei Freiheitsgraden (Efficiency Polyhedra, and Efficiency Dividers, cubic lattice complexes with less than three degrees of freedom) Dissertation, University Marburg/Lahn 1972 - Model 10/8–1, 28–404.
| Wikipedia |
Tendril perversion
Tendril perversion is a geometric phenomenon sometimes observed in helical structures in which the direction of the helix transitions between left-handed and right-handed.[1] [2] Such a reversal of chirality is commonly seen in helical plant tendrils and telephone handset cords.[3]
The phenomenon was known to Charles Darwin,[4] who wrote in 1865,
A tendril ... invariably becomes twisted in one part in one direction, and in another part in the opposite direction... This curious and symmetrical structure has been noticed by several botanists, but has not been sufficiently explained.[5]
The term "tendril perversion" was coined by Alain Goriely and Michael Tabor in 1998 based on the word perversion found in 19th-century science literature.[6][7] "Perversion" is a transition from one chirality to another and was known to James Clerk Maxwell, who attributed it to topologist J. B. Listing.[4][8]
Tendril perversion can be viewed as an example of spontaneous symmetry breaking, in which the strained structure of the tendril adopts a configuration of minimum energy while preserving zero overall twist.[2]
Tendril perversion has been studied both experimentally and theoretically. Gerbode et al. have made experimental studies of the coiling of cucumber tendrils.[9][10] A detailed study of a simple model of the physics of tendril perversion was made by McMillen and Goriely in the early 2000s.[2] Liu et al. showed in 2014 that "the transition from a helical to a hemihelical shape, as well as the number of perversions, depends on the height to width ratio of the strip's cross-section."[3]
Generalized tendril perversions were put forward by Silva et al., to include perversions that can be intrinsically produced in elastic filaments, leading to a multiplicity of geometries and dynamical properties.[11]
See also
• Helical growth
• Hemihelix
• Tentacle erotica
References
1. Goriely, Alain (2017). The mathematics and mechanics of biological growth. New York: Springer. ISBN 0-387-87709-6. OCLC 989037743.
2. McMillen; Goriely (2002). "Tendril Perversion in Intrinsically Curved Rods". Journal of Nonlinear Science. 12 (3): 241. Bibcode:2002JNS....12..241M. CiteSeerX 10.1.1.140.352. doi:10.1007/s00332-002-0493-1. S2CID 18480860.
3. Liu, J.; Huang, J.; Su, T.; Bertoldi, K.; Clarke, D. R. (2014). "Structural Transition from Helices to Hemihelices". PLOS ONE. 9 (4): e93183. Bibcode:2014PLoSO...993183L. doi:10.1371/journal.pone.0093183. PMC 3997338. PMID 24759785.
4. Alain Goriely (2013). "Inversion, Rotation, and Perversion in Mechanical Biology: From Microscopic Anisotropy to Macroscopic Chirality" (PDF). p. 9.
5. Charles Darwin, "On the movements and habits of climbing plants", Journal of the Linnean Society, 1865.
6. Goriely, Alain; Tabor, Michael (1998-02-16). "Spontaneous Helix Hand Reversal and Tendril Perversion in Climbing Plants". Physical Review Letters. American Physical Society (APS). 80 (7): 1564–1567. doi:10.1103/physrevlett.80.1564. ISSN 0031-9007.
7. McMillen; Goriely (2002). "Tendril Perversion in Intrinsically Curved Rods". Journal of Nonlinear Science. 12 (3): 241. Bibcode:2002JNS....12..241M. CiteSeerX 10.1.1.140.352. doi:10.1007/s00332-002-0493-1. S2CID 18480860.
8. James Clerk Maxwell (1873). A Treatise of Electricity and Magnetism. Oxford: Clarendon Press. The operation of passing from one system to the other is called by Listing, Perversion. The reflection of an object in a mirror image is a perverted image of the object.
9. Gerbode, S. J.; Puzey, J. R.; McCormick, A. G.; Mahadevan, L. (2012). "How the Cucumber Tendril Coils and Overwinds". Science. 337 (6098): 1087–91. Bibcode:2012Sci...337.1087G. doi:10.1126/science.1223304. PMID 22936777. S2CID 17405225.
10. Geraint Jones (30 August 2012). "Scientists unwind the secrets of climbing plants' tendrils". The Guardian.
11. Silva, Pedro E. S.; Trigueiros, Joao L.; Trindade, Ana C.; Simoes, Ricardo; Dias, Ricardo G.; Godinho, Maria Helena; Abreu, Fernao Vistulo de (2016-03-30). "Perversions with a twist". Scientific Reports. 6: 23413. Bibcode:2016NatSR...623413S. doi:10.1038/srep23413. PMC 4812244. PMID 27025549.
External links
Look up tendril perversion in Wiktionary, the free dictionary.
• Media related to Tendril perversion at Wikimedia Commons
• A close-up image of a tendril perversion in a tendril of Bryonia dioica by Michael Becker
| Wikipedia |
Tennenbaum's theorem
Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).
Recursive structures for PA
A structure $M$ in the language of PA is recursive if there are recursive functions $\oplus $ and $\otimes $ from $\mathbb {N} \times \mathbb {N} $ to $\mathbb {N} $, a recursive two-place relation <M on $\mathbb {N} $, and distinguished constants $n_{0},n_{1}$ such that
$(\mathbb {N} ,\oplus ,\otimes ,<_{M},n_{0},n_{1})\cong M,$
where $\cong $ indicates isomorphism and $\mathbb {N} $ is the set of (standard) natural numbers. Because the isomorphism must be a bijection, every recursive model is countable. There are many nonisomorphic countable nonstandard models of PA.
Statement of the theorem
Tennenbaum's theorem states that no countable nonstandard model of PA is recursive. Moreover, neither the addition nor the multiplication of such a model can be recursive.
Proof sketch
This sketch follows the argument presented by Kaye (1991). The first step in the proof is to show that, if M is any countable nonstandard model of PA, then the standard system of M (defined below) contains at least one nonrecursive set S. The second step is to show that, if either the addition or multiplication operation on M were recursive, then this set S would be recursive, which is a contradiction.
Through the methods used to code ordered tuples, each element $x\in M$ can be viewed as a code for a set $S_{x}$ of elements of M. In particular, if we let $p_{i}$ be the ith prime in M, then $z\in S_{x}\leftrightarrow M\vDash p_{z}|x$. Each set $S_{x}$ will be bounded in M, but if x is nonstandard then the set $S_{x}$ may contain infinitely many standard natural numbers. The standard system of the model is the collection $\{S_{x}\cap \mathbb {N} :x\in M\}$. It can be shown that the standard system of any nonstandard model of PA contains a nonrecursive set, either by appealing to the incompleteness theorem or by directly considering a pair of recursively inseparable r.e. sets (Kaye 1991:154). These are disjoint r.e. sets $A,B\subseteq \mathbb {N} $ so that there is no recursive set $C\subseteq \mathbb {N} $ with $A\subseteq C$ and $B\cap C=\emptyset $.
For the latter construction, begin with a pair of recursively inseparable r.e. sets A and B. For natural number x there is a y such that, for all i < x, if $i\in A$ then $p_{i}|y$ and if $i\in B$ then $p_{i}\nmid y$. By the overspill property, this means that there is some nonstandard x in M for which there is a (necessarily nonstandard) y in M so that, for every $m\in M$ with $m<_{M}x$, we have
$M\vDash (m\in A\to p_{m}|y)\land (m\in B\to p_{m}\nmid y)$
Let $S=\mathbb {N} \cap S_{y}$ be the corresponding set in the standard system of M. Because A and B are r.e., one can show that $A\subseteq S$ and $B\cap S=\emptyset $. Hence S is a separating set for A and B, and by the choice of A and B this means S is nonrecursive.
Now, to prove Tennenbaum's theorem, begin with a nonstandard countable model M and an element a in M so that $S=\mathbb {N} \cap S_{a}$ is nonrecursive. The proof method shows that, because of the way the standard system is defined, it is possible to compute the characteristic function of the set S using the addition function $\oplus $ of M as an oracle. In particular, if $n_{0}$ is the element of M corresponding to 0, and $n_{1}$ is the element of M corresponding to 1, then for each $i\in \mathbb {N} $ we can compute $n_{i}=n_{1}\oplus \cdots \oplus n_{1}$ (i times). To decide if a number n is in S, first compute p, the nth prime in $\mathbb {N} $. Then, search for an element y of M so that
$a=\underbrace {y\oplus y\oplus \cdots \oplus y} _{p{\text{ times}}}\oplus n_{i}$
for some $i<p$. This search will halt because the Euclidean algorithm can be applied to any model of PA. Finally, we have $n\in S$ if and only if the i found in the search was 0. Because S is not recursive, this means that the addition operation on M is nonrecursive.
A similar argument shows that it is possible to compute the characteristic function of S using the multiplication of M as an oracle, so the multiplication operation on M is also nonrecursive (Kaye 1991:154).
References
• Boolos, George; Burgess, John P.; Jeffrey, Richard (2002). Computability and Logic (4th ed.). Cambridge University Press. ISBN 0-521-00758-5.
• Kaye, Richard (1991). Models of Peano arithmetic. Oxford University Press. ISBN 0-19-853213-X.
• Kaye, Richard (Sep 2011). "Tennenbaum's Theorem for Models of Arithmetic". In Juliette Kennedy and Roman Kossak (ed.). Set theory, arithmetic, and foundations of mathematics - theorems, philosophies (PDF). Lecture Notes in Logic. Vol. 36. ISBN 9781107008045.
• Tennenbaum, Stanley (1959). "Non-Archimedean models for arithmetic". Notices of the American Mathematical Society. 6: 270.
| Wikipedia |
Tennis ball theorem
In geometry, the tennis ball theorem states that any smooth curve on the surface of a sphere that divides the sphere into two equal-area subsets without touching or crossing itself must have at least four inflection points, points at which the curve does not consistently bend to only one side of its tangent line.[1] The tennis ball theorem was first published under this name by Vladimir Arnold in 1994,[2][3] and is often attributed to Arnold, but a closely related result appears earlier in a 1968 paper by Beniamino Segre, and the tennis ball theorem itself is a special case of a theorem in a 1977 paper by Joel L. Weiner.[4][5] The name of the theorem comes from the standard shape of a tennis ball, whose seam forms a curve that meets the conditions of the theorem; the same kind of curve is also used for the seams on baseballs.[1]
The tennis ball theorem can be generalized to any curve that is not contained in a closed hemisphere. A centrally symmetric curve on the sphere must have at least six inflection points. The theorem is analogous to the four-vertex theorem according to which any smooth closed plane curve has at least four points of extreme curvature.
Statement
Precisely, an inflection point of a doubly continuously differentiable ($C^{2}$) curve on the surface of a sphere is a point $p$ with the following property: let $I$ be the connected component containing $p$ of the intersection of the curve with its tangent great circle at $p$. (For most curves $I$ will just be $p$ itself, but it could also be an arc of the great circle.) Then, for $p$ to be an inflection point, every neighborhood of $I$ must contain points of the curve that belong to both of the hemispheres separated by this great circle. The theorem states that every $C^{2}$ curve that partitions the sphere into two equal-area components has at least four inflection points in this sense.[6]
Examples
The tennis ball and baseball seams can be modeled mathematically by a curve made of four semicircular arcs, with exactly four inflection points where pairs of these arcs meet.[7] A great circle also bisects the sphere's surface, and has infinitely many inflection points, one at each point of the curve. However, the condition that the curve divide the sphere's surface area equally is a necessary part of the theorem. Other curves that do not divide the area equally, such as circles that are not great circles, may have no inflection points at all.[1]
Proof by curve shortening
One proof of the tennis ball theorem uses the curve-shortening flow, a process for continuously moving the points of the curve towards their local centers of curvature. Applying this flow to the given curve can be shown to preserve the smoothness and area-bisecting property of the curve. Additionally, as the curve flows, its number of inflection points never increases. This flow eventually causes the curve to transform into a great circle, and the convergence to this circle can be approximated by a Fourier series. Because curve-shortening does not change any other great circle, the first term in this series is zero, and combining this with a theorem of Sturm on the number of zeros of Fourier series shows that, as the curve nears this great circle, it has at least four inflection points. Therefore, the original curve also has at least four inflection points.[8][9]
Related theorems
A generalization of the tennis ball theorem applies to any simple smooth curve on the sphere that is not contained in a closed hemisphere. As in the original tennis ball theorem, such curves must have at least four inflection points.[5][10] If a curve on the sphere is centrally symmetric, it must have at least six inflection points.[10]
A closely related theorem of Segre (1968) also concerns simple closed spherical curves, on spheres embedded into three-dimensional space. If, for such a curve, $o$ is any point of the three-dimensional convex hull of a smooth curve on the sphere that is not a vertex of the curve, then at least four points of the curve have osculating planes passing through $o$. In particular, for a curve not contained in a hemisphere, this theorem can be applied with $o$ at the center of the sphere. Every inflection point of a spherical curve has an osculating plane that passes through the center of the sphere, but this might also be true of some other points.[4][5]
This theorem is analogous to the four-vertex theorem, that every smooth simple closed curve in the plane has four vertices (extreme points of curvature). It is also analogous to a theorem of August Ferdinand Möbius that every non-contractible smooth curve in the projective plane has at least three inflection points.[2][9]
References
1. Chamberland, Marc (2015), "The Tennis Ball Theorem", Single digits: In praise of small numbers, Princeton University Press, Princeton, NJ, p. 114, doi:10.1515/9781400865697, ISBN 978-0-691-16114-3, MR 3328722
2. Martinez-Maure, Yves (1996), "A note on the tennis ball theorem", American Mathematical Monthly, 103 (4): 338–340, doi:10.2307/2975192, MR 1383672
3. Arnol'd, V. I. (1994), "20. The tennis ball theorem", Topological invariants of plane curves and caustics, University Lecture Series, vol. 5, Providence, RI: American Mathematical Society, pp. 53–58, doi:10.1090/ulect/005, ISBN 0-8218-0308-5, MR 1286249
4. Segre, Beniamino (1968), "Alcune proprietà differenziali in grande delle curve chiuse sghembe", Rendiconti di Matematica, 1: 237–297, MR 0243466
5. Weiner, Joel L. (1977), "Global properties of spherical curves", Journal of Differential Geometry, 12 (3): 425–434, MR 0514446. For the tennis ball theorem (applicable more generally to curves that are not contained in a single hemisphere), see Theorem 2, p. 427
6. Thorbergsson, Gudlaugur; Umehara, Masaaki (1999), "A unified approach to the four vertex theorems II", in Tabachnikov, Serge (ed.), Differential and Symplectic Topology of Knots and Curves, Amer. Math. Soc. Transl. Ser. 2, vol. 190, Amer. Math. Soc., Providence, RI, pp. 229–252, doi:10.1090/trans2/190/12, MR 1738398. See in particular pp. 242–243.
7. Juillet, Nicolas (April 5, 2013), "Voyage sur une balle de tennis", Images des mathématiques (in French), CNRS
8. Ovsienko, V.; Tabachnikov, S. (2005), Projective differential geometry old and new: From the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge Tracts in Mathematics, vol. 165, Cambridge: Cambridge University Press, p. 101, ISBN 0-521-83186-5, MR 2177471
9. Angenent, S. (1999), "Inflection points, extatic points and curve shortening" (PDF), Hamiltonian systems with three or more degrees of freedom (S'Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, Dordrecht: Kluwer Acad. Publ., pp. 3–10, MR 1720878
10. Pak, Igor (April 20, 2010), "Theorems 21.22–21.24, p. 203", Lectures on Discrete and Polyhedral Geometry
External links
• Weisstein, Eric W., "Tennis Ball Theorem", MathWorld
| Wikipedia |
Tensor-hom adjunction
In mathematics, the tensor-hom adjunction is that the tensor product $-\otimes X$ and hom-functor $\operatorname {Hom} (X,-)$ form an adjoint pair:
$\operatorname {Hom} (Y\otimes X,Z)\cong \operatorname {Hom} (Y,\operatorname {Hom} (X,Z)).$
This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.
General statement
Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):
${\mathcal {C}}=\mathrm {Mod} _{S}\quad {\text{and}}\quad {\mathcal {D}}=\mathrm {Mod} _{R}.$
Fix an $(R,S)$-bimodule $X$ and define functors $F\colon {\mathcal {D}}\rightarrow {\mathcal {C}}$ and $G\colon {\mathcal {C}}\rightarrow {\mathcal {D}}$ as follows:
$F(Y)=Y\otimes _{R}X\quad {\text{for }}Y\in {\mathcal {D}}$
$G(Z)=\operatorname {Hom} _{S}(X,Z)\quad {\text{for }}Z\in {\mathcal {C}}$
Then $F$ is left adjoint to $G$. This means there is a natural isomorphism
$\operatorname {Hom} _{S}(Y\otimes _{R}X,Z)\cong \operatorname {Hom} _{R}(Y,\operatorname {Hom} _{S}(X,Z)).$
This is actually an isomorphism of abelian groups. More precisely, if $Y$ is an $(A,R)$-bimodule and $Z$ is a $(B,S)$-bimodule, then this is an isomorphism of $(B,A)$-bimodules. This is one of the motivating examples of the structure in a closed bicategory.[1]
Counit and unit
Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. Using the notation from the previous section, the counit
$\varepsilon :FG\to 1_{\mathcal {C}}$
has components
$\varepsilon _{Z}:\operatorname {Hom} _{S}(X,Z)\otimes _{R}X\to Z$
given by evaluation: For
$\phi \in \operatorname {Hom} _{S}(X,Z)\quad {\text{and}}\quad x\in X,$
$\varepsilon (\phi \otimes x)=\phi (x).$
The components of the unit
$\eta :1_{\mathcal {D}}\to GF$
$\eta _{Y}:Y\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)$
are defined as follows: For $y$ in $Y$,
$\eta _{Y}(y)\in \operatorname {Hom} _{S}(X,Y\otimes _{R}X)$
is a right $S$-module homomorphism given by
$\eta _{Y}(y)(t)=y\otimes t\quad {\text{for }}t\in X.$
The counit and unit equations can now be explicitly verified. For $Y$ in ${\mathcal {D}}$,
$\varepsilon _{FY}\circ F(\eta _{Y}):Y\otimes _{R}X\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)\otimes _{R}X\to Y\otimes _{R}X$
is given on simple tensors of $Y\otimes X$ by
$\varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes x.$
Likewise,
$G(\varepsilon _{Z})\circ \eta _{GZ}:\operatorname {Hom} _{S}(X,Z)\to \operatorname {Hom} _{S}(X,\operatorname {Hom} _{S}(X,Z)\otimes _{R}X)\to \operatorname {Hom} _{S}(X,Z).$
For $\phi $ in $\operatorname {Hom} _{S}(X,Z)$,
$G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )$
is a right $S$-module homomorphism defined by
$G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )(x)=\varepsilon _{Z}(\phi \otimes x)=\phi (x)$
and therefore
$G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )=\phi .$
The Ext and Tor functors
The Hom functor $\hom(X,-)$ commutes with arbitrary limits, while the tensor product $-\otimes X$ functor commutes with arbitrary colimits that exist in their domain category. However, in general, $\hom(X,-)$ fails to commute with colimits, and $-\otimes X$ fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.
See also
• Currying
• Ext functor
• Tor functor
• Change of rings
References
1. May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5.
• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
Tensor (intrinsic definition)
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
For an introduction to the nature and significance of tensors in a broad context, see Tensor.
In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally.
Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article.
Definition via tensor products of vector spaces
Given a finite set { V1, ..., Vn } of vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, an element of which is termed a tensor.
A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:
$V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}$
where V∗ is the dual space of V.
If there are m copies of V and n copies of V∗ in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant order n and total order m + n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V∗ (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m, n) is denoted
$T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.$
Example 1. The space of type (1, 1) tensors, $T_{1}^{1}(V)=V\otimes V^{*},$ is isomorphic in a natural way to the space of linear transformations from V to V.
Example 2. A bilinear form on a real vector space V, $V\times V\to F,$ corresponds in a natural way to a type (0, 2) tensor in $T_{2}^{0}(V)=V^{*}\otimes V^{*}.$ An example of such a bilinear form may be defined, termed the associated metric tensor, and is usually denoted g.
Tensor rank
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor (Hackbusch 2012, pp. 4)) is a tensor that can be written as a product of tensors of the form
$T=a\otimes b\otimes \cdots \otimes d$
where a, b, ..., d are nonzero and in V or V∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors. The rank of a tensor T is the minimum number of simple tensors that sum to T (Bourbaki 1989, II, §7, no. 8).
The zero tensor has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is dn−1 when each product is of n vectors from a finite-dimensional vector space of dimension d.
The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:
$A=vw^{\mathrm {T} }.$
The rank of a matrix A is the smallest number of such outer products that can be summed to produce it:
$A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.$
In indices, a tensor of rank 1 is a tensor of the form
$T_{ij\dots }^{k\ell \dots }=a_{i}b_{j}\cdots c^{k}d^{\ell }\cdots .$
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix (Halmos 1974, §51), and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very hard to determine, and low rank decompositions of tensors are sometimes of great practical interest (de Groote 1987). Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms
$z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}$
for given inputs xi and yj. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known (Knuth 1998, pp. 506–508).
Universal property
The space $T_{n}^{m}(V)$ can be characterized by a universal property in terms of multilinear mappings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for free modules, and the "universal" approach carries over more easily to more general situations.
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces
$f:V_{1}\times \cdots \times V_{N}\to F$
is multilinear if it is linear in each argument. The space of all multilinear mappings from V1 × ... × VN to W is denoted LN(V1, ..., VN; W). When N = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V; W).
The universal characterization of the tensor product implies that, for each multilinear function
$f\in L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};W)$
(where $W$ can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function
$T_{f}\in L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};W)$
such that
$f(\alpha _{1},\ldots ,\alpha _{m},v_{1},\ldots ,v_{n})=T_{f}(\alpha _{1}\otimes \cdots \otimes \alpha _{m}\otimes v_{1}\otimes \cdots \otimes v_{n})$
for all $v_{i}\in V$ and $\alpha _{i}\in V^{*}.$
Using the universal property, it follows that the space of (m,n)-tensors admits a natural isomorphism
$T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};F)\cong L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};F).$
Each V in the definition of the tensor corresponds to a V* inside the argument of the linear maps, and vice versa. (Note that in the former case, there are m copies of V and n copies of V*, and in the latter case vice versa). In particular, one has
${\begin{aligned}T_{0}^{1}(V)&\cong L(V^{*};F)\cong V\\T_{1}^{0}(V)&\cong L(V;F)=V^{*}\\T_{1}^{1}(V)&\cong L(V;V)\end{aligned}}$
Tensor fields
Main article: tensor field
Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.
References
• Abraham, Ralph; Marsden, Jerrold E. (1985), Foundations of Mechanics (2 ed.), Reading, Mass.: Addison-Wesley, ISBN 0-201-40840-6.
• Bourbaki, Nicolas (1989), Elements of Mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9.
• de Groote, H. F. (1987), Lectures on the Complexity of Bilinear Problems, Lecture Notes in Computer Science, vol. 245, Springer, ISBN 3-540-17205-X.
• Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4.
• Jeevanjee, Nadir (2011), "An Introduction to Tensors and Group Theory for Physicists", Physics Today, 65 (4): 64, Bibcode:2012PhT....65d..64P, doi:10.1063/PT.3.1523, ISBN 978-0-8176-4714-8
• Knuth, Donald E. (1998) [1969], The Art of Computer Programming vol. 2 (3rd ed.), pp. 145–146, ISBN 978-0-201-89684-8.
• Hackbusch, Wolfgang (2012), Tensor Spaces and Numerical Tensor Calculus, Springer, p. 4, ISBN 978-3-642-28027-6.
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
This article is about tensors on a single vector space. It is not to be confused with Vector field or Tensor field.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.[1]
Definition
Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.
As multidimensional arrays
A tensor may be represented as an (potentially multidimensional) array. Just as a vector in an n-dimensional space is represented by a one-dimensional array with n components with respect to a given basis, any tensor with respect to a basis is represented by a multidimensional array. For example, a linear operator is represented in a basis as a two-dimensional square n × n array. The numbers in the multidimensional array are known as the components of the tensor. They are denoted by indices giving their position in the array, as subscripts and superscripts, following the symbolic name of the tensor. For example, the components of an order 2 tensor T could be denoted Tij , where i and j are indices running from 1 to n, or also by T i
j
. Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. Thus while Tij and T i
j
can both be expressed as n-by-n matrices, and are numerically related via index juggling, the difference in their transformation laws indicates it would be improper to add them together.
The total number of indices (m) required to identify each component uniquely is equal to the dimension or the number of ways of an array, which is why an array is sometimes referred to as an m-dimensional array or an m-way array. The total number of indices is also called the order, degree or rank of a tensor,[2][3][4] although the term "rank" generally has another meaning in the context of matrices and tensors.
Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see Covariance and contravariance of vectors), where the new basis vectors $\mathbf {\hat {e}} _{i}$ are expressed in terms of the old basis vectors $\mathbf {e} _{j}$ as,
$\mathbf {\hat {e}} _{i}=\sum _{j=1}^{n}\mathbf {e} _{j}R_{i}^{j}=\mathbf {e} _{j}R_{i}^{j}.$
Here R ji are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article.[Note 1] The components vi of a column vector v transform with the inverse of the matrix R,
${\hat {v}}^{i}=\left(R^{-1}\right)_{j}^{i}v^{j},$
where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse of the change of basis. In contrast, the components, wi, of a covector (or row vector), w, transform with the matrix R itself,
${\hat {w}}_{i}=w_{j}R_{i}^{j}.$
This is called a covariant transformation law, because the covector components transform by the same matrix as the change of basis matrix. The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). If the transformation matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript).
As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array $T$ that transforms under a change of basis matrix $R=\left(R_{i}^{j}\right)$ by ${\hat {T}}=R^{-1}TR$. For the individual matrix entries, this transformation law has the form ${\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}$ so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1).
Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above:
$\mathbf {v} ={\hat {v}}^{i}\,\mathbf {\hat {e}} _{i}=\left(\left(R^{-1}\right)_{j}^{i}{v}^{j}\right)\left(\mathbf {e} _{k}R_{i}^{k}\right)=\left(\left(R^{-1}\right)_{j}^{i}R_{i}^{k}\right){v}^{j}\mathbf {e} _{k}=\delta _{j}^{k}{v}^{j}\mathbf {e} _{k}={v}^{k}\,\mathbf {e} _{k}={v}^{i}\,\mathbf {e} _{i}$,
where $\delta _{j}^{k}$ is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices (j into k in this example). This shows several features of the component notation: the ability to re-arrange terms at will (commutativity), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like ${v}^{i}\,\mathbf {e} _{i}$ can immediately be seen to be geometrically identical in all coordinate systems.
Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. That is, the components $(Tv)^{i}$ are given by $(Tv)^{i}=T_{j}^{i}v^{j}$. These components transform contravariantly, since
$\left({\widehat {Tv}}\right)^{i'}={\hat {T}}_{j'}^{i'}{\hat {v}}^{j'}=\left[\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}\right]\left[\left(R^{-1}\right)_{k}^{j'}v^{k}\right]=\left(R^{-1}\right)_{i}^{i'}(Tv)^{i}.$
The transformation law for an order p + q tensor with p contravariant indices and q covariant indices is thus given as,
${\hat {T}}_{j'_{1},\ldots ,j'_{q}}^{i'_{1},\ldots ,i'_{p}}=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}$ $T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}$ $R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.$
Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or type (p, q). The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalization in other definitions), p + q in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type (p, q) is also called a (p, q)-tensor for short.
This discussion motivates the following formal definition:[5][6]
Definition. A tensor of type (p, q) is an assignment of a multidimensional array
$T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}[\mathbf {f} ]$
to each basis f = (e1, ..., en) of an n-dimensional vector space such that, if we apply the change of basis
$\mathbf {f} \mapsto \mathbf {f} \cdot R=\left(\mathbf {e} _{i}R_{1}^{i},\dots ,\mathbf {e} _{i}R_{n}^{i}\right)$
then the multidimensional array obeys the transformation law
$T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}$ $T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]$ $R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.$
The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.[1]
An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space. If $\mathbf {f} =(\mathbf {f} _{1},\dots ,\mathbf {f} _{n})$ is an ordered basis, and $R=\left(R_{j}^{i}\right)$ is an invertible $n\times n$ matrix, then the action is given by
$\mathbf {f} R=\left(\mathbf {f} _{i}R_{1}^{i},\dots ,\mathbf {f} _{i}R_{n}^{i}\right).$
Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL(n). Let W be a vector space and let $\rho $ be a representation of GL(n) on W (that is, a group homomorphism $\rho :{\text{GL}}(n)\to {\text{GL}}(W)$ :{\text{GL}}(n)\to {\text{GL}}(W)} ). Then a tensor of type $\rho $ is an equivariant map $T:F\to W$. Equivariance here means that
$T(FR)=\rho \left(R^{-1}\right)T(F).$
When $\rho $ is a tensor representation of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds,[7] and readily generalizes to other groups.[5]
As multilinear maps
Main article: Multilinear map
A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V, which is usually taken to be a particular vector space of some geometrical significance like the tangent space to a manifold.[8] In this approach, a type (p, q) tensor T is defined as a multilinear map,
$T:\underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ copies}}}\times \underbrace {V\times \dots \times V} _{q{\text{ copies}}}\rightarrow \mathbf {R} ,$
where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the real numbers, ℝ. More generally, V can be taken over any field F (e.g. the complex numbers), with F replacing ℝ as the codomain of the multilinear maps.
By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗,
$T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\equiv T\left({\boldsymbol {\varepsilon }}^{i_{1}},\ldots ,{\boldsymbol {\varepsilon }}^{i_{p}},\mathbf {e} _{j_{1}},\ldots ,\mathbf {e} _{j_{q}}\right),$
a (p + q)-dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.
In viewing a tensor as a multilinear map, it is conventional to identify the double dual V∗∗ of the vector space V, i.e., the space of linear functionals on the dual vector space V∗, with the vector space V. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V∗ against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual.
Using tensor products
Main article: Tensor (intrinsic definition)
For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property as explained here and here.
A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces,[9][10]
$T\in \underbrace {V\otimes \dots \otimes V} _{p{\text{ copies}}}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{q{\text{ copies}}}.$
A basis vi of V and basis wj of W naturally induce a basis vi ⊗ wj of the tensor product V ⊗ W. The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {ei} for V and its dual basis {εj}, i.e.
$T=T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\;\mathbf {e} _{i_{1}}\otimes \cdots \otimes \mathbf {e} _{i_{p}}\otimes {\boldsymbol {\varepsilon }}^{j_{1}}\otimes \cdots \otimes {\boldsymbol {\varepsilon }}^{j_{q}}.$
Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor. Moreover, the universal property of the tensor product gives a one-to-one correspondence between tensors defined in this way and tensors defined as multilinear maps.
This 1 to 1 correspondence can be archived the following way, because in the finite dimensional case there exists a canonical isomorphism between a vectorspace and its double dual:
$U\otimes V\cong \left(U^{**}\right)\otimes \left(V^{**}\right)\cong \left(U^{*}\otimes V^{*}\right)^{*}\cong \operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)$
The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from $\operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)$ and $\operatorname {Hom} \left(U^{*}\otimes V^{*};\mathbb {F} \right)$.[11]
Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space V and its dual, as above.
Tensors in infinite dimensions
This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are naturally isomorphic.[Note 2] Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to vector bundles or coherent sheaves.[12] For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see topological tensor product). In some applications, it is the tensor product of Hilbert spaces that is intended, whose properties are the most similar to the finite-dimensional case. A more modern view is that it is the tensors' structure as a symmetric monoidal category that encodes their most important properties, rather than the specific models of those categories.[13]
Tensor fields
Main article: Tensor field
In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space. This was the setting of Ricci's original work. In modern mathematical terminology such an object is called a tensor field, often referred to simply as a tensor.[1]
In this context, a coordinate basis is often chosen for the tangent vector space. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions,
${\bar {x}}^{i}\left(x^{1},\ldots ,x^{n}\right),$
defining a coordinate transformation,[1]
${\hat {T}}_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}\left({\bar {x}}^{1},\ldots ,{\bar {x}}^{n}\right)={\frac {\partial {\bar {x}}^{i'_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial {\bar {x}}^{i'_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial {\bar {x}}^{j'_{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial {\bar {x}}^{j'_{q}}}}T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\left(x^{1},\ldots ,x^{n}\right).$
History
The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century.[14] The word "tensor" itself was introduced in 1846 by William Rowan Hamilton[15] to describe something different from what is now meant by a tensor.[Note 3] Gibbs introduced Dyadics and Polyadic algebra, which are also tensors in the modern sense.[16] The contemporary usage was introduced by Woldemar Voigt in 1898.[17]
Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci-Curbastro in 1892.[18] It was made accessible to many mathematicians by the publication of Ricci-Curbastro and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications).[19] In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense. [16]
In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann.[20] Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect:
I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.
— Albert Einstein[21]
Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics, and Hassler Whitney popularized the tensor product. [16]
From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem).[22] Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. Multilinear algebra can be developed in greater generality than for scalars coming from a field. For example, scalars can come from a ring. But the theory is then less geometric and computations more technical and less algorithmic.[23] Tensors are generalized within category theory by means of the concept of monoidal category, from the 1960s.[24]
Examples
See also: Dyadic tensor
An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol $\varepsilon _{ijk}$ nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems.
This table shows important examples of tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, M)-entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor.
Example tensors on vector spaces and tensor fields on manifolds
m
0 1 2 3 ⋯ M ⋯
n 0 Scalar, e.g. scalar curvature Covector, linear functional, 1-form, e.g. dipole moment, gradient of a scalar field Bilinear form, e.g. inner product, quadrupole moment, metric tensor, Ricci curvature, 2-form, symplectic form 3-form E.g. octupole moment E.g. M-form i.e. volume form
1 Euclidean vector Linear transformation,[25] Kronecker delta E.g. cross product in three dimensions E.g. Riemann curvature tensor
2 Inverse metric tensor, bivector, e.g., Poisson structure E.g. elasticity tensor
⋮
N Multivector
⋮
Raising an index on an (n, m)-tensor produces an (n + 1, m − 1)-tensor; this corresponds to moving diagonally down and to the left on the table. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. Contraction of an upper with a lower index of an (n, m)-tensor produces an (n − 1, m − 1)-tensor; this corresponds to moving diagonally up and to the left on the table.
Orientation defined by an ordered set of vectors.
Reversed orientation corresponds to negating the exterior product.
Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its n − 1-dimensional boundary and on which side the interior is.[26][27]
Properties
Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. Changing the basis transforms the values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a tensor. Compare this to the array representing $\varepsilon _{ijk}$ not being a tensor, for the sign change under transformations changing the orientation.
Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, vectors: n (contravariant indices) and dual vectors: m (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law. The order of a tensor is the sum of these two numbers.
The order (also degree or rank) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices needed to label each component in that array. For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stress tensor, taking one vector and returning another 1 + 1 = 2. The $\varepsilon _{ijk}$-symbol, mapping two vectors to one vector, would have order 2 + 1 = 3.
The collection of tensors on a vector space and its dual forms a tensor algebra, which allows products of arbitrary tensors. Simple applications of tensors of order 2, which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product should not be confused with this.
Notation
There are several notational systems that are used to describe tensors and perform calculations involving them.
Ricci calculus
Ricci calculus is the modern formalism and notation for tensor indices: indicating inner and outer products, covariance and contravariance, summations of tensor components, symmetry and antisymmetry, and partial and covariant derivatives.
Einstein summation convention
The Einstein summation convention dispenses with writing summation signs, leaving the summation implicit. Any repeated index symbol is summed over: if the index i is used twice in a given term of a tensor expression, it means that the term is to be summed for all i. Several distinct pairs of indices may be summed this way.
Penrose graphical notation
Penrose graphical notation is a diagrammatic notation which replaces the symbols for tensors with shapes, and their indices by lines and curves. It is independent of basis elements, and requires no symbols for the indices.
Abstract index notation
The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. This notation captures the expressiveness of indices and the basis-independence of index-free notation.
Component-free notation
A component-free treatment of tensors uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the tensor product of vector spaces.
Operations
There are several operations on tensors that again produce a tensor. The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the scaling of a vector. On components, these operations are simply performed component-wise. These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type.
Tensor product
Main article: Tensor product
The tensor product takes two tensors, S and T, and produces a new tensor, S ⊗ T, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.,
$(S\otimes T)(v_{1},\ldots ,v_{n},v_{n+1},\ldots ,v_{n+m})=S(v_{1},\ldots ,v_{n})T(v_{n+1},\ldots ,v_{n+m}),$
which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, i.e.,
$(S\otimes T)_{j_{1}\ldots j_{k}j_{k+1}\ldots j_{k+m}}^{i_{1}\ldots i_{l}i_{l+1}\ldots i_{l+n}}=S_{j_{1}\ldots j_{k}}^{i_{1}\ldots i_{l}}T_{j_{k+1}\ldots j_{k+m}}^{i_{l+1}\ldots i_{l+n}}.$
If S is of type (l, k) and T is of type (n, m), then the tensor product S ⊗ T has type (l + n, k + m).
Contraction
Main article: Tensor contraction
Tensor contraction is an operation that reduces a type (n, m) tensor to a type (n − 1, m − 1) tensor, of which the trace is a special case. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Components for which those two indices are different are discarded. For example, a (1, 1)-tensor $T_{i}^{j}$ can be contracted to a scalar through $T_{i}^{i}$. Where the summation is again implied. When the (1, 1)-tensor is interpreted as a linear map, this operation is known as the trace.
The contraction is often used in conjunction with the tensor product to contract an index from each tensor.
The contraction can also be understood using the definition of a tensor as an element of a tensor product of copies of the space V with the space V∗ by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from V∗ to a factor from V. For example, a tensor $T\in V\otimes V\otimes V^{*}$ can be written as a linear combination
$T=v_{1}\otimes w_{1}\otimes \alpha _{1}+v_{2}\otimes w_{2}\otimes \alpha _{2}+\cdots +v_{N}\otimes w_{N}\otimes \alpha _{N}.$
The contraction of T on the first and last slots is then the vector
$\alpha _{1}(v_{1})w_{1}+\alpha _{2}(v_{2})w_{2}+\cdots +\alpha _{N}(v_{N})w_{N}.$
In a vector space with an inner product (also known as a metric) g, the term contraction is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse. For example, a (2, 0)-tensor $T^{ij}$ can be contracted to a scalar through $T^{ij}g_{ij}$ (yet again assuming the summation convention).
Raising or lowering an index
Main article: Raising and lowering indices
When a vector space is equipped with a nondegenerate bilinear form (or metric tensor as it is often called in this context), operations can be defined that convert a contravariant (upper) index into a covariant (lower) index and vice versa. A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index. This operation is quite graphically known as lowering an index.
Conversely, the inverse operation can be defined, and is called raising an index. This is equivalent to a similar contraction on the product with a (2, 0)-tensor. This inverse metric tensor has components that are the matrix inverse of those of the metric tensor.
Applications
Continuum mechanics
Important examples are provided by continuum mechanics. The stresses inside a solid body or fluid[28] are described by a tensor field. The stress tensor and strain tensor are both second-order tensor fields, and are related in a general linear elastic material by a fourth-order elasticity tensor field. In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently represented as a 3 × 3 array. The three faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number. Thus, 3 × 3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment. Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, a second-order tensor is needed.
If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2, 0), in linear elasticity, or more precisely by a tensor field of type (2, 0), since the stresses may vary from point to point.
Other examples from physics
Common applications include:
• Electromagnetic tensor (or Faraday tensor) in electromagnetism
• Finite deformation tensors for describing deformations and strain tensor for strain in continuum mechanics
• Permittivity and electric susceptibility are tensors in anisotropic media
• Four-tensors in general relativity (e.g. stress–energy tensor), used to represent momentum fluxes
• Spherical tensor operators are the eigenfunctions of the quantum angular momentum operator in spherical coordinates
• Diffusion tensors, the basis of diffusion tensor imaging, represent rates of diffusion in biological environments
• Quantum mechanics and quantum computing utilize tensor products for combination of quantum states
Computer vision and optics
The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix.
The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:
${\frac {P_{i}}{\varepsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots .\!$
Here $\chi ^{(1)}$ is the linear susceptibility, $\chi ^{(2)}$ gives the Pockels effect and second harmonic generation, and $\chi ^{(3)}$ gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.
Machine learning
The properties of Tensors (machine learning), especially tensor decomposition, have enabled their use in machine learning to embed higher dimensional data in artificial neural networks.
Generalizations
Tensor products of vector spaces
The vector spaces of a tensor product need not be the same, and sometimes the elements of such a more general tensor product are called "tensors". For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense,[29] and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces.[30] A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more general sense. The concept of tensor product can be extended to arbitrary modules over a ring.
Tensors in infinite dimensions
The notion of a tensor can be generalized in a variety of ways to infinite dimensions. One, for instance, is via the tensor product of Hilbert spaces.[31] Another way of generalizing the idea of tensor, common in nonlinear analysis, is via the multilinear maps definition where instead of using finite-dimensional vector spaces and their algebraic duals, one uses infinite-dimensional Banach spaces and their continuous dual.[32] Tensors thus live naturally on Banach manifolds[33] and Fréchet manifolds.
Tensor densities
Main article: Tensor density
Suppose that a homogeneous medium fills R3, so that the density of the medium is described by a single scalar value ρ in kg⋅m−3. The mass, in kg, of a region Ω is obtained by multiplying ρ by the volume of the region Ω, or equivalently integrating the constant ρ over the region:
$m=\int _{\Omega }\rho \,dx\,dy\,dz,$
where the Cartesian coordinates x, y, z are measured in m. If the units of length are changed into cm, then the numerical values of the coordinate functions must be rescaled by a factor of 100:
$x'=100x,\quad y'=100y,\quad z'=100z.$
The numerical value of the density ρ must then also transform by 100−3 m3/cm3 to compensate, so that the numerical value of the mass in kg is still given by integral of $\rho \,dx\,dy\,dz$. Thus $\rho '=100^{-3}\rho $ (in units of kg⋅cm−3).
More generally, if the Cartesian coordinates x, y, z undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density. To model a non-constant density, ρ is a function of the variables x, y, z (a scalar field), and under a curvilinear change of coordinates, it transforms by the reciprocal of the Jacobian of the coordinate change. For more on the intrinsic meaning, see Density on a manifold.
A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition:[34]
$T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left|\det R\right|^{-w}\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.$
Here w is called the weight. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor.[35][36] An example of a tensor density is the current density of electromagnetism.
Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index. These come from the rational representations of the general linear group. But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still semisimple representations. A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation,[37] consisting of an (x, y) ∈ R2 with the transformation law
$(x,y)\mapsto (x+y\log \left|\det R\right|,y).$
Geometric objects
The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms). This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes.[38] Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles.[39][40]
Spinors
Main article: Spinor
When changing from one orthonormal basis (called a frame) to another by a rotation, the components of a tensor transform by that same rotation. This transformation does not depend on the path taken through the space of frames. However, the space of frames is not simply connected (see orientation entanglement and plate trick): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other. It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1.[41] A spinor is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.[42][43]
Succinctly, spinors are elements of the spin representation of the rotation group, while tensors are elements of its tensor representations. Other classical groups have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.
See also
• The dictionary definition of tensor at Wiktionary
• Array data type, for tensor storage and manipulation
Foundational
• Cartesian tensor
• Fibre bundle
• Glossary of tensor theory
• Multilinear projection
• One-form
• Tensor product of modules
Applications
• Application of tensor theory in engineering
• Continuum mechanics
• Covariant derivative
• Curvature
• Diffusion tensor MRI
• Einstein field equations
• Fluid mechanics
• Gravity
• Multilinear subspace learning
• Riemannian geometry
• Structure tensor
• Tensor Contraction Engine
• Tensor decomposition
• Tensor derivative
• Tensor software
Explanatory notes
1. The Einstein summation convention, in brief, requires the sum to be taken over all values of the index whenever the same symbol appears as a subscript and superscript in the same term. For example, under this convention $B_{i}C^{i}=B_{1}C^{1}+B_{2}C^{2}+\cdots +B_{n}C^{n}$
2. The double duality isomorphism, for instance, is used to identify V with the double dual space V∗∗, which consists of multilinear forms of degree one on V∗. It is typical in linear algebra to identify spaces that are naturally isomorphic, treating them as the same space.
3. Namely, the norm operation in a vector space.
References
Specific
1. Kline, Morris (1990). Mathematical Thought From Ancient to Modern Times. Vol. 3. Oxford University Press. ISBN 978-0-19-506137-6.
2. De Lathauwer, Lieven; De Moor, Bart; Vandewalle, Joos (2000). "A Multilinear Singular Value Decomposition" (PDF). SIAM J. Matrix Anal. Appl. 21 (4): 1253–1278. doi:10.1137/S0895479896305696.
3. Vasilescu, M.A.O.; Terzopoulos, D. (2002). "Multilinear Analysis of Image Ensembles: TensorFaces" (PDF). Computer Vision — ECCV 2002. Lecture Notes in Computer Science. Vol. 2350. pp. 447–460. doi:10.1007/3-540-47969-4_30. ISBN 978-3-540-43745-1.
4. Kolda, Tamara; Bader, Brett (2009). "Tensor Decompositions and Applications" (PDF). SIAM Review. 51 (3): 455–500. Bibcode:2009SIAMR..51..455K. doi:10.1137/07070111X. S2CID 16074195.
5. Sharpe, R.W. (2000). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer. p. 194. ISBN 978-0-387-94732-7.
6. Schouten, Jan Arnoldus (1954), "Chapter II", Tensor analysis for physicists, Courier Corporation, ISBN 978-0-486-65582-6
7. Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley Interscience, ISBN 978-0-471-15733-5
8. Lee, John (2000), Introduction to smooth manifolds, Springer, p. 173, ISBN 978-0-387-95495-0
9. Dodson, C.T.J.; Poston, T. (2013) [1991]. Tensor geometry: The Geometric Viewpoint and Its Uses. Graduate Texts in Mathematics. Vol. 130 (2nd ed.). Springer. p. 105. ISBN 9783642105142.
10. "Affine tensor", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
11. "Why are Tensors (Vectors of the form a⊗b...⊗z) multilinear maps?". Mathematics Stackexchange. June 5, 2021.{{cite web}}: CS1 maint: url-status (link)
12. Bourbaki, N. (1998). "3". Algebra I: Chapters 1-3. Springer. ISBN 978-3-540-64243-5. where the case of finitely generated projective modules is treated. The global sections of sections of a vector bundle over a compact space form a projective module over the ring of smooth functions. All statements for coherent sheaves are true locally.
13. Joyal, André; Street, Ross (1993), "Braided tensor categories", Advances in Mathematics, 102: 20–78, doi:10.1006/aima.1993.1055
14. Reich, Karin (1994). Die Entwicklung des Tensorkalküls. Science networks historical studies. Vol. 11. Birkhäuser. ISBN 978-3-7643-2814-6. OCLC 31468174.
15. Hamilton, William Rowan (1854–1855). Wilkins, David R. (ed.). "On some Extensions of Quaternions" (PDF). Philosophical Magazine (7–9): 492–9, 125–137, 261–9, 46–51, 280–290. ISSN 0302-7597. From p. 498: "And if we agree to call the square root (taken with a suitable sign) of this scalar product of two conjugate polynomes, P and KP, the common TENSOR of each, … "
16. Guo, Hongyu (2021-06-16). What Are Tensors Exactly?. World Scientific. ISBN 978-981-12-4103-1.
17. Voigt, Woldemar (1898). Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung [The fundamental physical properties of crystals in an elementary presentation]. Von Veit. pp. 20–. Wir wollen uns deshalb nur darauf stützen, dass Zustände der geschilderten Art bei Spannungen und Dehnungen nicht starrer Körper auftreten, und sie deshalb tensorielle, die für sie charakteristischen physikalischen Grössen aber Tensoren nennen. [We therefore want [our presentation] to be based only on [the assumption that] conditions of the type described occur during stresses and strains of non-rigid bodies, and therefore call them "tensorial" but call the characteristic physical quantities for them "tensors".]
18. Ricci Curbastro, G. (1892). "Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique". Bulletin des Sciences Mathématiques. 2 (16): 167–189.
19. Ricci & Levi-Civita 1900.
20. Pais, Abraham (2005). Subtle Is the Lord: The Science and the Life of Albert Einstein. Oxford University Press. ISBN 978-0-19-280672-7.
21. Goodstein, Judith R. (1982). "The Italian Mathematicians of Relativity". Centaurus. 26 (3): 241–261. Bibcode:1982Cent...26..241G. doi:10.1111/j.1600-0498.1982.tb00665.x.
22. Spanier, Edwin H. (2012). Algebraic Topology. Springer. p. 227. ISBN 978-1-4684-9322-1. the Künneth formula expressing the homology of the tensor product...
23. Hungerford, Thomas W. (2003). Algebra. Springer. p. 168. ISBN 978-0-387-90518-1. ...the classification (up to isomorphism) of modules over an arbitrary ring is quite difficult...
24. MacLane, Saunders (2013). Categories for the Working Mathematician. Springer. p. 4. ISBN 978-1-4612-9839-7. ...for example the monoid M ... in the category of abelian groups, × is replaced by the usual tensor product...
25. Bamberg, Paul; Sternberg, Shlomo (1991). A Course in Mathematics for Students of Physics. Vol. 2. Cambridge University Press. p. 669. ISBN 978-0-521-40650-5.
26. Penrose, R. (2007). The Road to Reality. Vintage. ISBN 978-0-679-77631-4.
27. Wheeler, J.A.; Misner, C.; Thorne, K.S. (1973). Gravitation. W.H. Freeman. p. 83. ISBN 978-0-7167-0344-0.
28. Schobeiri, Meinhard T. (2021). "Vector and Tensor Analysis, Applications to Fluid Mechanics". Fluid Mechanics for Engineers. Springer. pp. 11–29.
29. Maia, M. D. (2011). Geometry of the Fundamental Interactions: On Riemann's Legacy to High Energy Physics and Cosmology. Springer. p. 48. ISBN 978-1-4419-8273-5.
30. Hogben, Leslie, ed. (2013). Handbook of Linear Algebra (2nd ed.). CRC Press. pp. 15–7. ISBN 978-1-4665-0729-6.
31. Segal, I. E. (January 1956). "Tensor Algebras Over Hilbert Spaces. I". Transactions of the American Mathematical Society. 81 (1): 106–134. doi:10.2307/1992855. JSTOR 1992855.
32. Abraham, Ralph; Marsden, Jerrold E.; Ratiu, Tudor S. (February 1988). "5. Tensors". Manifolds, Tensor Analysis and Applications. Applied Mathematical Sciences. Vol. 75 (2nd ed.). Springer. pp. 338–9. ISBN 978-0-387-96790-5. OCLC 18562688. Elements of Trs are called tensors on E, [...].
33. Lang, Serge (1972). Differential manifolds. Addison-Wesley. ISBN 978-0-201-04166-8.
34. Schouten, Jan Arnoldus, "§II.8: Densities", Tensor analysis for physicists
35. McConnell, A.J. (2014) [1957]. Applications of tensor analysis. Dover. p. 28. ISBN 9780486145020.
36. Kay 1988, p. 27.
37. Olver, Peter (1995), Equivalence, invariants, and symmetry, Cambridge University Press, p. 77, ISBN 9780521478113
38. Haantjes, J.; Laman, G. (1953). "On the definition of geometric objects. I". Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A: Mathematical Sciences. 56 (3): 208–215.
39. Nijenhuis, Albert (1960), "Geometric aspects of formal differential operations on tensor fields" (PDF), Proc. Internat. Congress Math.(Edinburgh, 1958), Cambridge University Press, pp. 463–9, archived from the original (PDF) on 2017-10-27, retrieved 2017-10-26.
40. Salviori, Sarah (1972), "On the theory of geometric objects", Journal of Differential Geometry, 7 (1–2): 257–278, doi:10.4310/jdg/1214430830.
41. Penrose, Roger (2005). The road to reality: a complete guide to the laws of our universe. Knopf. pp. 203–206.
42. Meinrenken, E. (2013). "The spin representation". Clifford Algebras and Lie Theory. Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Vol. 58. Springer. pp. 49–85. doi:10.1007/978-3-642-36216-3_3. ISBN 978-3-642-36215-6.
43. Dong, S. H. (2011), "2. Special Orthogonal Group SO(N)", Wave Equations in Higher Dimensions, Springer, pp. 13–38
General
• Bishop, Richard L.; Samuel I. Goldberg (1980) [1968]. Tensor Analysis on Manifolds. Dover. ISBN 978-0-486-64039-6.
• Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Springer. ISBN 978-1-4020-1015-6.
• Jeevanjee, Nadir (2011). An Introduction to Tensors and Group Theory for Physicists. Birkhauser. ISBN 978-0-8176-4714-8.
• Lawden, D. F. (2003). Introduction to Tensor Calculus, Relativity and Cosmology (3/e ed.). Dover. ISBN 978-0-486-42540-5.
• Lebedev, Leonid P.; Cloud, Michael J. (2003). Tensor Analysis. World Scientific. ISBN 978-981-238-360-0.
• Lovelock, David; Rund, Hanno (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
• Munkres, James R. (1997). Analysis On Manifolds. Avalon. ISBN 978-0-8133-4548-2. Chapter six gives a "from scratch" introduction to covariant tensors.
• Ricci, Gregorio; Levi-Civita, Tullio (March 1900). "Méthodes de calcul différentiel absolu et leurs applications". Mathematische Annalen. 54 (1–2): 125–201. doi:10.1007/BF01454201. S2CID 120009332.
• Kay, David C (1988-04-01). Schaum's Outline of Tensor Calculus. McGraw-Hill. ISBN 978-0-07-033484-7.
• Schutz, Bernard F. (28 January 1980). Geometrical Methods of Mathematical Physics. Cambridge University Press. ISBN 978-0-521-29887-2.
• Synge, John Lighton; Schild, Alfred (1969). Tensor Calculus. Courier Corporation. ISBN 978-0-486-63612-2.
• This article incorporates material from tensor on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
External links
Wikimedia Commons has media related to Tensors.
• Weisstein, Eric W. "Tensor". MathWorld.
• Bowen, Ray M.; Wang, C.C. (1976). Linear and Multilinear Algebra. Introduction to Vectors and Tensors. Vol. 1. Plenum Press. hdl:1969.1/2502. ISBN 9780306375088.
• Bowen, Ray M.; Wang, C.C. (2006). Vector and Tensor Analysis. Introduction to Vectors and Tensors. Vol. 2. hdl:1969.1/3609. ISBN 9780306375095.
• Kolecki, Joseph C. (2002). "An Introduction to Tensors for Students of Physics and Engineering". Cleveland, Ohio: NASA Glenn Research Center. 20020083040.
• Kolecki, Joseph C. (2005). "Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity" (PDF). Cleveland, Ohio: NASA Glenn Research Center. 20050175884.
• A discussion of the various approaches to teaching tensors, and recommendations of textbooks
• Sharipov, Ruslan (2004). "Quick introduction to tensor analysis". arXiv:math.HO/0403252.
• Feynman, Richard (1964–2013). "31. Tensors". The Feynman Lectures. California Institute of Technology.
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
Authority control: National
• Japan
| Wikipedia |
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below).
The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.
Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.
Construction
Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times:
$T^{k}V=V^{\otimes k}=V\otimes V\otimes \cdots \otimes V.$
That is, TkV consists of all tensors on V of order k. By convention T0V is the ground field K (as a one-dimensional vector space over itself).
We then construct T(V) as the direct sum of TkV for k = 0,1,2,…
$T(V)=\bigoplus _{k=0}^{\infty }T^{k}V=K\oplus V\oplus (V\otimes V)\oplus (V\otimes V\otimes V)\oplus \cdots .$
The multiplication in T(V) is determined by the canonical isomorphism
$T^{k}V\otimes T^{\ell }V\to T^{k+\ell }V$
given by the tensor product, which is then extended by linearity to all of T(V). This multiplication rule implies that the tensor algebra T(V) is naturally a graded algebra with TkV serving as the grade-k subspace. This grading can be extended to a Z grading by appending subspaces $T^{k}V=\{0\}$ for negative integers k.
The construction generalizes in a straightforward manner to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ring, one can still perform the construction for any R-R bimodule M. (It does not work for ordinary R-modules because the iterated tensor products cannot be formed.)
Adjunction and universal property
The tensor algebra T(V) is also called the free algebra on the vector space V, and is functorial; this means that the map $V\mapsto T(V)$ extends to linear maps for forming a functor from the category of K-vector spaces to the category of associative algebra. Similarly with other free constructions, the functor T is left adjoint to the forgetful functor that sends each associative K-algebra to its underlying vector space.
Explicitly, the tensor algebra satisfies the following universal property, which formally expresses the statement that it is the most general algebra containing V:
Any linear map $f:V\to A$ from V to an associative algebra A over K can be uniquely extended to an algebra homomorphism from T(V) to A as indicated by the following commutative diagram:
Here i is the canonical inclusion of V into T(V). As for other universal properties, the tensor algebra T(V) can be defined as the unique algebra satisfying this property (specifically, it is unique up to a unique isomorphism), but this definition requires to prove that an object satisfying this property exists.
The above universal property implies that T is a functor from the category of vector spaces over K, to the category of K-algebras. This means that any linear map between K-vector spaces U and W extends uniquely to a K-algebra homomorphism from T(U) to T(W).
Non-commutative polynomials
If V has finite dimension n, another way of looking at the tensor algebra is as the "algebra of polynomials over K in n non-commuting variables". If we take basis vectors for V, those become non-commuting variables (or indeterminates) in T(V), subject to no constraints beyond associativity, the distributive law and K-linearity.
Note that the algebra of polynomials on V is not $T(V)$, but rather $T(V^{*})$: a (homogeneous) linear function on V is an element of $V^{*},$ for example coordinates $x^{1},\dots ,x^{n}$ on a vector space are covectors, as they take in a vector and give out a scalar (the given coordinate of the vector).
Quotients
Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain quotient algebras of T(V). Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
Coalgebra
The tensor algebra has two different coalgebra structures. One is compatible with the tensor product, and thus can be extended to a bialgebra, and can be further be extended with an antipode to a Hopf algebra structure. The other structure, although simpler, cannot be extended to a bialgebra. The first structure is developed immediately below; the second structure is given in the section on the cofree coalgebra, further down.
The development provided below can be equally well applied to the exterior algebra, using the wedge symbol $\wedge $ in place of the tensor symbol $\otimes $; a sign must also be kept track of, when permuting elements of the exterior algebra. This correspondence also lasts through the definition of the bialgebra, and on to the definition of a Hopf algebra. That is, the exterior algebra can also be given a Hopf algebra structure.
Similarly, the symmetric algebra can also be given the structure of a Hopf algebra, in exactly the same fashion, by replacing everywhere the tensor product $\otimes $ by the symmetrized tensor product $\otimes _{\mathrm {Sym} }$, i.e. that product where $v\otimes _{\mathrm {Sym} }w=w\otimes _{\mathrm {Sym} }v.$
In each case, this is possible because the alternating product $\wedge $ and the symmetric product $\otimes _{\mathrm {Sym} }$ obey the required consistency conditions for the definition of a bialgebra and Hopf algebra; this can be explicitly checked in the manner below. Whenever one has a product obeying these consistency conditions, the construction goes through; insofar as such a product gave rise to a quotient space, the quotient space inherits the Hopf algebra structure.
In the language of category theory, one says that there is a functor T from the category of K-vector spaces to the category of K-associative algebras. But there is also a functor Λ taking vector spaces to the category of exterior algebras, and a functor Sym taking vector spaces to symmetric algebras. There is a natural map from T to each of these. Verifying that quotienting preserves the Hopf algebra structure is the same as verifying that the maps are indeed natural.
Coproduct
The coalgebra is obtained by defining a coproduct or diagonal operator
$\Delta :TV\to TV\boxtimes TV$
Here, $TV$ is used as a short-hand for $T(V)$ to avoid an explosion of parentheses. The $\boxtimes $ symbol is used to denote the "external" tensor product, needed for the definition of a coalgebra. It is being used to distinguish it from the "internal" tensor product $\otimes $, which is already being used to denote multiplication in the tensor algebra (see the section Multiplication, below, for further clarification on this issue). In order to avoid confusion between these two symbols, most texts will replace $\otimes $ by a plain dot, or even drop it altogether, with the understanding that it is implied from context. This then allows the $\otimes $ symbol to be used in place of the $\boxtimes $ symbol. This is not done below, and the two symbols are used independently and explicitly, so as to show the proper location of each. The result is a bit more verbose, but should be easier to comprehend.
The definition of the operator $\Delta $ is most easily built up in stages, first by defining it for elements $v\in V\subset TV$ and then by homomorphically extending it to the whole algebra. A suitable choice for the coproduct is then
$\Delta :v\mapsto v\boxtimes 1+1\boxtimes v$
and
$\Delta :1\mapsto 1\boxtimes 1$
where $1\in K=T^{0}V\subset TV$ is the unit of the field $K$. By linearity, one obviously has
$\Delta (k)=k(1\boxtimes 1)=k\boxtimes 1=1\boxtimes k$
for all $k\in K.$ It is straightforward to verify that this definition satisfies the axioms of a coalgebra: that is, that
$(\mathrm {id} _{TV}\boxtimes \Delta )\circ \Delta =(\Delta \boxtimes \mathrm {id} _{TV})\circ \Delta $
where $\mathrm {id} _{TV}:x\mapsto x$ is the identity map on $TV$. Indeed, one gets
$((\mathrm {id} _{TV}\boxtimes \Delta )\circ \Delta )(v)=v\boxtimes 1\boxtimes 1+1\boxtimes v\boxtimes 1+1\boxtimes 1\boxtimes v$
and likewise for the other side. At this point, one could invoke a lemma, and say that $\Delta $ extends trivially, by linearity, to all of $TV$, because $TV$ is a free object and $V$ is a generator of the free algebra, and $\Delta $ is a homomorphism. However, it is insightful to provide explicit expressions. So, for $v\otimes w\in T^{2}V$, one has (by definition) the homomorphism
$\Delta :v\otimes w\mapsto \Delta (v)\otimes \Delta (w)$
Expanding, one has
${\begin{aligned}\Delta (v\otimes w)&=(v\boxtimes 1+1\boxtimes v)\otimes (w\boxtimes 1+1\boxtimes w)\\&=(v\otimes w)\boxtimes 1+v\boxtimes w+w\boxtimes v+1\boxtimes (v\otimes w)\end{aligned}}$
In the above expansion, there is no need to ever write $1\otimes v$ as this is just plain-old scalar multiplication in the algebra; that is, one trivially has that $1\otimes v=1\cdot v=v.$
The extension above preserves the algebra grading. That is,
$\Delta :T^{2}V\to \bigoplus _{k=0}^{2}T^{k}V\boxtimes T^{2-k}V$
Continuing in this fashion, one can obtain an explicit expression for the coproduct acting on a homogenous element of order m:
${\begin{aligned}\Delta (v_{1}\otimes \cdots \otimes v_{m})&=\Delta (v_{1})\otimes \cdots \otimes \Delta (v_{m})\\&=\sum _{p=0}^{m}\left(v_{1}\otimes \cdots \otimes v_{p}\right)\;\omega \;\left(v_{p+1}\otimes \cdots \otimes v_{m}\right)\\&=\sum _{p=0}^{m}\;\sum _{\sigma \in \mathrm {Sh} (p,m-p)}\;\left(v_{\sigma (1)}\otimes \dots \otimes v_{\sigma (p)}\right)\boxtimes \left(v_{\sigma (p+1)}\otimes \dots \otimes v_{\sigma (m)}\right)\end{aligned}}$
where the $\omega $ symbol, which should appear as ш, the sha, denotes the shuffle product. This is expressed in the second summation, which is taken over all (p, m − p)-shuffles. The shuffle is
${\begin{aligned}\operatorname {Sh} (p,q)=\{\sigma :\{1,\dots ,p+q\}\to \{1,\dots ,p+q\}\;\mid \;&\sigma {\text{ is bijective}},\;\sigma (1)<\sigma (2)<\cdots <\sigma (p),\\&{\text{and }}\;\sigma (p+1)<\sigma (p+2)<\cdots <\sigma (m)\}.\end{aligned}}$ :\{1,\dots ,p+q\}\to \{1,\dots ,p+q\}\;\mid \;&\sigma {\text{ is bijective}},\;\sigma (1)<\sigma (2)<\cdots <\sigma (p),\\&{\text{and }}\;\sigma (p+1)<\sigma (p+2)<\cdots <\sigma (m)\}.\end{aligned}}}
By convention, one takes that Sh(m,0) and Sh(0,m) equals {id: {1, ..., m} → {1, ..., m}}. It is also convenient to take the pure tensor products $v_{\sigma (1)}\otimes \dots \otimes v_{\sigma (p)}$ and $v_{\sigma (p+1)}\otimes \dots \otimes v_{\sigma (m)}$ to equal 1 for p = 0 and p = m, respectively (the empty product in $TV$). The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements $v_{k}$ is preserved in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.
Equivalently,
$\Delta (v_{1}\otimes \cdots \otimes v_{n})=\sum _{S\subseteq \{1,\dots ,n\}}\left(\prod _{k=1 \atop k\in S}^{n}v_{k}\right)\boxtimes \left(\prod _{k=1 \atop k\notin S}^{n}v_{k}\right)\!,$
where the products are in $TV$, and where the sum is over all subsets of $\{1,\dots ,n\}$.
As before, the algebra grading is preserved:
$\Delta :T^{m}V\to \bigoplus _{k=0}^{m}T^{k}V\boxtimes T^{(m-k)}V$
Counit
The counit $\epsilon :TV\to K$ is given by the projection of the field component out from the algebra. This can be written as $\epsilon :v\mapsto 0$ for $v\in V$ and $\epsilon :k\mapsto k$ for $k\in K=T^{0}V$. By homomorphism under the tensor product $\otimes $, this extends to
$\epsilon :x\mapsto 0$
for all $x\in T^{1}V\oplus T^{2}V\oplus \cdots $ It is a straightforward matter to verify that this counit satisfies the needed axiom for the coalgebra:
$(\mathrm {id} \boxtimes \epsilon )\circ \Delta =\mathrm {id} =(\epsilon \boxtimes \mathrm {id} )\circ \Delta .$
Working this explicitly, one has
${\begin{aligned}((\mathrm {id} \boxtimes \epsilon )\circ \Delta )(x)&=(\mathrm {id} \boxtimes \epsilon )(1\boxtimes x+x\boxtimes 1)\\&=1\boxtimes \epsilon (x)+x\boxtimes \epsilon (1)\\&=0+x\boxtimes 1\\&\cong x\end{aligned}}$
where, for the last step, one has made use of the isomorphism $TV\boxtimes K\cong TV$, as is appropriate for the defining axiom of the counit.
Bialgebra
A bialgebra defines both multiplication, and comultiplication, and requires them to be compatible.
Multiplication
Multiplication is given by an operator
$\nabla :TV\boxtimes TV\to TV$
which, in this case, was already given as the "internal" tensor product. That is,
$\nabla :x\boxtimes y\mapsto x\otimes y$
That is, $\nabla (x\boxtimes y)=x\otimes y.$ The above should make it clear why the $\boxtimes $ symbol needs to be used: the $\otimes $ was actually one and the same thing as $\nabla $; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor product $\otimes $ of the tensor algebra corresponds to the multiplication $\nabla $ used in the definition of an algebra, whereas the tensor product $\boxtimes $ is the one required in the definition of comultiplication in a coalgebra. These two tensor products are not the same thing!
Unit
The unit for the algebra
$\eta :K\to TV$
is just the embedding, so that
$\eta :k\mapsto k$
That the unit is compatible with the tensor product $\otimes $ is "trivial": it is just part of the standard definition of the tensor product of vector spaces. That is, $k\otimes x=kx$ for field element k and any $x\in TV.$ More verbosely, the axioms for an associative algebra require the two homomorphisms (or commuting diagrams):
$\nabla \circ (\eta \boxtimes \mathrm {id} _{TV})=\eta \otimes \mathrm {id} _{TV}=\eta \cdot \mathrm {id} _{TV}$
on $K\boxtimes TV$, and that symmetrically, on $TV\boxtimes K$, that
$\nabla \circ (\mathrm {id} _{TV}\boxtimes \eta )=\mathrm {id} _{TV}\otimes \eta =\mathrm {id} _{TV}\cdot \eta $
where the right-hand side of these equations should be understood as the scalar product.
Compatibility
The unit and counit, and multiplication and comultiplication, all have to satisfy compatibility conditions. It is straightforward to see that
$\epsilon \circ \eta =\mathrm {id} _{K}.$
Similarly, the unit is compatible with comultiplication:
$\Delta \circ \eta =\eta \boxtimes \eta \cong \eta $
The above requires the use of the isomorphism $K\boxtimes K\cong K$ in order to work; without this, one loses linearity. Component-wise,
$(\Delta \circ \eta )(k)=\Delta (k)=k(1\boxtimes 1)\cong k$
with the right-hand side making use of the isomorphism.
Multiplication and the counit are compatible:
$(\epsilon \circ \nabla )(x\boxtimes y)=\epsilon (x\otimes y)=0$
whenever x or y are not elements of $K$, and otherwise, one has scalar multiplication on the field: $k_{1}\otimes k_{2}=k_{1}k_{2}.$ The most difficult to verify is the compatibility of multiplication and comultiplication:
$\Delta \circ \nabla =(\nabla \boxtimes \nabla )\circ (\mathrm {id} \boxtimes \tau \boxtimes \mathrm {id} )\circ (\Delta \boxtimes \Delta )$
where $\tau (x\boxtimes y)=y\boxtimes x$ exchanges elements. The compatibility condition only needs to be verified on $V\subset TV$; the full compatibility follows as a homomorphic extension to all of $TV.$ The verification is verbose but straightforward; it is not given here, except for the final result:
$(\Delta \circ \nabla )(v\boxtimes w)=\Delta (v\otimes w)$
For $v,w\in V,$ an explicit expression for this was given in the coalgebra section, above.
Hopf algebra
The Hopf algebra adds an antipode to the bialgebra axioms. The antipode $S$ on $k\in K=T^{0}V$ is given by
$S(k)=k$
This is sometimes called the "anti-identity". The antipode on $v\in V=T^{1}V$ is given by
$S(v)=-v$
and on $v\otimes w\in T^{2}V$ by
$S(v\otimes w)=S(w)\otimes S(v)=w\otimes v$
This extends homomorphically to
${\begin{aligned}S(v_{1}\otimes \cdots \otimes v_{m})&=S(v_{m})\otimes \cdots \otimes S(v_{1})\\&=(-1)^{m}v_{m}\otimes \cdots \otimes v_{1}\end{aligned}}$
Compatibility
Compatibility of the antipode with multiplication and comultiplication requires that
$\nabla \circ (S\boxtimes \mathrm {id} )\circ \Delta =\eta \circ \epsilon =\nabla \circ (\mathrm {id} \boxtimes S)\circ \Delta $
This is straightforward to verify componentwise on $k\in K$:
${\begin{aligned}(\nabla \circ (S\boxtimes \mathrm {id} )\circ \Delta )(k)&=(\nabla \circ (S\boxtimes \mathrm {id} ))(1\boxtimes k)\\&=\nabla (1\boxtimes k)\\&=1\otimes k\\&=k\end{aligned}}$
Similarly, on $v\in V$:
${\begin{aligned}(\nabla \circ (S\boxtimes \mathrm {id} )\circ \Delta )(v)&=(\nabla \circ (S\boxtimes \mathrm {id} ))(v\boxtimes 1+1\boxtimes v)\\&=\nabla (-v\boxtimes 1+1\boxtimes v)\\&=-v\otimes 1+1\otimes v\\&=-v+v\\&=0\end{aligned}}$
Recall that
$(\eta \circ \epsilon )(k)=\eta (k)=k$
and that
$(\eta \circ \epsilon )(x)=\eta (0)=0$
for any $x\in TV$ that is not in $K.$
One may proceed in a similar manner, by homomorphism, verifying that the antipode inserts the appropriate cancellative signs in the shuffle, starting with the compatibility condition on $T^{2}V$ and proceeding by induction.
Cofree cocomplete coalgebra
Main article: Cofree coalgebra
One may define a different coproduct on the tensor algebra, simpler than the one given above. It is given by
$\Delta (v_{1}\otimes \dots \otimes v_{k}):=\sum _{j=0}^{k}(v_{0}\otimes \dots \otimes v_{j})\boxtimes (v_{j+1}\otimes \dots \otimes v_{k+1})$
Here, as before, one uses the notational trick $v_{0}=v_{k+1}=1\in K$ (recalling that $v\otimes 1=v$ trivially).
This coproduct gives rise to a coalgebra. It describes a coalgebra that is dual to the algebra structure on T(V∗), where V∗ denotes the dual vector space of linear maps V → F. In the same way that the tensor algebra is a free algebra, the corresponding coalgebra is termed cocomplete co-free. With the usual product this is not a bialgebra. It can be turned into a bialgebra with the product $v_{i}\cdot v_{j}=(i,j)v_{i+j}$ where (i,j) denotes the binomial coefficient for ${\tbinom {i+j}{i}}$. This bialgebra is known as the divided power Hopf algebra.
The difference between this, and the other coalgebra is most easily seen in the $T^{2}V$ term. Here, one has that
$\Delta (v\otimes w)=1\boxtimes (v\otimes w)+v\boxtimes w+(v\otimes w)\boxtimes 1$
for $v,w\in V$, which is clearly missing a shuffled term, as compared to before.
See also
• Braided vector space
• Braided Hopf algebra
• Monoidal category
• Multilinear algebra
• Stanisław Lem's Love and Tensor Algebra
• Fock space
References
• Bourbaki, Nicolas (1989). Algebra I. Chapters 1-3. Elements of Mathematics. Springer-Verlag. ISBN 3-540-64243-9. (See Chapter 3 §5)
• Serge Lang (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (3rd ed.), Springer Verlag, ISBN 978-0-387-95385-4
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| Wikipedia |
Tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M. [1]
Not to be confused with the Tensor product of fields.
Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor field as it associates a tensor to each point of a Riemannian manifold, which is a topological space.
Geometric introduction
Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface.
Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field $g$, such that given any two vectors $v,w$ at point $x$, their inner product is $g_{x}(v,w)$. The field $g$ could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is Tissot's indicatrix.
In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.
Via coordinate transitions
Following Schouten (1951) and McConnell (1957), the concept of a tensor relies on a concept of a reference frame (or coordinate system), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems.[2]
For example, coordinates belonging to the n-dimensional real coordinate space $\mathbb {R} ^{n}$ may be subjected to arbitrary affine transformations:
$x^{k}\mapsto A_{j}^{k}x^{j}+a^{k}$
(with n-dimensional indices, summation implied). A covariant vector, or covector, is a system of functions $v_{k}$ that transforms under this affine transformation by the rule
$v_{k}\mapsto v_{i}A_{k}^{i}.$
The list of Cartesian coordinate basis vectors $\mathbf {e} _{k}$ transforms as a covector, since under the affine transformation $\mathbf {e} _{k}\mapsto A_{k}^{i}\mathbf {e} _{i}$. A contravariant vector is a system of functions $v^{k}$ of the coordinates that, under such an affine transformation undergoes a transformation
$v^{k}\mapsto (A^{-1})_{j}^{k}v^{j}.$
This is precisely the requirement needed to ensure that the quantity $v^{k}\mathbf {e} _{k}$ is an invariant object that does not depend on the coordinate system chosen. More generally, a tensor of valence (p,q) has p downstairs indices and q upstairs indices, with the transformation law being
${T_{i_{1}\cdots i_{p}}}^{j_{1}\cdots j_{q}}\mapsto A_{i_{1}}^{i'_{1}}\cdots A_{i_{p}}^{i'_{p}}{T_{i'_{1}\cdots i'_{p}}}^{j'_{1}\cdots j'_{q}}(A^{-1})_{j'_{1}}^{j_{1}}\cdots (A^{-1})_{j'_{q}}^{j_{q}}.$
The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be smooth (or differentiable, analytic, etc). A covector field is a function $v_{k}$ of the coordinates that transforms by the Jacobian of the transition functions (in the given class). Likewise, a contravariant vector field $v^{k}$ transforms by the inverse Jacobian.
Tensor bundles
A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle.
The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold M. For example, a vector space of one dimension depending on an angle could look like a Möbius strip or alternatively like a cylinder. Given a vector bundle V over M, the corresponding field concept is called a section of the bundle: for m varying over M, a choice of vector
vm in Vm,
where Vm is the vector space "at" m.
Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Starting with the tangent bundle (the bundle of tangent spaces) the whole apparatus explained at component-free treatment of tensors carries over in a routine way – again independently of coordinates, as mentioned in the introduction.
We therefore can give a definition of tensor field, namely as a section of some tensor bundle. (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space
$V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*},$
where V is the tangent space at that point and V∗ is the cotangent space. See also tangent bundle and cotangent bundle.
Given two tensor bundles E → M and F → M, a linear map A: Γ(E) → Γ(F) from the space of sections of E to sections of F can be considered itself as a tensor section of $\scriptstyle E^{*}\otimes F$ if and only if it satisfies A(fs) = fA(s), for each section s in Γ(E) and each smooth function f on M. Thus a tensor section is not only a linear map on the vector space of sections, but a C∞(M)-linear map on the module of sections. This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are.
Notation
The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle TM = T(M) might sometimes be written as
$T_{0}^{1}(M)=T(M)=TM$
to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M. This should not be confused with the very similar looking notation
$T_{0}^{1}(V)$;
in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold M.
Curly (script) letters are sometimes used to denote the set of infinitely-differentiable tensor fields on M. Thus,
${\mathcal {T}}_{n}^{m}(M)$
are the sections of the (m,n) tensor bundle on M that are infinitely-differentiable. A tensor field is an element of this set.
The C∞(M) module explanation
There is another more abstract (but often useful) way of characterizing tensor fields on a manifold M, which makes tensor fields into honest tensors (i.e. single multilinear mappings), though of a different type (although this is not usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (C∞) vector fields on M, ${\mathcal {T}}(M)$ (see the section on notation above) as a single space — a module over the ring of smooth functions, C∞(M), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over any commutative ring.
As a motivating example, consider the space ${\mathcal {T}}^{*}(M)$ of smooth covector fields (1-forms), also a module over the smooth functions. These act on smooth vector fields to yield smooth functions by pointwise evaluation, namely, given a covector field ω and a vector field X, we define
(ω(X))(p) = ω(p)(X(p)).
Because of the pointwise nature of everything involved, the action of ω on X is a C∞(M)-linear map, that is,
(ω(fX))(p) = f(p)ω(p)(X(p)) = (fω)(p)(X(p)) = (fω(X))(p)
for any p in M and smooth function f. Thus we can regard covector fields not just as sections of the cotangent bundle, but also linear mappings of vector fields into functions. By the double-dual construction, vector fields can similarly be expressed as mappings of covector fields into functions (namely, we could start "natively" with covector fields and work up from there).
In a complete parallel to the construction of ordinary single tensors (not tensor fields!) on M as multilinear maps on vectors and covectors, we can regard general (k,l) tensor fields on M as C∞(M)-multilinear maps defined on l copies of ${\mathcal {T}}(M)$ and k copies of ${\mathcal {T}}^{*}(M)$ into C∞(M).
Now, given any arbitrary mapping T from a product of k copies of ${\mathcal {T}}^{*}(M)$ and l copies of ${\mathcal {T}}(M)$ into C∞(M), it turns out that it arises from a tensor field on M if and only if it is multilinear over C∞(M). Thus this kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to a function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously.
A frequent example application of this general rule is showing that the Levi-Civita connection, which is a mapping of smooth vector fields $(X,Y)\mapsto \nabla _{X}Y$ taking a pair of vector fields to a vector field, does not define a tensor field on M. This is because it is only R-linear in Y (in place of full C∞(M)-linearity, it satisfies the Leibniz rule, $\nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y$)). Nevertheless, it must be stressed that even though it is not a tensor field, it still qualifies as a geometric object with a component-free interpretation.
Applications
The curvature tensor is discussed in differential geometry and the stress–energy tensor is important in physics, and these two tensors are related by Einstein's theory of general relativity.
In electromagnetism, the electric and magnetic fields are combined into an electromagnetic tensor field.
It is worth noting that differential forms, used in defining integration on manifolds, are a type of tensor field.
Tensor calculus
In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus. Even to formulate such equations requires a fresh notion, the covariant derivative. This handles the formulation of variation of a tensor field along a vector field. The original absolute differential calculus notion, which was later called tensor calculus, led to the isolation of the geometric concept of connection.
Twisting by a line bundle
An extension of the tensor field idea incorporates an extra line bundle L on M. If W is the tensor product bundle of V with L, then W is a bundle of vector spaces of just the same dimension as V. This allows one to define the concept of tensor density, a 'twisted' type of tensor field. A tensor density is the special case where L is the bundle of densities on a manifold, namely the determinant bundle of the cotangent bundle. (To be strictly accurate, one should also apply the absolute value to the transition functions – this makes little difference for an orientable manifold.) For a more traditional explanation see the tensor density article.
One feature of the bundle of densities (again assuming orientability) L is that Ls is well-defined for real number values of s; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a half-density, the case where s = ½. In general we can take sections of W, the tensor product of V with Ls, and consider tensor density fields with weight s.
Half-densities are applied in areas such as defining integral operators on manifolds, and geometric quantization.
The flat case
When M is a Euclidean space and all the fields are taken to be invariant by translations by the vectors of M, we get back to a situation where a tensor field is synonymous with a tensor 'sitting at the origin'. This does no great harm, and is often used in applications. As applied to tensor densities, it does make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion.
Cocycles and chain rules
As an advanced explanation of the tensor concept, one can interpret the chain rule in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of tensor giving rise to tensor fields.
Abstractly, we can identify the chain rule as a 1-cocycle. It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, which come from applying functorial properties of tensor constructions to the chain rule itself; this is why they also are intrinsic (read, 'natural') concepts.
What is usually spoken of as the 'classical' approach to tensors tries to read this backwards – and is therefore a heuristic, post hoc approach rather than truly a foundational one. Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses. The construction of tensor densities is a 'twisting' at the cocycle level. Geometers have not been in any doubt about the geometric nature of tensor quantities; this kind of descent argument justifies abstractly the whole theory.
Generalizations
Tensor densities
Main article: Tensor density
The concept of a tensor field can be generalized by considering objects that transform differently. An object that transforms as an ordinary tensor field under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian of the inverse coordinate transformation to the wth power, is called a tensor density with weight w.[3] Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle such as the (1-dimensional) space of n-forms (where n is the dimension of the space), as opposed to taking their values in just R. Higher "weights" then just correspond to taking additional tensor products with this space in the range.
A special case are the scalar densities. Scalar 1-densities are especially important because it makes sense to define their integral over a manifold. They appear, for instance, in the Einstein–Hilbert action in general relativity. The most common example of a scalar 1-density is the volume element, which in the presence of a metric tensor g is the square root of its determinant in coordinates, denoted ${\sqrt {\det g}}$. The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition:
$\det(g')=\left(\det {\frac {\partial x}{\partial x'}}\right)^{2}\det(g),$
which is the transformation law for a scalar density of weight +2.
More generally, any tensor density is the product of an ordinary tensor with a scalar density of the appropriate weight. In the language of vector bundles, the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles w times. While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value. Non-integral powers of the (positive) transition functions of the bundle of densities make sense, so that the weight of a density, in that sense, is not restricted to integer values. Restricting to changes of coordinates with positive Jacobian determinant is possible on orientable manifolds, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of n-forms are distinct. For more on the intrinsic meaning, see density on a manifold.
See also
• Jet bundle – fiber bundle whose fibers are spaces of jets of sections of a fiber bundlePages displaying wikidata descriptions as a fallback
• Ricci calculus – Extension of vector calculus to tensors
• Spinor field
Notes
1. O'Neill, Barrett. Semi-Riemannian Geometry With Applications to Relativity
2. The term "affinor" employed in the English translation of Schouten is no longer in use.
3. "Tensor density", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
References
• O'neill, Barrett (1983). Semi-Riemannian Geometry With Applications to Relativity. Elsevier Science. ISBN 9780080570570.
• Frankel, T. (2012), The Geometry of Physics (3rd edition), Cambridge University Press, ISBN 978-1-107-60260-1.
• Lambourne [Open University], R.J.A. (2010), Relativity, Gravitation, and Cosmology, Cambridge University Press, ISBN 978-0-521-13138-4.
• Lerner, R.G.; Trigg, G.L. (1991), Encyclopaedia of Physics (2nd Edition), VHC Publishers.
• McConnell, A. J. (1957), Applications of Tensor Analysis, Dover Publications, ISBN 9780486145020.
• McMahon, D. (2006), Relativity DeMystified, McGraw Hill (USA), ISBN 0-07-145545-0.
• C. Misner, K. S. Thorne, J. A. Wheeler (1973), Gravitation, W.H. Freeman & Co, ISBN 0-7167-0344-0{{citation}}: CS1 maint: multiple names: authors list (link).
• Parker, C.B. (1994), McGraw Hill Encyclopaedia of Physics (2nd Edition), McGraw Hill, ISBN 0-07-051400-3.
• Schouten, Jan Arnoldus (1951), Tensor Analysis for Physicists, Oxford University Press.
• Steenrod, Norman (5 April 1999). The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-00548-5. OCLC 40734875.
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
• Differentiable/Smooth manifold
• Differential structure
• Smooth atlas
• Submanifold
• Riemannian manifold
• Smooth map
• Submersion
• Pushforward
• Tangent space
• Differential form
• Vector field
Main results (list)
• Atiyah–Singer index
• Darboux's
• De Rham's
• Frobenius
• Generalized Stokes
• Hopf–Rinow
• Noether's
• Sard's
• Whitney embedding
Maps
• Curve
• Diffeomorphism
• Local
• Geodesic
• Exponential map
• in Lie theory
• Foliation
• Immersion
• Integral curve
• Lie derivative
• Section
• Submersion
Types of
manifolds
• Closed
• (Almost) Complex
• (Almost) Contact
• Fibered
• Finsler
• Flat
• G-structure
• Hadamard
• Hermitian
• Hyperbolic
• Kähler
• Kenmotsu
• Lie group
• Lie algebra
• Manifold with boundary
• Oriented
• Parallelizable
• Poisson
• Prime
• Quaternionic
• Hypercomplex
• (Pseudo−, Sub−) Riemannian
• Rizza
• (Almost) Symplectic
• Tame
Tensors
Vectors
• Distribution
• Lie bracket
• Pushforward
• Tangent space
• bundle
• Torsion
• Vector field
• Vector flow
Covectors
• Closed/Exact
• Covariant derivative
• Cotangent space
• bundle
• De Rham cohomology
• Differential form
• Vector-valued
• Exterior derivative
• Interior product
• Pullback
• Ricci curvature
• flow
• Riemann curvature tensor
• Tensor field
• density
• Volume form
• Wedge product
Bundles
• Adjoint
• Affine
• Associated
• Cotangent
• Dual
• Fiber
• (Co) Fibration
• Jet
• Lie algebra
• (Stable) Normal
• Principal
• Spinor
• Subbundle
• Tangent
• Tensor
• Vector
Connections
• Affine
• Cartan
• Ehresmann
• Form
• Generalized
• Koszul
• Levi-Civita
• Principal
• Vector
• Parallel transport
Related
• Classification of manifolds
• Gauge theory
• History
• Morse theory
• Moving frame
• Singularity theory
Generalizations
• Banach manifold
• Diffeology
• Diffiety
• Fréchet manifold
• K-theory
• Orbifold
• Secondary calculus
• over commutative algebras
• Sheaf
• Stratifold
• Supermanifold
• Stratified space
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